3
Albert S. Kobayashi, Satya N. Atluri
In this chapter we consider certain useful fundamental topics from the vast panorama of the
analytical mechanics of solids, which, by itself, has
been the subject of several handbooks. The specific topics that are briefly summarized include:
elementary theories of material response such as
elasticity, dynamic elasticity, viscoelasticity, plasticity, viscoplasticity, and creep; and some useful
analytical results for boundary value problems in
elasticity.
1.1
Elementary Theories
of Material Responses ...........................
1.1.1 Elasticity .....................................
1.1.2 Viscoelasticity ..............................
1.1.3 Plasticity .....................................
1.1.4 Viscoplasticity and Creep ...............
7
9
Boundary Value Problems in Elasticity ....
1.2.1 Basic Field Equations ....................
1.2.2 Plane Theory of Elasticity...............
1.2.3 Basic Field Equations for the State
of Plane Strain .............................
1.2.4 Basic Field Equations for the State
of Plane Stress .............................
1.2.5 Infinite Plate with a Circular Hole...
1.2.6 Point Load on a Semi-Infinite Plate
11
11
12
Summary .............................................
14
References ..................................................
14
1.2
4
4
6
1.3
12
12
13
13
Herein, we employ Cartesian coordinates exclusively.
We use a fixed Cartesian system with base vectors
ei (i = 1, 2, 3). The coordinates of a material particle before and after deformation are xi and yi , respectively.
The deformation gradient, denoted as Fij , is defined to
be
∂yi
≡ yi, j .
(1.1)
∂x j
A wide variety of other strain measures may be derived [1.1–4].
Let ( da) be a differential area in the deformed body,
and let n i be direction cosines of a unit outward normal
to ( da). If the differential force acting on this area is
d f i , the true stress or Cauchy stress τij is defined from
the relation
The displacement components will be denoted by
u i (= yi − xi ), such that
Thus τij is the stress per unit area in the deformed body.
The nominal stress (or the transpose of the so-called
first Piola–Kirchhoff stress) tij and the second Piola–
Kirchhoff stress Sij are defined through the relations
Fij = δij + u i, j ,
(1.2)
where δij is a Kronecker delta. The Green–Lagrange
strain tensor εij is given by
1
1
εij = (Fki Fk j − δij ) ≡ (u i, j + u j,i + u k,i u k, j ) .
2
2
(1.3)
When displacements and their gradients are infinitesimal, (1.3) may be approximated as
1
(1.4)
εij = (u i, j + u j,i ) ≡ u (i, j) .
2
d f i = ( da)n j τij .
d f i = ( da)N j t ji
= ( dA)N j S jk yi,k ,
(1.5)
(1.6)
(1.7)
where ( dA)N j is the image in the undeformed configuration, of the oriented vector area ( da)n j in the
deformed configuration. Note that both t ji and S ji are
stresses per unit area in the undeformed configuration,
and t ji is unsymmetric, while S ji is, by definition, symmetric [1.3,4]. It should also be noted that a wide variety
of other stress measures may be derived [1.3, 4].
Part A 1
Analytical Me
1. Analytical Mechanics of Solids
4
Part A
Solid Mechanics Topics
Part A 1.1
From the geometric theory of deformation [1.5], it
follows that
∂xi
(1.8)
,
( da)n j = (J)( dA)Nk
∂y j
where
ρ0
dv
(1.9)
=
.
J=
d∀
ρ
In the above dv is a differential volume in the deformed body, and d∀ is its image in the undeformed
body. From (1.5) through (1.9), it follows that
∂x j
∂xi
∂xi
τm j and Sij = J
τmn
. (1.10)
tij = J
∂ym
∂ym
∂yn
Another useful stress tensor is the so-called Kirchhoff
stress tensor, denoted by σij and defined as
σij = Jτij .
(1.11)
When displacements and their gradients are infinitesimal, J ≈ 1, ∂xi /∂yk = δik and so on, and
thus the distinction between all the stress measures
largely disappears. Hence, in an infinitesimal deformation theory, one may speak of the stress tensor σij .
For more on finite deformation mechanism of solids
see [1.6, 7].
1.1 Elementary Theories of Material Responses
The mathematical characterization of the behavior of
solids is one of the most complex aspects of solid
mechanics. Most of the time, the general behavior of
a material defies our mathematical ability to characterize it. The theories discussed below must be viewed
simply as idealizations of regimes of material response
under specific types of loading and/or environmental
conditions.
invariants of εij . These invariants may be defined as
1.1.1 Elasticity
where eijk is equal to 1 if (ijk) take on values 1, 2, 3
in a cyclic order, and equal to −1 if in anticyclic order,
and is zero if two of the indices take on identical values.
Sometimes, invariants J1 , J2 , and J3 , defined as
In this idealization, the underlying assumption is that
stress is a single-valued function of strain and is independent of the history of straining. Also, for such
materials, one may define a potential for stress in terms
of strain, in the form of a strain-energy density function,
denoted here by W.
It is customary [1.4] to measure W per unit of the
undeformed volume. In the general case of finite deformations, different stress measures are related to the
derivative of W with respect to specific strain measures,
labeled as conjugate strain measures, i. e., strains conjugate to the appropriate form of stress. Thus it may be
shown [1.3] that
tij =
∂W
,
∂F ji
Sij =
∂W
.
∂εij
(1.12)
Note that, for finite deformations, the Cauchy stress τij
does not have a simple conjugate strain measure.
When W does not depend on the location of the material
particle (in the undeformed conjugation), the material
is said to be homogeneous. A material is said to be
isotropic if W depends on εij only through the basic
I1 = 3 + 2εkk ,
I2 = 3 + 4εkk + 2(εkk εmm − εkm εkm ) ,
and
I3 = det|δmn + 2εmn |
≡ 1 + 2εkk + 2(εkk εmm − εkm εkm ) ,
J1 = (I1 − 3), J2 = (I2 − 2I1 + 3) ,
J3 = (I3 − I2 + I1 − 1)
(1.13)
(1.14)
are also used. When the material is isotropic, the
Kirchhoff stress tensor may be shown to be the derivative of W with respect to a certain logarithmic strain
measure [1.4]. Also, by decomposing the deformation
gradient Fij into pure stretch and rigid rotation [1.4, 8],
one may derive certain other useful stress measures,
such as the Biot–Lure stress, Jaumann stress, and so
on [1.4].
