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and Mechanics of Solids
Existence, Uniqueness and Stability in Linear Cosserat Elasticity for Weakest
Curvature Conditions
Jena Jeong and Patrizio Neff
Mathematics and Mechanics of Solids 2010 15: 78 originally published online 17 September 2008
DOI: 10.1177/1081286508093581
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Existence, Uniqueness and Stability in Linear Cosserat Elasticity
for Weakest Curvature Conditions
J ENA J EONG
Ecole Spéciale des Travaux Publics du Bâtiment et de l’Industrie (ESTP), 28 avenue du
Président Wilson, 94234 Cachan Cedex, France
PATRIZIO N EFF
AG6, Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7,
64289 Darmstadt, Germany
(Received 13 March 20081 accepted 23 April 2008)
Abstract: We investigate the weakest possible constitutive assumptions on the curvature energy in linear
Cosserat models still providing for existence, uniqueness and stability. The assumed curvature energy is
1L 2c 1dev sym 2axl A12 where axl A is the axial vector of the skewsymmetric microrotation A 3 12233
and dev is the orthogonal projection on the Lie-algebra 13233 of trace free matrices. The proposed Cosserat
parameter values coincide with values adopted in the experimental literature by R. S. Lakes. It is observed
that unphysical stiffening for small samples is avoided in torsion and bending while size effects are still
present. The number of Cosserat parameters is reduced from six to four. One Cosserat coupling parameter
1c 4 0 and only one length scale parameter L c 4 0. Use is made of a new coercive inequality for conformal
Killing vectorfields. An interesting point is that no (controversial) essential boundary conditions on the
microrotations need to be specified1 thus avoiding boundary layer effects. Since the curvature energy is the
weakest possible consistent with non-negativity of the energy, it seems that the Cosserat couple modulus 1c 4
0 remains a material parameter independent of the sample size which is impossible for stronger curvature
expressions.
Key Words: Polar-materials, microstructure, parameter-identification, structured continua, solid mechanics, variational
methods
1. INTRODUCTION
We establish well-posedness of the linear elastic Cosserat model in parameter ranges hitherto
not considered, extending and precising previous work of the second author [1].
General continuum models involving independent rotations have been introduced by
the Cosserat brothers [2] at the beginning of the last century. Their originally nonlinear, geometrically exact development has been largely forgotten for decades only to be rediscovered
in a restricted linearized setting in the early 1960s [3–10]. Since then, the original Cosserat
concept has been generalized in various directions, notably by Eringen and his co-workers
Mathematics and Mechanics of Solids 15: 78–95, 2010
1
42010
SAGE Publications
Los Angeles, London, New Delhi and Singapore
DOI: 10.1177/1081286508093581
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EXISTENCE, UNIQUENESS AND STABILITY 79
who extended the Cosserat concept to include also microinertia effects and to rename it subsequently into micropolar theory. For an overview of these so-called microcontinuum
theories we refer to [11–16].
The Cosserat model includes in a natural way size effects, i.e. small samples behave
comparatively stiffer than large samples. These effects have recently received new attention
in conjunction with nano-devices. The mathematical analysis establishing well-posedness
for the infinitesimal strain, Cosserat elastic solid is presented in [17–21] and in [22–24] for
so-called linear microstretch models. This analysis has always been based on the uniform
positivity of the free energy of the linear elastic Cosserat solid.
The second author has extended the existence results for both the Cosserat model and
the more general micromorphic models to the geometrically exact, finite-strain case, see e.g.
[25–27].
The important problem of the determination of Cosserat material parameters for continuous solids with random microstructure is still an open problem. Usually, a series of
experiments with specimens of different slenderness is performed in order to determine the
Cosserat parameters [28, 29]. By using the traditional curvature energy complying with
pointwise definiteness, one observes, however, an unphysical unbounded stiffening behavior for slender specimens which seems to make it impossible to arrive at consistent values for
the Cosserat parameters: the value for the parameters will depend strongly on the smallest
investigated specimen size. This inconsistency may be in part responsible for the facts that
(1) (linear) Cosserat parameters for continuous solids have never gained general acceptance
even in the “Cosserat community” and (2) that the linear elastic Cosserat model has never
been really accepted by a majority of applied scientists as a useful model to describe size
effects in continuous solids.
As a possible way out of this problem we propose to use a weaker curvature energy of
the type
Wcurv 22axl A3 5 1 L 2c 1dev sym 2axl A12
(1.1)
which is not pointwise positive. But now it is not at all clear that a suitable mathematical
model can be built with this energy. This is the question we address here.
