Balázs Tóth Three-field dual-mixed variational

University of Miskolc
Faculty of Mechanical Engineering and Informatics
Department of Mechanics
Balázs Tóth
Three-field dual-mixed variational formulation
and hp finite element model for elastodynamic
analysis of axisymmetric shells
PhD dissertation
Supervisor: Edgár Bertóti, DSc
István Sályi Doctoral School
Head: Miklós Tisza, DSc
Miskolc-Egyetemváros
2012
Table of contents
Acknowledgements
iii
Declaration
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List of notations
v
1 Introduction
1.1 State of the art . . . . . . . . . . . . . .
1.1.1 Variational principles in elasticity
1.1.2 Shell theories and models . . . .
1.2 Objectives . . . . . . . . . . . . . . . . .
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1
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2 Differential geometry of surfaces
2.1 Surface coordinate system and its base vectors . . . . . . .
2.2 First fundamental form . . . . . . . . . . . . . . . . . . . .
2.3 Second and third fundamental forms. Principal curvatures
2.4 Surface Christoffel symbols . . . . . . . . . . . . . . . . . .
2.5 Covariant derivatives on a surface . . . . . . . . . . . . . .
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3 Differential geometry of shells
3.1 Undeformed configuration of a shell . . . . .
3.2 Local coordinate system at an arbitrary shell
3.3 Relationships between covariant derivatives .
3.4 Volume and surface elements of the shell . .
3.5 Geometry of shells of revolution . . . . . . .
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The shifter
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4 Three-field dual-mixed functional for elastodynamics
4.1 Governing equations and boundary conditions . . . . . . . . . . . . . . .
4.2 Complementary Hamilton’s principle for continua . . . . . . . . . . . . .
4.3 Dual-mixed functionals for elastodynamics using non-symmetric stresses .
4.3.1 Four-field functional and variational principle . . . . . . . . . . .
4.3.2 Three-field functional and variational principle . . . . . . . . . . .
4.4 Three-field dual-mixed functional for axisymmetric shells . . . . . . . . .
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5 Dimensionally reduced axisymmetric shell model
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5.1 Three-dimensional translational equations of motion for axisymmetric shells 39
5.2 Three-dimensional constitutive equations . . . . . . . . . . . . . . . . . . . . 40
i
ii
Table of contents
5.3
5.4
5.5
5.6
5.7
Stress resultant force- and couple vectors . . . . . . . . . . . . .
Approximation of the fundamental variables along the thickness
Initial and boundary conditions . . . . . . . . . . . . . . . . . .
Euler–Lagrange equations and natural boundary conditions . . .
5.6.1 The first variation of the complementary strain energy .
5.6.2 The first variation of the boundary integral term . . . . .
5.6.3 The first variation of the Lagrangian multiplier terms . .
Fundamental equation system of the axisymmetric shell model .
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41
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48
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6 Finite element formulation for elastodynamic problems
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6.1 Dual-mixed variational principle for axisymmetric shells . . . . . . . . . . . . 57
6.2 Matrix notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3 Dual-mixed hp finite element model . . . . . . . . . . . . . . . . . . . . . . . 62
7 Numerical examples
7.1 Elastostatic problems . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Clamped cylindrical shell with end loads . . . . . . .
7.1.2 Clamped cylindrical shell loaded by normal pressures
7.2 Dynamic analysis of a clamped cylindrical shell . . . . . . .
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8 Summary and theses
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9 Publications
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A Analitic solutions of Koiter’s cylindrical shell model
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Bibliography
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Acknowledgements
First of all, I would like to express my gratitude to Professor Edgár Bertóti for his
guidance as my supervisor, inspirations and continuous mental support during this research.
He has always known when and how to give me a little push in the forward direction when
I needed it.
I owe my thanks to Professor György Szeidl, who made me like Mechanics and
also for reviewing my dissertation. I am thankful to Professor István Páczelt for his
encouragements and to Professor Tamás Szabó, who expressed his interest in my work,
for providing useful advices and for sharing his numerical experiences with me.
Moreover, I am grateful to Professor István Ecsedi and Professor Csaba Takács
for the helpful hints on the theoretical analysis. I especially thank my colleague Lajos
György Kocsán for the countless discussions in a friendly environment. We exchanged
many valuable thoughts on both theoretical and numerical aspects of the work.
My dearest thanks go to my parents and my sister for their care, patience and constant
moral support, and to Krisztina who has been my guiding light and love over the three
last years. Without them this work would never have come into existence.
Furthermore I acknowledge the financial support provided by the Hungarian Scientific Research Fund under Grant No. OTKA T49427 and the TÁMOP-4.2.2/B-10/1-20100008 project in the framework of the New Hungarian Development Plan, and also by the
TÁMOP-4.2.1.B-10/2/KONV-2010-0001 project. The realization of these projects is supported by the European Union, co-financed by the European Social Fund.
Last, but not least, this research was also supported by the European Union and the
State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP
4.2.4.A/2-11-1-2012-0001 ’National Excellence Program’.
iii
Declaration
The author hereby declares that the work in this dissertation contains no material previously
published or written by another person and no part of the dissertation has been submitted,
either in the same or different form, to this or any other university for a PhD degree.
The author confirms that the work presented in this dissertation is his own and the appropriate credit has been given where reference has been made to the work of the others.
Miskolc, September 3, 2012.
Balázs Tóth
iv
List of notations
Although all symbols will be defined precisely when first introduced, most of the mathematical/mechanical quantities and notations are listed below. Throughout the dissertation,
the indicial notation of tensors and the usual summation convention is used. The range of
the Latin indices is 1, 2, 3 and that of the Greek indices is 1 and 2, except when special
operations are interpreted.
Symbol
Description
Greek symbols
αk , λ r
k
Γsmℓ , Γsm
γ±
γ0
δ kℓ
ǫαβ , ǫκλ
ǫkℓm , ǫpqr
ǫαβ , ǫ κλ
ǫkℓm , ǫ pqr
εkℓ
η
κ1 , κ2
µ
µ±
k
µkℓ , (µ−1 ) ℓ
ν
ξ3
ξα
Ò
Π
Ò
Π
H
ρ
σ
σ kℓ
kℓ
k3
iσ , 2σ
Lagrangian multipliers
(middle) surface Christoffel symbols of the first and the second kind
boundary curves of the upper and lower surfaces ω ±
reference curve, boundary curve of the reference surface ω0
Kronecker-delta
two-dimensional permutation tensors at an arbitrary shell point
three-dimensional permutation tensors at an arbitrary shell point
two-dimensional permutation tensors on the (shell middle) surfaceS0
three-dimensional permutation tensors on the (shell middle) surface
strain tensor
local coordinate of the standard element ωst
principal curvatures of the (shell middle) surface S0
determinant of the shifter µkℓ
determinant of the shifter µkℓ on the top and bottom surfaces S ±
shifter and its inverse
Poisson ratio
arc-length along the normal to the (shell middle) surface S0
curvilinear coordinates on the (shell middle) surface S0
total complementary energy functional
functional of the complementary Hamilton’s principle
density of the material
generalized stress vector
stress tensor
stress coefficients (i = 0, 1)
v
List of notations
φs
r
iφ
i Φpq , 2 Φp3
ψkℓ
ω
ω±
ω×
ω0
ω low
ωel
ωref
ωst
Ω
∂Ω
∇
∇0
∇
vi
axial vector of the skew-symmetric rotation tensor ψkℓ
rotation coefficients (i = 0, 1)
rotation resultants (i = 0, 1)
skew-symmetric rotation tensor
circular frequency of the forced vibration
upper and lower surfaces of the reference domain Ω
edge boundary of the reference domain Ω
reference surface
lowest natural frequency
e-th cylinder element
reference solution for the lowest natural frequency ω low
master element
reference domain
boundary of the reference domain Ω
three-dimensional differential operator at an arbitrary shell point
two-dimensional differential operator on the (shell middle) surface
three-dimensional differential operator on the (shell middle) surface
Latin symbols
a
A1
akℓ , apq
ak , ap
bk
bαβ , bκβ
(Bk )κα
p
ib
c
c0
cαβ
C
d
D kℓrs , Ckℓrs
E
elow
ek
eb
i Epq , 2 Ep3
pq
ie
i ep
fk
× k
× k
0 (f ) , 1 (f )
k
k
0f , 1f
f
f±
determinant of the (shell middle) surface covariant metric tensor akℓ
ratio of the infinitesimal increments dξ 1 and dx1
metric tensors of the (shell middle) surface coordinate system
base vectors of the (shell middle) surface coordinate system
density of the body forces
covariant and mixed curvature tensors of the (shell middle) surface
κ
coefficients in the power series expansion of the inverse shifter (µ−1 ) α
body force density coefficients (i = 0, 1)
smooth curve, defined on the lateral surface, for which ξ 3 = const.
smooth curve on the shell middle surface S0
third fundamental form of the (shell middle) surface S0
constitutive matrix
thickness of the reference domain Ω and the (axisymmetric) shell
fourth-order material stiffness tensor and elastic compliance tensor
elasticity modulus
relative error measured in the lowest natural frequency ω low
unit base vectors of the right-handed orthogonal Cartesian frame
relative error measured in energy norm
strain resultants (i = 0, 1)
relative error in maximum norm of the stresses i σ pq (i = 0, 1)
relative error in maximum norm of the displacements i up (i = 0, 1)
traction vector on the surface part Sσ
vector-valued load coefficients on the lateral surface S L
vector-valued load coefficients on the outer and inner surfaces S ±
generalized vector of prescribed displacements on S 0
prescribed traction vectors on the outer and inner surfaces S ±
List of notations
f×
F
G
g
gkℓ , g pq
g, h
g e , he
gk , g m
H, K
H, H e
H
HR
Je
K ,K
L
L, N
ℓ±
ℓ0
ℓ0 (0) , ℓ0 (L)
ℓe
l, m
Ò
L
M κα
M,
Me
M
n
N κα , Q α
N1 , N2
Nj
n±
n×
p
pk
Pj
t
p, t t
pe , t e
t
q
qe
R
R0 , RL
R1 , R2
Rmin
t
r
re
r
r0
vii
prescribed traction vector on the lateral surface S L
dual-mixed functional for elastodynamics
shear modulus, Lamé constant
determinant of the covariant metric tensor at an arbitrary shell point
metric tensors at an arbitrary shell point
generalized load vectors
generalized load vectors of the e-th cylinder element ωel
covariant and contravariant base vectors at an arbitrary shell point
mean and Gaussian curvature of the (shell middle) surface S0
global and local flexibility matrices
complementary Hamiltonian function
three-field dual-mixed functional of elastostatics
Jacobi determinant of the coordinate transformation η → ξ 1
total kinetic energy of the continua and its complementary function
length of the axisymmetric shell
interpolation matrices
boundary curves of the top and bottom surfaces S ± of the shell
boundary curve of the shell middle surface S0
boundary curves of the axisymmetric shell middle surface
length of the e-th cylinder element ωel
stress resultant force and couple vectors
complementary Lagrangian function
stress couple resultants: bending and twisting moments
global and local consistent mass matrices
mass matrix
number of elements
stress resultants: membrane and shear forces
external shape functions
internal shape functions (j = 3, 4, ...)
outward unit normal vectors to the top and bottom surfaces S ±
outward normal vector to the lateral surface S ×
degree of the polynomial approximation
impulse vector
Legendre polynomials (j = 0, 1, ...)
global load vectors
nodal load vectors of the e-th cylinder element ωel
global displacement vector
nodal displacement vector of the e-th cylinder element ωel
meridian curve
radii of the circles ℓ0 (0) and ℓ0 (L)
principal radii of curvature of the (shell middle) surface S0
least principal radius of curvature of the (shell middle) surface S0
global vector of prescribed displacements on S 0
nodal vector of prescribed displacements of the e-th element ωel
position vector of an arbitrary shell point
position vector of an arbitrary (shell middle) surface point
List of notations
S
s
S 0, S L
S±
S×
S0
s0
Sσ
sL
Su
t
s
S, U
se
Se, U e
t
t0 , t1
t×
T , TÒ
uk
0
uk , 0 vk
UÒ
UÒ b-s
U
u
Ò
U
b-s
Ò
U, U
ue k
e
i Up
0
i up
p
iu
V
vk
0
i vp
xk
∆t
viii
the boundary surface of the (axisymmetric) shell
arc-length along the lateral surface curve c for which ξ 3 = const.
lateral surfaces of the axisymmetric shell
top and bottom surfaces of the (axisymmetric) shell
lateral surface of the shell
middle surface of the (axisymmetric) shell
arc-length along the curve c0 of the shell middle surface S0
surface part, on which the traction vector f k is only prescribed
length of the meridian curve R
surface part, on which the displacement vector uk is only prescribed
global stress vector
global differential operator matrices
nodal stress vector of the e-th cylinder element ωel
differential operator matrices of the e-th cylinder element ωel
time coordinate
two arbitrary instants of the time t
tangent vector to the lateral surface curve c
kinetic energy density and its complementary function
displacement vector
initial conditions for the displacements uk and the velocities vk
total complementary strain energy of the continua
total complementary bending strain energy of the shell
differential operator matrix
generalized displacement vector
complementary bending-shearing strain energy density
strain energy density and its complementary function
displacement vector prescribed on the surface part Su
prescribed displacement resultants (i = 0, 1)
initial displacement coefficients (i = 0, 1)
displacement coefficients (i = 0, 1)
undeformed configuration of the (axisymmetric) shell, volume
velocity vector
initial velocity coefficients (i = 0, 1)
coordinate lines of the right-handed orthogonal Cartesian frame
size of time steps during the time integration
Chapter 1
Introduction
1.1
1.1.1
State of the art
Variational principles in elasticity
The most common approach to construct finite element models for linear elasticity problems is to consider the pure displacement formulation, where the displacement is the only
variable and the solution is based on the global minimization of the total potential energy
functional. In fact, this leads to the principle of virtual work ensuring the satisfaction of
the translational equilibrium equation and the traction boundary condition in a weak sense,
while the kinematic equation and the displacement boundary condition are enforced exactly.
Therefore it is also named as the compatible displacement finite element model, for which
the existence and uniqueness of the solution can be guaranteed, namely the system matrix
is symmetric and positive definite. The simplest classical finite elements, such as the beam,
the triangular and the rectangular element, were first published in the 1950’s. These elements use linear piecewise polynomials to approximate the solution, and achieve increased
accuracy with mesh refinement. This philosophy of using low-order polynomials over successively finer meshes called h-type approximation technique. It had been considered as
the predominant one by researchers for many years, and was the one in the direction of
which the development of the finite element method proceeded with much success through
the 1970’s. More information on the formulation and its applications can be found in many
books, see for instance [32, 67, 88, 115, 130].
The widely used standard displacement finite element technique has some significant
disadvantages. Stability problems can be expected at nearly incompressible materials,
when the Poisson ratio is close to the incompressibility limit of 0.5, i.e., when one of
the Lamé constants tends to infinity. The displacements can be inaccurately computed
and even worse values can be obtained for the sum of the normal stresses. This is the
well-known incompressibility locking effect [28, 29]. Furthermore, the conventional finite
element models provide slow convergences and low accuracy in the evaluation of the stress
field, which is often more important in many engineering applications than the knowledge
of displacements. Namely, the stresses are obtained through the numerical differentiation
of the approximated displacements, thus leading to this decrease in accuracy.
Experiments of the research group conducted by Barna Szabó in the mid-1970’s indicated that an alternative strategy might hold great promise. Their idea was to keep
1
Chapter 1. Introduction
2
the coarse mesh fixed, and the convergence is achieved solely by increasing the polynomial
degree p of the approximated displacements. This finite element technique is called as pversion to be distinguished from the classical method, which is labeled as h-version, where p
is fixed and the mesh is refined. The computational results obtained for elasticity problems
indicated that the new philosophy was always competitive with, and often out-performed,
the conventional h-version. The first theoretical paper on the p-version was published by
Babuška, Szabó and Katz [31] in 1981. It was proven that the rate of convergence is
double than that is possible with h-version, and exponential for the solution. The cooperation between Szabó and Babuška also led to the development of the so-called hp-version
finite element models, which is the combination of both strategies, i.e., both p and the
characteristic element size h are changed. It was shown that an exponential convergence
rate could be achieved even for unsmooth solutions, i.e., in the presence of corner singularities by properly chosen mesh refinement coupled with the increase of the polynomial
degree p. Then Babuška and Dorr analyzed these finite element approximation techniques theoretically in [25], as well as Babuška and Suri tested numerically in [30]. A
detailed survey of the method can be found in [24] or [154]. Although the higher-order displacement elements give optimal accuracy for the displacements, the stresses recovered by
postprocessing from the displacement field are not free from the incompressibility locking.
This was proven analytically and numerically by Suri [151], as well as Szabó, Babuška
and Chayapathy [156].
One of the strategies to overcome the incompressibility locking effect for low-order hversion finite elements is to use a primal1 -mixed2 variational principle introduced in [86]
by Herrmann, where the incompressibility constraint equation is satisfied only approximately, in a weak sense. This method is known as the displacement-pressure formulation
for nearly incompressible materials and as the velocity-pressure formulation for the Stokes
problem in fluid mechanics. Nevertheless, in contrast to the compatible displacement finite element models, mixed formulations lead to indefinite algebraic systems, i.e., the approximation spaces have to satisfy two criterions, namely the so-called coercivity and the
Ladyzhenskaya–Babuška–Brezzi conditions. The mathematical theory of them is based on
the works of Babuška [23], Brezzi [51], as well as Ladyzhenskaya [101]. Several papers
and books dealt with the stability and convergence properties of mixed methods. Brezzi
and Fortin [52], Roberts and Thomas [136], as well as Stenberg and Suri [146] analyzed it theoretically and pointed out that low-order mixed h-elements are stable and robust
in contrast to standard h-version with respect to incompressibility locking. Primal-mixed
p and hp finite element models based on this approach were developed for linear elasticity
problems as well as for Stokes and non-Newtonian flow by Chilton and Suri [62–64],
Kouhia and Stenberg [99], Schwab and Suri [142], as well as Stenberg and Suri
[146]. Computational results presented in [62–64, 99, 142, 146] showed that primal-mixed
p and hp elements have excellent locking-free approximation properties not only for the
displacements but also for the stresses.
An alternative way to avoid the incompressibility locking effect for low-order h-type
finite elements is to apply dual3 -mixed variational principles in the framework of linear
1
‘primal’ means that the displacement is the primary variable and its gradient appears in the formulation.
‘mixed’ indicates that the number of the independent fields is at least two in the variational principle.
3
‘dual’ implies that the primary unknown is the stress field and its divergence appears in the formulation.
2
1.1. State of the art
3
elasticity. These methods are less, or not sensitive to the material properties and deliver
better convergence rates and higher accuracy for the stresses, than pure displacement formulations and strain energy-based primal-mixed methods mentioned previously. The weak
solution of the classical dual formulation is based on the global maximization of the total
complementary energy functional in terms of the stress field. In fact, this is equivalent
to the principle of complementary virtual work, i.e., Castigliano’s principle, satisfying the
strain compatibility equation and the displacement boundary condition in a weak sense.
The traction boundary condition, as well as the translational and rotational equilibrium
equation are enforced in the strong sense using the second-order stress function tensor. It
is the so-called equilibrium method proposed first by Fraeijs de Veubeke [73]. Later
Gallagher [77], Sarigul and Gallagher [140], Vallabhan and Azene [162], as well
as Watwood and Hartz [165] analyzed it numerically for two-dimensional elasticity problems. The related finite element schemes require, however, C 1 continuous approximation of
the second-order stress functions. This requirement, which is primarily due to the a priori
satisfaction of the symmetry condition for the stress tensor, makes it rather difficult and
complicated to establish a numerically efficient and well manageable equilibrium method
and stress elements for general problems.
One of the possibilities to overcome the difficulties mentioned in connection with the
development of numerically efficient stress-based formulation is to incorporate the symmetry
condition for the stress tensor into the total complementary energy functional using the
rotations as Lagrangian multipliers. It means that the rotational equilibrium equation is
not enforced exactly in the finite element procedure, but come out as one of the Euler–
Lagrange equations in the variational formulation. Applying this two-field dual-mixed
variational principle named after Fraeijs de Veubeke, the translational equilibrium equation
has to be satisfied a priori with the first-order stress function tensor, which has only
six independent components [48]. The most important advantage of the finite element
methods based on the Fraeijs de Veubeke principle is that the approximation of the firstorder stress function tensor requires only C 0 continuity at the interelement boundaries.
The bibliographic chain for this research direction commences with the pioneering papers
by Fraeijs de Veubeke [74, 75]. After the appearance of them, assumed nonsymmetric
stress elements were developed for two-dimensional nonlinear elasticity problems by Atluri
and Murakawa in [21, 111, 112]. Subsequently Cazzani and Atluri [56], Fraeijs de
Veubeke and Millard [76], as well as Bertóti [43, 44] elaborated finite element models
based on the linearized version of this principle for two-dimensional elasticity problems.
The nonsymmetric stress-based complementary energy principle of Fraeijs de Veubeke was
extended to large strain rate problems of classical elastoplasticity by Atluri [20], as well
as Reed and Atluri [131]. Mathematical analysis of triangular elements based on Fraeijs
de Veubeke’s variational principle was given, for instance, by Amara and Thomas [8], as
well as Farhloul and Fortin [71].
Other alternative strategy to construct complementary energy-based finite element models with optimal convergence rates for stresses and displacements, is to satisfy the translational equilibrium equation weakly using the displacement field as Lagrangian multiplier
and letting the stress field be a priori symmetric. Thus the classical two-field variational
principle proposed originally by Hellinger [84] and Reissner [133–135] are obtained,
in which the stress tensor and the displacement vector are simultaneously approximated
as independent variables. Primal-mixed h-version elements based on this approach were
Chapter 1. Introduction
4
developed for three-dimensional and plane elasticity problems, among others, by Mijuca
[108], Punch and Atluri [123], as well as Xue, Karlovitz and Atluri [166]. Without
completeness the application of the pure dual formulation and the two-field primal-mixed
Hellinger–Reissner variational principle, which is complementary to the Hamilton’s principle, were extended to general dynamic analysis of continuous media by Ignaczak [89],
Karnopp [93, 94], Minagawa [109], Schaefer [141], Tabarrok [157, 158], Tabarrok
and Rimrott [159], as well as Washizu [164].
Since in primal functionals the gradient operator is applied to the displacement vector,
the displacements have to be C 0 continuous, while the normal components of the stress tensor are discontinuous across the element interfaces. However, it has the major shortcoming
that the order of convergence for the stresses is lower by one than that is possible with
the use of finite element models based on Fraeijs de Veubeke’s or classical dual principles.
To avoid this, the divergence operator is shifted from the displacement vector to the stress
tensor obtaining the two-field dual-mixed Hellinger–Reissner functional. In this case the
normal components of the stress tensor have to be C 0 continuous and the displacements are
discontinuous at the interelement boundaries. It has a decisive influence on the construction
of the related finite elements. Namely, the simultaneous strong enforcement of the symmetry and the continuity condition causes substantial difficulties in finite element modeling. A
number of dual-mixed element models of this type were devised and thoroughly studied for
plane elasticity problems by Arnold and Awanou [10], Arnold and Winther [14, 15],
Arnold, Douglas and Gupta [17], Lovadina and Stenberg [105], Pitkäranta and
Stenberg [118], Stenberg [144], as well as Suri [150]. The two-field Hellinger–Reissner
formulations and their alternatives are discussed in a comprehensive paper by Arnold [9].
The lack of simple stable and efficient dual-mixed three-dimensional elements for the
two-field dual-mixed Hellinger–Reissner functional has led to the construction of modified
Hellinger–Reissner-type variational principles in which the symmetry of the stresses is fulfilled in a weak sense via the rotations as Lagrangian multipliers retaining the basic stress
and displacement variables. The resulting formulation has three independent fields: the
stress tensor, the displacement vector and the rotation vector. This approach has numerous advantages over the traditional displacement finite element models. It provides more
accurate approximation and higher convergence rate for the stress field which is usually the
variable of primary interest in engineering practice. Furthermore, it helps to overcome several locking phenomena. Dual-mixed elements was presented for linear elasticity problems
by Arnold and Falk [12], Arnold, Brezzi and Douglas [16], Arnold, Falk and
Winther [18], Brink, Klaas, Niekamp and Stein [54], Klaas, Schröder, Stein
and Miehe [96], Qiu and Demkowicz [125], Rolfes [137], Stein and Rolfes [143],
as well as Stenberg [145]. Besides, the three-field primal-mixed Hellinger–Reissner-type
variational principle was applied to static and dynamic analysis of flexible beam structures
and multibody systems by Iura and Atluri [90, 91], as well as Quadrelli and Atluri
[126, 127]. Moreover Cazzani and Atluri [56], Cazzani and Lovadina [57], as well
as Iura and Atluri [92] analyzed the methods of this type numerically for membrane
problems.
1.1. State of the art
1.1.2
5
Shell theories and models
The classical shell theories are usually derived by applying the so-called dimensional reduction procedure, where the original three-dimensional shell problem is replaced with an
approximate two-dimensional one. The classical dimensionally reduced shell models can be
derived from the power series expansions of the differential equations of three-dimensional
elasticity with respect to the thickness coordinate. Thereby it is possible to generate a
hierarchy of families of shell models, of which the well-known Naghdi-model [113] is the
lowest-order member. This theory assumes that the displacement components are linear
functions of the thickness coordinate except the constant transverse deflection. It is the
first-order shear deformation theory named for plates after Mindlin [110] and Reissner
[132].
The other, often used classical shell model is the Koiter-model [98], which is essentially
analogous to the Kirchhoff-model [95] of plates. It is based on the Kirchhoff–Love
hypothesis [95, 106] which implies further restrictions on the deformation of the normal to
the shell mid-surface. The latter is the asymptotic limit of the three-dimensional elasticity
equations as the thickness tends to zero. It was proven mathematically for shells in the case
of special boundary conditions by Ciarlet [65]. Namely, the asymptotic character of the
shell deformation depends strongly on the geometry of the shell, the kinematical constraints
and even on the type of loading. The asymptotic behavior of the Reissner–Mindlin plate
model was investigated by Alessandrini, Arnold, Falk and Madureira [3], as well
as Babuška and Li [26, 27]. The asymptotic validity of the Naghdi-model has not been
verified yet. Historically, several variants of the classical shell theories were developed by
Flügge [72], Novozhilov [114], Sanders [139], Vlasov [163] etc., but the differences
in these formulations seem to be rather insignificant.
In the theoretical analysis of shells the Koiter-model is used most often. However,
the Naghdi-model is more advantageous for the development of displacement-based finite
element algorithms, since the strain energy in this model contains only first derivatives
of the displacements, i.e., the related finite element formulation requires C 0 continuity of
the displacements. The Koiter-model is less ‘user-friendly’ for finite element applications,
since it involves second derivatives of the transverse deflection, i.e., C 1 continuous elements
are needed. These two models use modified constitutive equations, therefore the strains
and stresses normal to the mid-surface cannot be taken into account. This problem can
be avoided by the application of higher-order or hierarchical displacement-based plate and
shell elements, see the works of Babuška and Li [26, 27], Ovaskainen and Pitkäranta
[116], Rank, Düster, Nübel, Preusch and Bruhns [129], Szabó and Babuška [154],
as well as Szabó and Sahrmann [155].
The standard dimensionally reduced shell models and the related low-order finite elements suffer from several numerical problems. The basic source of these numerical difficulties arises from the small value of the thickness, i.e., when the shell becomes thin. This
phenomenon is the well-known numerical locking effect. The standard low-order h-version
triangular and quadrilateral shell elements with Naghdi-type kinematics are characterized
by the most serious locking phenomena.
There are several types of numerical locking in dimensionally reduced shell modeling.
The transverse shear locking appears in bending dominated cases when the bending energy
is restrained and nearly all the strain energy is stored in transverse shear energy terms
Chapter 1. Introduction
6
for small thicknesses. This is partly caused by the fact that the finite element solution
of the Naghdi-model is forced to satisfy the Kirchhoff–Love hypothesis, as the value of
the shell thickness tends to zero. The even more severe membrane locking occurs when
the bending energy is restrained and the total strain energy is stored in membrane energy
terms for small thickness values. The numerical locking in boundary layers is a more
hidden effect, but it can also causes significant errors in local mechanical quantities such
as stress maxima. One of the effective strategies, to overcome this singular behavior, i.e.,
the deterioration of the finite element solution, is the use of local mesh refinement near
the boundary. The shear- and membrane locking had been first manifested by Stolarski
and Belytschko [148, 149], later Bletzinger, Bischoff and Ramm [50], Gerdes,
Matache and Schwab [78], Hakula, Leino and Pitkäranta [81], Pitkäranta [117],
as well as Suri, Babuška and Schwab [153] analyzed the locking phenomena theoretically
and numerically.
There are several attempts to circumvent transverse shear locking. One of them is to
modify the principle of virtual work to enforce the Kirchhoff-Love constraint in a weak sense.
This leads to a primal-mixed functional which approximates the transverse shear stresses as
independent variables. For low-order h-type analyses, such reduced constraint methods seem
to be promising to avoid this locking effect. Numerous shear locking-free formulations of this
type were proposed and analyzed for plates and shells by Arnold and Falk [13], Bathe,
Brezzi and Fortin [34], Brezzi, Fortin and Stenberg [53], Dvorkin and Bathe [68],
as well as Pitkäranta and Suri [119, 120].
Another way to overcome this kind of locking is the application of the reduced or selective
integrations. Nevertheless, the use of these methods is often accompanied by spurious zero
energy modes for certain boundary conditions and mesh-layouts. Hence, Arnold and
Brezzi [11], Gruttmann and Wagner [79, 80], as well as Chapelle and Stenberg
[60, 61] developed stabilization techniques to regain the correct rank of the element stiffness
matrix. Membrane locking can be treated similarly as shear locking in the case of bending
dominated shell problems. However, the shell elements developed in such way do not
perform very well up to now, when they are applied to membrane dominated cases, see
[11, 78, 117, 148, 149].
One of the most reliable locking removal techniques is based on the mixed interpolation
of the strain tensor components (MITC). The theoretical basis of this method had been
first published for 4-node and 8-node quadrilateral plate/shell finite elements by Dvorkin
and Bathe [68–70] and was later extended to 9- and 16-node general shell elements by
Bucalem and Bathe [55]. This technique was also applied to the development of triangular plate and shell elements by Bathe, Brezzi and Cho [33], as well as Lee and Bathe
[102]. The numerical efficiency of these elements was tested through representative model
problems, among others, by Bathe, Iosilevich and Chapelle [35, 36], Chapelle and
Paris Suarez [59], Hernández, Hervella-Nieto and Rodrı́guez [85], as well as Lee
and Bathe [102–104].
Another method to construct shear locking-free plate and shell bending finite elements
is the so-called discrete Kirchhoff technique (DKT). The independent variables of the element formulations based on this approach are usually the transverse deflection and two
infinitesimal ‘rotation’ components, which require only C 0 continuous approximations at
the interelement boundaries. Furthermore the transverse shear energy terms are neglected,
i.e., the bending strain energy is only considered, as well as the Kirchhoff–Love constraints
1.1. State of the art
7
are imposed in a discrete way on the element sides. The numerical behavior of the DKT
elements of various shapes was studied and discussed, among others, by Batoz [37], Batoz
and Ben Tahar [38], Batoz and Lardeur [39], Batoz, Bathe and Ho [40], as well as
Batoz, Zheng and Hammadi [41].
Additional efficient strategy for avoiding locking problems in the displacement-based finite element formulation is to use high-order methods and p-version finite elements. These
p elements are verified to be locking-free in the energy norm computations for general shells,
but the numerical results obtained for stresses are not exempt from locking, see for example
the works of Chapelle and Bathe [58], as well as Pitkäranta, Leino, Ovaskainen
and Piila [121]. A combination of the reduced constraint method with higher-order finite
element techniques was proposed by Stenberg and Suri [147], as well as Suri [151].
Another solution for locking problems is the application of stress-based variational principles. The design of this kind of hp-version shell elements is complicated and more demanding task than the construction of classical displacement-based shell finite element models.
A general shell model in terms of symmetric stresses had been constructed without the use
of the classical kinematical hypotheses by Kozák [100]. Subsequently, a complementary
energy-based cylindrical shell model, using the second-order stress function tensor, was
derived by Bertóti [47, 49]. Applying the two-field dual-mixed variational principle of
Fraeijs de Veubeke, locking-free hp-version plate and shell elements were presented for oneand two-dimensional elastostatic problems by Bertóti [45, 46] and Kocsán [97]. The twofield primal- and dual-mixed variational principle of Hellinger–Reissner was applied to the
elastostatic problems of plates by Alessandrini, Arnold, Falk and Madureira [3], as
well as Prager [122]. The application of these kinds of variational principles and the pure
dual formulation was also extended to the vibration analysis of plates and axisymmetric
shells by Abbas [1], Altman and Bismarck-Nasr [4], Altman and Iguti [5], Altman
and Neto [6], Altman and Venancio-Filho [7], as well as Sakaguchi and Tabarrok
[138]. Using the three-field primal-mixed variational principle of Hellinger–Reissner-type,
h-version plate and shell elements were developed for linear and nonlinear problems by
Atluri [19], Atluri, Iura and Suetake [22], as well as Punch and Atluri [124].
Nevertheless, these plate and shell finite element models are based on the Kirchhoff–Love
hypothesis and modified constitutive equations. To the author’s knowledge, dimensionally
reduced axisymmetric shell model based on the three-field dual-mixed variational principle
of Hellinger–Reissner-type, without the classical hypotheses, cannot be found in the current
literature.
Beside the use of the two-dimensional shell finite element models, the application of
the so-called degenerated elements derived from the three-dimensional shell theories are
also widespread. These shell elements are also based on the Naghdi-type kinematics,
but the integrations with respect to the thickness coordinate are carried out numerically.
Nevertheless, not only the numerical problems that occur in the dimensionally reduced
finite element shell models but also additional numerical locking effects (curvature locking, dilatation locking, thickness locking, trapezoidal locking, incompressibility locking, see
[82, 83, 88, 107, 128]) appear in the case of the degenerated shell elements.
Chapter 1. Introduction
1.2
8
Objectives
The developments presented in the dissertation have been motivated by (i) the lack of
general locking-free shell finite element that is robust and reliable in both bending- and
membrane dominated situations, for both h- and p-extensions, and give accurate and reliable numerical results not only for the displacements but also for the stresses, and (ii)
the limited applicability of the principle of complementary virtual work to time-dependent
problems. In view of this, the goals of the dissertation are:
• to modify and extend the dual-mixed variational formulation of Hellinger–Reissnertype to elastodynamic problems of linearly elastic solids under isothermal conditions,
neglecting the electric- and magnetic effects,
• to derive a new dimensionally reduced axisymmetric shell model for elastodynamic
analysis from the three-field dual-mixed variational principle of Hellinger–Reissnertype without the application of the classical kinematical hypotheses regarding the
deformation of the normal to the shell mid-surface,
• to deduce the Euler–Lagrange equations and the natural boundary conditions of the
shell model, considering axisymmetrically loaded thin shells of revolution,
• the derivation of the related differential equation system and boundary conditions
for the fundamental variables using the unmodified inverse stress-strain relations for
linearly elastic, homogeneous and isotropic materials,
• the development of a new hp-type finite element model for the elastodynamic problems
of thin cylindrical shells and the related numerical algorithms and finite element codes,
• to implement the new dual-mixed hp finite element model and performing numerical
static, frequency and transient vibration analysis for cylindrical shells, as well as
to compare the computational results obtained to displacement-based models, their
analytical and finite element solutions,
• to prove through numerical examples that the h- and p-version dual-mixed cylindrical
shell elements developed are locking-free for nearly incompressible materials and well
suited for the numerical analysis of both thin and moderately thick cylindrical shells.
The text of the dissertation is organized into nine chapters. After the overview of the
literature, a detailed differential geometric description of surfaces and shells is given in
Chapter 2 and Chapter 3, respectively.
In Chapter 4, a four-field dual-mixed variational principle are derived for elastodynamic problems of linearly elastic continua. After eliminating the impulse field from
this principle, a three-field dual-mixed variational formulation in terms of non-symmetric
stresses, displacements and rotations as independent variables is obtained. The special form
of its functional for thin shells of revolution is also derived in Chapter 4.
Chapter 5 is devoted to the dimensional reduction procedure. Firstly, the threedimensional translational equations of motion and constitutive equations are written down
in the curvilinear coordinate system attached to the axisymmetric shell mid-surface. Thereafter employing truncated power series expansions, the independent variables, i.e., the
1.2. Objectives
9
stresses, the rotations and the displacements are approximated by linear and quadratic polynomials in the thickness direction. After inserting the expanded variables into the three-field
dual-mixed variational principle for thin shells of revolution, the Euler–Lagrange equations
and the natural boundary conditions are deduced assuming axisymmetrical loads. Then
the related differential equations and boundary conditions are derived using the unmodified
inverse stress-strain relations for linearly elastic, homogeneous and isotropic materials.
A dual-mixed hp-type finite element formulation with appropriately chosen polynomial
stress- and displacement interpolation and C 0 continuous normal components of stresses is
constructed and presented for the bending-shearing static and dynamic problems of cylindrical shells in Chapter 6. The h- and p-version cylindrical shell elements proposed herein
are tested through static and dynamic problems for nearly incompressible, thin and moderately thick cylindrical shells in Chapter 7. The dissertation closes with a brief summary
and the list of the most important theoretical and numerical results, i.e., the theses, as well
as the author’s publications.
Chapter 2
Differential geometry of surfaces
2.1
Surface coordinate system and its base vectors
Let ω0 ⊂ R2 be a two-dimensional simply-connected domain bounded by the curve γ0 and
ξ α define the Cartesian coordinates of a generic point P0 on ω0 . Let
ω 0 → S0 ,
γ 0 → ℓ0
(2.1.1)
be a sufficiently smooth and injective mapping, where S0 is a simply-connected surface
and ℓ0 defines its boundary line (see Figure 2.1). The pair of ξ 1 and ξ 2 then furnish the
ξ
xk (ξ α )
ξ2
2
ξ3
x3
γ0
S0
ω0
e3
ξ1
e1
P0
ξ1
x
r0
O
1
a3
a2
P0
ξ1
a1
ℓ0
ξ2
e2
x2
Figure 2.1: Surface coordinate system and its covariant base vectors.
curvilinear coordinates for parametrizing the surface S0 . Naturally, there are infinitely
many ways of defining curvilinear coordinates ξ α , depending on how the domain ω0 and
mapping are chosen. For instance, a sphere surface may be parametrized by Cartesian,
spherical, or stereographic coordinates. The position vector of each point P0 ∈ S0 can be
unambiguously given by
r0 = r0 (ξ α ) = xk ek ,
(2.1.2)
where xk are identified with a set of coordinate lines of the fixed right-handed orthogonal
Cartesian frame in an Euclidean three-dimensional space, the corresponding unit base vectors are denoted by ek (see Figure 2.1). The relation between the Cartesian coordinates xk
and the surface coordinates ξ α can briefly be written as
xk = xk (ξ α ) .
10
(2.1.3)
2.2. First fundamental form
11
If it is assumed that the function (2.1.3) is differentiable with respect to the surface coordinate ξ α , the tangent vectors to the curvilinear coordinate lines ξ α at any point P0 ∈ S0
are given by
∂r0
aα = α .
(2.1.4)
∂ξ
The unit normal vector to the surface S0 at an arbitrary point P0 is obtained as1
a3 = a3 =
a1 × a2
.
|a1 × a2 |
(2.1.5)
Let the coordinate ξ 3 be the arc-length measured along the normal a3 to the surface S0 and
ξ 3 = 0 holds true at every point P0 . The vectors ak defined by (2.1.4)–(2.1.5) interpret the
covariant bases of the so-called surface coordinate system at an arbitrary point P0 ∈ S0 .
2.2
First fundamental form
Taking into account (2.1.2) and (2.1.4), the infinitesimal line element on the surface S0 can
be written in the form
∂r0
dr0 = α dξ α = aα dξ α .
(2.2.1)
∂ξ
The square of the line element (2.2.1), called the first fundamental form of the surface S0 ,
is given by2