An isotropic nonlinearly elastic material may be
characterized, in its behavior at finite deformations, by
W=
∞
Crst (I1 − 3)r (I2 − 3)s (I3 − 1)t ,
r,s,t=0
C000 = 0 .
(1.15)
The ratio of volume change due to deformation, dv/ d∀,
is given, for finite deformations, by I3 . Thus, for
Analytical Mechanics of Solids
W̄ = W(εij ) + p(I3 − 1) ,
(1.16)
that is,
tij =
∂W
∂I3
+p
,
∂F ji
∂F ji
Sij =
∂W
∂I3
+p
. (1.17)
∂εij
∂εij
For isotropic, incompressible, elastic materials,
W(εij ) = W(I1 , I2 ) .
(1.18)
Thus (1.17) and (1.18) yield, for instance,
∂W
∂W
δij + 4[δij (1 + εmm ) − δim δ jn εmn ]
∂I1
∂I2
+ p[δij (1 + 2εmm ) − 2δim δ jn εmn
+ 2eimn e jrs εmr εns ] .
(1.19)
Sij = 2
A well-known representation of (1.18) is due to
Mooney [1.8], where
W(I1 , I2 ) = C1 (I1 − 3) + C2 (I2 − 3) .
(1.20)
So far, we have discussed isotropic materials. In general, for a homogeneous solid, one may write
W = E ij εij + 12 E ijmn εij εmn
+ 13 E ijmnrs εij εmn εrs + · · · .
(1.21)
We use, for convenience of presentation, Sij and εij as
conjugate measures of stress and strain. Since Sij and εij
are both symmetric, one must have
E ij = E ji , E ijmn = E jinm = E ijnm = E mnij ,
E ijmnrs = E jimnrs = E ijnmrs
= E ijmnsr = · · · = Ersijmn = · · · .
(1.22)
Thus
Sij = E ij + E ijmn εmn + E ijmnrs εmn εrs + · · · . (1.23)
Henceforth, we will consider the case when deformations are infinitesimal. Thus εij ≈ (1/2)(u i, j + u j,i ).
Further, the differences in the definitions of various
stress measures disappear, and one may speak of the
stress σij . Thus (1.23) may be rewritten as
σij = E ij + E ijmn εmn + E ijmnrs εmn εrs + · · · . (1.24)
A material is said to be linearly elastic if a linear approximation of (1.24) is valid for the magnitude of strains
under consideration. For such a material,
σij = E ij + E ijmn εmn .
(1.25)
The stress at zero strain (i. e., E ij ) most commonly is
due to temperature variation from a reference state. The
simplest assumption in thermal problems is to set
E ij = −βij ΔT
where ΔT [= (T − T0 )] is the temperature increment
from the reference value T0 .
For an anisotropic linearly elastic solid, in view of
the symmetries in (1.23), one has 21 independent elastic constants E ijkl and six constants βij . In the case of
isotropic linearly elastic materials, an examination of
(1.13) through (1.15) reveals that the number of independent elastic constants E ijkl is reduced to two, and the
number of independent βs to one. Thus, for an isotropic
elastic material,
σij = λεkk δij + 2μεij − βΔT δij ,
(1.26a)
where λ and μ are Lamé parameters, which are related
to the Young’s modulus E and the Poisson’s ratio ν
through
E
Eν
, μ=
.
λ=
(1 + ν)(1 − 2ν)
2(1 + ν)
The bulk modulus K is defined as
3λ + 2μ
.
K=
3
The inverse of (1.26a) is
ν
1+ν
(1.26b)
σij + αΔT δij ,
εij = − σmn δij +
E
E
where β and α are related through
Eα
β=
(1.26c)
1 − 2ν
and α is the linear coefficient of thermal expansion.
The state of plane strain is characterized by the conditions that u r = u r (X s ), r, s = 1, 2, and u 3 = 0. Thus
ε3i = 0, i = 1, 2, 3. In plane strain,
1 − ν2
ν
σ11 −
σ22 + α(1 + ν)ΔT ,
ε11 =
E
1−ν
ε22 =
1 − ν2
E
σ22 −
(1.27a)
ν
σ11 + α(1 + ν)ΔT ,
1−ν
(1.27b)
1+ν
ε12 =
σ12 ,
E
σ33 = ν(σ11 + σ ) − αEΔT .
(1.27c)
(1.27d)
5
Part A 1.1
incompressible materials, I3 = 1. For incompressible
materials, stress is determined from strain only to within
a scalar quantity (function of material coordinates)
called the hydrostatic pressure. For such materials, one
may define a modified strain-energy function, say W̄, in
which the incompressibility condition, I3 = 1, is introduced as a constraint through the Lagrange multiplier p.
Thus
1.1 Elementary Theories of Material Responses
6
Part A
Solid Mechanics Topics
Part A 1.1
The state of plane stress is characterized by the conditions that σ3k = 0, k = 1, 2, 3. Here one has
1
(1.28a)
ε11 = (σ11 − νσ22 ) + αΔT ,
E
1
ε22 = (σ22 − νσ11 ) + αΔT ,
(1.28b)
E
1+ν
ε12 =
(1.28c)
σ12 ,
E
ν
ε33 = − (σ11 + σ22 ) + αΔT .
(1.28d)
E
Note that in (1.27c) and (1.28c), ε12 is the tensor component of strain. Sometimes it is customary to use
the engineering strain component γ12 = 2ε12 . Note also
that, in the case of a linearly elastic material, the strainenergy density W is given by
When the bulk modulus k → ∞ (or ν → 12 ), it is seen
that εkk → 3αΔT and is independent of the mean stress.