This contribution is then organized as follows: first, we recall the linear elastic static
isotropic Cosserat model in variational form and re-derive the necessary conditions for nonnegativity of the energy. Then we focus on weaker conditions than the uniform positivity
of the strain and curvature energy which still lead to a well-posed boundary value problem.
In the proof, which is based on the direct methods of the calculus of variations, we also
re-derive a new coercive inequality for vector fields in 13 . The decisive new observation is
that pointwise positivity is not really necessary for existence and stability. Rather, only after
integration over the domain 5 does uniform convexity hold true. This mirrors the case of
linear elasticity, which itself is not pointwise positive in the displacement gradient but gets
so only after integration and use of Korn’s inequality. We believe that the usually assumed
pointwise positivity of the Cosserat curvature energy is responsible for the fact that material
parameters for the Cosserat solid have not been successfully determined. Thus, relaxing the
curvature energy might allow for a new chance of parameter determination, notably of the
Cosserat couple modulus 1c .
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J. JEONG and P. NEFF
1.1. Notation
We use the following notation. Let 5 6 13 be a bounded domain with Lipschitz boundary
65 and let 7 be a smooth subset of 65 with non-vanishing two-dimensional Hausdorff
measure. For a8 b 3 13 we let 7a8 b813 denote the scalar product on 13 with associated
vector norm 1a1213 5 7a8 a813 . We denote by 2393 the set of real 3 9 3 second-order
tensors, written with1capital
letters. The standard Euclidean scalar product on 2393 is given
2
by 7X8 Y 82393 5 tr X Y T , and thus the Frobenius tensor norm is 1X 12 5 7X8 X 82393 . In
the following we omit the index 13 8 2393 . The identity tensor on 2393 will be denoted by
11, so that tr [X ] 5 7X8 118. We set sym 2X 3 5 12 2X T X 3 and skew 2X 3 5 12 2X X T 3
such that X 5 sym 2X3 skew 2X3. For X 3 2393 we set for the deviatoric part dev X 5
X 13 tr [X] 11 3 13233 where 13233 is the Lie-algebra of traceless matrices. The Lie-algebra
of SO233 :5 X 3 GL233 X T X 5 118 det X 5 1 is given by the set 12233 :5 X 3
2393 X T 5 X of all skew symmetric tensors. The canonical identification of 12233 and
13 is denoted by axl A 3 13 for A 3 12233.
2. THE LINEAR ELASTIC ISOTROPIC COSSERAT MODEL REVISITED
This section does not contain any new results, rather it serves to accommodate the widespread
notation used in Cosserat elasticity and to introduce the problem.
2.1. The Linear Elastic Cosserat Model in Variational Form
For the displacement u : 5 6 13 13 and the skew-symmetric infinitesimal microrotation A : 5 6 13 12233 we consider the two-field minimization problem
3
I 2u8 A3 5
3
7S
5
Wmp 293 Wcurv 22axl A3 7 f8 u8 7M8 A8 dx
7 f S 8 u8 7M S 8 A8 dS min w.r.t. 2u8 A38
(2.1)
under the following constitutive requirements and boundary conditions:1
9 5 2u A8
u 7 5 u d 8
Wmp 293 5 1 1sym 912 1c 1skew 912 22
1
tr sym 9
2
5 1 1sym 2u12 1c 1skew 22u A312 22
1
tr sym 2u
2
5 1 1dev sym 2u12 1c 1skew 22u A312 5 1 1sym 2u12 strain energy
22
21 3 1
tr sym 2u
6
1c
1curl u 2axl A1213 2Div u32 8
2
2
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EXISTENCE, UNIQUENESS AND STABILITY 81
:5
axl A 3 13 8
4 5 28
1curl 1213
5
41axl skew 21213 5 21skew 2122393 8
Wcurv 223
5
1sym 212 1skew 212 tr [2]2
2
2
2
curvature energy
5
3 2 3
1dev sym 212 1skew 212 tr [2]2
2
2
6
5
1212 728 2 T 8 tr [2]2
2
2
2
5
1sym 212 1curl 1213 2Div 32 2
4
2
(2.3)
Here, f8 M are volume force and volume couples, respectively1 f s 8 M S are surface tractions
and surface couples at 7 S 6 65, respectively, while u d are Dirichlet boundary conditions
for displacement at 7 6 65.2 The strain energy Wmp and the curvature energy Wcurv are
the most general isotropic, centro-symmetric quadratic forms in the non-symmetric strain
tensor 9 5 2u A and the micropolar curvature tensor 4 5 2axl A (curvature-twist
tensor). The parameters 18 [MPa] are the classical Lamé moduli and 8 8 are additional
micropolar moduli with dimension [Pa m2 ] 5 [N] of a force. It is usually clearer to write
8 8 1 L 2c 8 1 L 2c 8 1 L 2c with corresponding non-dimensional parameters 8 8 and a material length scale L c 4 0 [m].