dr0 · dr0 = (aα dξ α ) · aβ dξ β = aαβ dξ α dξ β ,
(2.2.2)
where the surface covariant metric tensor
aαβ = aβα = aα · aβ
(2.2.3)
δ κα = aκβ aβα ,
(2.2.4)
aκβ = aβκ = aκ · aβ
(2.2.5)
aκ = aκβ aβ .
(2.2.6)
aα3 = a3α = aα · a3 = a3 · aα = 0 ,
(2.2.7)
a33 = a3 · a3 = 1
(2.2.8)
satisfies
in which
is the contravariant metric tensor on the surface S0 and δ κα = aκ · aα is the Kronecker-delta.
The contravariant base vectors aκ of the surface coordinate system can be obtained by
Due to equation (2.1.5), the formulas
and
1
2
The symbol × designates the cross product of two tensorial quantities.
The centered dot indicates the scalar product of two tensorial quantities.
12
Chapter 2. Differential geometry of surfaces
are also valid. The determinants of the surface covariant and contravariant metric tensors
are interpreted, respectively, by
a = det(aαβ ) = a11 a22 − (a12 )2 ,
and
(2.2.9)
1
.
(2.2.10)
a
The covariant and contravariant permutation tensors on the surface S0 are defined, respectively, as
√
√
ǫαβ = ǫαβ3 = a eαβ3 = a eαβ ,
(2.2.11)
1
1
ǫ κλ = ǫ κλ3 = √ eκλ3 = √ eκλ ,
(2.2.12)
a
a

det aκλ =
where
8
αβ
eαβ = e
>
<
=>
:
1,
even
−1 , if αβ is odd permutation of the numbers 1, 2 ,
0,
not a
(2.2.13)
for which the truth of the identities
eαβ eκλ = δ κα δ λβ − δ λα δ κβ ,
(2.2.14)
eαβ eκβ = δ κα
(2.2.15)
and
can easily be verified. The surface contravariant metric tensor, on account of (2.2.4), can
be determined as the inverse of (2.2.3):
aκλ =
2.3
1 κβ λα
e e aαβ .
a
(2.2.16)
Second and third fundamental forms. Principal
curvatures

The infinitesimal line element in the normal direction a3 = a3 ξ β to the surface S0 can be
written as3
∂a3
da3 = β dξ β = a3,β dξ β .
(2.3.1)
∂ξ
The second fundamental form of the surface S0 may be defined by the scalar product of
the line elements (2.2.1) and (2.3.1):

−dr0 · da3 = − (aα dξ α ) · a3,β dξ β = −aα · a3,β dξ α dξ β = bαβ dξ α dξ β ,
3
(2.3.2)
A comma followed by an index m or β in the subscript means partial differentiation with respect to
the corresponding coordinates ξ m or ξ β .
2.3. Second and third fundamental forms. Principal curvatures
13
where, by (2.1.4) and (2.2.7),
bαβ = −aα · a3,β = aα,β · a3 =
∂ 2 r0
= bβα
∂ξ α ∂ξ β
(2.3.3)
interprets the covariant curvature tensor of the surface S0 . Applying (2.2.6) to (2.3.3), the
following expressions are obtained:
bκβ = aκλ bλβ = aκλ (−aλ · a3,β ) = −aκ · a3,β ,
(2.3.4)
bκβ = aκλ bλβ = aκλ (aλ,β · a3 ) = aκ,β · a3 .
(2.3.5)
a3,β = (a3,β · aα ) aα = −bαβ aα ,
a3,β = (a3,β · aκ ) aκ = −bκβ aκ ,
(2.3.6)
(2.3.7)
Making use of (2.3.3) and (2.3.4) we have
which are known as Weingarten’s formulas, see [66]. The characteristic equation of the
eigenvalue problem of the curvature tensor bκβ may be written as
κ2 − 2Hκ + K = 0 ,
(2.3.8)
where

1 α 1 α β
1
1 1
tr b β = b β δ α = bαα =
b 1 + b22
2
2
2
2
is the mean curvature of the surface S0 and
H=

K = det bαβ =
1
eαβ eκλ bακ bβλ = b11 b22 − b12 b21
2
(2.3.9)
(2.3.10)
is the Gaussian curvature of the surface S0 . It can clearly be seen from (2.3.9)–(2.3.10) that
both H and K are surface invariants which depend only on the curvature tensor, but the
Gaussian curvature K can also be expressed entirely in terms of the surface covariant and
contravariant metric tensors and their derivatives, see [113]. The roots κ = κ1 and κ = κ2
of (2.3.8) are the principal curvatures of the surface S0 at the point P0 in question and the
corresponding directions are called principal directions. It can be proven that the roots of
(2.3.8) are always real at any point P0 ∈ S0 . The principal radii of curvature are defined
by the forms [113]
1
1
R1 = − ,
R2 = − ,
(2.3.11)
κ1
κ2
where by (2.3.8),
κ1 + κ2 = −
1
1
−
= 2H ,
R1 R2
κ1 κ2 =
1
=K
R1 R2
(2.3.12)
hold true. The sign convention in (2.3.11) is according to the rule that R1 and R2 are positive
(negative) if the unit normal vector to the surface S0 is directed away from (toward) the
center of curvature. Depending on the sign of the Gaussian curvature K, a surface point
P0 is said to be
elliptic
parabolic
hyperbolic
9
8
>
=
>
<
>
;
if
>
:
K > 0 , e.g. ellipsoid or spherical surfaces,
K = 0 , e.g. conical or cylindrical surfaces,
K < 0 , e.g. hyperbolic paraboloid or ring surface.
(2.3.13)
14
Chapter 2. Differential geometry of surfaces
The third fundamental form, with the aid of (2.3.4)–(2.3.7), is defined as the scalar product
da3 · da3 = a3,α · a3,β dξ α dξ β = bλα bκβ aλ · aκ dξ α dξ β = bλα bκβ δ λκ dξ α dξ β
= bαλ bλβ dξ α dξ β = bαλ aλκ bκβ dξ α dξ β = cαβ dξ α dξ β ,
(2.3.14)
cαβ = cβα = bαλ aλκ bκβ = bαλ bλβ .
(2.3.15)
in which
2.4
Surface Christoffel symbols
The surface Christoffel symbols of the first- and second kind4 are given, respectively, by
Γsmℓ = as,m · aℓ = as,m · ak akℓ ,
(2.4.1)
Γsm = as,m · ak = −ak,m · as = Γsmℓ aℓk .
(2.4.2)
k
Upon substitution of the formulas for the curvature tensors from (2.3.3) and (2.3.5) into
(2.4.2) we have
κ
Γαβ = aα,β · aκ = −aκ,β · aα ,
(2.4.3)
Γκλ = aκ,λ · a3 = −a3,λ · aκ = bκλ ,
3
(2.4.4)
Γ3κ = a3,κ · aµ = −aµ,κ · a3 = −bµκ ,
µ
(2.4.5)
Γλ3 = aλ,3 · aµ = −aµ,3 · aλ = 0 ,
µ
(2.4.6)
Γ33 = a3,3 · am = −am,3 · a3 = 0 ,
m
(2.4.7)
3
(2.4.8)
3
(2.4.9)
Γ3κ = a3,κ · a3 = −a3,κ · a3 = 0 ,
Γλ3 = aλ,3 · a3 = −a3,3 · aλ = 0 .
It follows unambiguously from relations (2.1.2)–(2.1.5) that the Christoffel symbols (2.4.5)–
(2.4.6) are not symmetric with respect to the interchange of their subscripts, however the
Christoffel symbols (2.4.3) retain this type of symmetry property. In view of this and
(2.2.3), as well as (2.4.1), the partial differentiation of the surface covariant metric tensor
aαλ with respect to the surface coordinates ξ β can be expressed by
aαλ,β = (aα · aλ ),β = aα,β · aλ + aα · aλ,β = Γαβλ + Γλβα .
(2.4.10)
By cyclically permuting the free indices α, β, λ, we obtain two further equivalent equations
aλβ,α = Γλαβ + Γβαλ ,
aβα,λ = Γβλα + Γαλβ .
4
(2.4.11)
(2.4.12)
A bar over a tensorial variable denotes its tensorial components related to the base vectors ak of the
surface S0 .
2.5. Covariant derivatives on a surface
15
Multiplying (2.4.10)–(2.4.12) by 1/2 and subtracting (2.4.12) from the sum of (2.4.10)–
(2.4.11), and taking into account the symmetry properties of both Γαβλ and aαλ , the following formula can be obtained:
Γαβλ =
1
(aαλ,β + aλβ,α − aβα,λ ) .
2
(2.4.13)
Contracting both sides of the previous equation with aκλ , the Christoffel symbols (2.4.3)
can be written in the form
1
κ
Γαβ = aκλ (aαλ,β + aλβ,α − aβα,λ ) .
2
(2.4.14)
It means that the surface Christoffel symbols (2.4.3)–(2.4.5) can be unambiguously determined with knowledge of the metric- and the curvature tensors of the surface S0 .
2.5
Covariant derivatives on a surface
Let
T = T (ξ α ) = T
kℓ
(ξ α ) ak (ξ α ) aℓ (ξ α )
(2.5.1)
be an arbitrary second-order tensor field defined on the surface S0 and
∇0 =
∂ α
a
∂ξ α
(2.5.2)
interpret the two-dimensional differential operator on the surface S0 . With the aid of (2.4.2),
the two-dimensional covariant derivative of (2.5.1) on the surface S0 can be written as
0
kℓ

T∇ = T ak aℓ
where
T
kℓ
|α
=T
kℓ
,α

∂ α
kℓ
a = T |α ak aℓ aα ,
∂ξ α
k
+ Γsα T
sℓ
ℓ
+ Γrα T
kr
.
(2.5.3)
(2.5.4)
On account of (2.4.3)–(2.4.9), (2.5.4) can be separated into
κλ
|α
3λ
T |α
κ3
T |α
33
T |α
T
where
T
κλ
kα
= T
= T
= T
= T
=T
κλ
3λ
κ3
κ
− bλα T ,
kα − b α T
3λ
κλ
33
− bλα T ,
kα + bκα T
κ3
κλ
33
− bκα T ,
kα + bλα T
33
κ3
3λ
+ bλα T
,α + bκα T
κλ
,α
κ
+ Γβα T
βλ
λ
+ Γµα T
κµ
,
(2.5.5)
(2.5.6)
(2.5.7)
(2.5.8)
(2.5.9)
and
T
T
3λ
kα
κ3
kα
=T
=T
3λ
,α
κ3
,α
λ
3β
κ
β3
+ Γβα T
+ Γβα T
,
(2.5.10)
(2.5.11)
Chapter 2. Differential geometry of surfaces
16
κλ
are, respectively, the two-dimensional covariant derivatives of the surface tensor field T
3λ
κ3
and the surface vector fields T , T with respect to the surface metric. Analogously to
(2.5.1)–(2.5.3), the two-dimensional covariant derivative of an arbitrary vector field
u = u (ξ α ) = uk (ξ α ) ak (ξ α)
(2.5.12)
defined on the surface S0 can be given by
0

k
u∇ = uk a
i.e.,

∂ α
ℓ
=
u
−
Γ
u
ak aα = uk|α ak aα ,
a
k,α
ℓ
kα
α
∂ξ
ℓ
uk|α = uk,α − Γkα uℓ ,
(2.5.13)
(2.5.14)
from which, with the aid of (2.4.3)–(2.4.9), relations similar to (2.5.5)–(2.5.8) can be generated in the following forms:
uκ|α = uκkα − bκα u3 ,
u3|α = u3,α + bκα uκ ,
where
λ
uκkα = uκ,α − Γκα uλ
(2.5.15)
(2.5.16)
(2.5.17)
is the two-dimensional covariant derivative of the surface vector field uκ with respect to the
surface metric.
Chapter 3
Differential geometry of shells
3.1
Undeformed configuration of a shell
The shell is considered as a three-dimensional body, for which it is assumed that one of its
dimensions is much smaller than any other of its characteristic lengths. Let xk refer to a
fixed right-handed orthogonal Cartesian frame, the corresponding base vectors are denoted
by ek . Let Ω ⊂ R3 be a three-dimensional simply-connected reference domain bounded by
two planar surfaces ω + ⊂ R2 , ω − ⊂ R2 and the lateral surface ω × , i.e., ∂Ω = ω ± ∪ ω × .
The surfaces ω + and ω − are symmetrically situated with respect to a planar reference
surface ω0 ⊂ R2 – shown in green in Figure 3.1 – and are called upper and lower surfaces.
The lateral surface ω × intersects the reference surface ω0 in the reference curve γ0 . The
S
ξ3
+
V
S0
xk (ξ m)
+ ξ
ω0 ω
Ω γ+
P0
ξ1
γ0
ω×
ℓ+
P0
3
ω
−
γ−
ξ
ξ1
x3
2
e1
ℓ0
ℓ
e3
O e2
x1
S×
−
2
S− ξ
x2
Figure 3.1: Undeformed configuration of a shell.
boundary curves of the upper and lower surfaces ω ± are denoted by γ ± . Let the reference
domain Ω be parametrized by a set of normal coordinates ξ m : the Cartesian coordinates
ξ α are defined on ω0 and the rectilinear coordinate ξ 3 is measured along the normal to the
reference surface ω0 , and ξ 3 = 0 is valid on ω0 . The distance d, between the upper and
lower surfaces ω ± , measured along the normal to ω0 is the thickness. It will be assumed
that d is constant. Let
Ω→V :
r = r(ξ m ) ,
17
xk = xk (ξ m ) ,
(3.1.1)
18
Chapter 3. Differential geometry of shells
r0 = r0 (ξ α ) ,
ω 0 → S0 :
xk = xk (ξ α ) ,
(3.1.2)
and
γ 0 → ℓ0
(3.1.3)
be a sufficiently smooth and injective mapping, where the simply-connected surface S0 is
called the mid-surface of the shell, ℓ0 defines its boundary line and r is the position vector
of an arbitrary shell point. If r is located on the shell mid-surface then it is denoted by
r0 , see Figure 3.2 for details. The pair of ξ 1 and ξ 2 furnish the curvilinear coordinates on
the shell mid-surface S0 . Furthermore it is assumed that the thickness d does not change
and the coordinate line ξ 3 remains rectilinear and normal to S0 after the transformation
(3.1.1)–(3.1.3). Thus the undeformed configuration of the shell is defined by
¨
d
V = ξ | ξ ∈ S0 , |ξ | 6
2
m
α
«
3
.
(3.1.4)
The boundary of V , denoted by S, consists of the top and bottom surfaces
¨
d
ω → S = ξ | ξ ∈ S0 , ξ = ±
2
±
±
m
α
«
3
,
(3.1.5)
,
(3.1.6)
and the lateral surface
¨
d
ω → S = ξ | ξ ∈ ℓ0 , |ξ | <
2
×
m
×
α
«
3
i.e., ∂Ω → S = S ± ∪ S × (see Figure 3.1). The boundary curves of the top and bottom
surfaces S ± are defined by
¨
d
γ → ℓ = ξ | ξ ∈ ℓ0 , ξ = ±
2
±
±
m
α
3
«
.
(3.1.7)
The covariant base vectors ak of the mid-surface coordinate system can be determined from
expressions (2.1.4)–(2.1.5).
3.2
Local coordinate system at an arbitrary shell point.
The shifter
The position vector of any shell point P can be written as
r(ξ m ) = r0 (ξ α ) + ξ 3 a3 (ξ α ) ,
(3.2.1)
(see Figure 3.2). Taking into account (2.1.4) and (3.2.1), the covariant base vectors of the
local coordinate system at P can be given as
∂r
∂r0
∂a3
= α + ξ 3 α = aα + ξ 3 a3,α ,
α
∂ξ
∂ξ
∂ξ
∂r
=
= a3 = a3 .
3
∂ξ
gα =
(3.2.2)
g3
(3.2.3)
3.2. Local coordinate system at an arbitrary shell point. The shifter
19
ξ3
g3
P
g1
ξ
g2
1
ξ2
a3
P0 a2
a1
r
ξ1
ξ2
x3
S0
r0
O
x1
x2
Figure 3.2: Local coordinate systems and their base vectors.
Substitution of Weingarten’s formula (2.3.7) into (3.2.2) yields

gα = aα − ξ 3 bκα aκ = 0 gα + 1 gα ξ 3 = δ κα − ξ 3 bκα aκ = µκα aκ ,
(3.2.4)
where
0 gα
= aκ ,
1 gα
= bκα aκ ,
(3.2.5)
and the tensor
µκα = δ κα − bκα ξ 3
(3.2.6)
is the shifter. The matrix of (3.2.6) can be represented by

[µκα ]
=
1 − b11 ξ 3 −b21 ξ 3
−b12 ξ 3 1 − b22 ξ 3

.
(3.2.7)
µ33 = 1 .
(3.2.8)
It follows from (3.2.3)–(3.2.4) that
µκ3 = 0 ,
µ3α = 0 ,
Using equations (2.2.13)–(2.2.14), (2.3.9)–(2.3.10) and (3.2.6)–(3.2.8), we obtain the formula