Note also from (1.26c) that β → ∞ as ν → 12 . For
such materials, the mean stress is indeterminate from
deformation alone. In this case, the relation (1.26a) is
replaced by
(1.31a)
with the constraint
(1.31b)
where ρ is the hydrostatic pressure and εij is the deviator of the strain. Note that the strain-energy density of
a linearly elastic incompressible material is
W = μεij εij − p(εkk − 3αΔT ) ,
σij (t) = εkl (0 )E ijkl +
E ijkl (t − τ)
0
∂εkl
dτ
∂τ
(1.33a)
= E kl (0+ )εijkl +
t
εijkl (t − τ)
0
(1.29)
From (1.26b) it is seen that, for linearly elastic isotropic
materials,
1 − 2ν
σmm
σmm + 3αΔT ≡
+ 3αΔT . (1.30)
εkk =
E
3k
εkk = 3αΔT ,
t
∂E kl
dτ .
∂τ
In the above, it has been assumed that σkl = εkl = 0 for
t < 0 and that εij (t) and E ij (t) are piecewise continuous.
E ijkl (t) is called the relaxation tensor for an anisotropic
material. Conversely, one may write
= 12 (σ11 ε11 + σ22 ε22 + σ33 ε33
+ 2ε12 σ12 + 2ε13 σ13 + 2ε23 σ23 )
σij = −ρδij + 2μεij
+
(1.33b)
W = 12 σij εij
≡ 12 (σ11 ε11 + σ22 ε22 + σ33 ε33
+ γ12 σ12 + γ23 σ23 + γ13 σ13 ) .
materials are those for which the current deformation
is a function of the entire history of loading, and conversely, the current stress is a function of the entire
history of straining. Linearly viscoelastic materials are
those for which the hereditary relations are expressed
in terms of linear superposition integrals, which, for
infinitesimal strains, take the forms
(1.32)
wherein p acts as a Lagrange multiplier to enforce
(1.31b).
1.1.2 Viscoelasticity
A linearly elastic solid, by definition, is one that has
the memory of only its unstrained state. Viscoelastic
+
t
εij (t) = σkl (0 )Cijkl +
Cijkl (t − τ)
0
∂σkl
dτ ,
∂τ
(1.34)
where Cijkl (t) is called the creep compliance tensor.
For isotropic linearly viscoelastic materials,
E ijkl = μ(t)(δik δ jl + δlk δ jk ) + λ(t)δij δkl ,
(1.35)
where μ(t) is the shear relaxation modulus and
B(t) ≡ [3α(t) + 2μ(t)]/3 is the bulk relaxation modulus. It is often assumed that B(t) is a constant; so that
the material is assumed to have purely elastic volumetric change. In viscoelasticity, a Poisson function
corresponding to the strain ratio in elasticity does not
exists. However, for every deformation history there is
computable Poisson contraction or expansion behavior.
For instance, in a uniaxial tension test, let the stress be
σ11 , the longitudinal strain ε11 , and the lateral strain ε22 .
For creep at constant stress, the ratio of lateral contraction, denoted by νc (t), is νc (t) = −ε22 (t)/ε11 (t). On the
other hand, under relaxation at constant strain, ε11 , the
lateral contraction ratio is ν R (t) = −ε22 (t)/ε11 .
It is often convenient, though not physically correct, to assume that Poisson’s ratio is a constant, which
renders B(t) proportional to μ(t). A constant bulk modulus provides a much better and simple approximation
for the material behavior than a constant Poisson’s ratio
when properties over the whole time range are needed.
Analytical Mechanics of Solids
σ ij ( p) = pE ijkl ( p)εkl ( p)
(1.36a)
and
deformation is assumed to be insensitive to hydrostatic
pressure, the yield function is assumed, in general, to
depend on the stress deviator, σij = σij − 13 σmm δij . The
commonly used yield functions are
von Mises
εij ( p) = pC ijkl ( p)σ kl ( p) ,
(1.36b)
where (·) is the Laplace transform of (·) and p is the
Laplace variable. From (1.36a) and (1.36b), it follows
that
p E ijkl C klmn = δim δn j .
2
M
μm exp(−μm t) ,
m=1
M
B(t) = B0 +
Bm exp(−βm t) .
m=1
1.1.3 Plasticity
Most structural metals behave elastically for only very
small values of strain, after which the materials yield.
During yielding, the apparent instantaneous tangent
modulus of the material is reduced from those in the
prior elastic state. Removal of load causes the material to unload elastically with the initial elastic modulus.
Such materials are usually labeled as elastic–plastic.
Observed phenomena in the behavior of such materials
include the so-called Bauschinger effect (a specimen
initially loaded in tension often yields at a much reduced
stress when reloaded in compression), cyclic hardening, and so on [1.9, 10]. (When a specimen a specimen
is subjected to cyclic straining of amplitude −ε to +ε,
the stress for the same value of tensile strain ε, prior to
unloading, increases monotonically with the number of
cycles and eventually saturates.) Various levels of sophistication of elastic-plastic constitutive theories are
necessary to incorporate some or all of these observed
phenomena. Here we give a rather cursory review of this
still burgeoning literature.
In most theories of metal plasticity, it is assumed
that plastic deformations are entirely distortional in nature, and that volumetric strain is purely elastic. The
elastic limit of the material is assumed to be specified
by a yield function, which is a function of stress (or
of strain, but most commonly of stress). Since plastic
J2 = 12 σij σij ,
(1.38)
Tresca
f (σij ) = (σ1 − σ2 )2 − 4k2 (σ2 − σ3 )2 − 4k2
(1.39)
× (σ1 − σ3 )2 − 4k2 = 0 .
(1.37)
It is also customary to represent the relaxation moduli,
μ(t) and B(t), in series form, as
μ(t) = μ0 +
f (σij ) = J2 − k2 ,
In (1.38) and (1.39), k may be a function of the plastic strain. Both (1.38) and (1.39) represent a surface,
which is defined as the yield surface, in the stress space.
The two equations also imply the equality of the tensile
and compressive yield stresses at all times – so-called
isotropic hardening.
The yield surface expands while its center remains
fixed in the stress space.√
The relation of k to test data follows: in (1.38), k = σ̄ / 3, where σ̄ is the yield stress
in uniaxial tension, which may be a function of plas√
tic strain for strain-hardening materials, or k = τ̄/ 2,
where τ̄ is the yield stress in pure shear; or in (1.39),
k = σ̄ /2 or τ̄. Experimental data appear to favor the use
of the Mises condition [1.11, 12].