The additional parameter 1c 0[MPa] in the strain energy is the Cosserat couple
modulus. For 1c 5 0 the two fields of displacement and microrotations decouple and one is
left formally with classical linear elasticity for the displacement u.
2.2. The Linear Elastic Cosserat Balance Equations: Hyperelasticity
Taking free variations of the energy in (2.1) w.r.t. both displacement u 3 13 and infinitesimal
microrotation A 3 12233, one arrives at the equilibrium system (the Euler–Lagrange equations of (2.1))
Div 5 f8
Div m 5 41c axl skew 9 axl skew 2M38
9 5 2u A8
5 21 sym 9 21c skew 9 tr [9] 11
5 21 1c 3 9 21 1c 3 9 T tr [9] 118
m 5 2 2 T tr [2] 118
u 7 5 u d 8
n 7 S 5 f S 8
n 6527S 73 5 08
m
n 7S 5
5 axl A8
1
axl 2skew 2M S 338
2
m
n 6527S 73 5 0
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(2.4)
82
J. JEONG and P. NEFF
Here, m is the couple stress tensor. For comparison, in [12, p. 111] or [29–31] the elastic
moduli in our notation are defined to be 1 5 1 2 8 1c 5 2 .3 But in this last definition (see
[32]), 1 cannot be regarded as one of the classical Lamé constants.485 We note that under
the usual positivity requirements on the curvature energy, the couple stress/ curvature relation
can be pointwise inverted. The case which we have in mind is characterized by 5 4 0
and 5 23 . Thus, the constant cannot be considered to to be a “spring constant”-like
quantity!
3. CONSTITUTIVE RESTRICTIONS FOR COSSERAT HYPERELASTICITY
3.1. Pointwise Positivity of the Micropolar Energy
For a mathematical treatment in the hyperelastic case it is often assumed that for arbitrary
nonzero strain and curvature 98 4 3 2393 one has the local positivity condition
98 4 5 0 :
Wmp 293 4 08
Wcurv 243 4 0
(3.1)
This condition is most often invoked as the basis of uniqueness proofs in static micropolar
elasticity, see e.g. [12, 17, 19, 33]. By splitting 9 in its deviatoric and volumetric part, i.e.
writing
1
9 5 dev sym 9 skew 9 tr [9] 11
3
(3.2)
and inserting this into the energy Wmp one gets
Wmp 293 5 1 1dev sym 912 1c 1skew 912 21 3
tr [9]2 6
(3.3)
Since all three contributions in (3.2) can be chosen independent of each other, one obtains
from (3.1) the pointwise positive-definiteness condition
1 4 08
21 3 4 08
4 08
1c 4 08
2 3 3 4 08
4 08
2 4 038
(3.4)
where the argument pertaining to the curvature energy Wcurv is exactly similar, see [23, 2.9].
In effect, one ensures uniform convexity of the integrand w.r.t 98 4. In this case, then, the
stress/strain and couple-stress/curvature relation can be inverted, simplifying the mathematical treatment considerably.
By a thermodynamical stability argument [12] one may similarly infer the nonnegativity of the energy (material stability), leading only to
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EXISTENCE, UNIQUENESS AND STABILITY 83
1 08
21 3 08
08
1c 08
2 3 3 08
08
2 038
(3.5)
which allows for classical linear elasticity but which condition alone is not strong enough to
guarantee existence and uniqueness of the corresponding boundary value problem. Nevertheless, all constitutive restrictions on a linear Cosserat solid must at least be consistent with
(3.5) from a purely physical point of view.