2
1
µ = det(µkℓ ) = eκβ eαλ µκα µβλ = 1 − 2Hξ 3 + K ξ 3
2
(3.2.9)
for the determinant of the shifter µkℓ . In view of (3.2.6), the partial derivative of the shifter
with respect to the thickness coordinate ξ 3 can be written as
µκα,3 = −bκα .
Let us assume that
3
ξ < |Rmin | ,
(3.2.10)
(3.2.11)
where Rmin stands for the least principal radius of curvature of the shell mid-surface S0 . If
this condition is satisfied then (a) the shifter is not singular and (b) it possesses a unique
20
Chapter 3. Differential geometry of shells
k
inverse which is denoted by (µ−1 ) ℓ , see [113]. With the application of the Cayley–Hamilton
theorem, [87], the inverse shifter can be given by

µ−1
κ
α
and
=

1 κ
1 κ κ
1 κ
δ α + b α − bλλ δ κα ξ 3 =
δ α + (bκα − 2Hδ κα ) ξ 3 ,
[δ α tr (µκα ) − µκα ] =
µ
µ
µ
(3.2.12)

µ−1
κ
3

µ−1
=0,
3
α

µ−1
=0,
κ
(µ−1 ) α
which is due to (3.2.8). Another representation of
expansion with respect to the thickness coordinate [113]:

µ−1
κ
α

= δ κα + bκα ξ 3 + bκλ bλα ξ 3
2

+ bκλ bλβ bβα ξ 3
3
+ ... =
3
3
=1,
(3.2.13)
is the following power series
∞ h
X

(Bk )κα ξ 3
k i
,
(3.2.14)
k=0
where
(B0 )κα = δ κα , (B1 )κα = bκα , (B2 )κα = bκλ bλα , . . . ,
i.e.,
(Bk )κα = bκλ (Bk−1 )λα ,
k ≥1.
(3.2.15)
(3.2.16)
In view of (3.2.3)–(3.2.4), the relations between the covariant base vectors at P and P0 ∈ S0 ,
being on the same normal to the shell mid-surface, can be written in the form
gℓ = µkℓ ak ,

aℓ = µ−1
k
ℓ
gk ,
(3.2.17)
and the relationship between the covariant metric tensors at P and P0 ∈ S0 are obtained
as
(3.2.18)
gkℓ = gk · gℓ = (µmk am ) · (µnℓ an ) = µmk µnℓ amn .
Taking into account (2.2.7)–(2.2.8), as well as the third fundamental form (2.3.15) and
(3.2.6)–(3.2.8), (3.2.18) can be separated into

gαβ = µκα µλβ aκλ = aαβ − 2 bαβ ξ 3 + cαβ ξ 3
2
,
(3.2.19)
and
gα3 = 0 ,
g3β = 0 ,
g33 = 1 .
Using the multiplication rule for the determinant of (3.2.18) we have
√
√
det(gkℓ ) = g = µ2 a ,
g =µ a.
(3.2.20)
(3.2.21)
Taking into account (3.2.18), from (3.2.17) we obtain the formulas
am = µmp gp ,

gm = µ−1
m
p
ap .
(3.2.22)
With the application of (3.2.17) and (3.2.22), an arbitrary tensor field T = T(ξ α, ξ 3 ) defined
at the point P of the shell can be shifted to the point P0 of the shell mid-surface:
n
T = Tkℓ gk gℓ = T m am an ,
(3.2.23)
3.3. Relationships between covariant derivatives
where

n
T m = Tkℓ µ−1
21
k
m
µnℓ
(3.2.24)
are the shifted tensor components. We can also write:
kℓ
T = T ak aℓ = T pq gp gq ,
in which
T pq = T
(3.2.25)
kℓ −1 p −1 q
µ
µ ℓ
k
(3.2.26)
are the tensorial components of T related to the covariant base vectors of the local coordinate system at P . Similarly to (3.2.23)–(3.2.26), the components of an arbitrary vector
field
(3.2.27)
u = up gp = uk gk = uq aq = u ℓ aℓ
defined at the shell point P can be transformed as

up = uq µqp ,
uq = up µ−1
u ℓ = uk µℓk ,
uk =
p
q

ℓ
−1 k
u µ
ℓ
,
(3.2.28)
.
(3.2.29)
Taking into account (3.2.21), the relations between the permutation tensors at P and
P0 ∈ S0 can be given as
√
√
ǫαβ = ǫαβ3 = g eαβ3 = µ a eαβ3 = µǫαβ3 = µǫαβ ,
(3.2.30)
1
1
1
1
ǫκλ = ǫκλ3 = √ eκλ3 = √ eκλ3 = ǫ κλ3 = ǫ αβ ,
g
µ a
µ
µ
(3.2.31)
where
8
kℓm
e
1,
even
= ekℓm = > −1 , if kℓm is odd permutation of the numbers 1, 2, 3 .
:
0,
not a
>
<
(3.2.32)
In addition the following identities hold:
ekℓm ekqr = δ qℓ δ rm − δ rℓ δ qm ,
(3.2.33)
ekℓm ekℓr = 2 δ rm .
(3.2.34)
and
3.3
Relationships between covariant derivatives
Having deduced the transformation relations (3.2.24), (3.2.26) and (3.2.28)–(3.2.29), we
now proceed to ascertain relationships between the three-dimensional covariant derivatives
at the arbitrary shell point P and mid-surface point P0 . Let
∇=
∂ m
∂
∂
∂
∂
g = α gα + 3 g3 = α gα + 3 a3 ,
m
∂ξ
∂ξ
∂ξ
∂ξ
∂ξ
(3.3.1)
22
Chapter 3. Differential geometry of shells
and
∂
∂
∂
∂ m
a = α aα + 3 a3 = ∇0 + 3 a3
(3.3.2)
m
∂ξ
∂ξ
∂ξ
∂ξ
interpret, respectively, the three-dimensional differential operators at P and P0 ∈ S0 , being
on the same normal to the shell mid-surface. With the aid of the definitions (2.4.2) and
(3.3.2), the three-dimensional covariant derivative of the second-order tensor (3.2.25) on
the shell mid-surface S0 can be written as
∇=
kℓ

T∇ = T ak aℓ
in which
T
kℓ
|m
=T
kℓ
,m

∂ m
kℓ
a
= T |m ak aℓ am ,
m
∂ξ
k
+ Γsm T
sℓ
ℓ
+ Γrm T
kr
,
(3.3.3)
(3.3.4)
on account of (2.4.6)–(2.4.9), can be separated into the two-dimensional covariant derivative
T
kℓ
|ν
=T
kℓ
,ν
k
sℓ
+ Γsν T
ℓ
+ Γrν T
kr
,
(3.3.5)
detailed in Section 2.5, and the partial derivative
kℓ
|3
T
=T
kℓ
,3
.
(3.3.6)
Analogously to (3.3.3), the gradient of the vector field (3.2.27) on the shell mid-surface S0
can be given by

∂ m
ℓ
u∇ = uℓ a
a
= uℓ |m aℓ am ,
(3.3.7)
∂ξ m
where
s
uℓ |m = uℓ,m − Γℓ m us ,
(3.3.8)
from which, with the aid of (2.4.6)–(2.4.9), the relations similar to (3.3.5)–(3.3.6) can be
generated in the following forms
s
uℓ |λ = uℓ,λ − Γℓλ us ,
uℓ |3 = uℓ,3 .
(3.3.9)
(3.3.10)
In view of (2.4.2), (3.2.17) and (3.3.1), the three-dimensional covariant derivative of the
second-order tensor (3.2.25) at an arbitrary shell point P can be given by

∂ m
= T pq;m gp gq gm ,
T∇ = (T gp gq )
g
m
∂ξ
pq
or

∂ m
kℓ
−1 p
−1 q
T∇ = T ak aℓ
g
=
T
µ
µ
g g gm .
|m
k
ℓ p q
∂ξ m
From the comparison of the latter two equations, it follows that
kℓ
(3.3.11)
T pq;m = T

kℓ
−1 p
−1 q
µ
|m µ
k
ℓ
,
(3.3.12)
(3.3.13)
expressing the relationship between the three-dimensional covariant derivatives of a secondorder tensor at P and P0 ∈ S0 being on the same normal to the shell mid-surface. Having
repeated (3.3.11) and (3.3.12) with the vector field (3.2.27), the following transformation
relations are obtained:
uk;m = uℓ |m µℓk ,

uk;m = u ℓ|m µ−1
k
ℓ
.
(3.3.14)
3.4. Volume and surface elements of the shell
3.4
23
Volume and surface elements of the shell
We now review the fundamental formulas relating to the volume and surface elements of the
shell. In regard to (3.2.21), the volume element of the shell at an arbitrary point P (ξ α, ξ 3 )
and the vectorial mid-surface element are defined, respectively, by the scalar triple product
√
√
dV = ( g1 × g2 ) · g3 dξ 1 dξ 2 dξ 3 = g dξ 1 dξ 2 dξ 3 = µ a dξ 1 dξ 2 dξ 3 ,
(3.4.1)
and the cross product

dS 0 = dξ 1 a1 × dξ 2 a2 =
in which
dS0 =
√
√
a dξ 1 dξ 2 a3 ,
a dξ 1 dξ 2
(3.4.2)
(3.4.3)
is the scalar mid-surface element of the shell. With the aid of (3.4.3), (3.4.1) can be put in
the form
dV = µ dS0 dξ 3 .
(3.4.4)
Taking into account (3.2.21), the scalar area elements of the top and bottom surfaces S ±
are also expressed in terms of the mid-surface element (3.4.3) as
dS ± = µ± dS0 ,
where

d
µ =µ ±
2

±
= 1 ∓ Hd +
(3.4.5)
Kd 2
4
(3.4.6)
is the value of the determinant (3.2.9) at ξ 3 = ±d/2.
We now proceed with the derivation of the lateral surface element. Let c0 , with arclength s0 , be a smooth curve on the shell mid-surface S0 , and let the normal vectors a3
to S0 along c0 generate the lateral surface S × of the shell. If the arc-length of the curve
ξ3
a 3 a3 = g3
dS ×
P
×
c
t× = t
c0
n× = n
s
s0
Figure 3.3: Lateral surface element with its orthogonal base vectors.
24
Chapter 3. Differential geometry of shells
c, defined by the intersection of S × and the surface ξ 3 = const., are denoted by s, the
coordinate curves on the lateral surface S × can be taken as ξ 3 and s with orthogonal base
vectors a3 and t× (see Figure 3.3). Recalling (3.2.1) and since
dξ 3
=0,
ds
(3.4.7)
the tangent vector to the curve s passing through the point P × (ξ α ∈ S × , ξ 3 ) is given by
t× = t =
dr
κ
= tα gα = tα µκα aκ = t aκ ,
ds
where
tα =
∂ξ α
,
∂s
for which
(3.4.8)
(3.4.9)
α
tα ds = t ds0
(3.4.10)
is valid. The outward normal vector to the lateral surface S × at P × , with the use of
(3.2.30)–(3.2.31) and (3.4.8), can be written as
n× = n = t × g3 = ǫα3β tα gβ = ǫβα tα gβ = nβ gβ
κ
κ
= t × a3 = ǫκ3λ t aλ = ǫλκ t aλ = nλ aλ
(3.4.11)
(see Figure 3.3). In view of this and (3.4.8), the vectorial area element of S × at P × can be
given by

dr
dS = n dS = n ds dξ = (t × g3 ) ds dξ =
× g3 ds dξ 3 = (dr × g3 ) dξ 3 .
ds
(3.4.12)
Recalling (3.2.21)–(3.2.22) and (3.2.30), since dr = dξ β gβ , (3.4.12) becomes
×
×
3
3
dS × = dξ β dξ 3 (gβ × g3 ) = dξ β dξ 3 ǫβ3α gα

α
√
√
= dξ β dξ 3 g eαβ gα = µ a eαβ µ−1 λ aλ dξ β dξ 3 .
3.5
(3.4.13)
Geometry of shells of revolution
Let us consider now a shell of revolution as a three-dimensional body, the rotation axis of
which is interpreted by
0 ≤ x1 ≤ L ,
(3.5.1)
where L is the length of the axisymmetric shell (see Figure 3.4). In accordance with the
notations and the definitions introduced in Section 3.1, the axisymmetric shell is defined as
a region of space bounded by two axisymmetric surfaces S + and S − called the outer and
inner surfaces which are symmetrically situated with respect to the mid-surface S 0 , and the
lateral surfaces S 0 and S L , the intersections of which with S0 are the circles ℓ0 (0) and ℓ0 (L)
with radius R0 and RL , respectively, i.e., the boundary of the axisymmetric shell domain
V is defined by S = S ± ∪ S 0 ∪ S L (see Figure 3.4). The mid-surface of the axisymmetric
3.5. Geometry of shells of revolution
25
e2
x2
S
0
SL
ℓ0(L)
S0
S+
ℓ0(0)
S−
ξ 1 a3
R0
r0
R
a2 a1
P0
ξ3
RL
x1
ξ2
e1
ξ1
x3
Rξ 2
e3
L
Figure 3.4: Axisymmetric shell.
shell can be generated by rotating a curve about the x1 -axis. This curve is the meridian
curve defined by the twice-differentiable function

R = R x1 > 0 ,
for which

R x1 = 0 = R0 ,
(3.5.2)

R x1 = L = RL
(3.5.3)
(see Figure 3.5). The axisymmetric shell is parametrized by the thickness coordinate ξ 3
ξ3
R(x1)
a3
ξ
1
a1
P0
R(x1 )
R0
RL
r0
x1
Figure 3.5: Meridian curve.
and the mid-surface coordinates ξ α . The curvilinear coordinate ξ 1 is measured along the
meridian curve and the polar angle
0 ≤ ξ 2 < 2π
(3.5.4)
26
Chapter 3. Differential geometry of shells
is measured along the latitude circles lying in the planes perpendicular to the rotation axis.
With knowledge of function (3.5.2), relation (2.1.3) between the Cartesian coordinates xk
and the curvilinear coordinates ξ α are given by

x1 = f ξ 1 ,
x2 = R sin ξ 2 ,
x3 = R cos ξ 2 ,
(3.5.5)
on the basis of which the position vector (2.1.2) of an arbitrary point P0 ∈ S0 can be written
in the Cartesian coordinate frame as
r0 = xk ek = x1 e1 + R sin ξ 2 e2 + R cos ξ 2 e3 .
(3.5.6)
Applying equations (2.2.1)–(2.2.2) to the vector field (3.5.6), the square of the infinitesimal
arc-length ds0 = |dr0 | on the mid-surface of the axisymmetric shell can be expressed as

(ds0 )2 = dr0 · dr0 = 1 + (R′ )
2
dx1
2
where1

+ R2 dξ 2
2
= (ds01 )2 + (ds02 )2 ,
(3.5.7)
q
1
ds01 = dξ =
1 + (R′ )2 dx1
(3.5.8)
denotes the infinitesimal arc-length along ξ 1 and
ds02 = A2 dξ 2 = R dξ 2
(3.5.9)
stands for the infinitesimal circular arc-length along ξ 2 on the mid-surface S0 . Introducing
the notation
q

(3.5.10)
A1 x1 = 1 + (R′ )2 ≥ 1 ,
and using equation (3.5.8), we obtain the integral

ξ 1 = ξ 1 x1 =
Z x1
x̂1 =0

A1 x̂1 dx̂1
(3.5.11)
for the inverse of the function x1 = f (ξ 1 ). From the combination of (3.5.8) and (3.5.10), it
follows that
dξ 1
dx1
1
=
A
,
=
.
(3.5.12)
1
1
1
dx
dξ
A1
Applying (2.1.4)–(2.1.5) to (3.5.6) and making use of (3.5.12), the orthogonal base vectors
of the mid-surface coordinate system are obtained as
∂r0
1
1 ′
1 ′
=
e1 +
R sin ξ 2 e2 +
R cos ξ 2 e3 ,
1
∂ξ
A1
A1
A1
∂r0
=
= R cos ξ 2 e2 − R sin ξ 2 e3 ,
∂ξ 2

a1 × a2
1
=
=
sin ξ 2 e2 + cos ξ 2 e3 − R′ e1 .
|a1 × a2 |
A1
a1 =
(3.5.13)
a2
(3.5.14)
a3
(3.5.15)
With knowledge of (3.5.13)–(3.5.14), the covariant components of the metric tensor on the
mid-surface S0 can be computed on the basis of (2.2.3) as
a11 = a1 · a1 = 1 ,
1
a22 = a2 · a2 = R2 ,
a12 = a1 · a2 = a2 ·a1 = a21 = 0 .
(3.5.16)
The prime in the superscript indicates differentiation with respect to the Cartesian coordinate x1 .
3.5. Geometry of shells of revolution
27
In view of this and (2.2.3), the matrices of the covariant and contravariant metric tensors
on the axisymmetric shell mid-surface can be represented by

[aαβ ] =
1 0
0 R2
2

1
0
1
0
R2

aκλ = 4
,
3
5
,
(3.5.17)
with the use of which the contravariant base vectors of the mid-surface coordinate system,
on account of (2.1.5) and (2.2.6), can be written in the form
a1 = a1α aα = a11 a1 + a12 a2 = a1 ,
1
a2 = a2α aα = a21 a1 + a22 a2 = 2 a2 ,
R
3
a3 = a .
(3.5.18)
(3.5.19)
(3.5.20)
Substituting (3.5.13)–(3.5.15) into (2.3.3) and carrying out the differentiations, after taking
into account (3.5.12), the covariant components of the curvature tensor of the mid-surface
S0 can be written as
b11 = a1,1 · a3 =
R′′
,
A31
b22 = a2,2 · a3 = −
b12 = a1,2 · a3 = 0 ,
R
,
A1
b21 = a2,1 · a3 = 0 .
(3.5.21)
(3.5.22)
Accordingly, the matrix of the covariant curvature tensor takes the following form:
2
6
[bαβ ] = 6
6
3
R′′
A31
4
0
R
−
A1
0
7
7
7
5
.
(3.5.23)
Using the results (3.5.17) and (3.5.23), from equation (2.3.5) the mixed components of the
curvature tensor of S0 are obtained as
R′′
,
A31
1
= −
,
RA1
b11 = a1λ bλ1 =
(3.5.24)
b22 = a2λ bλ2
(3.5.25)
and
b12 = a1λ bλ2 = 0 ,
b21 = a2λ bλ1 = 0 .
(3.5.26)
The principal curvatures of the axisymmetric shell mid-surface, on account of (3.5.26), can
be given by
R′′
1
κ1 = b11 = 3 ,
κ2 = b22 = −
,
(3.5.27)
A1
RA1
from which, recalling (2.3.11),
R1 = −
1
A3
= − 1′′ ,
κ1
R
R2 = −
1
= A1 R
κ2
(3.5.28)
28
Chapter 3. Differential geometry of shells
are obtained for the principal radii of curvature. In regard to (3.5.24)–(3.5.26), the matrix
of the mixed curvature tensor of the mid-surface S0 can be represented by
2

3
R′′
A31
6
6
bκβ = 6
4
0
1
−
RA1
0
7
7
7
5
.
(3.5.29)
Applying (2.4.13)–(2.4.14) to (3.5.17) and making use of (3.5.12), the mid-surface Christoffel symbols of the second kind are obtained as
2
R′
,
RA1
RR′
= Γ221 a11 = −a22,1 a11 = −
,
A1
2
Γ22 = Γ222 a22 = −a22,2 a22 = 0 ,
2
Γ12 = Γ21 = Γ122 a22 = a22,1 a22 =
1
Γ22
1
1
Γ12 = Γ21 = Γ121 a11 = a11,2 a11 = 0 ,
1
Γ11 = Γ111 a11 = −a11,1 a11 = 0 ,
2
Γ11 = Γ112 a22 = −a11,2 a22 = 0 .
(3.5.30)
(3.5.31)
(3.5.32)
With the aid of (3.5.23) and (3.5.29), it follows from (2.4.4)–(2.4.9) that
R′′
,
A31
R′′
= −b11 = − 3 ,
A1
3
3
Γ11 = b11 =
1
Γ31
R
,
A1
1
= −b22 =
,
RA1
Γ22 = b22 = −
2
Γ32
3
3
3
3
1
2
Γ12 = Γ21 = b12 = b21 = 0 ,
Γ32 = Γ31 = −b12 = −b21 = 0 ,
µ
Γ3κ = Γλ3 = 0 ,
m
Γλ3 = 0 ,
Γ33 = 0 .
(3.5.33)
(3.5.34)
(3.5.35)
In view of (2.2.7)–(2.2.8) and (3.5.17), the covariant and contravariant metric tensors of
the mid-surface coordinate system can be written in the matrix forms
2
[akℓ ] = 6
4
1 0 0
0 R2 0
0 0 1
2
3
7
5
,
6
[apq ] = 6
4
1
0 0
1
0
0
R2
0 0 1
3
7
7
5
,
(3.5.36)
from which, using (2.2.9), the determinant of the mid-surface covariant metric tensor is
a = R2 . Inserting the results (3.5.24)–(3.5.26) into (3.2.7) and taking into account (3.2.8),
the matrix representation of the shifter is obtained as
2

µkℓ

=
6
6
6
6
6
4
3
R′′ 3
1− 3 ξ
A1
0
0
0
1+
0
1 3
ξ 0
RA1
0
1
7
7
7
7
7
5
,
(3.5.37)
the determinant of which, with the use of (2.3.9)–(2.3.10), (3.2.9) and (3.5.27), can be put
in the form

R′′
1
R′′ 3 2
3
µ=1−
−
ξ
−
ξ
,
(3.5.38)
A31 RA1
RA41
i.e.,
R′′
1
R′′
H=
−
,
K
=
−
(3.5.39)
2A31 2RA1
RA41
3.5. Geometry of shells of revolution
29
are, respectively, the mean and Gaussian curvatures of the axisymmetric shell mid-surface.
The relations between the covariant base vectors at an arbitrary shell point P and a midsurface point P0 , being on the same normal to the mid-surface S0 , can be written on the
basis of (3.2.17) and (3.5.37) as

R′′
g1 =
= 1 − 3 ξ 3 a1 ,
A1
1 3
2
g2 = µ 2 a2 = 1 +
ξ a2 ,
RA1
g3 = µ33 a3 = a3 = a3 = g3 .
µ11 a1
(3.5.40)
(3.5.41)
(3.5.42)
In the case of axisymmetric shells, the outward unit normal vectors to the outer and inner
surfaces S ± and the lateral surfaces S 0 and S L , on account of (3.4.11) and (3.5.42), can be
given by
¨
±
±
± 3
±
± 3
±
n = n a3 = n a = n g3 = n g ,
±
n =n =
and
¨
×
× 1
×
×
n = n a = n a1 ,
n =
1 on S + ,
−1 on S − ,
−1 on S 0 ,
1 on S L .
(3.5.43)
(3.5.44)
The following assumptions will be used:
1. We will prescribe traction boundary conditions only on the outer and inner surfaces
S ± and lateral surface S L , and displacement boundary condition only on the lateral
surface S 0 .
2. We assume that the tensorial quantities (such as the base vectors, the stress and
strain tensors and the displacement vector) are multiple times differentiable functions
and can be expanded into Taylor series with respect to the thickness coordinate ξ 3 .
3. For thin shells of revolution, the thickness is assumed to be small, i.e., the relations
d
≪1,
L
d
≪1
Rmin
(3.5.45)
hold true, where
Rmin = 1min {|R1 | , |R2 |} .
x ∈[0;L]
(3.5.46)
Thus, from (3.2.6)–(3.2.9),

µkℓ ∼
= µ−1
i.e.,
k
ℓ
∇∼
=∇,
∼
= δ kℓ ,
µ=1,
gα ∼
= aα .
(3.5.47)
(3.5.48)
In view of the previous assumptions, for thin shells of revolution the volume element (3.4.1)
and the surface elements (3.4.3), (3.4.5), and (3.4.13), on account of (2.2.13) and (3.5.44),
can be defined by
dV = R dξ 1 dξ 2 dξ 3 ,
(3.5.49)
Chapter 3. Differential geometry of shells
30
dS0 = dS ± = R dξ 1 dξ 2 ,
(3.5.50)
dS 0 = R0 dξ 2 dξ 3 ,
(3.5.51)
dS L = RL dξ 2 dξ 3 .
(3.5.52)
Chapter 4
Three-field dual-mixed functional for
elastodynamics
4.1
Governing equations and boundary conditions
Let us consider a three-dimensional linearly elastic body occupying the simply connected
domain V bounded by the surface S = Sσ ∪ Su , Sσ ∩ Su = Ø. We suppose that the body
is loaded by a body force density bk in V and surface tractions f k on Sσ , whereas the
displacements uek are prescribed on the remaining surface part Su . The field equations of
the linear elastodynamic continua are the translational equations of motion
σ kℓ;ℓ + b k = ṗ k
in V ,
(4.1.1)
where σ kℓ is the stress tensor and p k is the impulse vector1 , the symmetry conditions
ǫpqr σ pq = 0
in V ,
(4.1.2)
and the strain-displacement relations
εkℓ = u(k;ℓ ) =
1
(uk;ℓ + uℓ;k )
2
in V ,
(4.1.3)
where εkℓ is the strain tensor and uk is the displacement vector, as well as the relations
p k = ρv k
in V ,
(4.1.4)
in V
(4.1.5)
and
vk = u̇k
stand for the impulses p k and the velocities v k , and ρ denotes the density of the material.
The boundary conditions can be written in the forms:
f k = σ kℓ nℓ
on Sσ ,
(4.1.6)
and
uk = uek
1
on Su ,
A dot over a tensor variable denotes differentiation with respect to time.
31
(4.1.7)
Chapter 4. Three-field dual-mixed functional for elastodynamics
32
where nℓ is the outward unit normal to the surface S. The initial conditions can be given
for the displacements and the velocities by
uk (ξ m , t = 0) = uk (ξ m , t0 ) = 0 uk
in V ,
(4.1.8)
vk (ξ m , t = 0) = vk (ξ m , t0 ) = 0 vk
in V .
(4.1.9)
and
For linearly elastic material the stress-strain relations and their inverses can be written as
σ kℓ = D kℓrs εrs
εkℓ = Ckℓrs σ rs
in V ,
in V ,
(4.1.10)
(4.1.11)
in which the fourth-order tensors D kℓrs and Ckℓrs with symmetry properties
D kℓrs = D kℓsr = D rskℓ(= D ℓkrs) ,
(4.1.12)
Ckℓrs = Ckℓsr = Crskℓ (= Cℓkrs )
(4.1.13)
as well as
are, respectively, the material stiffness tensor and the elastic compliance tensor. The expression
(4.1.4)
for the impulses p k admits the existence of a quadratic kinetic energy density

T p k and its complementary function TÒ (vk ) such that
∂T
∂T
δp k = vk δp k −→
= vk
k
∂p
∂ pk
∂ TÒ
∂ TÒ
δvk = p k δvk −→
= pk
δ TÒ (vk ) =
∂ vk
∂ vk

δ T pk =
in V ,
(4.1.14)
in V
(4.1.15)
hold true2 . From the linear constitutive equations (4.1.10)–(4.1.11) it follows that there
Ò
exists a quadratic strain energy density U (εkℓ ) and its complementary function U
σ kℓ
such that
∂U
∂U
δεkℓ −→
= σ kℓ
∂ εkℓ
∂ εkℓ
Ò

∂ UÒ
∂U
kℓ
Ò
δU
σ kℓ =
δσ
−→
= εkℓ
∂ σ kℓ
∂ σ kℓ
δ U (εkℓ ) =
in V ,
(4.1.16)
in V
(4.1.17)
hold true. The convex energy and complementary energy functions are not independent
of each other, i.e., the complementary energy densities can be obtained by the Legendre
transformations

Ò
U
σ kℓ = σ kℓ εkℓ − U (εkℓ )

TÒ (vk ) = vk p k − T p k
2

in V ,
(4.1.18)
in V .
(4.1.19)
The operator δ indicates an arbitrary variation (or change) of a mechanical quantity. The operational
properties of δ can be found, for example, in [42, 130, 159].
4.2. Complementary Hamilton’s principle for continua
4.2
33
Complementary Hamilton’s principle for continua
The complementary form of Hamilton’s principle for the elastodynamic problem of linearly elastic solids is derived from the complementary D’Alembert’s principle, which can be
written on the basis of [158] as
(vk − u̇k ) δp k = 0
in V ,
(4.2.1)
and δp k is arbitrary in V . Integrating (4.2.1) over the region V with respect to time t over
the closed time interval t ∈ [t0 , t1 ], where t0 and t1 are two arbitrary instants of time, as
well as making use of (4.1.14), the following equation is obtained:
Z t Z
1
t0

