Z. angew. Math. Phys. 60 (2009) 479–497
0044-2275/09/030479-19
DOI 10.1007/s00033-008-8090-2
c 2009 Birkhäuser Verlag, Basel
°
Zeitschrift für angewandte
Mathematik und Physik ZAMP
Mean field modeling of isotropic random Cauchy elasticity
versus microstretch elasticity
Patrizio Neff∗ , Jena Jeong, Ingo Münch and Hamid Ramezani
Abstract. We show that the averaged response of random isotropic Cauchy elastic material
can be described analytically. It leads to a higher gradient model with explicit expressions for
the dependence on the second derivatives of the mean field. A subsequent penalty formulation
coincides with a linear elastic micro-stretch model with specific choice of constitutive parameters,
depending only on the average cut-off length (the internal characteristic length scale Lc > 0).
Thus the microstretch displacement field can be interpreted as an approximated mean field
response for these parameter ranges. The mean field free energy in this micro-stretch formulation
is not uniformly pointwise positive, nevertheless, the model is well posed.
Mathematics Subject Classification (2000). 74A35, 74A30, 74C05, 74C10.
Keywords. Mean field, microstructure, couple-stress model, structured continua, solid mechanics, variational methods.
1. Introduction
Higher gradient or extended continuum models involve an increasing number of
constitutive parameters which cannot easily be interpreted and determined. In this
contribution we assume we are dealing with micro-heterogeneous random isotropic
Cauchy material for which we would like to determine a variational principle for
the averaged response over a given length scale Lc > 0. Averaging at scale Lc
introduces naturally a scale-dependence into the problem. It is clear that the
Lc -averaged (henceforth called mean field) displacement field u will acquire additional smoothness compared to linear elasticity. Thus we are prepared to allow
for second gradients of the mean field D2 u to appear in the variational formulation for the mean field problem. How should this additional dependence in the
energy on the second gradient look like? Infinitesimal invariance principles like
objectivity, meaning that the response is invariant under superposition of conb + bb, A
b ∈ so(3), bb ∈ R3 gives no restant infinitesimal rotations u 7→ u + A.x
striction for the second gradient. Descending from a finite strain development we
∗ Corresponding
author
480
P. Neff, J. Jeong, I. Münch and H. Ramezani
ZAMP
know [25, eq.(10.25)] that the strain energy of a non-simple material of grade two
is expressible as W = W (F T F, ∇x (F T F )) which, after linearization, reduces to
W = W (sym ∇u, ∇ sym ∇u) but gives us not much further insight.1
In our approach, we try to directly expand isotropic linear elasticity and to
read off the appearing higher gradient terms acting on the averaged response.
To do this, we need to invoke an orthogonality assumption (3.6), motivated by
assumed statistical micro-randomness. Moreover we assume that the third
gradient D3 u of the mean field is much smaller then its second gradient D2 u.
It was surprising for us to be able to determine the explicit formulas for the
higher gradient terms using established formulas for spherical averaging. Equally
surprising was the fact that the higher gradient contribution is not controlling
the full second gradient! The finally obtained second gradient model for the mean
field response will be modified by considering a better manageable second gradient
expression. For this modification then we offer a penalised model which can be
interpreted in a natural way as a microstretch model. The microstretch model
is a slight generalisation of the micropolar model in that it involves, besides the
microrotations A ∈ so(3) also a scalar microdilation p ∈ R as additional field
variable. Thus, the microstretch model is “nearly” a penalty formulation for our
mean-field response.
The mathematical analysis establishing well-posedness for the infinitesimal
strain, micro-stretch elastic model is given in [11, 9, 10]. This analysis has always been based on the uniform positivity of the free quadratic energy of the
microstretch solid. The first author has extended the existence results for the more
general micromorphic models [23, 18, 5] to the geometrically exact, finite-strain
case, see e.g. [21, 20, 22]. It is interesting in this respect to note that our penaltymicro-stretch formulation for the mean field is not uniformly pointwise positive.
Nevertheless one can show that the model is well-posed along the lines presented
in [12] but we abstain from presenting further details here.