3.2. Legendre–Hadamard Ellipticity Conditions for the Cosserat Model
For a dynamic problem, another condition, implying real wave speeds in wave propagation
problems, is useful. This is the Legendre–Hadamard ellipticity condition. Let us investigate
the restrictions which it imposes on the constitutive parameters of the Cosserat model. In
the following we treat the generic case of a quadratic form which can then be applied to the
balance of linear and angular momentum system. Our quadratic form is
W 223 :5 a1 1sym 212 a2 1skew 212 a3 tr [2]2 (3.6)
Replacing 2 by the rank one dyadic product we obtain
a1 1sym 12 a2 1skew 12 a3 tr [ ]2
5
5
5
a1
a2
1 12 1 12 a3 7 8 82
4
4
5
a1 4
2 1 12 2 7 8 8
4
5
a2 4
2 1 12 2 7 8 8 a3 7 8 82
4
5 a2 4
5
a1 4
2 1 12 112 2 7 8 82 2 1 12 112 2 7 8 82 a3 7 8 82
4
4
5
a1 a2
a1 a2 2 a3
1 12 112 7 8 82
2
2
5
a1 a2
a1 a2 2 a3
1 12 112 1 12 112 cos2 2
2
5
5
5
a1 a2
a1 a2 2 a3
1 12 112 2sin2 cos2 3 1 12 112 cos2 2
2
6
7
a1 a2 a1 a2 2 a3
a1 a2
2
2
2
1 1 11 sin 1 12 112 cos2 2
2
2
a1 a2
a1 2 a3
1 12 112 sin2 1 12 112 cos2 2
2
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(3.7)
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J. JEONG and P. NEFF
Thus
D 2 W 223
2 8 3 5 2a1 a2 3 1 12 112 sin2 2a1 2 a3 3 1 12 112 cos2 8 (3.8)
and Legendre–Hadamard ellipticity demands that the acoustic tensor Q2 3 : 13 13
defined through D 2 W 223
2 8 3 5 78 Q2 3
8 is strictly positive definite for any
nonzero wave direction 3 13 . We infer the necessary and sufficient conditions for strict
Legendre–Hadamard ellipticity of the quadratic form (3.6)
a1 a2 4 08
a1 2 a3 4 0
(3.9)
Applying this result to both the strain energy and curvature energy in (2.3) we obtain the
Legendre–Hadamard ellipticity condition for linear, isotropic Cosserat solids
1 1c 4 08
4 08
1 4 08
4 0
(3.10)
In the case of 1c 5 0 we recover the well known ellipticity condition for linear elasticity. It
is clear that (3.4) is sufficient for (3.10). But (3.10) alone is not sufficient for well-posedness
of the Cosserat boundary value problem.
3.3. Coercivity of the Micropolar Energy
What one really needs for a mathematical treatment of the boundary value problem in the
variational context, is, however, a coercivity condition, in the sense that a bounded energy I
implies a bound on the displacement u and the infinitesimal microrotation A in appropriate
Sobolev spaces. More precisely, for H 1 -coercivity it must hold that
I 2u8 A3 K 1 u 3 H 182 25 8 13 38 A 3 H 182 25 8 122333
(3.11)
In this contribution we consider a limit case of non-negativity
1 4 08
21 3 4 08
4 08
1c 4 08
2 3 3 5 08
5 08
2 4 038
(3.12)
in which the curvature energy is not pointwise positive definite, but only non-negative.
In terms of the non-dimensional polar ratio defined as 5 one has
:5
32 3
32 3
5
5
8
32 3
3 2 3 22 3
which leads with (3.12) to the restriction 5 32 .6 Our energy can be written as
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(3.13)
EXISTENCE, UNIQUENESS AND STABILITY 85
Wmp 22u8 A3 5 1 1sym 2u12 1c 1skew 22u A312 22
1
tr sym 2u 8
2
Wcurv 22axl A3 5 1 L 2c 1dev sym 2axl A12 (3.14)
A first observation is in order. Considering ever smaller samples means, by a simple scaling
argument, that L c . In this case, since the curvature energy must remain bounded, we
must have dev sym 2axl A 5 0. This does not imply, however, that A is constant7 or affine
linear. It is exactly this indeterminacy which is necessary for our purpose. For this curvature
energy, the couple stress/curvature relation in (2.4) cannot be inverted.
Here, dev sym X 5 0 implies that X 5 p 11 A with p 3 1 and A 3 12233. The kernel
of the linear operator dev sym : 2393 2393 is thus the Lie-algebra of the conformal
group which comprises all invertible deformations : 13 13 such that 2 3 1
SO233.
3.4. The Conformal Indeterminate Couple Stress Model
This model is formally obtained by setting 1c 5 , which enforces the constraint curl u 5
2axl A [5, 8, 34]. For the displacement u : 5 6 13 13 we consider therefore the
one-field minimization problem
3
1
Wmp 22u3 Wcurv 22curl u3 7 f8 u8 7axl 2M38 curl u8 dV
2
5
3
1
7 f S 8 u8 7axl 2M S 38 curl u8 dS min w.r.t. u8
(3.15)
2
7S
I 2u3 5
under the constitutive requirements and boundary conditions
Wmp 293 5 1 1sym 2u12 22
1
tr sym 2u 8
2
u 7 5 u d 8
1
1 L 2c
Wcurv 22curl u3 5 1 L 2c 1dev sym 2 curl u12 5
1sym 2curl u12 2
4
(3.16)
In this limit model, the curvature parameter , related to the spherical part of the couple stress
tensor m remains indeterminate, since tr [2] 5 Div 12 curl u axl A 5 Div 12 curl u 5 0. We
remark the intricate relation between 1c 8 and the indeterminacy of . Since our
curvature energy is much weaker than the one usually considered, the criticism which has
been raised against this limit model might not apply here. This will be subject of future
research.