δ T − u̇k δp k dV dt = 0 .
V
(4.2.2)
Making use of the time derivation rule

u̇k δp k = uk δp k − uk δ ṗ k ,
(4.2.3)
δ ṗ k = δσ kℓ;ℓ
(4.2.4)
where
is the first variation of the translational equations of motion (4.1.1), the variational equation
(4.2.2) can be written as
Z
Z
V
t1

δ T + uk δσ kℓ;ℓ dt − uk δp k
t0
t
1
t0
dV = 0 .
(4.2.5)
uk;ℓ δσ kℓ dV
(4.2.6)
Applying the divergence theorem
Z
Z
V
uk δσ kℓ;ℓ
dV =
Z
kℓ
Sσ ∪Su
uk δσ nℓ dS −
V
in (4.2.5) and taking into account the kinematic equations (4.1.3) and the displacement
boundary conditions (4.1.7), as well as the symmetry conditions (4.1.2) and the expression
(4.1.17), we obtain the variational equation
Ò
δ ΠH =
=
Z t Z
1
t0
V
Z t Z
1
t0
Z
kℓ
Ò
V
δ(T − U ) dV +
Z
Su
uek δσ nℓ dS dt
kℓ
Ò
δ L dV +
Su
uek δσ nℓ dS dt = 0
(4.2.7)
with the assumption that the impulses p k are prescribed at times t0 and t1 , i.e., δp k (t0 ) =
Ò
Ò
δp k (t1 ) = 0, and L
= T −U
is the complementary Lagrangian function [159]. Equation
(4.2.7) can be simplified into the form
Ò
δ ΠH = δ
Z t
1
t0

Ò
K −Π
dt = 0 ,
(4.2.8)
which can be considered as the first variation of the functional
Ò
ΠH =
Z t
1
t0

Ò
K −Π
dt ,
(4.2.9)
Chapter 4. Three-field dual-mixed functional for elastodynamics
where
1 Z p kp k
K (p ) =
T dV =
dV
V
2 V ρ
is the kinetic energy of the whole elastic body and
34
Z
k
Z
kℓ
Ò
Π(σ
)=
(4.2.10)
Z
Ò
U
dV −
V
Su
ue k σ kℓ nℓ dS
(4.2.11)
is the total complementary energy functional [42], as well as
Z
Ò
U =
V
Ò
U
dV
(4.2.12)
defines the complementary strain energy of the whole elastic body. Essentially the complementary Hamilton’s principle is the generalization of the total complementary energy
principle to linear elastodynamics [94, 157]. This is represented by the stationary condition
Ò
(4.2.8). The subsidiary conditions to Π
H are the equations of motion (4.1.1)–(4.1.2), the
impulse-velocity relations (4.1.4), the inverse stress-strain relations (4.1.11) and the stress
boundary conditions (4.1.6), as well as the initial conditions (4.1.8)–(4.1.9).
4.3
Dual-mixed functionals for elastodynamics using
non-symmetric stresses
4.3.1
Four-field functional and variational principle
To derive dual-mixed functionals based on the a priori non-symmetric stress field for elastodynamic problems of linearly elastic solids, the Lagrangian multiplier technique is applied.
The subsidiary conditions (4.1.1) and (4.1.2) are multiplied by the Lagrangian multipliers
αk and λr , respectively and these equations are then integrated over the domain V and
in time interval t ∈ [t0 , t1 ]. Adding these integrals to the functional (4.2.9), the following
four-field functional is obtained:
Ò
F ( p k , σ kℓ , αk , λr ) = Π
H +
Z t Z
1
t0

αk σ kℓ;ℓ + b k − ṗ k + λr ǫpqr σ pq dV dt
(4.3.1)
which, with the aid of (4.2.7)–(4.2.11), can be put in the form
k
kℓ
r
F ( p , σ , αk , λ ) =
+
Z t Z
1
t0
Z t Z
1
t0
L dV +
V

V
Z
kℓ
Ò

Su
uek σ nℓ dS dt

αk σ kℓ;ℓ + b k − ṗ k + λr ǫpqr σ pq dV dt .
(4.3.2)
The Lagrangian multipliers are determined from the stationary conditions of (4.3.2), which
can be written on the basis of [42, 130, 159] as
δσ F = 0 ,
δp F = 0 ,
δα F = 0 ,
δλ F = 0 .
(4.3.3)
Applying the differentiation rule (4.2.3) and the divergence theorem (4.2.6), after taking into
account (4.1.14) and (4.1.17), from (4.3.3) the following integral expressions are obtained:
δσ F =
Z t Z
1
t0
V
[λr ǫkℓr − εkℓ (σ rs ) − αk;ℓ ] δσ kℓ dV dt
4.3. Dual-mixed functionals for elastodynamics using non-symmetric stresses
+
Z t Z
1
t0
Z
δp F =
δα F =
δλ F =
Su
(αk + ue k ) δσ kℓ nℓ dS dt = 0 ,
Z
t1
V
t0
Z t Z

1
V
t0
Z t Z
1
t0
V

(vk + α̇k ) δp k dt − αk δp k
35
(4.3.4)
t
1
dV = 0 ,
t0

σ kℓ;ℓ + b k − ṗ k δαk dV dt = 0 ,
(4.3.5)
(4.3.6)
ǫpqr σ pq δλr dV dt = 0 .
(4.3.7)
If δσ kℓ , δp k , δαk and δλr are arbitrary in V and δσ kℓ is arbitrary on the surface part Su as
well, it follows from (4.3.4)–(4.3.7) that
λr ǫkℓr − εkℓ (σ rs ) − αk;ℓ = 0
αk + ue k = 0
vk + α̇k = 0
σ kℓ;ℓ + b k − ṗ k = 0
ǫpqr σ
pq
=0
in V ,
on Su ,
in V ,
(4.3.8)
(4.3.9)
(4.3.10)
in V ,
(4.3.11)
in V ,
(4.3.12)
and δp k (t0 ) = δp k (t1 ) = 0. On comparing equations (4.3.8)–(4.3.12) with (4.1.1)–(4.1.4),
(4.1.7) and (4.1.15) and making use of (3.2.34), the Lagrangian multipliers can be identified
as follows:
on Su ,
(4.3.13)
αk = −uk = −ue k
1
λs = φs = − ǫkℓs ψkℓ
in V ,
(4.3.14)
2
where φs is the axial vector of the skew-symmetric rotation tensor
ψkℓ = −ψℓk = u[k;ℓ ] =
1
(uk;ℓ − uℓ;k )
2
in V .
(4.3.15)
Substituting the results (4.3.13)–(4.3.14) into (4.3.1), the four-field dual-mixed functional
(4.3.2) can be written as
k
kℓ
F ( p , σ , uk , ψkℓ ) =
−
Z t Z
1
t0
L dV +
V
t0
Z t Z
1
Z
kℓ
Ò

V
Su
uek σ nℓ dS dt

uk σ kℓ;ℓ + b k − ṗ k + ψkℓ σ kℓ dV dt
(4.3.16)
which, since ψkℓ = −ǫkℓs φs , can be put in the form
k
kℓ
s
Ò
F ( p , σ , uk , φ ) = ΠH −
Z t Z
1
t0

uk σ kℓ;ℓ + b k − ṗ k − φs ǫkℓs σ kℓ dV dt .
(4.3.17)
In functional (4.3.17) we have generalized a Hellinger–Reissner-type dual-mixed variational
principle, wherein the translational- and rotational equations of motion (4.1.1)–(4.1.2), the
kinematic equations (4.1.3), the velocity-displacement relations (4.1.5) and the displacement boundary conditions (4.1.7) are weakly enforced, i.e., these equations are obtained as
the stationary conditions of functional (4.3.17). The generalization is on three accounts:
(1) extension to linear elastodynamic problems, (2) allowing independent variations for
impulses and displacements and (3) using a priori non-symmetric stress field.
36
Chapter 4. Three-field dual-mixed functional for elastodynamics
4.3.2
Three-field functional and variational principle

Transforming the kinetic energy density T p k into its complementary function TÒ (vk ) by
the inverse of the Legendre transformation (4.1.19) and using the time derivative rule

ṗ k uk = p k uk − p k u̇k ,
(4.3.18)
as well as satisfying a priori the relations (4.1.4)–(4.1.5), the following three-field functional
is obtained:
Z
kℓ

s
F (σ , uk , φ ) =
−
t
1
uk ρ u̇k t
0
V
Z t Z
1
t0

−
Z t
1
t0

H dt dV +

Z t Z
1
t0
Su
uek σ kℓ nℓ dS dt

uk σ kℓ;ℓ + b k − φs ǫkℓs σ kℓ dV dt ,
V
(4.3.19)
where H
= TÒ + UÒ is the complementary Hamiltonian function [94]. Since δuk (t0 ) =
δuk (t1 ) = 0 holds true and δp k (t0 ) and δp k (t1 ) are arbitrary at the time instants t0 and
t1 , (4.3.19) can be simplified into the form
F (σ kℓ , uk , φs ) =
where
Z t 1
H R −KÒ dt ,
t0
(4.3.20)
Z
1Z
K (u̇k ) =
T dV =
ρ u̇k u̇k dV
V
2 V
is the complementary kinetic energy of the whole elastic body and
Ò
Ò
Z
H R(σ kℓ , uk , φs ) = −
Z
−
V
(4.3.21)
Z
V

Ò
U
dV +

Su
ue k σ kℓ nℓ dS

uk σ kℓ;ℓ + b k − φs ǫkℓs σ kℓ dV
(4.3.22)
is the three-field dual-mixed Hellinger–Reissner-type functional [84, 133–135] of elastostatics, i.e., the three-field dual-mixed functional (4.3.20) can be considered as the sum of the
complementary kinetic energy functional and the three-field dual-mixed Hellinger–Reissner

Ò
type functional of elastostatics. The complementary strain energy density function U
σ kℓ
of the linearly elastic body can be given by
1 pq
σ εpq (σ kℓ )
in V ,
(4.3.23)
2
thus the three-field dual-mixed functional of Hellinger–Reissner-type for the dynamic problems of a linearly elastic body can be written as
UÒ (σ kℓ ) =
kℓ
s
F (σ , uk , φ ) = −
−
Z t 1
t0
Z t Z
1
t0
Z

1Z k
pq
kℓ
ρ u̇ u̇k + σ εpq (σ ) dV −
uek σ kℓ nℓ dS
2 V
su

uk σ kℓ;ℓ + b k − φs ǫkℓs σ kℓ dV dt ,
dt
(4.3.24)
the fundamental variables of which are the not a priori symmetric stress tensor σ kℓ , the
displacements uk and the rotations φs . According to the presented variational principle, the
solution of the linear elastodynamic problem can be characterized as the stationary point of
functional (4.3.24) over the space of all vector fields uk , φs and all a priori non-symmetric
tensor fields σ kℓ , satisfying the stress boundary conditions (4.1.6), the initial conditions
(4.1.8)–(4.1.9) and the inverse stress-strain relations (4.1.11).
4.4. Three-field dual-mixed functional for axisymmetric shells
4.4
37
Three-field dual-mixed functional for axisymmetric shells
In this section the three-field dual-mixed functional (4.3.24) is presented for shells of revolution. After using the transformation relations (3.2.24), (3.2.26), (3.2.28)–(3.2.29) and
(3.3.13)–(3.3.14), functional (4.3.24) for shell problems takes the form
pq
F (σ , up , ψpq ) = −
−
Z t Z
1
t0
Z t Z
1
n
V
t0
V
Z

1 p
pq
e σ pq n dS × dt
ρ u̇ u̇p + σ εpq dV −
u
p
q
Su
2
h

ψ pq σ pq + un µnk σ pq|ℓ µ−1
k
ℓ
µ−1
p
+b
q
io
m −1 k
µ
dV
m
dt . (4.4.1)
Taking into account (3.2.30) and (3.2.32), the skew-symmetric rotation tensor ψ pq for shells
of revolution can be written as
√
r
r
r
ψ pq = −ǫpqr φ = − a epqr φ = −R epqr φ ,
(4.4.2)
which can be represented in the matrix form
2

6
ψ pq = R 6
4
0
−φ
3
φ
0
2
1
−φ
φ
3
2
φ
1
−φ
0
3
7
7
5
,
(4.4.3)
i.e., the non-zero components of (4.4.2) are
1
3
ψ 32 = −ψ 23 = R φ ,
2
ψ 21 = −ψ 12 = R φ ,
ψ 13 = −ψ 31 = R φ .
(4.4.4)
Inserting the vectorial surface element (3.4.13) and the volume element (3.4.4) in (4.4.1)
and making use of (4.4.2), (4.4.1) becomes
r
pq
F (σ , up , φ ) = −
+
Z t
1
¨Z
t0
Su
− d2
t0
√
up σ µ a eαβ µ−1

pλ
e
Z t Z +d Z
1
2
S0
α
λ

1 p
r
ρ u̇ u̇p + σ pq εpq − σ pq ǫpqr φ µ dS0 dξ 3 dt
2
β
3
dξ dξ −
Z +d Z
2
− d2
h
S0
up
σ pq|ℓ

ℓ
µ−1 q
+b
p
i
µ dS0 dξ
«
3
dt .
(4.4.5)
Using the assumptions Sσ ≡ S ± ∪ S L , Su ≡ S 0 and taking into account (3.5.44), the
three-field dual-mixed functional (4.4.5) for shells of revolution can be written as
r
F (σ pq , up , φ ) = −
−
Z t Z + d Z 2π
1
2
t0
− d2
0
e
Z t Z +d Z
1
2
t0
p1
up σ µ
− d2

S0
1
µ−1 1

1 p
r
ρ u̇ u̇p + σ pq εpq − σ pq ǫpqr φ µ dS0 dξ 3 dt
2
Z
2
R0 dξ +
S0
h
up
σ pq|ℓ
which, on account of (3.5.47), can be put in the form

ℓ
µ−1 q
+b
p
i
µ dS0 dξ 3 dt (4.4.6)
Chapter 4. Three-field dual-mixed functional for elastodynamics
r
F (σ pq , up , φ ) = −
Z t Z +d Z
1
2
−
− d2
t0
S0
t0
0

1 p
r
ρ u̇ u̇p + σ pq εpq − σ pq ǫpqr φ dS0 dξ 3 dt
2
Z t Z + d Z 2π
1
2
− d2
38
Z
e
p1
2
up σ R0 dξ +
S0
up
σ pq|q
+b
p
dS0 dξ 3 dt (4.4.7)
for elastodynamic problems of thin axisymmetric shells. Functional (4.4.7) serves as a basis
for the developments presented in the subsequent chapters.
Chapter 5
Dimensionally reduced axisymmetric
shell model
5.1
Three-dimensional translational equations of motion for axisymmetric shells
In this chapter, a new dimensionally reduced axisymmetric shell model based on the threefield dual-mixed functional (4.4.7) will be derived for elastodynamic problems. Firstly,
the three-dimensional translational equations of motion are presented for shells of revolution. After applying transformations (3.2.28)–(3.2.29) and (3.3.13) to equation (4.1.1), the
translational equations of motion are obtained as

σ pq|ℓ µ−1
k
p
µ−1
ℓ
+b
q
m −1 k
µ
m
= ρ ü
n −1 k
µ
n
in V ,
(5.1.1)
which for thin shells, on account of the assumption (3.5.47), can be written in the simple
form
k
k
σ kℓ|ℓ + b = ρ ü
in V .
(5.1.2)
The three-dimensional mid-surface covariant derivatives σ kℓ|ℓ can be expressed, on the basis
of (3.3.5)–(3.3.6), as
σ kℓ|ℓ = σ kα|α + σ k3,3 ,
(5.1.3)
which, with the aid of (2.5.5)–(2.5.11), can be separated into
σ κℓ|ℓ = σ καkα − bκα σ 3α − bαα σ κ3 + σ κ3,3 ,
(5.1.4)
κ
(5.1.6)
σ 3ℓ |ℓ = σ 3αkα + bκα σ κα − bαα σ 33 + σ 33,3 ,
where
α
σ καkα = σ κα,α + Γβα σ βα + Γµα σ κµ ,
and
α
σ 3αkα = σ 3α,α + Γβα σ 3β .
(5.1.5)
(5.1.7)
Making use of (3.5.30)–(3.5.35), the three-dimensional scalar translational equations of
motion for thin shells of revolution can be written as
σ 11,1 −

R′′ 31
1 ′ 11
RR′ 22
1
1
13
12
13
σ
+
σ
+
σ
+
R
σ
+
σ
−
σ + σ 13,3 + b = ρ ü , (5.1.8)
,2
3
A1
RA1
A1
39
Chapter 5. Dimensionally reduced axisymmetric shell model
40
σ 21,1 −

R′′ 23
R′ 12
1 23
2
2
22
σ
+
σ
+
σ + 2σ 21 +
σ + σ 32 + σ 23,3 + b = ρ ü , (5.1.9)
,2
3
A1
RA1
RA1
σ 31,1 +

R′′ 11
1 ′ 31
R 22
3
3
33
32
33
σ
−
σ
+
σ
+
R
σ
+
σ
−
σ + σ 33,3 + b = ρ ü . (5.1.10)
,2
3
A1
RA1
A1
For axisymmetrically loaded shells of revolution the differentiation with respect to the angle
coordinate ξ 2 leads to zero, thus equations (5.1.8)–(5.1.10) can be simplified to

R′′ 31
1 ′ 11
RR′ 22
1
1
13
13
σ
+
σ
+
R
σ
+
σ
−
σ + σ 13,3 + b = ρ ü ,
3
A1
RA1
A1
(5.1.11)
σ 21,1 −

R′′ 23
R′ 12
1 23
2
2
σ
+
σ + 2σ 21 +
σ + σ 32 + σ 23,3 + b = ρ ü ,
3
A1
RA1
RA1
(5.1.12)
σ 31,1 +

R′′ 11
1 ′ 31
R 22
3
3
33
33
σ
−
σ
+
R
σ
+
σ
−
σ + σ 33,3 + b = ρ ü .
3
A1
RA1
A1
(5.1.13)
σ 11,1 −
These scalar translational equations of motion can be separated into two independent groups
according to the stress and acceleration components. Equations (5.1.11) and (5.1.13) for
1
3
the stress and acceleration components σ 11 , σ 22 , σ 13 , σ 31 , σ 33 and ü , ü describe the
bending-shearing problems (including tension-compression) of the axisymmetric shell and
equation (5.1.12) describes the torsion problem of the shell, the corresponding stress and
2
acceleration components are σ 12 , σ 21 , σ 23 , σ 32 and ü .
5.2
Three-dimensional constitutive equations
The a priori non-symmetric stress field σ rs appeared in the three-field dual-mixed formulation can be decomposed, as all arbitrary second-order tensor fields, into the sum of two
second-order tensors:
σ rs = σ (rs) + σ [rs] ,
(5.2.1)
where
σ (rs) =
σ rs + σ sr
,
2
(5.2.2)
and
σ rs − σ sr
(5.2.3)
2
are, respectively, the symmetric and the skew-symmetric part of the stress tensor σ rs . In
view of this and (4.1.11), as well as (4.1.13), the symmetric strain tensor εkℓ can be expressed
for linearly elastic continua by the symmetric part of the stress tensor, through the inverse
relations (Hooke’s law)
σ [rs] =
εkℓ = Ckℓrs σ rs = Ckℓrs σ (rs) = Ckℓrs
σ rs + σ sr
.
2
(5.2.4)
Assuming homogeneous and isotropic materials, the constitutive equations (5.2.4) can be
written in the form
1
ν
εkℓ (σ ) =
σ (rs) grk gsℓ −
σ mn gmn gkℓ
2G
1+ν
rs
,
(5.2.5)
5.3. Stress resultant force- and couple vectors
41
where ν is the Poisson ratio,
E
2 (1 + ν)
G=
(5.2.6)
is the shear modulus and E is the elasticity modulus. With the use of (5.2.2), the form
(5.2.5) of Hooke’s law yields
εkℓ (σ rs ) =
1
2ν
(σ rs + σ sr ) grk gsℓ −
σ mn gmn gkℓ
4G
1+ν
,
(5.2.7)
which, after applying the transformation relations (3.2.18) and (3.2.26), and recalling the
assumption (3.5.47), can be rewritten for thin shells as
εpq (σ rs ) =
1
2ν
(σ rs + σ sr ) arp asq −
σ kℓ akℓ apq
4G
1+ν
,
(5.2.8)
i.e., the scalar components of the strain field can be obtained as
ε11 =
ε22 =
ε33 =
ε31 =
ε21 =
ε32 =

1 11
σ − ν R2 σ 22 + σ 33 ,
E

R2 2 22
R σ − ν σ 11 + σ 33 ,
E

1 33
σ − ν σ 11 + R2 σ 22 ,
E

1 13
σ + σ 31 = ε13 ,
4G

R2 12
σ + σ 21 = ε12 ,
4G

R2 23
σ + σ 32 = ε23 .
4G
(5.2.9)
(5.2.10)
(5.2.11)
(5.2.12)
(5.2.13)
(5.2.14)
In spite of the fact that three-dimensional constitutive equations are being used, one of the
most important properties of the axisymmetric shell model is that the term 1/(1 − 2ν),
which is responsible for incompressibility locking in the classical displacement-based finite
element formulations when ν is close to 0.5, does not appear in the above expressions. This
indicates that dual-mixed finite element models based on the three-field functional (4.3.24)
are free from incompressibility locking.
5.3
Stress resultant force- and couple vectors
Let l and m denote, respectively, the resultant force and the resultant couple vectors,
each measured per unit length of an arbitrary curve c0 on the mid-surface S0 . Then, the
resultant force l acting on a portion of the curve c0 is equivalent to the resultant force due
to the stress vector acting over the corresponding portion of the surface S ∗ generated by
the normal vectors to the mid-surface S0 along c0 :
Z
c0
Z
l ds0 =
S∗
σ kα nα gk dS ∗ ,
(5.3.1)
Chapter 5. Dimensionally reduced axisymmetric shell model
42
and the resultant couple m acting on a portion of c0 is equal to the couple generated by
the stress vector over the corresponding portion of S ∗ , namely
Z
Z
m ds0 =
c0
S∗
ξ 3 g3 × gk σ kα nα dS ∗ ,
(5.3.2)
where nα dS ∗ is the vectorial area element of the normal surface S ∗ given by (3.4.12). The
latter two equations are serve to define the stress and couple resultants. Using relations
(3.2.17), (3.2.30) and (3.4.10)–(3.4.11), the right-hand side of (5.3.1) becomes
Z
Z
c0
l ds0 =
Z
Z +d
2
µ µℓk σ kα nα aℓ dξ 3 ds0 =
− d2
c0
Z +d
2
c0
− d2
Z

µ µκλ σ λα nα aκ + µ σ 3α nα a3 dξ 3 ds0 =
in which
N
κα
=
Z +d
2
− d2
µ µκλ σ λα dξ 3
c0
α
,
Q =−
(N κα aκ − Q α a3 ) nα ds0 , (5.3.3)
Z +d
2
− d2
µ σ 3α dξ 3
(5.3.4)
are, respectively, the membrane and shear forces as stress resultants. From (5.3.3) it follows
that for an arbitrary curve c0 on S0
κ
3
l = l aκ + l a3 = N κα nα aκ − Q α nα a3 ,
κ
l = N κα nα ,
3
l = −Q α nα .
(5.3.5)
(5.3.6)
Next, with the use of (3.4.10)–(3.4.11) and transformations (3.2.17), (3.2.30), the right-hand
side of equation (5.3.2) can be put in the form
Z
Z
c0
m ds0 =
c0
Z +d
2
− d2
Z
ǫκβ ξ
3
µ µκρ σ ρα nα aβ dξ 3 ds0
where
M κα =
Z +d
2
− d2
=
c0
ξ 3 µ µκρ σ ρα dξ 3
ǫκβ M κα nα aβ ds0 ,
(5.3.7)
(5.3.8)
defines the bending moments M 11 , M 22 and the twisting moments M 12 , M 21 as couple
resultants measured per unit length of coordinate curves on mid-surface S0 . For an arbitrary
curve c0 , (5.3.7) results in
m = mβ aβ = ǫκβ M κα nα aβ
(5.3.9)
mβ = ǫκβ M κα nα .
5.4
(5.3.10)
Approximation of the fundamental variables along
the thickness
In what follows, the components of the tensor variables shifted to the points of the axisymmetric shell mid-surface S0 are used without bars over them. The next step in the
derivation is the expansion of the fundamental variables, i.e., the stresses σ pq (ξ m , t), the
5.4. Approximation of the fundamental variables along the thickness
43
displacements up (ξ m , t) and the rotation components φr (ξ m , t), as well as the expansion of
the body force density vector bp into power series with respect to the thickness coordinate
ξ 3:
σ pq (ξ α , ξ 3, t) =
up (ξ α , ξ 3, t) =
φr (ξ α , ξ 3, t) =
∞
X
i=0
∞
X
i=0
∞
X
iσ
pq

(ξ α , t) ξ 3

upi (ξ α , t) ξ 3
iφ
r

i
i
(ξ α, t) ξ 3
,
,
i
(5.4.1)
(5.4.2)
,
(5.4.3)
i=0
as well as
b p (ξ α , ξ 3 , t) =
∞
X
ib
p

(ξ α , t) ξ 3
i
,
(5.4.4)
i=0
where the stress coefficients i σ pq (ξ α , t), the displacement coefficients i up (ξ α , t), the rotation coefficients i φr (ξ α , t) and the body force density coefficients i b p (ξ α , t) depend on the
time t and the mid-surface coordinates ξ α . Substituting the expanded stress components
(5.4.1), displacement components (5.4.2) and body force components (5.4.4) into the threedimensional equations of motion (5.1.8)–(5.1.10) and making separation with respect to the
powers of the thickness coordinate ξ 3 , the following translational equations of motion are
obtained for the stress and displacement coefficients:
11
i σ ,1 −
21
i σ ,1
31
i σ ,1
−
+

1 ′ 11
RR′ 22
R i σ + i σ 13 −
iσ
RA1
A1
+(i + 1) i+1 σ 13 + i b1 = ρ i ü1 ,
i = 0, 1, 2, ... ,
(5.4.5)

R′′ 23
1 23
R′ 12
22
21
32
σ
+
σ
+
σ
+
2
σ
+
σ
+
σ
i
i
i
i
i
i
,2
A31
RA1
RA1
23
2
2
+(i + 1) i+1σ + i b = ρ i ü ,
i = 0, 1, 2, ... ,
(5.4.6)
R′′
A31

R′′
A31

31
+ i σ 13 + i σ 12,2 +
iσ

iσ
11

1 ′ 31
R 22
R i σ + i σ 33 −
iσ
RA1
A1
+(i + 1) i+1σ 33 + i b3 = ρ i ü3 ,
i = 0, 1, 2, ... .