To account for the Bauschinger effect, one may use
the representation of the yield surface
f (σij − αij ) = 0 = 12 σij − αij σij − αij − 13 σ̄ 2
=0,
(1.40)
where αij represents the center of the yield surface
in the deviatoric stress space. The evolution equations
suggested for αij by Prager [1.13] and Ziegler [1.14],
respectively, are that the incremental dαij is proporp
tional to the incremental plastic strain dεij or
dαij = c dεij
(1.41)
dαij = dμ(σij − αij ) .
(1.42)
p
and
In the above, (·) denotes the deviatoric part of the
second-order tensor (·) where an additive decomposition of differential strain into elastic and plastic parts
p
(i. e., dεij = dεije + dεij ) was used.
Elastic processes (with no increase in plastic strain)
and plastic processes (with increase in plastic strain) are
defined [1.12] as
7
Part A 1.1
The Laplace transforms of (1.33) and (1.34) may be
written as
1.1 Elementary Theories of Material Responses
8
Part A
Solid Mechanics Topics
Part A 1.1
elastic process:
f < 0 or
f = 0 and
∂f
dσij ≤ 0 ,
∂σij
(1.43)
plastic process:
∂f
dσij > 0 .
(1.44)
∂σij
The flow rule for strain-hardening materials, arising
out of consideration of stress working in a cyclic process and stability of the process, often referred to as
Drucker’s postulates [1.15], is given by
f = 0 and
p
dεij
∂f
= dλ
.
∂σij
(1.45)
The scalar dλ is determined from the fact that d f = 0
during a plastic process, the so-called consistency condition. Using the isotropic-hardening (J2 flow) theory
for which f is given in (1.38), this consistency condition
leads to
9 σ dσmn
p
(1.46)
dεij = σij mn 2 ,
4
H σ̄
where H is the slope of the curve of stress versus plastic
strain in uniaxial tension (or, more correctly, the slope
of the curve of true stress versus logarithmic strain in
pure tension). On the other hand, for Prager’s linearly
kinematic hardening rules, given in (1.40) and (1.41),
the consistency condition leads to
3 p
σmn − αmn
dσmn σij − αij . (1.47)
dεij =
2
2cσ̄
For pressure-insensitive plasticity, the stress–strain laws
may be written as
dσmn = (3λ + 2μ) dεmn ,
p
dσij = 2μ dεij − dεij .
(1.48a)
(1.48b)
p
Choosing a parameter ζ such that ζ = 1 when dεij = 0
p
and α = 0 when dεij = 0, we have
9 σ dσmn
dσij = 2μ dεij − σij mn 2 α
(1.49)
4
H σ̄
for isotropic hardening, and
3
dσij = 2μ dεij −
(σ − αmn
)
2cσ̄ 2 mn
× dσmn (σij − αij )α
(1.50)
for Prager’s linearly kinematic hardening. Taking the
tensor product of both sides of (1.49) with σij (and
dσ noting that σmn dσmn = σmn
mn by definition), we
have
3α dσij σij = 2μ dεij σij −
σ dσ
(1.51a)
2H mn mn
or
2μH dσij σij = dε σ , when α = 1 . (1.51b)
H + 3μ ij ij
Use of (1.51b) in (1.49) results in
9α
dσij = 2μ dεij − 2μ
σ
σ
dε
.
4(H + 3μ)σ̄ 2 ij mn mn
(1.52)
Combining (1.48a) and (1.52), one may write the
isotropic-hardening elastic-plastic constitutive law in
differential form as
dσij = 2μδim δ jn + λδij δmn
9αμ
− 2μ
σ
σ
(1.53)
dεmn ,
(2H + 6μ)σ̄ 2 ij mn
dε ≡ σ dε
wherein σmn
mn has been noted. Simimn
mn
larly, by taking the tensor product of (1.50) with (σij −
αij ) and repeating steps analogous to those in (1.51)
through (1.53), one may write the kinematic-hardening
elastoplastic constitutive law as
3μ
dσij = 2δim δ jn + λδij δmn − 2μ
(c + 2μ)σ̄ 2
× (σij − αij )(σmn − αmn ) dεmn .
(1.54)
Note that all the developments above are restricted to
the infinitesimal strain and small-deformation case. Discussion of finite-deformation plasticity is beyond the
scope of this summary. (Even in small deformation plasticity, if the current tangent modulus of the stress–strain
relation are of the same order of magnitude as the current stress, one must use an objective stress rate, instead
of the material rate dσij , in (1.54).) Here the objectivity
of the stress–strain relation plays an important role. We
refer the reader to [1.4, 16, 17].
We now briefly examine the elastic-plastic stress–
strain relations in the isotropic-hardening case, for plane
strain and plane stress, leaving it to the reader to derive
similar relations for kinematic hardening. In the planestrain case, dε3n = 0, n = 1, 2, 3. Using this in (1.53),
we have
dσαβ = 2μδαθ δβν + λδαβ δθν
9αμ
− 2μ
σ
σ
(1.55)
dεθν
(2H + 6μ)σ̄ 2 αβ θυ
Analytical Mechanics of Solids
9αμ
σ σ dεϑν ,
dσ33 = λ dεθθ − 2μ
(2H + 6μ)σ̄ 2 33 θυ
α, β, θ, ν = 1, 2 .
(1.56)
Note that, in the plane-strain case, σ33 , as integrated
from (1.56), enters the yield condition. In the planestress case the stress–strain relation is somewhat tedious
to derive.
Noting that in the plane-stress case, dεαβ = dεeαβ +
p
dεαβ one may, using the elastic strain–stress relations as
given in (1.33), write
p
dεαβ = dεeαβ + dεαβ
1 ν
p
=
dσαβ −
dσθθ δαβ + dεαβ (1.57)
2μ
1+ν
1 ν
dσαβ −
=
dσθθ δαβ
2μ
1+ν
9
(σθυ
dσθυ )
,
(1.58)
+ σαβ
4H σ̄ 2
wherein (1.16) has been used. Equation (1.58) may be
inverted to obtain dσαβ in terms of dεθν . This 3 × 3
matrix inversion may be carried out, leading to the result [1.18]
2
Q
) + 2P dε11
dσ11 = (σ22
E
+ (−σ11
σ22
+ 2ν P) dε22
σ + νσ22 − 11
σ12 2 dε12 ,
1+ν
Q
σ22
+ 2ν P) dε11
dσ22 = (−σ11
E
2
+ (σ11 ) + 2P dε22
σ + σ11
2
− 22
dε12 ,
σ12
1+ν
σ + νσ22
Q
dσ12 = − 11
σ12 dε11
E
1+ν
+ νσ σ22
11
−
σ12 dε22
1
+
ν
R
2H +
+
(1 − ν)σ̄ dε12 ,
2(1 + ν)
9E
where
σ2
2H 2
P=
σ̄ + 12 , Q = R + 2(1 − ν2 )P ,
9H
1+ν
(1.59)
and
2 + 2νσ11
σ22 + σ22
2.