This paper is organised as follows. First we recall the microstretch model in
variational form together with its subvariants, the Cosserat or micropolar model
and the indeterminate couple stress model. Then we consider the averaged displacement field for linear isotropic elasticity and derive a second order correction
term. Subsequently we modify slightly the second gradient contribution, taking
care not to impart additional control on derivatives which is not originally present.
Finally, we present the penalty formulation and relate it to the microstretch model.
The notation and some detailed calculations concerning spherical averages and second derivatives are found in the appendix.
1 Note that all second derivatives D 2 u are expressible as linear combinations of strain gradient
derivatives ∇ sym ∇u, since ui,jk = εik,j + εij,k − εjk,i . This identity implies noteworthy the
existence of a constant C + > 0 such that kD2 uk2 ≤ C + k∇ sym ∇uk2 ; a fact sometimes used to
prove Korn’s inequality. Moreover, we see that W = W (sym ∇u, ∇ sym ∇u) = W̃ (sym ∇u, D2 u)
places in reality no condition on the higher gradient term.
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Mean field modeling of isotropic random Cauchy elasticity
481
2. The linear isotropic microstretch model
The investigation of microstretch and micromorphic continua (which are prominent
examples of so-called extended continua) dates back to Eringen’s pioneering works
in the mid 1960. The linear isotropic microstretch model of Eringen [2, p. 254] and
[3] features one additional degree of freedom as compared to a Cosserat or micropolar model: a scalar variable p for the “pressure”, also called microdilation. In a
variational format, the model can be shortly stated as follows: find the displacement u : Ω ⊂ R3 7→ R3 , the skew-symmetric infinitesimal microrotation
A : Ω ⊂ R3 7→ so(3) and the scalar pressure p : Ω ⊂ R3 7→ R3 minimising the
three-field problem
Z
Wstretch (ε, p) + Wcurv (∇ axl A, ∇p) − hf, ui dx
I(u, A, p) =
Ω
7→
min . w.r.t. (u, A, p),
(2.1)
under the following constitutive requirements and boundary conditions
ε = ∇u − (A + p 11),
generalised stretch tensor
u|Γ = ud ,
essential displacement boundary conditions
Kc
2
Wstretch (ε, p) = µe k dev sym εk2 + µc k skew εk2 +
Tr [ε]
2
´2
p
1 ³p
+
K0 Tr [ε] + λc Tr [p 11]
2
φ := axl A ∈ R3 ,
k curl φk2R3 = 4k axl skew ∇φk2R3 = 2k skew ∇φk2M3×3 ,
γ+β
γ−β
Wcurv (∇φ, ∇p) =
k dev sym ∇φk2 +
k skew ∇φk2
(2.2)
2
2
3α + (β + γ)
a0
2
+
Tr [∇φ] + k∇pk2 .
6
2
Here, f are given volume forces while ud are Dirichlet boundary conditions2 for
the displacement at Γ ⊂ ∂Ω, where Ω ⊂ R3 is the referential domain. Surface
tractions, volume couples and surface couples can be included in the standard
way. The strain energy Wstretch and the curvature energy Wcurv are isotropic
quadratic forms in the infinitesimal non-symmetric generalised strain tensor ε = ∇u − (A + p11), the micropolar curvature tensor k = ∇ axl A = ∇φ
(curvature-twist tensor) and the microdilation gradient ∇p. The parameters
µe , µc , Kc , K0 , λc [MPa] are constitutive parameters and α, β, γ, a0 are additional
microstretch curvature moduli with dimension [Pa · m2 ] = [N] of a force. Experimentally, the determination of the curvature moduli is extremely difficult compared to the first set of parameters [14].
2 Note that it is always possible to prescribe essential boundary values for the microrotations A
and the microdilation p but we abstain from such a prescription because the physical meaning
of this is doubtful. Choosing λc = K0 we are able to generate the term K20 Tr [∇u]2 .
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P. Neff, J. Jeong, I. Münch and H. Ramezani
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2.1. Non-negativity of the micro-stretch energy
From the representation of the energy in (2.2) we can read off immediately the
necessary and sufficient conditions for the non-negativity of this free energy. We
must have
µ ≥ 0,
γ + β ≥ 0,
µ c ≥ 0 , Kc ≥ 0 , K0 ≥ 0 , λ c ≥ 0 ,
γ − β ≥ 0 , 3α + (β + γ) ≥ 0 , a0 ≥ 0 .