− i σ 33 + i σ 32,2 +
(5.4.7)
In the case of axisymmetrically loaded thin shells of revolution, the differentiation with
respect to the polar coordinate ξ 2 leads to zero, thus the latter three equations can be
replaced by the translational equations of motion
11
i σ ,1
−
21
i σ ,1 −
R′′
A31

1 ′ 11
RR′ 22
R i σ + i σ 13 −
iσ
RA1
A1
+(i + 1) i+1 σ 13 + i b1 = ρ i ü1 ,
i = 0, 1, 2, ... ,
(5.4.8)

R′′ 23
R′ 12
1 23
σ
+
+ 2 i σ 21 +
+ i σ 32
iσ
iσ
3 i
A1
RA1
RA1
23
2
2
+(i + 1) i+1σ + i b = ρ i ü ,
i = 0, 1, 2, ... ,
(5.4.9)

iσ
31

+ i σ 13 +
44
Chapter 5. Dimensionally reduced axisymmetric shell model
31
i σ ,1 +
R′′
A31

1 ′ 31
R 22
R i σ + i σ 33 −
iσ
RA1
A1
+(i + 1) i+1 σ 33 + i b3 = ρ i ü3 ,
i = 0, 1, 2, ... .

iσ
11

− i σ 33 +
(5.4.10)
For thin cylindrical shells R = R0 = const., i.e., R′ = R′′ = 0 and A1 = 1, thus the
equations of motion (5.4.5)–(5.4.7) can be simplified to
11
i σ ,1
+ i σ 12,2 +
1 13
+ (i + 1) i+1 σ 13 + i b1 = ρ i ü1 ,
iσ
R0
1
( i σ 23 + i σ 32 ) + (i + 1) i+1 σ 23 + i b2 = ρ i ü2 ,
R0
1 33
31
32
22
+
+ (i + 1) i+1 σ 33 + i b3 = ρ i ü3 ,
i σ ,1 + i σ ,2 − R0 i σ
iσ
R0
21
i σ ,1
+ i σ 22,2 +
i = 0, 1, 2, ... ,
(5.4.11)
i = 0, 1, 2, ... ,
(5.4.12)
i = 0, 1, 2, ... .
(5.4.13)
Depending on the number of equations selected from the equations of motion (5.4.5)–(5.4.7),
a large number of dimensionally reduced axisymmetric shell models can be derived. In this
dissertation a shell model based on the equations of i = 0 and i = 1 is investigated, assuming
that the higher-order translational equations of motion are identically satisfied. In this case
the stress components σ kλ , σ k3 and the displacement components up are approximated,
respectively, by polynomials of first- and second-degree in ξ 3:
σ kλ (ξ α , ξ 3, t) =
0σ
kλ
σ k3 (ξ α , ξ 3, t) =
0σ
k3
(ξ α , t) + 1 σ kλ (ξ α, t) ξ 3 ,

(ξ α , t) + 1 σ k3 (ξ α , t) ξ 3 + 2 σ k3 (ξ α , t) ξ 3
2
(5.4.14)
,
(5.4.15)
and
up (ξ α , ξ 3 , t) = 0 up (ξ α , t) + 1 up (ξ α , t) ξ 3 ,
(5.4.16)
b p (ξ α , ξ 3, t) = 0 b p (ξ α, t) + 1 b p (ξ α , t) ξ 3
(5.4.17)
as well as
are valid for the components of the body force density vector. The corresponding translational equations of motion, for the 21 stress coefficients 0 σ kλ , 1 σ kλ , 0 σ k3 , 1 σ k3 , 2 σ k3 and 6
displacement coefficients 0 up , 1 up appeared in (5.4.14)–(5.4.16), are

1 ′ 11
RR′ 22
13
1
1
R 0 σ + 0 σ 13 −
0 σ + 1 σ + 0 b = ρ 0 ü ,
RA1
A1
(5.4.18)
′′
′

R
R
1
21
23
22
12
23
+ 2 0 σ 21 +
+ 0 σ 32 + 1 σ 23 + 0 b2 = ρ 0 ü2 ,
0 σ ,1 −
0 σ + 0 σ ,2 +
0σ
0σ
A31
RA1
RA1
(5.4.19)
′′

R
1
R
31
11
22
33
3
3
− 0 σ 33 + 0 σ 32,2 +
R′ 0 σ 31 + 0 σ 33 −
0 σ ,1 +
0σ
0 σ + 1 σ + 0 b = ρ 0 ü ,
A31
RA1
A1
(5.4.20)
′′
′

R
1
RR
11
31
22
13
1
1
+ 1 σ 13 + 1 σ 12,2 +
R′ 1 σ 11 + 1 σ 13 −
1 σ ,1 − 3 1 σ
1 σ + 2 2 σ + 1 b = ρ 1 ü ,
A1
RA1
A1
(5.4.21)
′′
′

R
R
1
12
21
23
21
σ 23 + 1 σ 22,2 +
+
+ 1 σ 32 + 2 2 σ 23 + 1 b2 = ρ 1 ü2 ,
1 σ ,1 −
1σ + 2 1σ
1σ
3 1
A1
RA1
RA1
(5.4.22)
11
0 σ ,1 −
R′′
A31

0σ
31

+ 0 σ 13 + 0 σ 12,2 +
5.5. Initial and boundary conditions
R′′
A31
45

1 ′ 31
R 22
33
3
3
R 1 σ + 1 σ 33 −
1 σ + 2 2 σ + 1 b = ρ 1 ü ,
RA1
A1
(5.4.23)
which, for cylindrical shells (R = R0 = const.), can be rewritten as
31
1 σ ,1 +

1σ
11

− 1 σ 33 + 1 σ 32,2 +
11
0 σ ,1
21
0 σ ,1
31
0 σ ,1
1
13
+ 1 σ 13 + 0 b1 = ρ 0 ü1 ,
0σ
R0
(5.4.24)
1
( 0 σ 23 + 0 σ 32 ) + 1 σ 23 + 0 b2 = ρ 0 ü2 ,
R0
(5.4.25)
1
33
+ 1 σ 33 + 0 b3 = ρ 0 ü3 ,
0σ
R0
(5.4.26)
+ 0 σ 12,2 +
+ 0 σ 22,2 +
+ 0 σ 32,2 − R0 0 σ 22 +
and
11
1 σ ,1
21
1 σ ,1
31
1 σ ,1
1
13
+ 2 2 σ 13 + 1 b1 = ρ 1 ü1 ,
1σ
R0
(5.4.27)
1
( 1 σ 23 + 1 σ 32 ) + 2 2 σ 23 + 1 b2 = ρ 1 ü2 ,
R0
(5.4.28)
1
33
+ 2 2 σ 33 + 1 b3 = ρ 1 ü3 .
1σ
R0
(5.4.29)
+ 1 σ 12,2 +
+ 1 σ 22,2 +
+ 1 σ 32,2 − R0 1 σ 22 +
With regard to (5.4.14)–(5.4.15) and the first Lagrangian multiplier term of (4.4.7), the
rotation components φr (ξ m , t) are approximated by polynomials of the first-degree in ξ 3 :
φr (ξ α , ξ 3 , t) = 0 φr (ξ α , t) + 1 φr (ξ α , t) ξ 3 .
(5.4.30)
Inserting the linear stress approximations (5.4.14) in (5.3.4) and (5.3.8), recalling the condition (3.5.47), after carrying out integrations with respect to the thickness coordinate ξ 3 ,
the relations between the stress and couple resultants N κα , Qα , M κα used in the classical
shell theories and the stress coefficients σ kλ for thin shells are obtained as:
N κα =
Z +d
2
− d2
σ κα dξ 3 = d 0 σ κα ,
and
M κα =
i.e.,
σ κα =
5.5
Qα = −
Z +d
2
− d2
Z +d
2
− d2
ξ 3 σ κα dξ 3 =
N κα 12M κα 3
+
ξ ,
d
d3
σ 3α dξ 3 = −d 0 σ 3α ,
d3 κα
1σ ,
12
0σ
3α
=−
(5.4.31)
(5.4.32)
Qα
.
d
(5.4.33)
Initial and boundary conditions
Taking into account the linear displacement approximations (5.4.16), the initial conditions
(4.1.8)–(4.1.9) for the displacement and velocity coefficients can be expressed as
up (ξ m , t = 0) = up (ξ m , t0 ) = 0 up (ξ α , t0 )+ 1 up (ξ α , t0 ) ξ 3 = 0 up = 00 up (ξ α )+ 01 up (ξ α ) ξ 3
in V ,
(5.5.1)
Chapter 5. Dimensionally reduced axisymmetric shell model
u̇p (ξ m , t = 0) = u̇p (ξ m , t0 ) = 0 u̇p (ξ α , t0 ) + 1 u̇p (ξ α , t0 ) ξ 3 = 0 vp = 00 vp (ξ α) + 01 vp (ξ α ) ξ 3
46
in V ,
(5.5.2)
i.e.,
0 up |t0
= 00 up (ξ α ) ,
1 up |t0
= 01 up (ξ α ) ,
0 u̇p |t0
= 00 vp (ξ α )
= 01 vp (ξ α ) ,
on S0 .
(5.5.3)
±
The stress boundary conditions on the outer and inner surfaces S ⊂ Sσ can be written in
the form
σ · n± = σ · (±a3 ) = ±σ k3 ak = f ± ,
1 u̇p |t0
±σ k3 = (f ± )k
on S ± ,
(5.5.4)
where f ± are prescribed surface tractions on S ± with outward unit normals n± defined by
(3.5.43). Substituting (5.4.15) into the stress boundary conditions (5.5.4) and taking into
account (3.5.43) (ξ 3 = ±d/2 on S ± ), we obtain the following equations:
k3
+ 1 σ k3
0σ
−0 σ k3 + 1 σ k3
d
+ 2 σ k3
2
d
− 2 σ k3
2
d2
= (f + )k ,
4
d2
= (f − )k .
4
(5.5.5)
(5.5.6)
By adding and subtracting the above two equations, the time-dependent vector-valued load
coefficients on S ± are obtained:

1 + k
(f ) + (f − )k ,
2

1
k α
(f + )k − (f − )k .
1 f (ξ , t) =
d
0f
k
(ξ α , t) =
(5.5.7)
(5.5.8)
Making use of equations (5.5.7)–(5.5.8), the stress boundary conditions (5.5.5)–(5.5.6) can
be manipulated into the forms
1σ
k3
=
2 k
0f ,
d
2σ
k3
=−
4 k3 2 k
+ 1f ,
0σ
d2
d
(5.5.9)
where
f k (ξ α, ξ 3 , t) = 0 f k + 1 f k ξ 3 ,
(5.5.10)
and
d k
(5.5.11)
1f .
2
Equations (5.5.9), considered as field equations on S0 , show that the stress coefficients 1 σ 13 ,
23
and 1 σ 33 are determined by the given surface loads (5.5.7), while 2 σ 13 , 2 σ 23 and 2 σ 33
1σ
can be expressed, respectively, by the stress coefficients 0 σ 13 , 0 σ 23 and 0 σ 33 and by the
given surface loads (5.5.8).
Let f × be the prescribed surface traction vector on the lateral surface S L ⊂ Sσ with
outward unit normal n× defined by equation (3.5.44) (Sσ = S ± ∪ S L ). The stress boundary
conditions on S L can be written in the form
f k (ξ α , ξ 3 = ±d/2, t) = (f ± )k = 0 f k ±
σ · n× = σ · a1 = σ k1 ak = f × ,
σ k1 = (f × )k
on S L .
(5.5.12)
5.6. Euler–Lagrange equations and natural boundary conditions
47
Expanding the prescribed lateral tractions (f × )k into a truncated power series with respect to ξ 3 and substituting (5.4.14) into (5.5.12), we obtain the following stress boundary
conditions on the lateral surface S L :
0σ
k1
i.e.,
+ 1 σ k1 ξ 3 = (f × )k = 0 (f × )k + 1 (f × )k ξ 3
0σ
k1 x1 =L
= 0 (f × )k ,
1σ
k1 x1 =L
= 1 (f × )k
on S L ,
(5.5.13)
on S L ,
(5.5.14)
where 0 (f × )k and 1 (f × )k are the time-dependent prescribed vector-valued load coefficients
on the lateral surface S L .
Considering the linear approximations (5.4.16), the displacement boundary conditions
on the lateral surface S 0 = Su with outward unit normal n× = −a1 defined by (3.5.44) can
be given as
up = 0 up + 1 up ξ 3 = ue p (ξ 2 , ξ 3 , t) = 0 ue p + 1 uep ξ 3
on S 0 ,
(5.5.15)
i.e.,
0 up |ξ 1 =0
= 0 ue p (ξ 2, t) ,
1 up |ξ 1 =0
= 1 ue p (ξ 2 , t)
on S 0 ,
(5.5.16)
in which 0 ue p and 1 ue p are time-dependent prescribed displacement coefficients on the lateral
surface S 0 .
5.6
Euler–Lagrange equations and natural boundary
conditions
Let us consider now an axisymmetrically loaded thin shells of revolution. In this case the
tensor variables are independent of the angle coordinate ξ 2 , i.e., the integration with respect
to ξ 2 can easily be carried out. Thus, in view of (3.5.4) and (3.5.50), the first variation of
(4.4.7) with respect to stresses σ pq , rotations φr and displacements up yields the following
variational equation:
1
δF (σ pq , up , φr , δσ pq , δup , δφr ) =
2π
−
−
Z t Z +d Z s
1
L
2
t0
Z t Z +d §
1
2
t0
− d2
− d2
0
(εpq δσ pq − φr ǫpqr δσ pq − σ pq ǫpqr δφr ) R dξ 1 dξ 3 dt
ue p δσ p1 R0 +
Z s h
L
0
up δσ pq|q + δup σ pq |q + bp − ρ üp
i
ª
R dξ 1 dξ 3 dt = 0 ,
(5.6.1)
where, on the basis of (3.5.10)–(3.5.11),
sL =
Z L
x̂1 =0

A1 x̂1 dx̂1
is the length of the meridian curve R (x1 ), x1 ∈ [0, L ].
(5.6.2)
48
Chapter 5. Dimensionally reduced axisymmetric shell model
5.6.1
The first variation of the complementary strain energy
Making use of the linear and quadratic stress approximations (5.4.14)–(5.4.15), the first
variation of the complementary strain energy of the thin axisymmetric shell can be written
as
−
1
2π
Z t
1
t0
δ UÒ dt = −
−
1
2π
Z t Z
1
V
t0
Z t Z +d Z s h
1
L
2
t0
− d2
0
Ò
δU
(σ pq , δσ pq ) dV dt = −
Z t Z +d Z s
1
L
2
t0
− 2d

εpq δσ pq R dξ 1 dξ 3 dt =
0
εpq δ0 σ pq + εpq δ1 σ pq ξ 3 + εp3 δ2 σ p3 ξ 3
2 i
R dξ 1 dξ 3 dt . (5.6.3)
Introducing the strain resultants
0 Epq
:=
Z +d
2
− d2
3
εpq dξ ,
1 Epq
:=
Z +d
2
− d2
3
3
εpq ξ dξ ,
2 Ep3
:=
Z +d
2
− d2

εp3 ξ 3
2
dξ 3 ,
(5.6.4)
and inserting them in (5.6.3), the following expression is obtained:
−
Z t Z +d Z s
1
L
2
t0
− d2
0
εpq δσ pq R dξ 1 dξ 3 dt =
−
Z t Z s
1
L
t0
0 Epq δ0 σ
0
pq

+ 1 Epq δ1 σ pq + 2 Ep3 δ2 σ p3 R dξ 1 dt . (5.6.5)
Substituting the constitutive equations (5.2.9)–(5.2.14) into (5.6.4), taking into account
(5.4.14)–(5.4.15) and the integration rule
Z +d
2
− d2
ξ3
8
m
<
dξ 3 = :

d
d
m+1 2
0,
m
,
if m is
even
positive number ,
odd
(5.6.6)
after carrying out integrations with respect to ξ 3 , the strain resultants (5.6.4) for linearly
elastic, homogeneous and isotropic materials can be expressed as

0 E11
=
0 E22
=
0 E33
=
0 E12
=
0 E23
=
0 E31
=
2 E33
=

d
d2 33
11
2
22
33
σ
−
νR
σ
−
ν
σ
+
,
0
0
0
2σ
E
12

dR2
d2 33
33
2
22
11
R 0σ − ν 0σ − ν 0σ +
,
2σ
E
12

d
d2 33
11
2
22
33
+
,
0σ
2 σ − ν 0 σ − νR 0 σ
E
12

dR2 12
+ 0 σ 21 = 0 E21 ,
0σ
4G

dR2
d2 23
23
32
+ 0σ +
= 0 E32 ,
0σ
2σ
4G
12

d
d2 13
13
31
+ 0σ +
= 0 E13 ,
0σ
2σ
4G
12

d3
3d2 33
33
11
2
22
− ν 0 σ − νR 0 σ +
,
0σ
2σ
12E
20
(5.6.7)
(5.6.8)
(5.6.9)
(5.6.10)
(5.6.11)
(5.6.12)
(5.6.13)
5.6. Euler–Lagrange equations and natural boundary conditions

5.6.2
2 E13
=
2 E23
=
1 E11
=
1 E22
=
1 E33
=
1 E12
=
1 E23
=
1 E31
=
49

d3
3d2 13
13
31
σ
+
σ
+
,
0
0
2σ
48G
20

d3 R 2
3d2 23
23
32
+ 0σ +
,
0σ
2σ
48G
20

d3 11
2
22
− ν 1 σ 33 ,
1 σ − νR 1 σ
12E

d3 R2 2 22
R 1 σ − ν 1 σ 11 − ν 1 σ 33 ,
12E

d3 33
11
− νR2 1 σ 22 ,
1σ − ν 1σ
12E

d3 R2 12
+ 1 σ 21 = 1 E21 ,
1σ
48G

d3 R2 23
+ 1 σ 32 = 1 E32 ,
1σ
48G

d3 13
+ 1 σ 31 = 1 E13 .
1σ
48G
(5.6.14)
(5.6.15)
(5.6.16)
(5.6.17)
(5.6.18)
(5.6.19)
(5.6.20)
(5.6.21)
The first variation of the boundary integral term
Making use of the linear stress approximation (5.4.14), the boundary integral appeared in
the variational equation (5.6.1) in terms of the variations of the stress coefficients can be
written as
−
Z t Z +d
1
2
− d2
t0
uep δσ p1 R0 dξ 3 dt = −
Z t Z +d
1
2
− d2
t0

uep δ0 σ p1 + ue p δ1 σ p1 ξ 3 R0 dξ 3 dt .
(5.6.22)
Introducing the prescribed displacement resultants
e
0 Up :=
Z +d
2
− d2
ue p dξ 3 ,
e
1 Up :=
Z +d
2
− d2
ue p ξ 3 dξ 3 ,
(5.6.23)
and substituting them into equation (5.6.22), the first variation of the boundary integral
term takes the form
−
Z t Z +d
1
2
t0
− d2
uep δσ p1 R0 dξ 3 dt = −
Z t
1
t0
e
0 Up R0 δ0 σ
p1

+ 1 Ue p R0 δ1 σ p1 dt .
(5.6.24)
Inserting (5.5.15) in (5.6.23) and carrying out integrations with respect to the thickness
coordinate ξ 3 , (5.6.23) yields the formulas
e
0 Up
5.6.3
= 0 uep d ,
e
1 Up
= 1 ue p
d3
.
12
(5.6.25)
The first variation of the Lagrangian multiplier terms
Using the variations of the expanded stress components (5.4.14)–(5.4.15) and rotation components (5.4.30), the second and third Lagrangian multiplier terms appeared in equation
(5.6.1) can be expressed, respectively, by
50
Chapter 5. Dimensionally reduced axisymmetric shell model
Z t Z +d Z s
1
L
2
t0
− 2d
0
φr ǫpqr δσ pq R dξ 1 dξ 3 dt =
Z t Z +d Z s h
1
L
2
− d2
t0

φr ǫpqr δ0 σ pq + φr ǫpqr δ1 σ pq ξ 3 + φr ǫp3r δ2 σ p3 ξ 3
0
2 i
R dξ 1 dξ 3 dt , (5.6.26)
and
Z t Z +d Z s
1
L
2
t0
− 2d
0
σ pq ǫpqr δφr R dξ 1 dξ 3 dt =
Z t Z +d Z s
1
L
2
− d2
t0

σ pq ǫpqr δ0 φr + σ pq ǫpqr δ1 φr ξ 3 R dξ 1 dξ 3 dt . (5.6.27)
0
Introducing the rotation resultants
0 Φpq
:= −
Z +d
2
− d2
r
3
φ ǫpqr dξ ,
1 Φpq
:= −
Z +d
2
− d2
r
3
3
φ ǫpqr ξ dξ ,
2 Φp3
:= −
Z +d
2
− d2

φr ǫp3r ξ 3
2
dξ 3 ,
(5.6.28)
and substituting them into (5.6.26), the following expression is obtained:
Z t Z +d Z s
1
L
2
t0
− 2d
0
φr ǫpqr δσ pq R dξ 1 dξ 3 dt =
−
Z t Z s
1
L
t0
0
0 Φpq δ0 σ
pq

+ 1 Φpq δ1 σ pq + 2 Φp3 δ2 σ p3 R dξ 1 dt . (5.6.29)
Inserting the expanded rotation components (5.4.30) in (5.6.28) and taking into account
(3.2.32), (4.4.2), as well as (5.6.6), after carrying out integrations with respect to ξ 3 , (5.6.28)
results in
0 Φ21 =
Z +d
2
− d2
φ3 R dξ 3 = dR 0 φ3 = −0 Φ12 ,
0 Φ32 =
Z +d
2
− d2
φ1 R dξ 3 = dR 0 φ1 = −0 Φ23 ,
(5.6.30)
0 Φ13
1 Φ32 =
=
Z +d
2
Z +d
2
2 Φ23
− d2
− d2
φ2 R dξ 3 = dR 0 φ2 = −0 Φ31 ,
φ1 R ξ 3dξ 3 =
=−
Z +d
2
− d2
1

φR ξ
d3 R 1
1 φ = −1 Φ23 ,
12

3 2
d3 R 1
dξ =
0φ ,
12
3
1 Φ21
=
1 Φ13 =
2 Φ13
=
Z +d
2
− d2
Z +d
2
− d2
Z +d
2
− d2
φ3 R ξ 3 dξ 3 =
φ2 R ξ 3 dξ 3 =

φ2 R ξ 3
2
3
dR 3
1 φ = −1 Φ12 ,
12
(5.6.31)
d3 R 2
1 φ = −1 Φ31 ,
12
(5.6.32)
dξ 3 =
d3 R 2
0φ .
12
(5.6.33)
Let us investigate the integral (5.6.27) appearing in (5.6.1). Substituting the expanded
stress components (5.4.14)–(5.4.15) into (5.6.27) and carrying out integrations with respect
to ξ 3 , the following equation is obtained:
Z t Z +d Z s
1
L
2
t0
− 2d
0
σ pq ǫpqr δφr R dξ 1 dξ 3 dt =
5.6. Euler–Lagrange equations and natural boundary conditions
Z t Z s
1
L t0
0
+
d

0σ
23

− 0 σ 32 δ0 φ1 +

0σ
31

− 0 σ 13 δ0 φ2 +

51
0σ
12

− 0 σ 21 δ0 φ3

d3 23
− 1 σ 32 δ1 φ1 + 1 σ 31 − 1 σ 13 δ1 φ2 + 1 σ 12 − 1 σ 21 δ1 φ3
1σ
12
©
+ 2 σ 23 δ0 φ1 − 2 σ 13 δ0 φ2 R2 dξ 1 dt . (5.6.34)
According to the variational equation (5.6.1), the coefficients of the rotation components
δ0 φr , δ1 φr in the above expression should be equal to zero:
1
δ0 φ :
0σ
23
δ0 φ2 :
0σ
13
=
δ0 φ3
δ1 φ1
δ1 φ2
δ1 φ3
0σ
12
1σ
23
1σ
13
1σ
12
=
=
=
=
:
:
:
:
=
d2 23
−
,
0σ
2σ
12
d2 13
31
,
−
2σ
0σ
12
21
,
0σ
32
,
1σ
31
,
1σ
21
.
1σ
32
(5.6.35)
(5.6.36)
(5.6.37)
(5.6.38)
(5.6.39)
(5.6.40)
It can easily be seen that equations (5.6.35)–(5.6.36) and (5.6.38)–(5.6.39) are equivalent
to the symmetry of the transverse shear stresses, satisfied in integral average sense. The
advantage of equations (5.6.35)–(5.6.40) is that 0 σ 23 , 0 σ 13 , 0 σ 12 , 1 σ 23 , 1 σ 13 and 1 σ 12 can be
expressed in terms of the stress coefficients 0 σ 32 and 2 σ 23 , 0 σ 31 and 2 σ 13 , 0 σ 21 , 1 σ 32 , 1 σ 31 and
21
1 σ , respectively. Thus, not only the number of rotation variables but also the number of
stress variables could further be reduced. Recalling the translational equations of motion
(5.4.8)–(5.4.10) and taking into account (5.4.14)–(5.4.16) and (5.6.6), after carrying out the
integration with respect to ξ 3, the last integral appeared in (5.6.1) can be written in the
form
Z t Z +d Z s
1
L
2
t0
− d2
Z t Z s 1
L
0
δup σ pq |q + bp − ρ üp R dξ 1 dξ 3 dt =
R′′
A31

1 ′ 11
RR′ 22
13
1
1
δ0 u1 d
+
R 0 σ + 0 σ 13 −
0σ + 0σ
0 σ + 1 σ + 0 b − ρ 0 ü
RA1
A1
t0
0

d3
R′′ 31
RR′ 22
1 ′ 11
11
13
13
13
1
1
+δ1 u1
−
R 1σ + 1σ
+ 2 2 σ + 1 b − ρ 1 ü
1 σ ,1 −
1σ + 1σ
1σ +
12
A1
RA1
A31

R′′
R′ 12
1 32
21
23
23
2
2
+ δ0 u2 d 0 σ 21,1 − 3 0 σ 23 +
σ
+
2
σ
+
σ
+
σ
+
σ
+
b
−
ρ
ü
0
0
0
0
1
0
0
A1
RA1
RA1

d3
R′′ 23
R′ 12
1 32
21
21
23
23
2
2
+ δ1 u2
σ
−
σ
+
σ
+
2
σ
+
σ
+
σ
+
2
σ
+
b
−
ρ
ü
1
1
1
1
1
1
2
1
1
,1
12
RA1
RA1
A31

d3
R′′ 33
R
1 ′ 31
31
11
22
33
33
3
3
+ δ1 u3
−
R 1σ + 1σ
+ 2 2 σ + 1 b − ρ 1 ü
1 σ ,1 −
1σ − 1σ
1σ +
12
A1
RA1
A31

R′′
1 ′ 31
R 22
33
3
3
+δ0 u3 d 0 σ 31,1 − 3 0 σ 33 − 0 σ 11 +
R 0 σ + 0 σ 33 −
σ
+
σ
+
b
−
ρ
ü
Rdξ 1 dt.
0
1
0
0
A1
RA1
A1
(5.6.41)
11
0 σ ,1 −

31
13

Chapter 5. Dimensionally reduced axisymmetric shell model
52
According to the variational equation (5.6.1), the coefficients of the displacement components δ0 up , δ1 up in the previous equation should be equal to zero:

1 ′ 11
RR′ 22
13
1
1
R 0 σ + 0 σ 13 −
0 σ + 1 σ + 0 b = ρ 0 ü ,
RA1
A1
(5.6.42)
′′
′

1
R
RR
31
13
22
11
−
R′ 1 σ 11 + 1 σ 13 + 2 2 σ 13 + 1 b1 = ρ 1 ü1 ,
1 σ ,1 − 3 1 σ + 1 σ
1σ +
A1
A1
RA1
(5.6.43)
′′
′

1
R
R
32
12
21
σ 23 +
+ 0 σ 23 +
+ 2 0 σ 21 + 1 σ 23 + 0 b2 = ρ 0 ü2 ,
0σ
0σ
0 σ ,1 −
3 0
A1
RA1
RA1
(5.6.44)
δ0 u1 :
δ1 u1 :
δ0 u2 :
11
0 σ ,1 −
R′′
A31
21
1 σ ,1 −
R′′ 23
R′
σ
+
1
A31
RA1

0σ
31

+ 0 σ 13 +
and
δ1 u2 :

1σ
12

+ 2 1 σ 21 +
1
RA1

1σ
32

+ 1 σ 23 + 2 2 σ 23 + 1 b2 = ρ 1 ü2 ,
(5.6.45)

1
R
22
33
3
3
33
δ0 u3 :
R′ 0 σ 31 + 0 σ 33 −
− 0 σ 11 +
0 σ + 1 σ + 0 b = ρ 0 ü ,
0σ
RA1
A1
(5.6.46)
′′

R
R 22
1
δ1 u3 : 1 σ 31,1 − 3 1 σ 33 − 1 σ 11 −
R′ 1 σ 31 + 1 σ 33 + 2 2σ 33 + 1 b3 = ρ 1 ü3 .
1σ +
A1
A1
RA1
(5.6.47)
Applying a similar procedure to the fifth integral appeared in (5.6.1) and taking into account
(3.5.12), (5.6.2) and the integration rule
R′′
31
σ
−
0
,1
A31
Z s
L
0

p1
1
i up δi σ ,1 R dξ =

p1
i up δi σ R
ξ 1 =s
L
ξ 1 =0
−
Z s
L
Z s
L
0
0
p1
1
i up,1 δi σ R dξ −
p1
i up δi σ
R′ 1
dξ ,
A1
i = 0, 1 ,
(5.6.48)
the following expression is obtained:
Z t Z +d Z s
1
L
2
t0
Z t
1
− 2d
®
0
up δσ pq|q R dξ 1 dξ 3 dt =
ξ 1 =s
L

d3 R
11
21
31
+ dR 0 u1 δ0 σ 11 + 0 u2 δ0 σ 21 + 0 u3 δ0 σ 31
1 u1 δ1 σ + 1 u2 δ1 σ + 1 u3 δ1 σ
t0
12
ξ 1 =0
Z s ¨
′