4. A NEW COERCIVE INEQUALITY
In order to show existence, uniqueness and stability we make use of a recent observation in
[35]. Let 5 6 13 be a three-dimensional domain with Lipschitz boundary. The following
theorems hold.
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86
J. JEONG and P. NEFF
Theorem 4.1. (Weak coercivity for n 5 3)
3
C D 4 0 3 H 1 25 8 13 3 :
5
1dev sym 212 112 dx C D 112H 1 25813 3 (4.1)
The proof of (4.1) is given in [35]. In fact, however, the inequality follows already from
a general result of Necas on the weak coercivity of formally positive quadratic forms. For
the convenience of the reader we include here an independent proof, based on the following
statement of Necas.
Theorem 4.2. Let N : 2393 2393 be a constant coeff icient linear operator and let
5 6 13 be a C 181 -domain. The formally positive quadratic form 1N
2u122393 is weakly
coercive, i.e.
3
C 4 0 3 H 1 25 8 13 3 :
1N
2122393 11213 dx 112H 1 25 813 3 8
(4.2)
5
if for all 3 33 8 5 0 the system N
2 u3 5 0 implies u 5 0.
Proof. This statement is a simple consequence of Theorem 3.2 in [36].
1
Let us now use this result and re-prove (4.1).
Proof. (Weak coercivity.) The quadratic form 1dev sym 212 is formally positive with
constant coefficients. Using Theorem 4.2 we check first the necessary (but not sufficient)
Legendre–Hadamard ellipticity condition (equivalent to allow only for 8 u 3 13 ) to the
extent that for all 8 u 3 13 with u 5 0
dev sym u 5 0 5 0
(4.3)
To see this, it is enough to compute
1dev sym u12 5 1sym u12 22
1 1
tr u
3
1
1
1
1 12 1u12 1 12 1u12 5 1 12 1u12 2
3
6
(4.4)
Thus 5 0 if dev sym u 5 0 and u 5 0. This suffices to obtain (4.1) in the case of the
space H01 253, i.e. zero boundary values throughout, which is also included in [36]. In order
to strengthen the result we need to show the same for complex-valued vectors. Thus we have
to show that for all complex-valued vectors 8 u 3 33 with 5 0
dev sym u 5 0 u 5 0
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EXISTENCE, UNIQUENESS AND STABILITY 87
The corresponding complex linear problem in matrix format reads
8
2 1
2 3
9
9
1
0 9 2
9
9
9 1 2 2 3 9
9
9 3
0
1 9
9
9 1 2 2 3 0
3
8 0
9 9 8 90
9 u1
9 9 90
9u 2 5 9 9 90
9 u3
9 90
2
(4.6)
0
Our remaining task is to evaluate whether the 693 matrix has full rank, for a nonzero 3 33 .
Elementary operations lead to the equivalent question whether the following 593 matrix
has rank three for a nonzero 3 33
8
2
1
0
9
9 0 2
9 1
9
9
0
1 9 3
9
9
9 0
3 2 0
2 3
(4.7)
We proceed by case distinction, as follows.
Let first 3 5 0. If then 22 23 5 0 (the determinant of rows 38 48 5), we are done. If
2
2 23 5 0, consider the determinant of rows 28 38 4, which is 3 2 21 22 3. If 21 22 5 0
we are done. Assume thus that 21 22 5 0 and consider finally the determinant of rows
18 38 4 which is 2 1 2 3 5 0 since we are in the case 21 22 5 0 and 22 23 5 0 and
3 5 0. Thus, for 3 5 0 the rank is three.
Assume now that 2 5 0. If the determinant of rows 18 48 5 : 2 2 23 22 3 5 0 we are
done. If 2 23 22 3 5 0 consider the determinant of rows 18 28 4. If 2 2 22 21 3 5 0 we are
done. If not, then 22 21 5 0 and consider the determinant of rows 18 38 4 : 2 1 2 3 .
This determinant is now non-vanishing since we are in the case that both 2 23 22 3 5 0 and
22 21 5 0 and 2 5 0. Thus, for 2 5 0 the rank is three.
Assume finally that 1 5 0. If the determinant of rows 18 28 3 : 1 2 22 21 3 5 0 we are
done. If 2 21 22 3 5 0 consider the determinant of rows 28 48 5 : 1 2 22 23 3. If 22 23 5 0
we are done. If 22 23 5 0 consider the determinant of rows 18 38 4 : 2 1 2 3 which
must be nonzero now. Thus, for 1 5 0 the rank is also three. This finishes the proof.