(2.3)
Certain of these inequalities need to be strict in order for the well-posedness of
the model. However, the uniform pointwise positivity (strict inequalities everywhere) is not necessary, although it is assumed most often in treatments of linear
microstretch elasticity.
2.2. The micro-stretch balance equations: strong form
For our choice of energy we collect the induced balance equations. Taking independent variations of the energy in (2.5) w.r.t. displacement u ∈ R3 , infinitesimal
microrotation A ∈ so(3) and microdilation p ∈ R, one arrives at the equilibrium
system (the Euler-Lagrange equations of (2.5))
Div σ = f ,
balance of linear momentum
− Div m = 4 µc · axl skew ε , balance of angular momentum
³p
´ p
p
p
a0 ∆p =
K0 Tr [ε] + λc Tr [p 11] 3 ( λc − K0 ) − 3 Kc Tr [ε]
(2.4)
balance of micro-dilational momentum
σ = 2µ · dev sym ε + 2µc · skew ε
h
³p
´i
p
p
+ Kc Tr [ε] + K0
K0 Tr [ε] + λc Tr [p 11] · 11 ,
m = γ ∇φ + β ∇φT + α Tr [∇φ] · 11
3α + (γ + β)
Tr [∇φ] 11 ,
2
= 0.
= (γ + β) dev sym ∇φ + (γ − β) skew ∇φ +
φ = axl A ,
u|Γ = ud ,
m.~n|∂Ω = 0 ∇p.~n|∂Ω
Here, m is the (second order) moment stress tensor which is given as a linear
function of the curvature ∇φ = ∇ axl A. For more on the microstretch model, its
applications and parameter determination we refer to [13, 17, 14]. The Cosserat
or micropolar model can be obtained formally by letting λc → ∞, in which case
the bulk modulus is K = Kc + K0 .
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Mean field modeling of isotropic random Cauchy elasticity
483
2.3. The linear elastic isotropic Cosserat model in variational form
For the displacement u : Ω ⊂ R3 7→ R3 and the skew-symmetric infinitesimal
microrotation A : Ω ⊂ R3 7→ so(3) we consider the two-field minimisation
problem
Z
I(u, A) = Wmp (ε) + Wcurv (∇ axl A) − hf, ui dx 7→ min . w.r.t. (u, A), (2.5)
Ω
under the following constitutive requirements and boundary conditions
ε = ∇u − A,
first Cosserat stretch tensor
K
2
Tr [∇u] ,
(2.6)
2
γ+β
γ−β
3α + (β + γ)
2
Wcurv (∇φ) =
k dev sym ∇φk2 +
k skew ∇φk2 +
Tr [∇φ] .
2
2
6
Wmp (ε) = µ k dev sym ∇uk2 + µc k skew(∇u − A)k2 +
Here, the strain energy Wmp and the curvature energy Wcurv are the most general
isotropic quadratic forms in the infinitesimal non-symmetric first Cosserat
strain tensor ε = ∇u−A and the micropolar curvature tensor k = ∇ axl A =
∇φ (curvature-twist tensor). The parameters µ, K[MPa] are the classical shear and
bulk modulus, respectively. The parameter µc ≥ 0[MPa] in the strain energy is
the Cosserat couple modulus. For µc = 0 the two fields of displacement and
microrotations decouple and one is left formally with classical linear elasticity for
the displacement u. Next, we obtain the indeterminate couple stress model by
letting formally µc → ∞.
2.4. The indeterminate couple stress problem
The indeterminate couple stress problem [19, 24, 15, 26] is characterised by the
identification 12 curl u = axl A = φ which can be formally obtained from the genuine
Cosserat model by setting µc = ∞. Since here the infinitesimal microrotations A ∈
so(3) cease to be an independent field the model has the advantage of conceptional
simplicity and improved physical transparency.