L
dR
11
21
31
d 0 u1,1 δ0 σ 11 + 0 u2,1 δ0 σ 21 + 0 u3,1 δ0 σ 31 +
−
0 u1 δ0 σ + 0 u2 δ0 σ + 0 u3 δ0 σ
0
RA1

d3
d3 R ′
11
11
+
+ 1 u2,1 δ1 σ 21 + 1 u3,1 δ1 σ 31 +
+ 1 u2 δ1 σ 21 + 1 u3 δ1 σ 31
1 u1,1 δ1 σ
1 u1 δ1 σ
12
12RA1

′′

1 ′ 11
R
RR′
13
31
13
22
13
− 0 u1 d
R δ0 σ + δ0 σ − 3 δ0 σ + δ0 σ −
δ0 σ + δ1 σ
RA1
A1
A1

d3
1 ′ 11
RR′
R′′ 31
13
22
13
13
− 1 u1
R δ1 σ + δ1 σ −
δ1 σ + 2 δ2 σ − 3 δ1 σ + δ1 σ
12 RA1
A1
A1

′′

R′ 12
R
1
21
23
23
32
23
− 0 u2 d
δ0 σ + 2 δ0 σ + δ1 σ − 3 δ0 σ +
δ0 σ + δ0 σ
RA1
A1
RA1

d3 R′ 12
R′′
1 32
21
23
23
23
− 1 u2
δ1 σ + 2 δ1 σ − 3 δ1 σ +
δ1 σ + δ1 σ + 2 δ2 σ
12 RA1
A1
RA1
5.6. Euler–Lagrange equations and natural boundary conditions

1 ′ 31
R′′
R
− 0 u3 d
R δ0 σ + δ0 σ 33 + δ1 σ 33 − 3 δ0 σ 33 − δ0 σ 11 −
δ0 σ 22
RA1
A1
A1

«
¸

R′′ 33
d3
1 ′ 31
R
22
33
11
33
1
− 1 u3
R δ1 σ + δ1 σ −
δ1 σ − 3 δ1 σ − δ1 σ + 2 δ2 σ
R dξ dt .
12 RA1
A1
A1
(5.6.49)
Keeping in mind expressions (5.6.1), (5.6.5), (5.6.24), (5.6.29) and (5.6.49), the coefficients
of δ0 σ pq , δ1 σ pq and δ2 σ p3 should be equal to zero, yielding the kinematic equations
δ0 σ 11 :
d 0 u1,1 −
δ0 σ 12 :
−
δ0 σ 13 :
δ0 σ 21 :
δ0 σ 22 :
δ0 σ 23 :
δ0 σ 31 :
δ0 σ 32 :
δ0 σ 33 :
δ1 σ 11 :
δ1 σ 12 :
δ1 σ 13 :
δ1 σ 21 :
δ1 σ 22 :
δ1 σ 23 :
δ1 σ 31 :
δ1 σ 32 :
dR′′
0 u3 − 0 E11 = 0 ,
A31
dR′
0 u2 − 0 Φ12 − 0 E12 = 0 ,
RA1

R′′
1
d
−
0 u1 − 0 Φ13 − 0 E13 = 0 ,
A31 RA1
dR′
d 0 u2,1 +
0 u2 − 0 Φ21 − 0 E21 = 0 ,
RA1
RR′
dR
d
0 u1 +
0 u3 − 0 E22 = 0 ,
A1
A1

1
R′′
−
d
0 u2 − 0 Φ23 − 0 E23 = 0 ,
A31 RA1
dR′′
d 0 u3,1 + 3 0 u1 − 0 Φ31 − 0 E31 = 0 ,
A1
d
−
0 u2 − 0 Φ32 − 0 E32 = 0 ,
RA1

R′′
1
−
d
0 u3 − 0 E33 = 0 ,
A31 RA1

d3
R′′
1 u1,1 −
1 u3 − 1 E11 = 0 ,
12
A31
d3 R ′
−
1 u2 − 1 Φ12 − 1 E12 = 0 ,
12 RA1

d3 R′′
1
−
1 u1 − d 0 u1 − 1 Φ13 − 1 E13 = 0 ,
12 A31 RA1

d3
R′
1 u2 − 1 Φ21 − 1 E21 = 0 ,
1 u2,1 −
12
RA1

d3 RR′
R
1 u1 +
1 u3 − 1 E22 = 0 ,
12 A1
A1

d3 R′′
1
−
1 u2 − d 0 u2 − 1 Φ23 − 1 E23 = 0 ,
12 A31 RA1

d3
R′′
1 u3,1 +
1 u1 − 1 Φ31 − 1 E31 = 0 ,
12
A31
d3
−
1 u2 − 1 Φ32 − 1 E32 = 0 ,
12RA1
(5.6.50)
(5.6.51)
(5.6.52)
(5.6.53)
(5.6.54)
(5.6.55)
(5.6.56)
(5.6.57)
(5.6.58)
(5.6.59)
(5.6.60)
(5.6.61)
(5.6.62)
(5.6.63)
(5.6.64)
(5.6.65)
(5.6.66)
Chapter 5. Dimensionally reduced axisymmetric shell model

δ1 σ
33
d3
12
d3
−
6
d3
−
6
d3
−
6
:
δ2 σ 13 :
δ2 σ 23 :
δ2 σ 33 :
R′′
1
−
3
A1 RA1
54

1 u3
− d 0 u3 − 1 E33 = 0 ,
(5.6.67)
1 u1
− 2 Φ13 − 2 E13 = 0 ,
(5.6.68)
1 u2
− 2 Φ23 − 2 E23 = 0 ,
(5.6.69)
1 u3
− 2 E33 = 0 ,
(5.6.70)
and the natural boundary conditions
δ0 σ p1 :
δ1 σ p1 :
d 0 up |ξ1 =0 − 0 Uep = 0 ,
(5.6.71)
3
d
e
1 up |ξ 1 =0 − 1 Up = 0 .
12
(5.6.72)
The Euler–Lagrange equations of the three-field dual-mixed functional (4.4.7) for thin shells
of revolution loaded axisymmetrically are the rotational and translational equations of motion, (5.6.35)–(5.6.40) and (5.6.42)–(5.6.47), respectively, the kinematic equations (5.6.50)–
(5.6.70) and the displacement boundary conditions (5.6.71)–(5.6.72).
5.7
Fundamental equation system of the axisymmetric shell model
Inserting the stress resultants (5.6.7)–(5.6.21), the rotation resultants (5.6.30)–(5.6.33) and
the prescribed displacement resultants (5.6.25), respectively, in the kinematic equations
(5.6.50)–(5.6.70) and the natural boundary conditions (5.6.71)–(5.6.72), the fundamental
equation system and the displacement boundary conditions of the axisymmetric shell model
for linearly elastic, homogeneous and isotropic materials can be written in terms of the
independent stress, rotation and displacement coefficients as
R′′
1
u
−
u −
0 1,1
3 0 3
A1
E
−

0σ
11
2
− νR 0 σ
22
d2 33
σ
+
0
2σ
12
−ν

1
0 u1 − R 0 φ −
4G

2
=0,
+
0σ

(5.7.2)
+ 0σ
31
d2 13
+
=0,
2σ
12

R′
R2 12
3
+ 0 σ 21 = 0 ,
0 u2 − R 0 φ −
0σ
RA1
4G

R′′
1
−
3
A1
RA1
R2
u
+
R
φ
−
0 2
0
4G
1
(5.7.1)

13
R′
1
1
R2 0 σ 22 − ν 0 σ 11 − ν
0 u1 +
0 u3 −
RA1
RA1
E

R′
R2 12
3
+ 0 σ 21 = 0 ,
0 u2 + R 0 φ −
0σ
RA1
4G
R′′
1
−
3
A1 RA1
0 u2,1

0σ
33

0σ
23
+ 0σ
32
d2 33
+
2σ
12
(5.7.3)
(5.7.4)

d2 23
+
2σ
12
=0,
(5.7.5)
=0,
(5.7.6)

5.7. Fundamental equation system of the axisymmetric shell model

R′′
1
u
+
u + R 0 φ2 −
0 3,1
3 0 1
A1
4G
1
R2
1
−
u
−
R
φ
−
0 2
0
RA1
4G

1
R′′
−
3
A1 RA1

1
0 u3 −
E
1 u1,1 −

0σ
13
+ 0σ

31
d2 13
+
2σ
12
55

=0,

0σ
23
+ 0σ
32
d2 23
+
=0,
2σ
12
0σ
R′′
1
u −
3 1 3
A1
E
d2 33
− ν 0 σ 11 − νR2 0 σ 22 = 0 ,
+
2σ
12

1σ
11

− νR2 1 σ 22 − ν 1 σ 33 = 0 ,

R′
R2 12
3
u
+
R
φ
−
+ 1 σ 21 = 0 ,
1 2
1
1σ
RA1
4G

12
R′′
1
1 13
31
2
u
−
σ
+
σ
=0,
−
u
−
R
φ
−
1
1
1
1
0
1
1
A31 RA1
d2
4G

R′
R2 12
3
21
=0,
1 u2,1 −
1 u2 − R 1 φ −
1σ + 1σ
RA1
4G

1
1 2 22
R′
R 1 σ − ν 1 σ 11 − ν 1 σ 33 = 0 ,
1 u1 +
1 u3 −
RA1
RA1
E

R′′
1
−
3
A1 RA1
1 u2
1 u3,1 +

−

R2 23
12
1
32
u
+
R
φ
−
σ
+
σ
=0,
0
2
1
1
1
d2
4G

R′′
1 13
u + R 1 φ2 −
+ 1 σ 31 = 0 ,
1σ
3 1 1
A1
4G

1
R2 23
1
u
−
R
φ
−
+ 1 σ 32 = 0 ,
1 2
1
1σ
RA1
4G

12
1 33
− ν 1 σ 11 − νR2 1 σ 22 = 0 ,
1 u3 − 2 0 u3 −
1σ
d
E
−
R′′
1
−
3
A1
RA1

1
−2 1 u1 − R 0 φ −
4G

2
R2
−2 1 u2 + R 0 φ −
4G
11
0 σ ,1 −
R′′
A31
11
1 σ ,1
−
R′′
A31
21
0 σ ,1
−
R′′
A31
0σ
13
0σ
23
3d2 13
+ 0σ +
2σ
20
(5.7.9)
(5.7.10)
(5.7.11)
(5.7.12)
(5.7.13)
(5.7.14)
(5.7.15)
(5.7.16)
(5.7.17)
(5.7.18)

31
=0,
(5.7.19)
32
3d2 23
+
=0,
2σ
20
(5.7.20)

1
1
−2 1 u3 −
E
(5.7.8)

33
−

(5.7.7)

+ 0σ

0σ
33
− ν 0σ
11
2
− νR 0 σ
22
3d2 33
+
=0,
2σ
20
(5.7.21)

1 ′ 11
RR′ 22
R 0 σ + 0 σ 13 −
+ 1 σ 13 + 0 b1 = ρ 0 ü1 ,
0σ
RA1
A1
(5.7.22)
′

RR 22
1
31
+ 1 σ 13 −
+
R′ 1 σ 11 + 1 σ 13 + 2 2 σ 13 + 1 b1 = ρ 1 ü1 ,
1σ
1σ
A1
RA1
(5.7.23)

1 32
R′ 12
23
+
+ 0 σ 23 +
+ 2 0 σ 21 + 1 σ 23 + 0 b2 = ρ 0 ü2 ,
0σ
0σ
0σ
RA1
RA1
(5.7.24)

31
+ 0 σ 13 +
0σ
56
Chapter 5. Dimensionally reduced axisymmetric shell model
21
1 σ ,1 −
R′′ 23
R′
σ
+
1
A31
RA1

1σ
12

+ 2 1 σ 21 +
1
RA1

1σ
32

+ 1 σ 23 + 2 2 σ 23 + 1 b2 = ρ 1 ü2 ,
(5.7.25)
R′′
1
R 22
R′ 0 σ 31 + 0 σ 33 −
+ 1 σ 33 + 0 b3 = ρ 0 ü3 ,
0σ
RA1
A1
(5.7.26)

R′′ 33
R 22
1 ′ 31
31
11
33
−
R
σ
+
σ
+ 2 2 σ 33 + 1 b3 = ρ 1 ü3 ,
σ
−
σ
σ
+
1 σ ,1 −
1
1
1
1
1
3
A1
A1
RA1
(5.7.27)
31
0 σ ,1
−
A31
0σ
33

− 0 σ 11 +

d2 23
=0,
2σ
12
d2 13
31
13
σ
−
σ
−
=0,
0
0
2σ
12
12
21
=0,
0σ − 0σ
0σ
23
− 0 σ 32 +
1σ
23
1σ
31
1σ
12
− 1 σ 32 = 0 ,
(5.7.28)
(5.7.29)
(5.7.30)
(5.7.31)
− 1 σ 13 = 0 ,
(5.7.32)
− 1 σ 21 = 0 ,
(5.7.33)
and
0 up |ξ 1 =0
− 0 ue p = 0 ,
1 up |ξ 1 =0
− 1 ue p = 0 .
(5.7.34)
The number of the unknown functions occurred in this shell model is 33 including the 21
stress coefficients 0 σ κλ , 1 σ κλ , 0 σ p3 , 1 σ p3 , 2 σ p3 , the 6 rotation coefficients 0 φr , 1 φr and the
6 displacement coefficients 0 up , 1 up . The subsidiary conditions to the fundamental equations (5.7.1)–(5.7.33) are the stress boundary conditions (5.5.9), (5.5.14), the displacement
boundary conditions (5.7.34) and the initial conditions (5.5.3).
Chapter 6
Finite element formulation for
elastodynamic problems
6.1
Dual-mixed variational principle for axisymmetric
shells
In this section, a special form of the variational equation (5.6.1) is derived for axisymmetric
elastodynamic problems of thin shells of revolution. In view of (3.5.4), (3.5.49), (5.6.2) and
the constitutive equations (5.2.8), the first variation of the complementary strain energy
appearing in functional (4.4.7) can be given as
−
Z t Z + d Z 2π Z s
1
L
2
t0
− d2
−
0
π
2G
0
Ò (σ pq , δσ pq ) R dξ 1 dξ 2 dξ 3 dt = −2π
δU
Z t Z +d Z s
1
L
2
t0
− d2
− d2
t0
0
εpq δσ pq R dξ 1 dξ 3 dt =

(σ rs + σ sr ) arp asq −
0
Z t Z +d Z s
1
L
2
2ν
σ kℓ akℓ apq δσ pq R dξ 1 dξ 3 dt . (6.1.1)
1+ν
Substituting the linear and quadratic stress approximations (5.4.14)–(5.4.15) into (6.1.1)
and taking into account the integration rule (5.6.6), after carrying out integrations with
respect to ξ 3 , the complementary virtual work (6.1.1) of the internal forces can be expressed
in terms of the stress coefficients as
− 2π
Z t Z +d Z s
1
L
2
t0
− d2
0
Ò
δU
(σ pq , δσ pq ) R dξ 1 dξ 3 dt = −2π
− 2πd
+
Z t Z s 1
L
t0
2
R
0
0σ
21
1
E

0σ
11
Z t Z +d Z s
1
L
2
t0
− d2
0
εpq δσ pq R dξ 1 dξ 3 dt =
δ0 σ 11 + R4 0 σ 22 δ0 σ 22 + 0 σ 33 δ0 σ 33
δ0 σ 21 + 0 σ 21 δ0 σ 12 + 0 σ 12 δ0 σ 21 + 0 σ 12 δ0 σ 12

R2 d2 12 12
+ 1 σ 12 δ1 σ 21 + 1 σ 21 δ1 σ 12 + 1 σ 21 δ1 σ 21
+
1 σ δ1 σ
48G

R2 23 23
+
+ 0 σ 23 δ0 σ 32 + 0 σ 32 δ0 σ 23 + 0 σ 32 δ0 σ 32
0 σ δ0 σ
4G

1 13 13
+
+ 0 σ 13 δ0 σ 31 + 0 σ 31 δ0 σ 13 + 0 σ 31 δ0 σ 31
0 σ δ0 σ
4G

ν 11 33
2
22
33
−
+ R2 0 σ 22 δ0 σ 11
0 σ δ0 σ + R 0 σ δ0 σ
E
57
Chapter 6. Finite element formulation for elastodynamic problems
−
ν
E

0σ
33
δ0 σ 11 + R2 0 σ 11 δ0 σ 22 + R2 0 σ 33 δ0 σ 22
58

d2 13 13
+
+ 2 σ 13 δ0 σ 31 + 0 σ 13 δ2 σ 13 + 0 σ 31 δ2 σ 13
2 σ δ0 σ
48G

R2 d2 23 23
+
+ 2 σ 23 δ0 σ 32 + 0 σ 23 δ2 σ 23 + 0 σ 32 δ2 σ 23
2 σ δ0 σ
48G
+
d2
12E

1σ
11
δ1 σ 11 + R4 1 σ 22 δ1 σ 22 + 1 σ 33 δ1 σ 33 + 2 σ 33 δ0 σ 33 + 0 σ 33 δ2 σ 33

d2 13 13
+ 1 σ 13 δ1 σ 31 + 1 σ 31 δ1 σ 13 + 1 σ 31 δ1 σ 31
1 σ δ1 σ
48G

R2 d2 23 23
+
+ 1 σ 23 δ1 σ 32 + 1 σ 32 δ1 σ 23 + 1 σ 32 δ1 σ 32
1 σ δ1 σ
48G

νd2 33 11
−
+ R2 2 σ 33 δ0 σ 22 + R2 0 σ 22 δ2 σ 33 + 0 σ 11 δ2 σ 33
2 σ δ0 σ
12E
νd2 11 33
−
+ R2 1 σ 33 δ1 σ 22 + R2 1 σ 22 δ1 σ 11
1 σ δ1 σ
12E

+R2 1 σ 11 δ1 σ 22 + 1 σ 33 δ1 σ 11 + R2 1 σ 22 δ1 σ 33
+

d4
d4
R2 d4 23 23
33
33
13
13
+
σ
δ
σ
+
σ
δ
σ
+
R dξ 1 dt . (6.1.2)
2
2
2
2
2 σ δ2 σ
80E
320G
320G
The number of the 21 independent stress coefficients is reduced by six through the a priori satisfaction of the six transformed traction boundary conditions (5.5.9) on the outer
and inner surfaces S ± . Inserting (5.5.9) in (5.6.35)–(5.6.40), the following expressions are
obtained:
3 3λ d λ
− 1f ,
0σ
2
4
2
κ3
= 1 σ 3κ = 0 f k ,
1σ
d
12
21
= 0σ ,
0σ
12
= 1 σ 21 .
1σ
0σ
λ3
(6.1.3)
=
(6.1.4)
(6.1.5)
(6.1.6)
The shear stresses 0 σ λ3 , 1 σ κ3 , the membrane stress 0 σ 12 and the torsional stress 1 σ 12 are
eliminated from the variational equation (5.6.1) by satisfying the above symmetry equations
in an integral average sense. Thereby, not only the number of the independent stress
variables are reduced further, but also the rotation coefficients 0 φr , 1 φr are eliminated.
Consequently, the second and third Lagrangian multiplier terms are vanishing from (5.6.1):
−
Z t Z
1
t0
pq
V
δ (ψpq σ ) dV dt = 2π
Z t Z +d Z s
1
L
2
|
+ 2π
− d2
t0
0
Z t Z +d Z s
1
L
2
|
t0
− d2
0
φr ǫpqr δσ pq R dξ 1 dξ 3 dt
{z
σ pq ǫpqr δφr R dξ 1 dξ 3 dt = 0 ,
{z
0
i.e., the variational equation (5.6.1) becomes
}
0
}
(6.1.7)
6.1. Dual-mixed variational principle for axisymmetric shells
Z t Z +d
1
1
2
δF (σ pq , up , δσ pq , δup ) = −
2π
t0
− d2
−
Z t Z +d Z s h
1
L
2
t0
− d2
0
Z s
L
0
59

εpq δσ pq R dξ 1 + ue p δσ p1 R0 dξ 3 dt
up δσ pq|q + δup σ pq |q + bp − ρ üp
i
R dξ 1 dξ 3 dt = 0 . (6.1.8)
Thus the number of the unknown functions is 15, including 9 stress- and 6 displacement coefficients. From the combination of the symmetry conditions (6.1.4) and the stress boundary
conditions (5.5.14) on the lateral surface S L , it follows that
× 3
1 (f ) =
2
1
f
0
x1 =L
d
on S L .
(6.1.9)
It means that the load coefficients 1 (f × )3 and 0 f 1 are not independent of each other.
Substituting the expanded displacement components (5.4.16) into the variational equation (6.1.8), taking into account (5.4.8)–(5.4.10), (5.4.14)–(5.4.16), (5.5.9), (5.6.6) and
(6.1.3)–(6.1.6), after carrying out integrations with respect to ξ 3 , the last two Lagrangian
multiplier terms appearing in (6.1.8) take the forms
2π
Z t Z +d Z s
1
L
2
t0
− d2
0
up δσ pq|q R dξ 1 dξ 3 dt =
Z t Z s 1
L
3
5R′′
R′
RR′
31
11
2πd
+
−
δ
σ
+
δ
σ
−
δ0 σ 22
0
0
2RA1
RA1
A1
2A31
t0
0
5
d2
3R′′
3R′
3R′
12
32
21
21
21
32
+ 0 u2 δ0 σ 21,1 +
−
δ
σ
+
δ
σ
+
u
δ
σ
+
δ
σ
−
δ
σ
0
0
1 2
1
1
0
,1
2RA1
RA1
12
RA1
d2
2A31
d2
d2 R′′
RR′
R′
12
R
8
11
22
33
+ 1 u1
δ1 σ 11,1 −
δ1 σ 22 +
δ1 σ 11 − 2 δ0 σ 31 +1 u3
δ
σ
−
δ
σ
−
δ
σ
1
1
0
12
A1
RA1
d
12 A31
A1
d2
R′
R′′
1
R′′
R
+ 0 u3 δ0 σ 31,1 +
δ0 σ 31 + 3 δ0 σ 11 +
− 3 δ0 σ 33 −
δ0 σ 22 R dξ 1 dt , (6.1.10)
RA1
RA1
A1
A1
A1
11
0 u1 δ0 σ ,1
and
2π
Z t Z +d Z s
1
L
2
t0
− d2
0
δup σ pq |q + bp − ρ üp R dξ 1 dξ 3 dt =
Z t Z s 1
L
δ0 u1
+
3
5R′′
−
2RA1
2A31
R′
RR′ 22
11
1
0σ −
0 σ − ρ 0 ü
RA1
A1
t0
0
2
dR′′
d
5
3R′′
3R′
1
1
21
32
21
2
+ 0f 1 +
−
f
+
b
+
δ
u
σ
+
−
1
0
0 2 0
0σ +
0 σ − ρ 0 ü
,1
3
3
d
4RA1
2RA1
RA1
4A1
2A1
2 2
dR′′
d
R′
R′′ 11
1
R′′
2
2
31
31
+ 0f +
−
− 3 0 σ 33
1 f + 0 b + δ0 u3 0 σ ,1 +
0σ +
0σ +
d
4RA1
RA1
RA1
4A31
A31
A1
2
′
′
R
2 3
R
RR
12 31
d
22
3
3
11
11
22
1
−
+ δ1 u1
0 σ − ρ 0 ü +
0f + 0b
1 σ ,1 +
1σ −
1 σ − 2 0 σ − ρ 1 ü
A1
d
12
RA1
A1
d
2
4R′′
6 1
d2
3R′
12 32
1
1
21
21
2
+
−
f
+
f
+
b
+
δ
u
σ
+
0
1
1
1 2
1
1 σ − 2 0 σ − ρ 1 ü
,1
3
dRA1 dA1
d
12
RA1
d
4
2R′′
6 2
d2 R′′ 11
R
8
2
2
22
33
3
+
−
+ δ1 u3
0f +
1f + 1b
1σ −
1 σ − 2 0 σ − ρ 1 ü
3
3
dRA1 dA1
d
12 A1
A1
d
2R′
2 1
1
R′′ 2 3 4 3
1
3
+
− 3
R dξ 1 dt . (6.1.11)
0f +
0 f ,1 +
0f +
1f + 1b
dRA1
d
RA1
A1 d
d
2πd
11
0 σ ,1
0σ
31
+
Chapter 6. Finite element formulation for elastodynamic problems
60
Making use of the linear approximations (5.4.14) and (5.5.15), as well as the integration
rule (5.6.6), after carrying out integrations with respect to the thickness coordinate ξ 3, the
boundary integral term of the variational equation (6.1.8) in terms of the variations of the
stress coefficients can be written as
2πR0
Z t Z +d
1
2
− d2
t0
uep δσ p1 dξ 3 dt = 2πdR0
+
Z t
1
t0
e 1 δ0 σ
0u
11
+ 0 ue 2 δ0 σ 21 + 0 ue 3 δ0 σ 31

d2
11
e 1 δ1 σ
+ 1 ue 2 δ1 σ 21 + 1 ue 3 δ1 σ 31 dt ,
1u
12
(6.1.12)
which, with the aid of (6.1.4), can be put in the form
2πR0
Z t Z +d
1
2
− d2
t0
uep δσ p1 dξ 3 dt = 2πdR0
Z t
1
t0
e 1 δ0 σ
0u
+
11
+ 0 ue 2 δ0 σ 21 + 0 ue 3 δ0 σ 31

d2
11
e 1 δ1 σ
+ 1 ue2 δ1 σ 21 dt.
1u
12
(6.1.13)
Inserting (5.5.9) and (6.1.3)–(6.1.6) in (6.1.2), the first variation of the complementary
strain energy is obtained as
−
Z t
1
t0
δ UÒ dt = −2π
Z t Z +d Z s
1
L
2
Z t Z s
1
L
t0
− d2
0
εpq δσ pq R dξ 1 dξ 3 dt =
d
8 33 33
3
33
− 2πd
+ R 0 σ δ0 σ +
+
0 σ δ0 σ
0 σ δ0 σ
1 f δ0 σ
t0
0
15
15E
21
d
21R2 32 32 dR2 2 32
31
31
1
31
−
+
+
0 σ δ0 σ
1 f δ0 σ
0 σ δ0 σ −
1 f δ0 σ
20G
40G
20G
40G

2ν 11 33
−
+ 0 σ 33 δ0 σ 11 + R2 0 σ 22 δ0 σ 33 + R2 0 σ 33 δ0 σ 22
0 σ δ0 σ
3E

d2 11 11
νR2 22 11
4
22
22
+
σ
δ
σ
+
R
σ
δ
σ
−
+ 0 σ 11 δ0 σ 22
1
1
1
1
0 σ δ0 σ
12E
E
2
2 2
R
R d
νd 3 11 νdR2 3 22
21
21
21
21
+
−
−
0 σ δ0 σ +
1 σ δ1 σ
1 f δ0 σ
1 f δ0 σ
G
12G
6E
6E

νd2
2
2R2 3 22
−
R2 1 σ 11 δ1 σ 22 + R2 1 σ 22 δ1 σ 11 + 0 f 3 δ1 σ 11 +
R dξ 1 dt . (6.1.14)
0 f δ1 σ
12E
d
d
1
E
11
11
4
22
22
Recalling that the equations of the torsional and the bending-shearing problems can be
separated from each other, (6.1.14) can be written for the bending-shearing problems (including tension-compression) of axisymmetric shells in the following form
−
Z t
1
t0
δ UÒ b-s dt = −2π
− 2πd
Z t Z s
1
L
t0
−
+
0
2ν
3E
d2
12E

Z t Z +d Z s
1
L
2
1
E
0σ
t0
− d2
0
11
11
0σ
11
δ0 σ
1
3
Ò
δU
b-s R dξ dξ dt =
4
22
+ R 0 σ δ0 σ
22
8 33 33
d
3
33
+
+
0 σ δ0 σ
1 f δ0 σ
15
15E
δ0 σ 33 + 0 σ 33 δ0 σ 11 + R2 0 σ 22 δ0 σ 33 + R2 0 σ 33 δ0 σ 22