1
Remark 4.3. The statement is not true in case of space-dimension n 5 2 together with
the definition of a two-dimensional deviator dev 2 X :5 X 12 tr [X ] 112 . In this case, the
quadratic form is not algebraically closed. Here, the corresponding system is
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88
J. JEONG and P. NEFF
1 2
2
Det
u1
1
u2
1 2
2
1
5
0
0
8
5 21 22 5 0 for 5 218 i3T 8
(4.8)
showing that the operator is not algebraically complete.
4.1. Existence
Without loss of generality, we consider the case without body and surface couples but with
given body forces f 3 L 2 25 8 13 3. Moreover, the infinitesimal microrotations A at the
boundary are left free (no essential boundary conditions for the microrotation field). We
fix the body at a part 7 6 65 which translates into the essential boundary conditions for
the displacement u 7 5 0. In addition, we assume 1 4 0 and 0 (this condition could
be weakened as in linear elasticity: only the bulk modulus K 5 21
3
needs to be strictly
3
positive). The task is to show that
3
I 2u8 A3
:5
5
1 1sym 2u12 1c 1skew 22u A312 22
1
tr sym 2u
2
1 L 2c 1dev sym 2axl A12 7 f8 u8 dx
(4.9)
admits minimizers in the set of admissible functions 1 with bounded energy I :
1 :5 u 3 H 1 25 8 13 38
A 3 H 1 25 8 1223338
I 2u8 A3 8
u 7 5 0 (4.10)
Since I is quadratic, it is clear that I is convex. Using the classical Korn’s first inequality of
linear elasticity (usually, for stronger curvature expressions, Korn’s inequality is not needed
for existence in linear Cosserat models!) we obtain the estimate
I 2u8 A3 1 C K
1u12H 1 253 1 f 1 L 2 253 1u1 L 2 253
3
1c 1skew 22u A312 1 L 2c 1dev sym 2axl A12 dx
5
1 C K
1u12H 1 253 1 f 1 L 2 253 1u1 H 1 253
3
1c 1skew 22u A312 1 L 2c 1dev sym 2axl A12 dx (4.11)
5
The first line of the last estimate gives us a quadratic inequality for 1u1 H 1 253 . Since I is
bounded on 1 we infer that u 3 1 is bounded in H 1 25 8 13 3. Since 1u1 H 1 253 is bounded and
the remaining terms in I are positive we infer that I is bounded below. Thus the value
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EXISTENCE, UNIQUENESS AND STABILITY 89
inf I 2u8 A3 5 m
1
(4.12)
exists. We choose minimizing sequences, i.e. sequences 2u k 8 Ak 3 3 1 with the property that
the lowest energy level is approximated:
lim I 2u k 8 Ak 3 5 m
k
(4.13)
The previous estimate shows that u k is bounded in H 1 25 8 13 3. Since 1c 4 0 we infer that
Ak is also bounded in L 2 253. Thus, for the microrotations we have that
3
1c 1Ak 12 1 L 2c 1dev sym 2axl Ak 12 dx
(4.14)
5
is bounded. Using the new coercivity estimate (4.1) shows that 1Ak 1 H 1 25 8122333 is bounded as
well. Extracting weakly convergent subsequences of displacements u k j u 3 H 1 25 8 13 3,
strongly converging in L 2 253 by Rellichs compact embedding, and similarly for microrotations Ak j A 3 H 1 25 8 122333 together with weak lower semicontinuity of the energy
I 2u8 A3 lim inf I 2u k j 8 Ak j 3 5 m
kj
(4.15)
implies that I 2u8 A3 m, which is only possible if I 2u8 A3 5 m. Thus the pair 2u8 A3 is a
minimizer.
4.2. Uniqueness
For uniqueness we have to show strict convexity of the energy. The energy is two-times
differentiable. Thus we may consider the second derivative of I , which is given by
3
D I 2u8 A3
2u8 A3
2
2
5
5
211sym 2u12 21c 1skew 22u A312
1
22
tr sym 2u 21 L 2c 1dev sym 2axl A12 dx
3
211sym 2u12 21c 1skew 22u A312
5
21 L 2c 1dev sym 2axl A12 dx
(4.16)
Obviously, D 2 I 2u8 A3
2u8 A32 5 0 implies that 2u8 A3 5 0 by using Korn’s first inequality on u and 1c 4 0. This shows that the second derivative is strictly positive, implying
strict convexity of I and uniqueness of the minimizer.
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90
J. JEONG and P. NEFF
4.3. Stability for all 1c
40
An interesting observation is that strict convexity of our energy does not immediately lead
to uniform convexity or stability. For this the second derivative must satisfy the uniform
inequality
4
5
D 2 I 2u8 A3
2u8 A32 C 1u12H 1 253 1 A12H 1 253
(4.17)
with a constant C 4 0 independent of 2u8 A3.