For the displacement u : Ω ⊂ R3 7→ R3 we consider therefore the one-field
second gradient minimisation problem
Z
I(u) = Wmp (∇u) + Wcurv (∇ curl u) dV 7→ min . w.r.t. u,
Ω
under the constitutive requirements and boundary conditions
λ
2
Tr [sym ∇u] , u|Γ = ud ,
2
γ+β
γ−β
Wcurv (∇ curl u) =
k dev sym ∇ curl uk2 +
k skew ∇ curl uk2
8
8
Wmp (∇u) = µ k sym ∇uk2 +
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P. Neff, J. Jeong, I. Münch and H. Ramezani
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3α + (β + γ)
2
Tr [∇ curl u]
24
γ+β
γ−β
=
k sym ∇ curl uk2 +
k skew ∇ curl uk2 .
(2.7)
8
8
In this limit model, the curvature parameter α, related to the spherical part of the
(higher order) couple stress tensor m remains indeterminate, since Tr [∇φ] =
Div axl A = Div 12 curl u = 0. Following [15], it is usually assumed that −1 < η :=
β
γ < 1 in order to guarantee uniform positive definiteness.
+
3. Formal homogenisation through averaging
Let us now turn to our homogenisation procedure. In order to approach the question of homogenisation we consider a rudimentary, formal, two-scale homogenisation method in which we use the word “homogenisation” in a loose sense. The
underlying assumption is that there exist two distinct levels in the body of interest: a discontinuous, heterogeneous microscopic one, consisting of matrix material,
voids and other inhomogeneities, and a continuous macroscopic one. The representative volume element RV E ] [4] defines the order of the scale of resolution of
the envisaged continuum model, effects below this scale do not appear explicitly
in the final model, cf. Figure 1. The classical continuum limit arises then as a
doubly asymptotic, namely
Figure 1. The basic situation of our multiscale approach. Here, the black points symbolise the
mesoscale.
• The RV E ] is big enough to be representative of the microstructure in a
statistical sense.
• The RV E ] is small enough compared to the actual sample size for it to be
considered to be infinitesimal.
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Mean field modeling of isotropic random Cauchy elasticity
485
Figure 2. Left: The RV E ] fill Ω. Each RV E ] represents a cluster of smaller RV E(0) which
themselves define the cutoff length (averaging scale). The mean field will be determined on a
coarser grid. Right: We assume that each RV E(0) consists of random Cauchy elastic material.
In our case it is not even clear whether a unique homogenised medium exists. We
assume to deal with statistically random Cauchy material in general. Nevertheless, there is a certain scale below which we are not interested in the displacement
details. Thus we cover the body of interest with RV E ] containing a representative
microstructure. The precise form of the RV E ] is irrelevant.
3.1. Formal homogenisation procedure: expansion to second order for
the mean field
In order to get rid of the fine scale details we average the inhomogeneous, random
response over a scale which is smaller then the scale induced by RV E ] . This scale
is given by a ball RV E(0) with diameter Lc . The mean field on the averaging
scale RV E(0) is defined as box-filtered quantity [8]
Z
1
u(x + ξ) dξ ,
(3.1)
u(x) :=
|B(0, Lc )| ξ∈B(0,Lc )
where B(x0 , Lc ) := {x ∈ R3 | kx−x0 k ≤ Lc }. Let us consider random linear elastic
Cauchy material inside RV E ] and consider the original linear elastic solution
(including all fine scale features)
Z
K(x)
2
Tr [∇u(x)] dx 7→ min . u
µ(x) k dev sym ∇u(x)k2 +
2
]
x∈RV E
Z
1
b,
∇x u(x) dx = B
average constraint over RV E ] or
|RV E ] | x∈RV E ]
b , B
b ∈ gl(3) , Dirichlet constraint .
(x) = B.x
(3.2)
u|
∂RV E ]
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P. Neff, J. Jeong, I. Münch and H. Ramezani
ZAMP
Figure 3. Left: Homogeneous deformation of the subgrid cluster RV E ] which is homogeneous
b
inside due to homogeneous boundary conditions y 7→ B.y.
Right: Inhomogeneous response
(micro-fluctuations) for the same homogeneous boundary conditions due to random Cauchy
material inside the RV E ] .
Here, µ(x), K(x) represent the random subgrid variation of the elastic moduli due
to the inhomogeneities on this scale. This problem is well-posed for µ(x), K(x) ≥
c0 > 0, but the solution u will in general develop irregular fluctuations with high
gradients inside RV E ] , see Fig. 3. Nevertheless, the solution u is in H 1 (RV E ] , R3 )
and the box-filtered u is an H 2 (RV E ] , R3 )-function.