11
11
+ R4 1 σ 22 δ1 σ 22 −
1 σ δ1 σ
νR2
E

0σ
22

δ0 σ 11 + 0 σ 11 δ0 σ 22

6.2. Matrix notation
61
21
d
νd 3 11 νdR2 3 22
31
31
1
31
σ
δ
σ
−
f
δ
σ
−
−
0
0
1
0
1 f δ0 σ
1 f δ0 σ
20G
40G
6E
6E

νd2
2 3 11 2R2 3 22
2
11
22
2
22
11
−
R 1 σ δ1 σ + R 1 σ δ1 σ + 0 f δ1 σ +
R dξ 1 dt . (6.1.15)
0 f δ1 σ
12E
d
d
+
Although the membrane-, shear- and bending energy parts are independent of each other,
the membrane stresses 0 σ 11 , 0 σ 22 and 0 σ 33 , the bending stresses 1 σ 11 , 1 σ 22 and the transverse shear stress 0 σ 31 are coupled, of course, through the translational equations of motion
appeared in the Lagrangian multiplier term (6.1.11).
In the remaining part of this chapter, the matrix formulation of the axisymmetric shell
model will be presented, then the hp-version finite element formulation and the applied
polynomial space will be described for the bending-shearing problems (including tensioncompression) of cylindrical shells.
6.2
Matrix notation
In order to write down the variational equation (6.1.8) in matrix notations in terms of
the stress and displacement coefficients, let us introduce the generalized vector σ of stress
unknowns,

(6.2.1)
σ T = 0 σ 22 0 σ 33 0 σ 11 1 σ 11 1 σ 22 0 σ 31 ,
the generalized displacement vector u,
uT =

0 u1
0 u3
1 u1
1 u3

,
(6.2.2)
the differential operator matrix U ,
2
U=
6
6
6
6
6
6
6
6
6
6
6
6
4
RR′
−
A1
R
−
A1
1
R
− 3
RA1 A1
∂
R′
+
∂ξ 1 RA1
R′′
A31
0
0
0
0
−
0
′′
2
3
0
0
0
d2
12
0

∂
R′
+
∂ξ 1 RA1
d2 R′′
12A31
0

d2 RR′
−
12A1
d2 R
−
12A1
3
5R′′
−
2RA1 2A31
∂
R′
+
∂ξ 1 RA1
−1
0
3
7
7
7
7
7
7
7
7
7
7
7
7
5
,
(6.2.3)
the constitutive matrix C,
2
C=
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
R4
2νR2
νR2
−
−
E
3E
E
2νR2
8
2ν
−
−
3E2
15E
3E
νR
2ν
1
−
−
E
3E
E
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
d2
νd2 R2
−
12 E
12E
νd2 R2
d2 R 4
−
12E
12E
0
0
0
0
0
21
20G
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
,
(6.2.4)
Chapter 6. Finite element formulation for elastodynamic problems
the mass matrix M ,
2
M=
6
6
6
6
6
6
6
4
ρ 0
0 ρ
0
0
ρd2
0 0
12
0 0 0
0
0
62
3
7
7
7
7
7
7
7
2 5
,
(6.2.5)
0
ρd
12
the generalized vectors g and h of the given surface loads (5.5.7)–(5.5.8) and the body
forces 0 bk , 1 bk ,
2
3
νdR2 3
6 −
1f 7
2
3

6
7
6E
d
2 1
dR′′
6
7
1
1
d
6
7
−
3
6
7
0f +
1f + 0b
6
7
1f
6
7
d
4A31
4RA1
6
7
15E
6
7
6
7
2
6
7
νd 3 7
3
3
6
6
7
f
+
b
6
7
0
0
−
f
1
6
7
6
7
d

6
7,
6E
′′
2
7, h = 6
g=6
7
dR
d
d 1 d 1
νd 3 7
6
1
6
7
6
7
−
f
+
b
f
+
f
−
1
1
0
6
7
0
3
6
7
6RA
3A
2
12
6
7
6E
1
1
6
7

2
6
7
′
′′
2
6
7
νdR
dR
d
1
R
d
d
d
4
3
6 −
7
1
1
3
3
3 5
f
+
−
0
6
7
0f +
0f
0f +
1f +
1b
6E
6
7
6RA1
6 ,1
RA1 A31 6
3
12
4
5
d
1
−
1f
40G
(6.2.6)
0
and the generalized vector f of prescribed displacements on S ,
f=
0 0
e 1 R0
0u
d2 R0
e1
0
1u
12
.
e 3 R0
0u
(6.2.7)
Making use of (6.2.1)–(6.2.7), taking into account (6.1.10)–(6.1.11), (6.1.13) and (6.1.15),
and since
p
1
p
3
(6.2.8)
i u1 = a1p i u = i u ,
i u3 = a3p i u = i u ,
(6.1.15) results in
−
Z t
1
t0
δ UÒ b-s dt = −
Z t Z
1
t0
V
Ò
δU
b-s dV dt = −2πd
Z t Z s
1
L
t0
0
δσ T (C σ + g) R dξ 1 dt ,
(6.2.9)
and the following matrix form is obtained for the variational equation (6.1.8):
δF =
Z t §Z s
1
L
t0
0

δσ T C σ + U T u + g + δuT (U σ + h − M ü) R dξ 1 + δσ T f
ª
dt = 0 .
(6.2.10)
The dimensionally reduced axisymmetric shell model developed can be considered as a tenfield model, it depends on 6 stress coefficients 0 σ 11 , 0 σ 22 , 0 σ 33 , 1 σ 11 , 1 σ 22 and 0 σ 31 as well
as 4 displacement coefficients 0 u1 , 0 u3 , 1 u1 and 1 u3 .
6.3
Dual-mixed hp finite element model
Let us consider now the longitudinal cross-section of the cylindrical shell mid-surface S0 (see
Figure 6.1). The master element ωst = {η | −1 ≤ η ≤ 1} is mapped onto the e-th cylinder
6.3. Dual-mixed hp finite element model
63
ξ 1(η)
x3 ξ 3
1
x =ξ =0
ξ1
e
1
ξe1
1
ξe+1
ξ1
e+1
x1 = ξ 1 = L
ℓe
1
ξe+1
R0
ξe1
−1
+1
η
x1
Figure 6.1: Coordinate transformation.
©
1
element ωel = ξ 1 | ξe1 ≤ ξ 1 ≤ ξe+1
with nodal points e and e + 1 by the transformation
1
ξ 1 (η) = N1 (η) ξe1 + N2 (η) ξe+1
,
(6.3.1)
where
1
1
(1 − η) ,
N2 (η) = (1 + η)
(6.3.2)
2
2
1
are the external shape functions and ξe1 and ξe+1
are the coordinates of the nodal points.
Using (6.3.2), the Jacobi determinant of the coordinate transformation (6.3.1) is given by
N1 (η) =
Je =
ξ 1 − ξe1
dξ 1
ℓe
= e+1
= ,
dη
2
2
(6.3.3)
where ℓe is the length of the e-th cylinder element ωel (see Figure 6.1). Since the variational
equation (6.1.8) contains only the first derivatives of the unknown variables 0 σ 11 , 1 σ 11 and
31
with respect to ξ 1 , they should be C 0 continuous in the direction of ξ 1 at each time
0σ
moment t. In view of this and taking into account (6.1.10)–(6.1.11), the unknown stresses
and displacements are approximated as
0σ
0σ
1σ
22
11
22

j=1
p+2
X

j=1
p
X
ξ 1 (η), t =

ξ 1 (η), t =

ξ 1 (η), t =
22
0 σj (t)Nj (η)
11
0 σj (t)Nj (η)
22
1 σj (t)Nj (η)
,
,
,
0σ
1σ
0σ
33
11
31

j=1
p+2
X

j=1
p+1
X
ξ 1 (η), t =

ξ 1(η), t =

ξ 1(η), t =
j=1
33
0 σj (t)Nj (η)
,
11
1 σj (t)Nj (η)
,
31
0 σj (t)Nj (η)
,
(6.3.4)
j=1
and

p+1
X

j=1
p+1
X
1
0 u1 ξ (η), t =
1 u1

1
ξ (η), t =
j=1
j
0 u1 (t)Nj (η) ,
j
1 u1 (t)Nj (η)
,

1 u3

1
p

j=1
p
X
1
0 u3 ξ (η), t =
ξ (η), t =
j=1
j
0 u3 (t)Nj (η)
,
(6.3.5)
j
1 u3 (t)Nj (η)
.
64
Chapter 6. Finite element formulation for elastodynamic problems
0σ
22
11
0σ
p+2
p
0σ
p
33
11
1σ
p+2
p
1σ
22
0σ
31
p+1
0 u1
0 u3
1 u1
1 u3
p+1
p
p+1
p
Table 6.1: Applied polynomial space.
The degrees of the applied polynomial space is summarized in Table 6.1. The coefficients
j
j
j
j
22
33
11
11
22
31
0 σj , 0 σj , 0 σj , 1 σj , 1 σj , 0 σj , 0 u1 , 0 u3 , 1 u1 and 1 u3 appeared in (6.3.4)–(6.3.5) are
unknown values and Nj (η) are hierarchic basis functions: N1 and N2 are defined by (6.3.2),
and
1
Nj (η) = È
[Pj−1 (η) − Pj−3 (η)] ,
j = 3, 4, ...
(6.3.6)
2 (2j − 3)
are the internal shape functions called ‘bubble modes’ in the h-version terminology, where
Pj−1 (η) and Pj−3 (η) are Legendre polynomials [154]. The dual-mixed finite element space
chosen in this way has led to reliable numerical solutions at every p level. The continuity
requirements are imposed on the corresponding stress coefficients through the external
shape functions (6.3.2). After introducing the generalized vectors sTe and q Te of the nodal
stresses and displacements as
sTe =

22
0 σ1
11
1 σ1
22
0 σ2
11
1 σ2
22
0 σp
...
...
33
0 σ1
11
1 σp+2
22
1 σ1
33
0 σ2
22
1 σ2
33
0 σp
...
...
11
0 σ1
22
1 σp
31
0 σ1
11
0 σ2
...
31
0 σ2
...
11
0 σp+2
31
0 σp+1

,
(6.3.7)
and
q Te =

1
0 u1
2
0 u1
p+1
0 u1
1
0 u3
2
0 u3
p
0 u3
1
1 u1
2
1 u1
p+1
1 u1
1
1 u3
2
1 u3
p
1 u3

,
(6.3.8)
the interpolations of the unknown vectors (6.2.1)–(6.2.2) along the coordinate ξ 1 within the
element e can be written in the following forms:
...
...
σ e = Lse ,
...
ue = N q e ,
...
(6.3.9)
where the interpolation matrices
2
L (η) =
6
6
6
6
6
6
6
6
4
3
N1 N2 ... Np 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0
0 0
0 N1 N2 ... Np 0 0
0 0 0
0 0 0
0 0 0
0 7
7
7
0 0
0 0 0
0 N1 N2 ... Np+2 0 0
0 0 0
0 0 0
0 7
7 ,
0 0
0 0 0
0 0 0
0 N1 N2 ... Np+2 0 0
0 0 0
0 7
7
7
0 0
0 0 0
0 0 0
0 0 0
0 N1 N2 ... Np 0 0
0 5
0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 N1 N2 ... Np+1
(6.3.10)
and
2
6
6
N (η) = 6
4
3
N1 N2 ... Np+1 0
0
0
0
0
0
0
0 0 0
0
0
0
N1 N2 ... Np 0
0
0
0
0 0 0 7
7
7
0
0
0
0
0
0 N1 N2 ... Np+1 0
0 0 0 5
0
0
0
0
0
0
0
0
0
N1 N2 ... Np
(6.3.11)
6.3. Dual-mixed hp finite element model
65
contain the hierarchic basis functions, introduced in (6.3.2) and (6.3.6). Making use of
(6.3.9), the variational equation (6.2.10) can be written in the following matrix form:
δF =
Z t
n
1 X
t0 e=1
Z t
1
t0

δsTe H e se + S Te q e + pe + r e + δq Te S e se + te −M
e q̈ e

δsT Hs + S T q + p + r + δq T Ss + t −M
q̈

dt = 0 ,

dt =
(6.3.12)
in which n denotes the number of the elements. The element’s flexibility and consistent
mass matrices are
Z +1
He =
LT CLJe dη ,
(6.3.13)
−1
and
M
e =
Z +1
−1
N T M N Je dη ,
(6.3.14)
the nodal load vectors from body forces and prescribed tractions on the element’s inner
and outer surfaces are
pe =
Z +1
−1
LT g e Je dη ,
te =
Z +1
−1
N T he Je dη ,
(6.3.15)
and the element’s differential operator matrix can be written as
Se =
Z +1
−1
N T Ue LJe dη .
(6.3.16)
Using the transformation relations (6.3.1)–(6.3.3), the matrices Ue and C as well as the
generalized load vectors g e and he are obtained for cylindrical shells (R = R0 = const.)
from (6.2.3)–(6.2.4) and (6.2.6) as
2
Ue =
2
C=
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
6
6
6
6
6
6
6
6
6
6
6
6
4
0
−R0
2 ∂
0
ℓe ∂η
1
0
R0
0
0
0
−
R04
2νR02
νR2
−
− 0
E
3E
E
2νR02
8
2ν
−
−
3E2
15E
3E
νR0
2ν
1
−
−
E
3E
E
0
0
0
0
0
0
0
0
0
2
3
0
0
0
0
0
d2 ∂
6ℓe ∂η
0
0
0
−
3
2R0
2 ∂
ℓe ∂η
d2 R0
12
0
0
0
0
0
0
0
0
d2
νd2 R02
−
12 E
12E
νd2 R02
d2 R04
−
12E
12E
0
0
0
0
0
21
20G
−1
0
3
7
7
7
7
7
7
7
7
7
7
7
7
5
,
(6.3.17)
3
2
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
,
ge =
3
νdR02 3
−
1f 7
7
6E
7
d
7
3
7
1f
7
15E
νd 3 7
7
7
−
1f
7
6E
,
νd 3 7
7
7
−
f
0
7
6E 2
7
νdR0 3 7
7
−
0f 7
6E
7
5
d
1
−
1f
40G
(6.3.18)
Chapter 6. Finite element formulation for elastodynamic problems
and
2
he =
6
6
6
6
6
6
6
6
6
6
6
4
d
2 1
1
1
0f −
1f + 0b
d
4R0
2 3
3
0f + 0b
d
d
d 1 d2 1
1
f
+
0
1f +
1b
6R0
2
12
d ∂ 0f 1
d
d 3 d2 3
3
+
0f +
1f +
1b
3ℓe ∂η
6R0
3
12
66
3
7
7
7
7
7
7
7
7
7
7
7
5
,
(6.3.19)
as well as r e = LT (η = −1) f , if x1 = ξ 1 = 0. Since δ ts and δ tq are arbitrary vectors along
ξ 1 , the resulting system of linear elastodynamic equations for the time-dependent global
stress and displacement vectors t s and t q has the following form at each time moment t:
t
δ ts 6= 0 : S ts + t t −M
q̈ = 0 ,
t
t
T t
t
(6.3.20)
t
δ q 6= 0 : H s + S q + p + r = 0 ,
(6.3.21)
which can be written in the following matrix form
"
0 0
0 M
#
t
q̇
t
q̈