R = σ11
(1.60)
As noted earlier, the classical plasticity theory has
several limitations. Intense research is underway to im-
prove constitutive modeling in cyclic plasticity, and so
on, some notable avenues of current research being
multisurface plasticity model, endochronic theories, and
related internal variable theories (see, e.g., [1.19–21]).
1.1.4 Viscoplasticity and Creep
A viscoplastic solid is similar to a viscous fluid, except
that the former can resist shear stress even in a rest configuration; but when the stresses reach critical values as
specified by a yield function, the material flows. Consider, for instance, the loading case of simple shear with
the only applied stress being σ12 . Restricting ourselves
to infinitesimal deformations and strains, let the shearstrain rate be ε̇12 = ( dε12 / dt). Then ε̇12 = 0 until the
magnitude of σ12 reaches a value k, called the yield
stress. When |σ12 | > k, ε̇12 , by definition for a simple
viscoplastic material, is proportional to |σ12 | − k and has
the same sign as σ12 . Thus, defining a function F 1 for
this one-dimensional problem as
|σ12 |
(1.61)
−1 ,
k
the viscoplastic property may be characterized by the
equation
F1 =
2ηε̇12 = k(F 1 )σ12 ,
where F 1 must have the property
⎧
⎨0
if F 1 < 0 ,
F 1 =
⎩ F 1 if F 1 ≥ 0 ,
(1.62)
(1.63)
and where η is the coefficient of viscosity.
The above relation for simple shear is due to Bing2 for simple shear,
ham [1.22]. Recognizing that J = σ12
a generalization of the above for three-dimensional case
was given by Hohenemser and Prager [1.23] as
νp
ηε̇ij = 2k
F 1 ∂F
,
∂σij
(1.64a)
where
1/2
1/2
σij σij /2
J2
F=
−1 =
−1 ,
k
k
(1.64b)
and the specific function F is defined similar to F 1 of (1.63).
For an elasto-viscoplastic solid undergoing infinitesimal straining, one may use the additive decomposition
p
ε̇ij = ε̇ije + ε̇ij
(1.65)
9
Part A 1.1
and
1.1 Elementary Theories of Material Responses
10
Part A
Solid Mechanics Topics
Part A 1.1
and the stress–strain rate relation
ν p
σ̇ij = E ijkl ε̇kl − ε̇kl ,
(1.66)
where E ijkl are the instantaneous elastic moduli. Note
that the viscoplastic strains in (1.64a) are purely deviatoric, since ∂F/∂σij = σij /2 is deviatoric. Thus, for an
isotropic solid, (1.66) may be written as
σ̇mn = (3λ + 2μ) ε̇mn
and
σ̇ij = 2μ ε̇ij − ε̇ij
.
n m
(1.68)
(1.69a)
or, equivalently,
ε̇c = g(σ, εc ) .
(1.69b)
The expression (1.69a) is often referred to as time hardening and (1.69b) as strain hardening. In as much as
(1.68), (1.69a), and (1.69b), are valid for constant stress,
(1.69a) and (1.69b), when integrated for variable stress
histories, do not necessarily give the same results. Usually, strain hardening leads to better agreement with
experimental findings for variable stresses.
In the study of creep at a given temperature and for
long times, called steady-state creep, the creep strain
rate in uniaxial loading is usually expressed as
ε̇c = f (σ, T ) ,
(1.70)
where T is the temperature. Assuming that the effects
of σ and T are separable, the relation
ε̇ = f 1 (σ) f 2 (T )
= Aσ n f 2 (T ) = Bσ n
3 σ σ
2 ij ij
and
ε̇ceq
=
2
ε̇ij ε̇ij
3
1/2
(1.73)
(1.67b)
where σ is the uniaxial stress and t is the time. The creep
rate may be written as
ε̇c = f (σ, t) ,
(1.72)
where the subscript “eq” denotes an “equivalent” quantity, defined, analogous to the case of plasticity, as
σ̇eq =
ε̇ = Aσ t ,
c
n
,
ε̇ceq = Bσeq
(1.67a)
On the other hand, for metals operating at elevated temperatures, the strain in uniaxial tension is known to
be a function of time, for a constant stress of magnitude even below the conventional elastic limit. Most
often, based on extensive experimental data [1.24], the
creep strain under constant stress, in uniaxial tests, is
expressed as
c
involve no volume change. Thus, in the multiaxial case,
ε̇ijc is a deviatoric tensor. The relation (1.71) may be
generalized to the multiaxial case as
(1.71a)
(1.71b)
is usually employed, with B denoting a function of the
temperature.
The steady-state creep strains are associated largely
with plastic deformations and are usually observed to
such that σeq ε̇ceq = σij ε̇ijc . Thus (1.72) implies that
ε̇ijc =
3
B(σeq )n−1 σij .
2
(1.74)
For the elastic-creeping solid, one may again write
ε̇ij = ε̇ije + ε̇ijc
(1.75)
and once again, the stress–strain rate relation may be
written as
(1.76)
σ̇ij = E ijkl ε̇kl − ε̇ckl .
In the above, the applied stress level has been assumed
to be such that the material remains within the elastic
limit. If the applied loads are of such a magnitude as
to cause the material to exceed its yield limit, one must
account for plastic or viscoplastic strains.
An interesting unified viscoplastic/plastic/creep
constitutive law has been proposed by Perzyna [1.25].