In order to see under what conditions we might be able to get such an inequality, we turn
back to the second derivative
D 2 I 2u8 A3
2u8 A32
3
1
22
5
211sym 2u12 21c 1skew 22u A312 tr sym 2u
5
21 L 2c 1dev sym 2axl A12 dx
3
211sym 2u12 21c 1skew 22u A312
5
21 L 2c 1dev sym 2axl A12 dx
3
21C K
12u12 21c 1skew 22u A312
5
21 L 2c 1dev sym 2axl A12 dx
3
4
5
21C K
12u12 21c 1skew 22u312 27skew 2u8 A8 1 A12
5
21 L 2c 1dev sym 2axl A12 dx
3
4
5
4
21C K
1sym 2u12 1skew 2u12 21c 1skew 22u312
5
5
27skew 2u8 A8 1 A12 21 L 2c 1dev sym 2axl A12 dx
3
21C K
1sym 2u12 221C K
21c 3 1skew 2u12 21c 27skew 2u8 A8
5
21c 1 A12 21 L 2c 1dev sym 2axl A12 dx
3
21C K
1sym 2u12 221C K
21c 3 1skew 2u12 21c 21skew 2u1 1 A1
5
21c 1 A12 21 L 2c 1dev sym 2axl A12 dx
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EXISTENCE, UNIQUENESS AND STABILITY 91
3
5
21C K
1sym 2u12 221C K
21c 3 1skew 2u12
6
7
1
1skew 2u12 91 A12 21c 1 A12 21 L 2c 1dev sym 2axl A12 dx
9
6
7
3
1
2
21C K 1sym 2u1 21C K 21c 21c
1skew 2u12
9
5
21c
221c 21c 93 1 A12 21 L 2c 1dev sym 2axl A12 dx
6
7
3
1
2
21C K 1sym 2u1 2 1C K 1c 1c
1skew 2u12 21c 21 93 1 A12
9
5
21 L 2c 1dev sym 2axl A12 dx
(4.18)
We need to choose 9 4 0 such that
6
1 9 4 0 and
1C K
1
1c 1c
9
7
0
(4.19)
This is satisfied whenever
149
1
1
1 C
1c K
(4.20)
Since the constant in Korn’s inequality C K
is strictly positive, this can always be achieved.
With such an 9 4 0 we obtain therefore the estimate
3
2
2
D I 2u8 A3
2u8 A3 21C K
1sym 2u12 21c 21 93 1 A12
5
21 L 2c 1dev sym 2axl A12 dx
212C K
32 1u12H 1 253 C D
1 A12H 1 253
4
5
min2212C K
32 8 C D
3 1u12H 1 253 1 A12H 1 253 (4.21)
4.4. Relations for Micropolar Constants
In the literature on Cosserat or micropolar solids the following abbreviations and definitions
are frequently encountered. As a convenience for the reader, we collect these technical constants here.
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92
J. JEONG and P. NEFF
3
8
non-dimensional polar ratio8 0 8
2
6
7
1
5
8
“characteristic length for torsion”8
21 2 1
:5
2t
:5
2b
5
5
8
2 221 3
41
p2
:5
2
8
N2
:5
1c
5
8
1 1c
2 21 3
5
21
“characteristic length for bending”8
:5 2 1c 8
Cosserat coupling number,
5
8
2 2 21 3
0 N 1
classical Poisson ratio
(4.22)
For every physical material, it is essential that small samples still have bounded rigidity. This
may or may not be true for Cosserat models, depending on the values of Cosserat parameters.
Based on analytic solution formulas for simple three-dimensional Cosserat boundary value
problems it has been shown in [1] that for bounded stiffness for arbitrary slender samples we
must have
1. torsion of a cylinder: 5 0 or 5 5 32 .
2. bending of a cylinder: 2 3 2 3 5 0.
Our curvature energy satisfies both requirements through 5 and 5 32 .9
Foams and bones have been identified by Lakes as prototype Cosserat solids. In order
to identify the material parameters, however, Lakes had to leave the traditionally admitted
parameter range motivated by strict pointwise positivity. In [29, 37] the value 5 32 has
been chosen in order to accommodate bounded stiffness with experimental findings. For a
syntactic foam [37] 5 has been taken for a best fit. In this case, the curvature energy
looks like Wcurv 223 5 1dev sym 212 with 4 0. With our new result it is shown that
this is a well-posed fit. For a polyurethane foam [37] 5 and the curvature energy looks
like Wcurv 223 5 1dev sym 212 1curl 12 . Our result can be easily adapted to
2
4
this case as well. The problem is well-posed.