−
H ST
S 0

t
s
t
q

=
t
p + tr
t
t

,
(6.3.22)
where H and M
are the global flexibility and mass matrices, S is the global differential
operator matrix, t p and t t are the time-dependent global load vectors computed from
body forces and prescribed tractions on the outer and inner surfaces S ± , and t r is the
time-dependent global vector of prescribed displacements on the lateral surface Su = S 0 .
The stress boundary conditions (5.5.14) prescribed on the lateral surface S L have to be
imposed a priori on the corresponding stress coefficients of ts at each time moment t. The
unconditionally stable Newmark -β time integration method [32, 88] will be applied to solve
the linear elastodynamic system (6.3.22) for a given time interval t ∈ [t0 , t1 ].
Chapter 7
Numerical examples
7.1
Elastostatic problems
In this section, the computational performance of the new hp-type dual-mixed finite element
model, presented in the previous chapter, are tested through two elastostatic problems. The
finite element solutions are obtained directly for the stresses and the displacements. The
numerical results are compared to the analytic solution of the related differential equation
system, see [97] and Section 5.7 for details. These comparisons are made through the
convergences of relative errors measured in energy norm and in maximum norm of stresses
and displacements, which are calculated, respectively, as
Í
eb =
F E Ò EX Ò
EX Ò
U −
U
U
,
pq
ie
=
EX
i
F E pq i σ ∞
σ pq −
k
EX pq
i σ k∞
,
i ep
=
EX
i
up −
FE
i up ∞
kEXi up k∞
, (7.1.1)
where
EX Ò
U = UÒ ( EXi σ pq ) ,
FE Ò
U = UÒ ( F Ei σ pq ) ,
(7.1.2)
and the indices EX and F E at left superscripts refer to the exact- and finite element
solution, respectively. Robustness and capability of the constructed dual-mixed element
in the stress and displacement approximations are analyzed for both p- and h-extensions
concentrating on nearly incompressible material, i.e., when the Poisson ratio ν is close to
0.5. The error measures (7.1.1) are computed for two different d/R0 values with elasticity
modulus E = 2 · 105 MPa. The relative errors of the analytic solution of Koiter’s cylindrical
shell model (see Appendix A) are also shown for both test problems in all the figures,
representing the difference between the two models.
7.1.1
Clamped cylindrical shell with end loads
Let us consider a thin cylindrical shell with radius R0 = 1000 mm and length L = 1000
mm. The lateral surface S 0 of the shell is clamped, i.e., the prescribed displacements 0 ue 1 ,
1
e3 , 1 u
e 1 and 1 u
e 3 at x
= ξ 1 = 0 are set to zero in (6.2.7), whereas the lateral surface
0u
L
S of the shell is subjected to the transverse shear force Q1 = −0 (f × )3 d and the bending
moment M 11 = 1 (f × )1 d3 /12, where 0 (f × )3 = 0.1 MPa and 1 (f × )1 = −0.01 MPa/mm (see
the equations (5.4.31)–(5.4.33), (5.5.14) and Figure 7.1).
67
68
Chapter 7. Numerical examples
x3 ξ 3
Q1
ξ1
M 11
R0
x1
L
Figure 7.1: Clamped cylindrical shell subjected to shear force and bending moment.
The main goal of the computation in this example is to show the numerical efficiency of
the developed dual-mixed element. The computational results presented for p-approximation
are obtained using a four-element mesh shown in Figure 7.2(a). Taking into account the
L − Lb
Lb
(a) First step: four-element mesh.
L − Lb
Lb
(b) Last step: sixteen-element mesh.
Figure 7.2: Clamped cylindrical shell with end loads – mesh refinement.
boundary layer effect, the length of the first two of the elements is set to (L − Lb )/2 and
the two others’ will be Lb /2, where Lb = 3.4 π/β (see Figure 7.2(a) and Appendix A). The
polynomial degree p is varying from 1 to 7 for all the four elements. During the h-extension,
the subdomains L − Lb and Lb are refined uniformly and simultaneously in seven steps from
the number of the elements n = 4 to 16 (see Figure 7.2). The polynomial degree of each
element is set to p = 2 and kept fixed.
Convergence curves for the relative errors measured in energy norm are plotted against
the number of degrees of freedom on a log-log scale in Figure 7.3 for d/R0 = d/L = 0.0001
as well as d/R0 = d/L = 0.01. In Figures 7.4–7.8, convergences of relative errors measured
in maximum norm of the transverse shear stress 0 σ 31 , membrane stresses 0 σ 33 , 0 σ 11 , 0 σ 22
and bending stresses 1 σ 11 , 1 σ 22 , as well as displacement components 0 u1 , 0 u3 , 1 u1 , 1 u3 are
shown for d/R0 = d/L = 0.0001 and 0.01.
It can be seen that the convergences are rapid for both p-approximation and h-extension,
7.1. Elastostatic problems
69
relative error in energy norm (%)
ν = 0.4999999
10
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approximation, d/R0 = d/L = 0.01
p-approximation, d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
4
103
102
101
100
4.675816 %
0.469902 %
10−1
10−2
102
102.2 102.4 102.6 102.8 103
number of degrees of freedom
Figure 7.3: Clamped shell with end loads – relative error in energy norm for p-approximation
on a four-element mesh and h-extension with p=2.
104
103
102
31
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
101
100
10−1
0.069215 %
10−2
10−3
0σ
, ν = 0.4999999
0.000695 %
102 102.2 102.4 102.6 102.8 103
number of degrees of freedom
104
relative error in maximum norm (%)
relative error in maximum norm (%)
0σ
103
102
33
, ν = 0.4999999
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter
100 %
101
100
10−1
102 102.2 102.4 102.6 102.8 103
number of degrees of freedom
Figure 7.4: Clamped shell with end loads – relative error of the stresses 0 σ 31 , 0 σ 33 in
maximum norm for p-approximation on a four-element mesh and h-extension with p=2.
70
Chapter 7. Numerical examples
0σ
6
105
104
10
0σ
, ν = 0.4999999
3
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter
100 %
102
101
100
10−1
10−2
10−3
relative error in maximum norm (%)
relative error in maximum norm (%)
10
11
10
5
10
4
103
, ν = 0.4999999
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
102
101
100
0.219591 %
10−1
10−2
10
10−4
102 102.2 102.4 102.6 102.8 103
number of degrees of freedom
22
0.002208 %
−3
102 102.2 102.4 102.6 102.8 103
number of degrees of freedom
Figure 7.5: Clamped shell with end loads – relative error of the stresses 0 σ 11 , 0 σ 22 in
maximum norm for p-approximation on a four-element mesh and h-extension with p=2.
1σ
1σ
22
, ν = 0.4999999
6
105
104
103
10
, ν = 0.4999999
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
2
101
100
10−1
10−2
10−3
0.132313 %
0.001361 %
10−4
102 102.2 102.4 102.6 102.8 103
number of degrees of freedom
relative error in maximum norm (%)
relative error in maximum norm (%)
10
11
105
104
103
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
102
101
1.548156 %
100
10−1
0.01551 %
10−2
102 102.2 102.4 102.6 102.8 103
number of degrees of freedom
Figure 7.6: Clamped shell with end loads – relative error of the stresses 1 σ 11 , 1 σ 22 in
maximum norm for p-approximation on a four-element mesh and h-extension with p=2.
7.1. Elastostatic problems
0 u3 ,
ν = 0.4999999
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
104
103
102
relative error in maximum norm (%)
relative error in maximum norm (%)
0 u1 ,
71
101
100
0.393813 %
10−1
10
−2
10
−3
0.006183 %
104
103
102
ν = 0.4999999
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
101
100
10
0.219168 %
−1
10−2
0.002208 %
10−3
102 102.2 102.4 102.6 102.8 103
number of degrees of freedom
102 102.2 102.4 102.6 102.8 103
number of degrees of freedom
Figure 7.7: Clamped shell with end loads – relative error of the displacements 0 u1 , 0 u3 in
maximum norm for p-approximation on a four-element mesh and h-extension with p=2.
ν = 0.4999999
1 u3 ,
105
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
105
104
103
relative error in maximum norm (%)
relative error in maximum norm (%)
1 u1 ,
102
101
100
0.166644 %
10−1
10−2
10
0.001667 %
104
103
102
ν = 0.4999999
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter
100 %
101
100
10−1
10−2
−3
10
2
2.2
2.4
2.6
2.8
10
10
10
10
10
number of degrees of freedom
3
10−3
102 102.2 102.4 102.6 102.8 103
number of degrees of freedom
Figure 7.8: Clamped shell with end loads – relative error of the displacements 1 u1 , 1 u3 in
maximum norm for p-approximation on a four-element mesh and h-extension with p=2.
72
Chapter 7. Numerical examples
independently of the thickness of the shell. As expected, for ν = 0.4999999, the new
dual-mixed hp element performs very well and gives excellent results for both thin (d/R0 =
0.0001) and moderately thick (d/R0 = 0.01) cylindrical shells. The convergence curves indicate clearly that the developed shell element is really free from shear- and incompressibility
locking [161].
7.1.2
Clamped cylindrical shell loaded by normal pressures
We consider a thin cylindrical shell clamped at x1 = ξ 1 = 0 with radius R0 = 1000 mm and
length L = 1000 mm. The shell is subjected to constant external pressure (f + )3 = −0.01
x3 ξ 3
(f +)3
ξ1
(f −)3
R0
L
x1
Figure 7.9: Clamped cylindrical shell loaded by external and internal pressure.
MPa and internal pressure (f − )3 = 0.02 MPa (see Figure 7.9). This problem is membrane
dominated for thin shells with a strong boundary layer at x1 = 0. The total complementary
strain energy is mainly stored in the membrane energy terms.
For the investigation of the p-approximation, a four-element mesh shown in Figure
7.10(a) is used. Accordingly, the two main regions Lb = 3.4 π/β and L − Lb are divided into
Lb
L − Lb
(a) First step: four-element mesh.
Lb
L − Lb
(b) Last step: fourteen-element mesh.
Figure 7.10: Clamped cylindrical shell loaded by normal pressures – mesh refinement.
two equal intervals considering the boundary layer effect (see Appendix A). The polynomial
degree p is ranging from 1 to 7 for all the four elements. During the h-extension, the regions
Lb and L − Lb are refined uniformly and simultaneously in six steps from the number of
7.1. Elastostatic problems
73
the elements n = 4 to 14 (see Figure 7.10) and the polynomial degree p = 2 is applied to
each element.
The relative errors in energy norm and in maximum norm of the stresses 0 σ 31 , 0 σ 33 0 σ 11 ,
22
11
22
and displacements 0 u1 , 0 u3 , 1 u1 , 1 u3 are investigated again for ν = 0.4999999
0σ , 1σ , 1σ
as well as for ratios d/R0 = d/L = 0.0001 and 0.01, and plotted as the function of the
number of degrees of freedom on log-log scales in Figures 7.11–7.16.
relative error in energy norm (%)
ν = 0.4999999
105
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approximation, d/R0 = d/L = 0.01
p-approximation, d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
104
103
102
11.932187 %
101
1.221954 %
100
10−1
10−2
102
102.2 102.4 102.6 102.8 103
number of degrees of freedom
Figure 7.11: Clamped shell loaded by pressures – relative error in energy norm for papproximation on a four-element mesh and h-extension with p=2.
104
103
102
31
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
101
1.589305 %
100
10−1
10
0σ
, ν = 0.4999999
0.015793 %
−2
10−3 2
10 102.2 102.4 102.6 102.8 103
number of degrees of freedom
relative error in maximum norm (%)
relative error in maximum norm (%)
0σ
105
104
103
102
33
, ν = 0.4999999
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter
100 %
101
100
10−1
10−2
102 102.2 102.4 102.6 102.8
number of degrees of freedom
Figure 7.12: Clamped shell loaded by pressures – relative error of the stresses 0 σ 31 , 0 σ 33 in
maximum norm for p-approximation on a four-element mesh and h-extension with p=2.
74
Chapter 7. Numerical examples
0σ
4
10
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter
103
100 %
102
10
1
10
0
10−1
10
0σ
6
105
10
, ν = 0.4999999
−2
10−3
relative error in maximum norm (%)
relative error in maximum norm (%)
10
11
22
, ν = 0.4999999
4
103
102
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
101
1.563995 %
10
0
10−1
0.0154046 %
10
−2
102 102.2 102.4 102.6 102.8
number of degrees of freedom
102 102.2 102.4 102.6 102.8
number of degrees of freedom
Figure 7.13: Clamped shell loaded by pressures – relative error of the stresses 0 σ 11 , 0 σ 22 in
maximum norm for p-approximation on a four-element mesh and h-extension with p=2.
1σ
, ν = 0.4999999
4
103
102
10
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
101
2.076678 %
100
10
−1
0.020503 %
10−2
10
1σ
−3
relative error in maximum norm (%)
relative error in maximum norm (%)
10
11
22
, ν = 0.4999999
5
104
103
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
102
101
2.330836 %
100
10−1
0.023003 %
10−2
102 102.2 102.4 102.6 102.8
number of degrees of freedom
102 102.2 102.4 102.6 102.8
number of degrees of freedom
Figure 7.14: Clamped shell loaded by pressures – relative error of the stresses 1 σ 11 , 1 σ 22 in
maximum norm for p-approximation on a four-element mesh and h-extension with p=2.
7.1. Elastostatic problems
10
10
5
104
103
102
ν = 0.4999999
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
101
10
3.228515 %
0
10−1
10−2
0.030126 %
10−3
10−4
10
0 u3 ,
104
relative error in maximum norm (%)
relative error in maximum norm (%)
0 u1 ,
6
75
−5
103
102
ν = 0.4999999
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
101
100
0.815176 %
10−1
10−2
102 102.2 102.4 102.6 102.8
number of degrees of freedom
0.008095 %
102 102.2 102.4 102.6 102.8
number of degrees of freedom
Figure 7.15: Clamped shell loaded by pressures – relative error of the displacements 0 u1 , 0 u3
in maximum norm for p-approximation on a four-element mesh and h-extension with p=2.
1 u1 ,
ν = 0.4999999
1 u3 ,
10
4
103
102
101
10
0
9.145104 %
0.947026 %
relative error in maximum norm (%)
relative error in maximum norm (%)
10
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter, d/R0 = d/L = 0.01
Koiter, d/R0 = d/L = 0.0001
ν = 0.4999999
5
104
103
102
h-extension, d/R0 = d/L = 0.01
h-extension, d/R0 = d/L = 0.0001
p-approx., d/R0 = d/L = 0.01
p-approx., d/R0 = d/L = 0.0001
Koiter
100 %
101
100
10−1
10−1
102 102.2 102.4 102.6 102.8
number of degrees of freedom
102 102.2 102.4 102.6 102.8
number of degrees of freedom
Figure 7.16: Clamped shell loaded by pressures – relative error of the displacements 1 u1 , 1 u3
in maximum norm for p-approximation on a four-element mesh and h-extension with p=2.
76
Chapter 7. Numerical examples
It is seen that the rates of convergence are high not only for p-approximation but also for
h-extension and insensitive to the thickness of the shell. As predicted, for ν = 0.4999999,
the new dual-mixed hp finite element model gives very good numerical results not only for
the transverse shear-, bending and membrane stresses but also for the transverse- and axial
displacement components. It has been verified numerically that the developed shell element
is free from the incompressibility locking effect.
It can also be observed that the p-convergence is much faster than the h-convergence,
when mesh refinement is used. This is a characteristic property of the p-version finite
elements, see for instance [154]. Furthermore, the relative errors of the analytic solution
of Koiter’s cylindrical shell model have also been indicated for both representative test
problems in all the figures, emphasizing its ‘ inaccuracy’ and certain modeling errors.
7.2
Dynamic analysis of a clamped cylindrical shell
The aim of this section is to test the new h- and p-version dual-mixed shell element by
investigating an elastodynamic problem. To show the numerical performance of the dualmixed hp element developed, first we investigate the convergence rates of relative errors
measured in the lowest natural frequency ω low for a clamped-free cylindrical shell. The
relative errors are computed on the basis of [88] as
elow
ω low
=
− 1
ωref
(7.2.1)
for two different d/R0 values with Poisson ratio ν = 0.3 and elasticity modulus E = 2 · 105
MPa. The reference solution ωref in (7.2.1) is obtained using a very fine mesh with highorder polynomial approximation of variables, since the exact solution for ωref is unknown
[104].
x3 ξ 3
ξ1
Q1(t)
R0
x1
L
Figure 7.17: Clamped cylindrical shell subjected to time-dependent shear force.
We next focus on the transient vibration analysis of a cylindrical shell problem (see
Figure 7.17). The related linear elastodynamic system (6.3.22) is solved for the small
7.2. Dynamic analysis of a clamped cylindrical shell
77
d/R0 = d/L = 0.01
10
0
h-extension
p-approximation
10−2
10−4
10−6
10−8
10−10
101.8 102 102.2 102.4 102.6
number of degrees of freedom
relative error in lowest natural frequency (%)
relative error in lowest natural frequency (%)
time interval t ∈ [0, 0.01] s, using the unconditionally stable Newmark -β time integration
technique [32, 88]. The number of the constant time step ∆t is equal to 105 . The computational results obtained for the displacements 0 u1 (x1 = L, t), 0 u3 (x1 = L, t), 1 u1 (x1 = L, t),
1
11 1
11 1
31 1
1 u3 (x = L, t) and stresses 0 σ (x = 0, t), 1 σ (x = 0, t), 0 σ (x = 0, t) are illustrated
as the function of time. The numerical solutions of the Naghdi-type shell model are also
shown in all figures using the finite element software ADINA [32].
The thin cylindrical shell with radius R0 = 1000 mm and length L = 1000 mm is
clamped at x1 = ξ 1 = 0, i.e., 0 ue1 (0, t) = 0 ue3 (0, t) = 1 ue1 (0, t) = 1 ue3 (0, t) = 0, and subjected
to a time-dependent shear force Q1 (t) = −0 (f × )3 d sin ωt at x1 = ξ 1 = L, where the load
coefficient 0 (f × )3 = 0.1 MPa and the circular frequency of the forced vibration is ω =
4000 rad/s < ω low (see Figure 7.17). The shell structure is initially undeformed and at rest,
i.e., 00 u1 (ξ 1 ) =00 u3 (ξ 1 ) = 01 u1 (ξ 1 ) =01 u3 (ξ 1) = 0 and 00 v1 (ξ 1 ) =00 v3 (ξ 1) = 01 v1 (ξ 1 ) =01 v3 (ξ 1) = 0.
The computational results presented for the p-convergence of the lowest natural frequency are obtained using a two-element mesh with element-length L/2. The polynomial
degree p is varying from 2 to 8 for both elements. During the h-extension, the main region
L is refined uniformly in seven steps from the number of elements n = 2 to 8. The polynomial degree of each element is set to p = 2 and kept fixed. The reference solution ωref
appearing in (7.2.1) is obtained using n = 50 elements with uniform mesh refinement and
the polynomial degree p = 10 is applied to each element.
Convergence curves for the relative errors measured in the lowest natural frequency
are plotted against the number of degrees of freedom on a log-log scale in Figure 7.18 for
d/R0 = d/L = 0.01, as well as d/R0 = d/L = 0.0001.
d/R0 = d/L = 0.0001
h-extension
p-approximation
10−1
10−2
101.8 102 102.2 102.4 102.6
number of degrees of freedom
Figure 7.18: Clamped-free cylindrical shell – relative error measured in lowest natural
frequency for p-approximation on a two-element mesh and h-extension with p=2.
78
Chapter 7. Numerical examples
Displacement 0 u1 (x1 = L), d/R0 = d/L = 0.01
−6
x 10
dual-mixed shell element, n = 10, p = 10
dual-mixed shell element, n = 20, p = 3
Naghdi-type axisymmetric shell element, n = 20, p = 3
Naghdi-type axisymmetric shell element, n = 300, p = 3
1.5
= L) [m]
1
1
0 u1 (x
0.5
0
−0.5
−1
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
0.01
Figure 7.19: Response curves for the displacement 0 u1 (L, t), d/R0 = d/L = 0.01.
Displacement 0 u3 (x1 = L), d/R0 = d/L = 0.01
−5
x 10
dual-mixed shell element, n = 10, p = 10
dual-mixed shell element, n = 20, p = 3
Naghdi-type axisymmetric shell element, n = 20, p = 3
Naghdi-type axisymmetric shell element, n = 300, p = 3
6
1
0 u3 (x = L) [m]
4
2
0
−2
−4
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
Figure 7.20: Response curves for the displacement 0 u3 (L, t), d/R0 = d/L = 0.01.
0.01
7.2. Dynamic analysis of a clamped cylindrical shell
Displacement 1 u1 (x1 = L), d/R0 = d/L = 0.01
−7
x 10
dual-mixed shell element, n = 10, p = 10
dual-mixed shell element, n = 20, p = 3
Naghdi-type axisymmetric shell element, n = 20, p = 3
Naghdi-type axisymmetric shell element, n = 300, p = 3
= L) [rad]
6
1
1 u1 (x
79
4
2
0
−2
−4
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
0.01
Figure 7.21: Response curves for the displacement 1 u1 (L, t), d/R0 = d/L = 0.01.
Displacement 1 u3 (x1 = L), d/R0 = d/L = 0.01
−8
x 10
dual-mixed shell element, n = 10, p = 10
dual-mixed shell element, n = 50, p = 10
Naghdi-type axisymmetric shell element
6
1
1 u3 (x
= L) [rad]
4
2
0
−2
−4
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
Figure 7.22: Response curves for the displacement 1 u3 (L, t), d/R0 = d/L = 0.01.
0.01
80
Chapter 7. Numerical examples
Displacement 0 u1 (x1 = L), d/R0 = d/L = 0.0001
−6
1.5
x 10
dual-mixed shell element, n = 10, p = 10
dual-mixed shell element, n = 20, p = 3
Naghdi-type axisymmetric shell element, n = 20, p = 3
Naghdi-type axisymmetric shell element, n = 300, p = 3
= L) [m]
0.5
1
0 u1 (x
1
0
−0.5
−1
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
0.01
Figure 7.23: Response curves for the displacement 0 u1 (L, t), d/R0 = d/L = 0.0001.
Displacement 0 u3 (x1 = L), d/R0 = d/L = 0.0001
−4
x 10
dual-mixed shell element, n = 10, p = 10
dual-mixed shell element, n = 20, p = 3
Naghdi-type axisymmetric shell element, n = 20, p = 3
Naghdi-type axisymmetric shell element, n = 300, p = 3
6
1
0 u3 (x = L) [m]
4
2
0
−2
−4
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
0.01
Figure 7.24: Response curves for the displacement 0 u3 (L, t), d/R0 = d/L = 0.0001.
7.2. Dynamic analysis of a clamped cylindrical shell
Displacement 1 u1 (x1 = L), d/R0 = d/L = 0.0001
−4
1.5
x 10
dual-mixed shell element, n = 10, p = 10
dual-mixed shell element, n = 20, p = 3
Naghdi-type axisymmetric shell element, n = 20, p = 3
Naghdi-type axisymmetric shell element, n = 300, p = 3
= L) [rad]
1
1 u1 (x
1
0.5
0
81
−0.5
−1
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
0.01
Figure 7.25: Response curves for the displacement 1 u1 (L, t), d/R0 = d/L = 0.0001.
Displacement 1 u3 (x1 = L), d/R0 = d/L = 0.0001
−7
x 10
dual-mixed shell element, n = 10, p = 10
dual-mixed shell element, n = 50, p = 10
Naghdi-type axisymmetric shell element
6
1
1 u3 (x = L) [rad]
4
2
0
−2
−4
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
0.01
Figure 7.26: Response curves for the displacement 1 u3 (L, t), d/R0 = d/L = 0.0001.
82
Chapter 7. Numerical examples
Membrane stress 0 σ11 (x1 = 0), d/R0 = d/L = 0.01
Naghdi-type axisymmetric shell element, n = 300, p = 3
dual-mixed shell element, n = 50, p = 10
(x1 = 0) [MPa]
0.1
0.05
0σ
11
0
−0.05
−0.1
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
0.01
Figure 7.27: Response curves for the membrane stress 0 σ 11 (0, t), d/R0 = d/L = 0.01.
Bending stress 1 σ11 (x1 = 0), d/R0 = d/L = 0.01
Naghdi-type axisymmetric shell element, n = 300, p = 3
dual-mixed shell element, n = 50, p = 10
1σ
11
(x1 = 0) [MPa/mm]
0.3
0.2
0.1
0
−0.1
−0.2
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
0.01
Figure 7.28: Response curves for the bending stress 1 σ 11 (0, t), d/R0 = d/L = 0.01.
7.2. Dynamic analysis of a clamped cylindrical shell
83
Transverse shear stress 0 σ31 (x1 = 0), d/R0 = d/L = 0.01
(x1 = 0) [MPa]
0.05
Naghdi-type axisymmetric shell element, n = 300, p = 3
dual-mixed shell element, n = 50, p = 10
0σ
31
0
−0.05
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
0.01
Figure 7.29: Response curves for the transverse shear stress 0 σ 31 (0, t), d/R0 = d/L = 0.01.
Membrane stress 0 σ11 (x1 = 0), d/R0 = d/L = 0.0001
−3
2
x 10
Naghdi-type axisymmetric shell element, n = 300, p = 3
dual-mixed shell element, n = 50, p = 10
1
0.5
0
0σ
11
(x1 = 0) [MPa]
1.5
−0.5
−1
−1.5
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
0.01
Figure 7.30: Response curves for the membrane stress 0 σ 11 (0, t), d/R0 = d/L = 0.0001.
84
Chapter 7. Numerical examples
Bending stress 1 σ11 (x1 = 0), d/R0 = d/L = 0.0001
Naghdi-type axisymmetric shell element, n = 300, p = 3
dual-mixed shell element, n = 50, p = 10
(x1 = 0) [MPa/mm]
−5
1σ
10
11
15
5
0
−10
−15
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
0.01
Figure 7.31: Response curves for the bending stress 1 σ 11 (0, t), d/R0 = d/L = 0.0001.
Transverse shear stress 0 σ31 (x1 = 0), d/R0 = d/L = 0.0001
−3
5
x 10
Naghdi-type axisymmetric shell element, n = 300, p = 3
dual-mixed shell element, n = 50, p = 10
4
0σ
31
(x1 = 0) [MPa]
3
2
1
0
−1
−2
−3
−4
0
0.001
0.002
0.003
0.004
0.005
t [s]
0.006
0.007
0.008
0.009
0.01
Figure 7.32: Response curves for the transverse shear stress 0 σ 31 (0, t), d/R0 =d/L=0.0001.
7.2. Dynamic analysis of a clamped cylindrical shell
85
It can be seen that the convergences are rapid for both h-extension and p-approximation,
independently of the thickness of the shell. The new dual-mixed hp element performs
very well and provides excellent results in the natural frequency computation for both thin
(d/R0 = 0.0001) and moderately thick (d/R0 = 0.01) cylindrical shells. As expected,
p-convergences are much faster than h-convergences.
In Figures 7.19–7.26, our finite element solution is compared with the solution of the
Naghdi-type shell finite element model throughout the response curves obtained for the
displacement components 0 u1 (L, t), 0 u3 (L, t), 1 u1 (L, t) and 1 u3 (L, t). The comparisons
are made for thin (d/R0 = d/L = 0.0001) and moderately thick (d/R0 = d/L = 0.01)
cylindrical shells. The domain L is uniformly refined in all cases. The reference solutions
of the Naghdi-model were calculated by the finite element software ADINA using a very
fine mesh (n = 300) with four-noded (p = 3) axisymmetric shell elements.
It can be observed in these figures that for very thin (d/R0 = d/L = 0.0001) cylindrical
shell, the presented dual-mixed shell model gives more accurate finite element solutions than
the Naghdi-type shell model for the polynomial degree p = 3 and relatively small number
of elements n = 20. Excellent numerical results are obtained also for high polynomial
degree p = 10 using only n = 10 elements, verifying the outstanding efficiency of the p-type
approximation technique. A certain modeling error of the Naghdi-type shell element is
clearly illustrated for d/R0 = d/L = 0.01 as well as d/R0 = d/L = 0.0001 in Figures 7.22
and 7.26.
Finally, the response curves obtained for the stresses 0 σ 11 (0, t), 1 σ 11 (0, t) and 0 σ 31 (0, t)
are represented for thin (d/R0 = d/L = 0.0001) and moderately thick (d/R0 = d/L = 0.01)
cylindrical shells in Figures 7.27–7.32, thereby comparing the dual-mixed hp-version finite
element solutions with the reference solutions of the Naghdi-model in terms of stresses. It
can be seen that the developed dual-mixed hp shell element performs very well not only
for displacements but also for stresses, independently of the thickness of the shell. It is
concluded that the constructed h- and p-version shell finite element models provide shearand incompressibility locking-free results for time-dependent cylindrical shell problems as
well.
Chapter 8
Summary and theses
The main goal of this dissertation was to derive a new dimensionally reduced shell model
and the related hp finite element formulation for elastodynamic problems of axisymmetric
shells using a three-field dual-mixed variational principle of Hellinger–Reissner-type.
An introduction and literature overview, summarizing the state of the art of mixed
variational formulations and finite element modeling of shells, have been given in Chapter
1. The brief differential geometric description of surfaces and shells in Chapter 2 and 3
were intended to help the understanding of the subsequent developments presented in the
dissertation.
In Chapter 4, a new four-field dual-mixed variational principle for three-dimensional
elastic bodies have been derived. After eliminating the impulse field from this principle, a
three-field dual-mixed variational formulation in terms of non-symmetric stresses, displacements and rotations as independent variables has been obtained. The functional of this
three-field principle can also be considered as a generalization of the classical Hellinger–
Reissner variational functional to elastodynamic problems of linear elastic continua. The
special form of the three-field functional for thin shells of revolution has also been derived
in Chapter 4.
The main steps of the dimensional reduction procedure for thin shells of revolution have
been presented in Chapter 5. The components of the three independent variables, i.e., the
stress tensor, the rotation tensor and the displacement vector, have been approximated by
polynomials of first- and second-degree in the thickness direction. Introducing the notion
of strain-, rotation- and displacement resultants, the Euler–Lagrange equations and the
natural boundary conditions of the shell model have been derived for axisymmetric elastodynamic problems. The fundamental differential equation system of the shell model, as
well as the corresponding initial and boundary conditions, have been derived for linearly
elastic, homogeneous and isotropic materials. The differential equation system presented
in Section 5.7 for the fundamental variables of the shell model can be used for analytical
investigations of axisymmetric shell problems, especially when closed-form solution for the
stress field is needed.
The derivation and the main steps of the implementation of a new dual-mixed hp finite
element model, based on the shell theory developed in the previous chapters of this dissertation, have been presented in Chapter 6. The numerical capabilities and the performance of
the new shell finite element model developed for static and dynamic problems of cylindrical
shells has been demonstrated in Chapter 7.
86
87
Thesis 1
I have extended the dual-mixed Hellinger–Reissner-type variational formulation to linear
elastodynamic problems. The independent fields of the new four-field variational principle
are the displacement vector, the non-symmetric stress tensor, the skew-symmetric rotation
tensor and the impulse vector. Eliminating the impulse field from the four-field functional,
I have derived a three-field dual-mixed variational principle for elastodynamic analyses of
three-dimensional time-dependent problems. This three-field functional served as a basis
for further developments presented in the dissertation and in the subsequent theses.
Thesis 2
I have derived the special form of the three-field dual-mixed functional of Thesis 1 for
thin shells of revolution, applying the differential geometric description of surfaces and
shells. I have developed a new dimensionally reduced shell model for linear elastodynamic
problems of axisymmetric shells. The fundamental (independent) variables of the shell
model developed are the displacement vector, the non-symmetric stress tensor and the
skew-symmetric rotation tensor. The shell model developed does not rely on the classical
kinematical hypotheses regarding the deformation of the normal to the shell middle surface
and, thus, uses unmodified three-dimensional constitutive equations.
Thesis 3
Starting from the special form of the three-field dual-mixed variational principle for axisymmetric shells of Thesis 2, I have derived the Euler–Lagrange equations and the natural
boundary conditions of the new dimensionally reduced shell model, assuming axisymmetric
loads and homogeneous isotropic materials. The variational equations have been formulated
in terms of one-dimensional variables defined on the middle surface of the shell and consist
of the special forms of the translational and rotational equations of motion, the kinematic
equations and the displacement boundary conditions. I have derived the fundamental differential equation system for the independent variables of the axisymmetric shell model.
Thesis 4
I have developed a new dual-mixed hp finite element model for axisymmetric elastodynamic
problems of thin cylindrical shells. This is based on a modified version of the three-field
dual-mixed variational principle. The modification has resulted in a two-field formulation
in which the symmetry conditions are satisfied in an integral average sense through the
related Euler–Lagrange equations. The polynomial interpolation spaces applied assume
C 0 -continuous approximation for the normal stress components and discontinuous approximation for the displacements at the element interfaces. The computational performance
of the new dual-mixed hp finite element model has been tested and demonstrated for static
and dynamic problems of cylindrical shells, for both h- and p-extensions. The numerical
results have been compared to the solutions of displacement-based models and analytical
solutions. I have proven through these computations and comparisons that the shell model
and the related hp-version shell finite elements developed in this dissertation are robust
and reliable, i.e., they are free from incompressibility- and shear locking-effects, not only
for the displacements, but also for the stress computations.
Chapter 9
Publications
The following publications were made in the topic of the dissertation:
Articles in journals:
• Tóth, B. A Hellinger–Reissner-féle variációs elv alakja forgáshéjakra. (The Hellinger–
Reissner variational principle for axisymmetric shells). GÉP, LVIII(5-6):48–53, 2007.
• Tóth, B. Axisymmetric shell model using a three-field dual-mixed variational principle. Journal of Computational and Applied Mechanics, Accepted: November 12, 2008.
unpublished full article.
• Tóth, B. Dual-mixed hp finite element model for elastic cylindrical shells. ZAMM,
Journal of Applied Mathematics and Mechanics, 92:236–252, 2012. (IF=0.948 ).
Conference papers:
• Tóth, B. Application of the Hellinger–Reissner variational principle to shells of revolution. In Proceedings of microCAD 2007 International Scientific Conference, pages 47–51,
University of Miskolc, Hungary, Section F. Applied Mechanics, March 22–23, 2007.
• Tóth, B. The Hellinger–Reissner variational principle for axisymmetric shells. In Proceedings of Ph.D. Students’ Forum, pages 158–166, University of Miskolc, Hungary, November
13, 2007.
• Tóth, B. Axisymmetric shell model using the three-field Hellinger–Reissner’s variational
principle. In Proceedings of microCAD 2008 International Scientific Conference, pages
47–51, University of Miskolc, Hungary, Section F. Applied Mechanics, March 20–21, 2008.
• Tóth, B. Dual-mixed finite element model for elastic cylindrical shells. In Proceedings of XXIII. microCAD International Scientific Conference, pages 51–60, University
of Miskolc, Hungary, Section F. Applied Mechanics, March 19–20, 2009.
• Tóth, B. Dual-mixed Hellinger–Reissner functional for elastodynamics. In Proceedings of XXIV. microCAD International Scientific Conference, pages 73–77, University of
Miskolc, Hungary, Section G. Applied Mechanics, March 18–20, 2010.
88
89
• Tóth, B. Vibration analysis of cylindrical shells based on a dual-mixed finite element
formulation. In Proceedings of XXV. microCAD International Scientific Conference,
pages 49–56, University of Miskolc, Hungary, Section E. Applied Mechanics, March 31 –
April 1, 2011.
Conference presentations:
• Tóth, B. A hárommezős Hellinger–Reissner-féle variációs elv alkalmazása forgáshéjakra
(The three-field Hellinger–Reissner variational principle for shells of revolution). X. Magyar Mechanikai Konferencia, University of Miskolc, Hungary, August 27–29, 2007.
• Tóth, B. A duál Hellinger–Reissner-féle variációs elven alapuló hp-verziós végeselemmodell körhengerhéjakra (An hp finite element model based on dual-mixed Hellinger–
Reissner variational principle for cylindrical shells). XI. Magyar Mechanikai Konferencia,
University of Miskolc, Hungary, August 29–31, 2011.
• Tóth, B. A hárommezős Hellinger–Reissner-féle variációs elv alkalmazása forgáshéjak
elasztodinamikai feladataira (Three-field Hellinger–Reissner variational principle for elastodynamic problems of axisymmetric shells). XI. Magyar Mechanikai Konferencia, University of Miskolc, Hungary, August 29–31, 2011.
• Tóth, B. Dual-mixed hp finite element model for elastodynamic problems of cylindrical
shells. 38th Solid Mechanics Conference, Warsaw, Poland, August 27–31, 2012.
• Tóth, B. and Bertóti, E. Dual-mixed finite element method for elastic cylindrical shells.
37th Solid Mechanics Conference, Warsaw, Poland, September 6–10, 2010.
• Tóth, B. and Bertóti, E. A three-field dual-mixed hp finite element model for cylindrical
shells. HOFEIM, Workshop on Higher Order Finite Element and Isogeometric Methods,
Cracow, Poland, June 27–29, 2011.
Talks:
• Tóth, B. Application of a three-field dual-mixed variational principle to axisymmetric
shells. Otto von Guericke University, Institute of Mechanics, Magdeburg, Germany,
December 17, 2007.
• Tóth, B. A Hellinger–Reissner-féle variációs elv alkalmazása forgáshéjakra (Application of the Hellinger–Reissner variational principle to axisymmetric shells). Doktoranduszok Szemináriuma (Ph.D. Students’ Seminar), Miskolci Egyetem (University of Miskolc,
Hungary), November 19, 2007.
• Tóth, B. Three-field dual-mixed finite element model for cylindrical shells. Otto von
Guericke University, Institute of Mechanics, Magdeburg, Germany, December 18, 2008.
• Tóth, B. A hárommezős duál Hellinger–Reissner-féle variációs elv alkalmazása körhengerhéjra, (Application of the three-field dual-mixed Hellinger–Reissner variational principle
to cylindrical shells). Doktoranduszok Szemináriuma (Ph.D. Students’ Seminar), Miskolci
Egyetem (University of Miskolc, Hungary), April 6, 2009.
Appendix A
Analitic solutions of Koiter’s
cylindrical shell model
The exact displacement components K0 u3 , K1 u1 and bending stress K1 σ 11 , as well as the
transverse shear stress K0 σ 31 of Koiter’s cylindrical shell model can be written for the
elastostatic problems investigated in Subsections 7.1.1–7.1.2, on the basis of [160] as
K
0 u3
= A1 exp β(x1 − L) cos β(L − x1 ) + A2 exp β(x1 − L) sin β(L − x1 )
+ A3 exp(−βx1 ) cos βx1 + A4 exp(−βx1 ) sin βx1 +
K
1 u1
P
0 u3
,
= (A2 − A1 )β exp β(x1 − L) sin β(L − x1 ) − (A1 + A2 )β exp β(x1 − L) cos β(L − x1 )
+ (A4 − A3 )β exp(−βx1 ) cos βx1 − (A3 + A4 )β exp(−βx1 ) sin βx1 ,
and
K 11
1σ
= 2E1 β 2 [A2 exp β(x1 − L) cos β(L − x1 ) − A1 exp β(x1 − L) sin β(L − x1 )]
− 2E1 β 2 [A3 exp(−βx1 ) sin βx1 − A4 exp(−βx1 ) cos βx1 ] ,
as well as
K 31
0σ
= −2I1 E1 β 3 /d A1 [exp β(x1 − L) sin β(L − x1 ) − exp β(x1 − L) cos β(L − x1 )]
− A2 [exp β(x1 − L) cos β(L − x1 ) + exp β(x1 − L) sin β(L − x1 )]
+ A3 [exp(−βx1 ) cos βx1 − exp(−βx1 ) sin βx1 ]
©
+A4 [exp(−βx1 ) cos βx1 + exp(−βx1 ) sin βx1 ] ,
where
I1 = d3 /12 ,
E1 = E/(1 − ν 2 ) ,
β=
È
4
È
3(1 − ν 2 )/(R0 d/R0 ) ,
and
P
0 u3
= R02 /E/d[(f + )3 + (f − )3 − νd/R0
K 11
0σ ],
K 11
0σ
=0,
as well as
A2 = M 11 /(2I1 E1 β 2 ) ,
A1 = −Q1 /(2I1 E1 β 3 ) − A2 ,
90
A3 = A4 = − P0 u3 .
91
The index K refers to the exact solutions of the Koiter-model. Using the previous equations,
the axial displacement K0 u1 and bending stress K1 σ 22 , as well as membrane stress K0 σ 22 can
be expressed as
K
0 u1
1
=
E1
Zx
1
K 11
0σ
x̂1 =0
ν
dx̂ −
R0
Zx
1
1
K
0 u3
dx̂1 ,
x̂1 =0
and
K 22
1σ
=ν
K 11
1σ
,
K 22
0σ
as well as
=ν
K 11
0σ
+ E/R0 K0 u3 .
In the case of small deformations, the exact value of the complementary strain energy is
equivalent to the exact value of the strain energy resulted by analytic solutions of Koiter’s
model, i.e.,
Z
Z
K Ò
U =
V
KÒ
U dV
=
K
V
U dV = πEd/R0
Z L

0
2
K
0 u3
1
dx + πR0 I1 /E1
Z L

K 11 2
1σ
dx1 .
Bibliography
A
[1] Abbas, S. A mixed finite element formulation for shells through a reduced Reissner’s
principle in elastodynamics. Acta Mechanica, 37:75–83, 1980.
[2] Alessandrini, S. M., Arnold, D. N., Falk, R. S. and Madureira, A. L.
Dimensional reduction for plates based on mixed variational principles. In Shells
(Santiago de Compostela, 1997), volume 105 of Cursos Congr. Univ. Santiago de
Compostela, pages 25–28. Univ. Santiago de Compostela, Santiago de Compostela,
1997.
[3] Alessandrini, S. M., Arnold, D. N., Falk, R. S. and Madureira, A. L.
Derivation and justification of plate models by variational methods. In Fortin, M.,
editor, Plates and Shells, volume 21 of Centre de Recherches Mathematiques, Proceedings and Lecture Notes, pages 1–20. Amer. Math. Soc., Providence, RI, 1999.
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