Under multiaxial conditions, the relation for inelastic
strain rate suggested in [1.26] is
ε̇ija = Aψ( f )
∂q
,
∂σij
(1.77)
where A is the fluidity parameter, the superscript “a” denotes an elastic strain rate, and f is a loading function,
expressed, analogous to the plasticity case, as
f (σij , k) = ϕ(σij ) − k = 0 ,
(1.78)
q is a viscoplastic potential defined as
q = q(σij ) ,
and ψ( f ) is a specific function such that
⎧
⎨0
if f < 0
ψ( f ) =
⎩ψ( f ) if f ≥ 0 .
(1.79)
(1.80)
Analytical Mechanics of Solids
ψ( f ) = f n
and
f =
3 σ σ
2 ij ij
(1.81a)
1/2
− σ̄ = σeq − σ̄ .
(1.81b)
By letting σ̄ = 0 and q = f , one may easily verify that
ε̇ija of (1.77) tends to the creep strain rate ε̇ijc of (1.74).
Letting σ̄ be a specified constant value and q = f ,
we obtain, using (1.81b) in (1.77), that
ε̇ija =
σij
3
A(σeq − σ̄)n
2
σeq
for f > 0
(i. e., σeq > σ̄) .
(1.82)
The equivalent inelastic strain may be written as
2 a a 1/2
= A(σeq − σ̄)n
(1.83a)
ε̇ij ε̇ij
ε̇aeq =
3
or
1 a 1/2
σeq − σ̄ =
.
(1.83b)
ε̇eq
A
Thus, if a stationary solution of the present inelastic model (i. e., when ε̇aeq → 0) is obtained, it
is seen that σeq → σ̄ . Thus a classical inviscid
plasticity solution is obtained. This fact has been
utilized in obtaining classical rate-independent plasticity solutions from the general model of (1.77),
by Zienkiewicz and Corneau [1.26]. An alternative
way of obtaining an inviscid plastic solution from
Perzyna’s model is to let A → ∞. This concept
has been implemented numerically by Argyris and
Kleiber [1.27].
Also, as seen from (1.83b), σeq or equivalently
the size of the yield surface is governed by isotropic
work-hardening effects as characterized by the dependence on viscoplastic work, and the strain-rate effect as
q
characterized by the term (ε̇eq )1/n . Thus rate-sensitive
plastic problems may also be treated by Perzyna’s
model [1.25].
Thus, by appropriate modifications, the general relation (1.77) may be used to model creep, rate-sensitive
plasticity, and rate-insensitive plasticity. By a linear
combination of strain rates resulting from these individual types of behavior, combined creep, plasticity, and
viscoplasticity may be modeled. However, such a model
is more or less formalistic and does not lead to any
physical insights into the problem of interactive effects
between creep, plasticity, and viscoplasticity. Modeling
of such interactions is the subject of a large number of
current research studies.
1.2 Boundary Value Problems in Elasticity
1.2.1 Basic Field Equations
When deformations are finite and the material is nonlinear, the field equations governing the motion of a solid
may become quite complicated. When the constitutive
equation is of a differential form, such as in plasticity,
viscoplasticity, and so on, it is often convenient to express the field equations in rate form as well. On the
other hand, when stress is a single-valued function of
strain as in nonlinear elasticity, the field equations may
be written in a total form. In general, when numerical procedures are employed to solve boundary/initial
value problems for arbitrary-shaped bodies, it is often
convenient to write the field equations in rate form,
for arbitrary deformations and general constitutive laws.
A wide variety of equivalent but alternative forms of
these equations is possible, since one may use a wide
variety of stress and strain measures, a wide variety of
rates of stress and strain, and a variety of coordinate
systems, such as those in the initial undeformed configuration (total Lagrangian), the currently deformed
configuration (updated Lagrangian), or any other known
intermediate configuration. For a detailed discussion
see [1.3, 4]. Each of the alternative forms may offer
advantages in specific applications.
It is beyond the scope of this chapter to discuss
the foregoing alternative forms. Here we state, for
a finitely deformed nonlinearly elastic solid, the relevant
field equations governing stress, strain, and deformation when the solid undergoes dynamic motion. For
this purpose, let x j denote the Cartesian coordinates of
a material particle in the undeformed solid. Let u k (xi )
be the arbitrary displacement of a material particle from
the undeformed to the deformed configuration. Let Sij
be the second Piola–Kirchhoff stress in the finitely deformed solid. Note that S jj is measured per unit area in
the undeformed configuration. Let the Green–Lagrange
strain tensor, which is work-conjugate to Sij [1.3],
11
Part A 1.2
If q ≡ f , one has the so-called associative law, and
if q = f , one has a nonassociative law. Perzyna [1.25]
suggests a fairly general form for ψ as
1.2 Boundary Value Problems in Elasticity
12
Part A
Solid Mechanics Topics
Part A 1.2
be εij . Let ρ0 be the mass density in the undeformed
solid; b j be body forces per unit mass; ti be tractions
measured per unit area in the undeformed solid, prescribed at surface St , of the undeformed solid; and let u i
be prescribed displacements at Su . The field equations
are [1.4]:
linear momentum balance
Sik (δ jk + u j,k ) + ρ0 b j = ρ0 ü j ,
(1.84)
angular momentum balance
Sij = S ji ,
(1.85)
strain displacement relation
εij = 12 (u i, j + u j,i + u k,i u k, j ) ,
constitutive law
(1.86)
∂W
,
∂εij
(1.87)
at Si ,
(1.88)
Sij =
traction boundary condition
n j Sik (δ jk + u j,k ) = t j
displacement boundary condition
u j = ū i at Su .
(1.89)
2
In the above, (·)k denotes ∂(·)/∂xk ; (¨·) denotes ∂ (·)/∂t 2 ;
n i are components of a unit normal to St ; and W is the
strain-energy density, measured per unit volume in the
undeformed body.
When the deformations and strains are infinitesimal,
the differences in the alternate stress and strain measures disappear. Further, considering only an isothermal
linearly elastic solid, the equations above simplify as
σij,i + ρ0 b j = ρ0 ü j ,
σij = σ ji ,
1
εij = (u i, j + u j,i ) ,
2
σij = E ijkl εkl ,
n i σij = t¯i at St ,
u i = ū i
at Su ,
(1.90)
(1.91)
(1.92)
(1.93)
(1.94)
(1.95)
and the initial condition,
u i (xk , 0) = u ∗ (xk ), u̇ i (xk , 0) = u̇ i∗ (xk , 0) at t = 0 .