4.5. Stress Concentration along a Cylindrical Hole
In [12, p. 222] or [33, p. 238] the analytical solution for the stress distribution around a
cylindrical hole with radius r 4 0 of an infinite plate is recalled. The stress concentration
factor K t , which classically is K t 5 3 turns for the linear Cosserat model into
Kt
5
3 F1
38 F1 5 8 21 3 N 2
1 F1
4
1
r2
c2
K 2r 3
2 rc K 01 2 rc 3
c
8 c2 :5
21 3
2b
8
5
221 3
N2
K 0 2 38 K 1 2 3 modified Bessel functions of the second kind
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(4.23)
EXISTENCE, UNIQUENESS AND STABILITY 93
In the genuine micropolar case with pointwise positive curvature, the stress intensities are
somewhat weakened: the Cosserat solid has the ability to distribute the stresses more
smoothly which is one of the salient features of the Cosserat model. For our relaxed curvature
energy, the same stress intensity formula still applies.10 This shows that the new conformal
linear Cosserat model still shares this classical feature of the linear Cosserat model.
5. CONCLUSION
That linear elastic Cosserat models may show singular stiffening behaviour has already been
observed previously. In [18, p. 17] we read “For some combinations of elastic constants,
the apparent modulus tends to infinity as the bar or plate size goes to zero. Large stiffening
effects might be seen in composite materials consisting of very stiff fibers or laminae in
a compliant matrix. However, infinite stiffening effects are unphysical. For very slender
specimens, it is likely that a continuum theory more general than Cosserat elasticity1 or use
of a discrete structural model, would be required to deal with the observed phenomena”.
But Lakes himself still used the linear Cosserat model, albeit within uncommon parameter ranges. Adopting the same values as Lakes did, pointwise strict positivity of the energy
is not true and has been criticized in [12]. However, the limit case 5 4 08 5 22 ,
does still allow for existence and uniqueness and stability. Usually, it is also a problem to determine some sound boundary conditions for the microrotations A. Even within the weaker
parameter range we need not, however, specify Dirichlet boundary conditions for the microrotations! Thus boundary layer effects can be completely avoided. In a sequel paper, we will
consider the FEM-implementation of the linear Cosserat model with the weakened curvature
energy. It is hoped that within the new parameter range, the Cosserat couple modulus 1c
can be identified with a material parameter, independent of the size of the sample - which
it cannot be in case of pointwise strict positivity of the curvature energy, as shown in [1].
We think that our relaxed curvature expression offers a fresh departure for the experimental
determination of the linear isotropic Cosserat constants.
NOTES
1. More detailed than strictly necessary in order to accommodate the different representations in the literature. Note that axl A 9 5 A
for all 3 13 , such that
8
0
9
axl 9
0
8 9 9 :5 8
0
Ai j 5
3
i j k 2axl A3k 8
(2.2)
k51
where i j k is the totally antisymmetric permutation tensor. Here, A
denotes the application of the
matrix A to the vector and a 9 b is the usual cross-product. Note that it is always possible to prescribe
essential boundary values for A but we abstain from such a prescription throughout.
2. For simplicity only we assume that 7 ! 7 S 5 " and that surface tractions and surface couples are
prescribed at the same portion of the boundary. Much more general combinations could be considered.
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94
J. JEONG and P. NEFF
3. In [12, 23] the Cauchy stress tensor is defined as 5 21 3 9 1 9 T tr [9] 11 with given
constants 1 8 8 and one must identify 1 5 1 1c 8 1 5 1 1c .
4. A simple definition of the Lamé constants in micropolar elasticity is that they should coincide with
the classical Lamé constants for symmetric situations. Equivalently, they are obtained by the classical
E
E
formula 1 5 221
3
8 5 21
32123
, where E and are uniquely determined from uniform traction
where Cosserat effects are absent.
5. Unfortunately, while authors are consistent in their usage of material parameters, one should be careful
when identifying the actually used parameters with his own usage. The different representations in (2.3)
might be useful for this purpose.
6. If we require only 23
1032 then the polar ratio may become negative.
7. A would be constant if the curvature energy was pointwise positive definite.
8. This formal limit 1c excludes that axl A is an independent field.
9. The additional conditions in [1] for bounded stiffness have been based on dimensionally reduced models
and must therefore be taken with care.
10. While the boundary conditions in this problem differ from those used in the existence part of this paper,
the analysis can be extended easily.
Acknowledgments. J. Jeong is grateful for the continued support of H. Adib-Ramezani, Ecole Polytech’Orléans. P.
Neff acknowledges helpful discussions with R. Lakes.
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