(1.96)
problems, unfortunately, are few and are limited to
semifinite or finite domains; homogeneous and isotropic
materials; and often relatively simple boundary conditions. In practice, however, it is common to encounter
problems with finite but complex shape in which the
material is neither homogeneous nor homogeneous and
the boundary conditions are complex. The rapid development and easy accessibility of large-scale numerical
codes in recent years are now providing engineers
with numerical tools to analyze these practical problems, which are mostly three dimensional in nature and
that commonly occur in engineering. Often, however,
some three-dimensional problems can be reduced, as
a first approximation, to two-dimensional problems for
which analytical solutions exist. The utility of such twodimensional solutions lies not in their elegant analysis
but in their use for physically understanding certain
classes of problems and, more recently, as benchmarks
for validating numerical modeling and computational
procedures. In the following, we will reformulate the
basic equations in two-dimensional Cartesian coordinates and provide analytical example solutions to two
simple problems in terms of a polar coordinate system.
1.2.3 Basic Field Equations
for the State of Plane Strain
A plane state of strain is defined as the situation with
zero displacement, say u 3 = 0. Equation (1.90) through
(1.96) recast with this definition are
σ11,1 + σ21,2 + ρ0 b1 = ρ0 ü 1
σ12,1 + σ22,2 + ρ0 b2 = ρ0 ü 2 ,
σ1,2 = σ2,1 ,
ε11 = u 1,1 ,
ε22 = u 2,2 ,
1 − ν2
(1.97)
(1.98)
ε12 =
1
2 (u 1,2 + u 2,1 ) ,
ν
σ11 −
σ22 ,
E
1−ν
1 − ν2
ν
ε22 =
σ22 −
σ11 ,
E
1−ν
1+ν
ε12 =
σ12 ,
E
σ33 = ν(σ11 + σ22 ) .
ε11 =
(1.99)
(1.100a)
(1.100b)
(1.100c)
(1.100d)
1.2.2 Plane Theory of Elasticity
The basic field equations and boundary conditions for
a three-dimensional boundary/initial value problem in
linear elasticity are given in (1.90) through (1.96). Analytical (exact) solutions to idealized three-dimensional
1.2.4 Basic Field Equations
for the State of Plane Stress
The plane state of stress is defined with zero surface traction, say σ33 = σ31 = σ32 = 0, parallel to the
Analytical Mechanics of Solids
1
(1.101a)
(σ11 − νσ22 ) ,
E
1
ε22 = (σ22 − νσ11 ) ,
(1.101b)
E
1+ν
ε12 =
(1.101c)
σ12 ,
E
ν
ε33 = − (σ11 + σ22 ) .
(1.102)
E
In the following, we cite three classical analytical solutions to the linear boundary value problem in Eqs. (1.97)
through (1.102) for specific cases that are often of interest.
ε11 =
1.2.5 Infinite Plate with a Circular Hole
Consider a plane problem of an infinite linearly elastic isotropic body containing a hole of radius a. Let the
body be subjected to uniaxial tension, say o∞
11 , along
the x1 axis. Let the Cartesian coordinate system be located at the center of the hole and let r and θ be the
corresponding polar coordinates with θ being the angle
measured from the x1 axis. The state of stress near the
hole is given by [1.28]
σ∞
a2
σrr = 11 1 − 2
2
r
∞
σ11
a4
a2
+
(1.103a)
1 − 4 2 + 3 4 cos 2θ ,
2
r
r
σ∞
σ∞
a2
a4
σθθ = 11 1 + 2 − 11 1 + 3 4 cos 2θ ,
2
2
r
r
σrθ = −
∞
σ11
2
1+2
a2
r2
−3
a4
r4
(1.103b)
sin 2θ .
(1.103c)
The solution for a biaxial stress state may be obtained
by superposition. For a compendium of solutions of
holes in isotropic and anisotropic bodies, and for shapes
of holes other than circular, such as elliptical holes, see
Savin [1.28].
A basic problem for a heterogeneous medium such
as a composite, is that of an inclusion (or inclusions).
Consider then a rigid inclusion of radius a and assume
a perfect bonding between the medium and the inclusion. The solution for stresses near the inclusion due to
a far-field uniaxial tension, say o∞
11 , are
σ∞
a2
σrr = 11 1 − ν 2
2
r
∞
σ11
a4
a2
+
1 − 2β 2 − 3δ 4 cos 2θ , (1.104a)
2
r
r
∞
∞
σ11
σ11
a2
a4
σθθ =
1+ν 2 −
1 − 3δ 4 cos 2θ ,
2
2
r
r
σrθ = −
∞
σ11
a2
a4
(1.104b)
(1.104c)
1 + β 2 + 3δ 4 sin 2θ ,
2
r
r
2(λ + μ)
μ
λ+μ
β=−
ν=−
δ=
,
λ + 3μ
λ+μ
λ + 3μ
(1.104d)
where λ and μ are Lamé constants.
1.2.6 Point Load on a Semi-Infinite Plate
Consider a concentrated vertical force P acting on
a horizontal straight edge of a semi-infinitely large
plate. The origin of a Cartesian coordinate is at the location of load application with x1 in the direction of the
force. Consider again a polar coordinate of r and θ with
θ being the angle measured from the x1 axis, positive
in the counterclockwise direction. The state of stress is
a very simple one given by
2P cos θ
π r
σθθ = σrθ = 0 .
σrr = −
(1.105a)
(1.105b)
This state of stress satisfies the natural boundary conditions as all three stress components vanish on the
straight boundary, i. e., θ = π/2, except at the origin,
where σrr → ∞ as r → 0. A contour integration of
the vertical component of σrr over a semicircular arc
from the origin yields the statically equivalent applied
force P and thus all boundary conditions are satisfied.
By using stress equations of transformations, which
can be found in textbooks on solid mechanics, the
stresses in polar coordinates can be converted into
Cartesian coordinates as
2P
(1.106a)
cos4 θ ,
σ11 = σrr cos2 θ = −
πa
13
Part A 1.2
x1 –x2 plane. This state shares the same stress equations of equilibrium and strain-displacement relations
with the state of plane strain, i. e. (1.97) through (1.99).
Equations (1.97) and (1.98) are necessary and sufficient to solve a two-dimensional elastostatic boundary
value problem when only tractions are prescribed on
the boundary. Thus the stresses of the plane-stress and
plane-strain solutions coincide while the strains differ.
The stress–strain relations for the state of plane
stress are
1.2 Boundary Value Problems in Elasticity
14
Part A
Solid Mechanics Topics
Part A 1
2P
(1.106b)
sin2 θ cos2 θ ,
πa
2P
σ12 = σrr sin θ cos θ = −
sin θ cos3 θ . (1.106c)
πa
σ22 = σrr sin2 θ = −
Since strain is what is actually being measured, the
corresponding strains can be computed by (1.100) for
the plane-strain state or by (1.101) for the plane-stress
state.
1.3 Summary
It is obviously impossible in this extremely brief review
even to mention all of the important subjects and recent developments in the theories of elasticity, plasticity,
viscoelasticity, and viscoplasticity. For further details,
readers are referred to the many excellent books and survey articles (see for example [1.29] and [1.30]), many of
which are referenced in the succeeding chapters, in each
of the disciplines.
References
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
C. Truesdell (Ed.): Mechanics of Solids. In: Encyclopedia of Physics, Vol. VIa/2 (Springer, Berlin, Heidelberg
1972)
C. Truesdell, W. Noll: The Nonlinear Field Theories of
Mechanics. In: Encyclopedia of Physics, Vol. III/3, ed.
by S. Flügge (Springer, Berlin, Heidelberg 1965)
S.N. Atluri: Alternate stress and conjugate strain
measures, and mixed foundations involving rigid
rotations for computational analysis of finitely deformed solids, with application to plates and shells.
I- Theory, Comput. Struct. 18(1), 93–116 (1986)
S.N. Atluri: On some new general and complementary energy theorems for rate problems in finite
strain, classical elastoplasticty, J. Struct. Mech. 8(1),
61–92 (1980)
A.C. Eringen: Nonlinear Theory of Continuous Media
(McGraw-Hill, New York 1962)
R.W. Ogden: Nonlinear Elastic Deformation (Dover,
New York 2001)
Y.C. Fung, P. Tong: Classical and Computational Solid
Mechanics (World Scientific, Singapore 2001)
M. Mooney: A theory of large elastic deformation, J.
Appl. Phys. 11, 582–592 (1940)
A. Abel, R.H. Ham: The cyclic strain behavior of crystal aluminum-4% copper, the Bauschinger effect,
Acta Metallur. 14, 1489–1494 (1966)
A. Abel, H. Muir: The Bauschinger effect and stacking
fault energy, Phil. Mag. 27, 585–594 (1972)
G.I. Taylor, H. Quinney: The plastic deformation of
metals, Phil. Trans. A 230, 323–362 (1931)
R. Hill: The Mathematical Theory of Plasticity (Oxford
Univ. Press, New York 1950)
W. Prager: A new method of analyzing stress and
strains in work-hardening plastic solids, J. Appl.
Mech. 23, 493–496 (1956)
H. Ziegler: A modification of Prager’s hardening rule,
Q. Appl. Math. 17, 55–65 (1959)
1.15
1.16
1.17
1.18
1.19
1.20
1.21
1.22
1.23
1.24
1.25
D.C. Drucker: A more fundamental approach to plane
stress-strain relations, Proc. 1st U.S. Nat. Congr.
Appl. Mech. (1951) pp. 487–491
S. Nemat-Nasser: Continuum bases for consistent
numerical foundations of finite strains in elastic
and inelastic structures. In: Finite Element Analysis of Transient Nonlinear Structural Behavior, AMD,
Vol. 14, ed. by T. Belytschko, J.R. Osias, P.V. Marcal
(ASME, New York 1975) pp. 85–98
S.N. Atluri: On constitutive relations in finite strain
hypoelasticity and elastoplasticity with isotropic or
kinematic hardening, Comput. Meth. Appl. Mech.
Eng. 43, 137–171 (1984)
Y. Yamada, N. Yoshimura, T. Sakurai: Plastic stressstrain matrix and its application to the solution
of elastic-plastic problems by the finite element
method, Int. J. Mech. Sci. 10, 343–354 (1968)
K. Valanis: Fundamental consequences of a new
intrinsic tune measure plasticity as a limit of the
endochronic theory, Arch. Mech. 32(2), 171–191 (1980)
Z. Mroz: An attempt to describe the behavior
of metals under cyclic loads using a more general workhardening model, Acta Mech. 7, 199–212
(1969)
O. Watanabe, S.N. Atluri: Constitutive modeling of
cyclic plasticity and creep using an internal time
concept, Int. J. Plast. 2(2), 107–134 (1986)
E.C. Bingham: Fluidity and Plasticity (McGraw-Hill,
New York 1922)
K. Hohenemser, W. Prager: Über die Ansätze der
Mechanik isotroper Kontinua, Z. Angew. Math.
Mech. 12, 216–226 (1932)
I. Finnie, W.R. Heller: Creep of Engineering Materials
(McGraw-Hill, New York 1959)
P. Perzyna: The constitutive equations for rate sensitive plastic materials, Quart. Appl. Mech. XX(4),
321–332 (1963)
Analytical Mechanics of Solids
1.27
O.C. Zienkiewicz, C. Corneau: Visco-plasticity, plasticity and creep in elastic solids – a unified
numerical solution approach, Int. J. Numer. Meth.
Eng. 8, 821–845 (1974)
J.H. Argyris, M. Keibler: Incremental formulation
in nonlinear mechanics and large strain elastoplasticity – natural approach – Part I, Comput.
Methods Appl. Mech. Eng. 11, 215–247 (1977)
1.28
1.29
1.30
G.N. Savin: Stress Concentration Around Holes (Pergamon, Elmsford, 1961)
A.S. Argon: Constitutive Equations in Plasticity (MIT
Press, Cambridge, 1975)
A.L. Anand: Constitutive equations for rate independent, isotropicelastic-plastic solid exhibitive
pressure sensitive yielding and plastic dilatancy, J.
Appl. Mech. 47, 439–441 (1980)
15
Part A 1
1.26
References