Next, the distance between the mean field gradient ∇u and the total field
gradient ∇u can be expressed to highest order as
Z
1
2
k∇u(x) − ∇u(x)k = k
∇u(x + ξ) dξ − ∇u(x)k2
|B(0, Lc )| ξ∈B(0,Lc )
Z
1
1
≈k
D2 u(x).ξ + D3 u(x).(ξ, ξ) + . . . dξk2
(3.3)
|B(0, Lc )| ξ∈B(0,Lc )
2
Z
1
1 3
≈k
D u(x).(ξ, ξ) + . . . dξk2 ∼ C + kD3 u(x)k2 L4c ,
|B(0, Lc )| ξ∈B(0,Lc ) 2
since here the second derivative D2 u is linear in ξ and does therefore not contribute
to the average.
Further, we want to relate the mean field energy (left) and the averaged energy
(right) through
Z
1
k dev sym ∇u(x)k2 + “Correction” =
k dev sym ∇u(x + ξ)k2 dξ ,
|B(0, Lc )| ξ∈B(0,Lc )
(3.4)
Vol. 60 (2009)
Mean field modeling of isotropic random Cauchy elasticity
487
2
and similarly for Tr [∇u(x)] . Thus we start from the right hand side and write
Z
1
k dev sym ∇u(x + ξ)k2 dξ
|B(0, Lc )| ξ∈B(0,Lc )
Z
1
=
k dev sym[∇u(x + ξ) − ∇u(x + ξ) + ∇u(x + ξ)]k2 dξ
|B(0, Lc )| ξ∈B(0,Lc )
Z
1
k dev sym[∇u(x + ξ) − ∇u(x + ξ)]k2
=
|B(0, Lc )| ξ∈B(0,Lc )
+ 2hdev sym[∇u(x + ξ) − ∇u(x + ξ)], ∇u(x + ξ)i
+ k dev sym ∇u(x + ξ)k2 dξ .
(3.5)
The last term in the integrand will be much simplified through expansion and
analytic integration in (3.8). For the middle scalar product term (which is in
principle of order L2c ) we invoke presently an orthogonality assumption on the
difference between average and total displacement gradient. This is the assumption
that these functions are statistically uncorrelated. More precisely, we assume
the orthogonality on the Lc -averaging scale3
Z
hdev sym[∇u(x + ξ) − ∇u(x + ξ)], ∇u(x + ξ)i dξ = 0 ,
ξ∈B(0,Lc )
Z
(3.6)
Tr [∇u(x + ξ) − ∇u(x + ξ)]] · Tr [∇u(x + ξ)] dξ = 0 .
ξ∈B(0,Lc )
Similar assumptions in a one dimensional setting are not uncommon [6]. It remains
to estimate the first term in (3.5). Here, we use for the integrand the highest order expansion in (3.3) which shows that the term is of order L4c (before and after
integration).
For the mean field u itself it is appropriate to assume that
∀ ξ ∈ S2 :
kD3 u.(ξ, ξ)k2M3×3 ¿ kD2 u.ξk2M3×3 ,
(3.7)
since averaging damps oscillations. This motivates to skip third gradients as compared to second gradients in the subsequent development. Applying these preliminaries to the generic energy expression k∇u(x)k2 results in
Z
1
k∇u(x + ξ)k2 dξ
|B(0, Lc )| ξ∈B(0,Lc )
Z
1
≈
k∇u(x) + D2 u(x).ξ + . . . k2 dξ
|B(0, Lc )| ξ∈B(0,Lc )
3
It would suffice to have an estimate ≤ C + L3c for these two terms. The condition is, effectively,
the requirement (g − g) g = 0 or g g = g g for a generic field g, where the overline denotes
averaging. In this setting, the well known micro-macro homogeneity condition [7] would read
g g = g g.
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P. Neff, J. Jeong, I. Münch and H. Ramezani
1
|B(0, Lc )|
ZAMP
Z
k∇u(x)k2 + 2h∇u(x), D2 u(x).ξi + kD2 u(x).ξk2 + . . . dξ
Z
1
2
= k∇u(x)k + 0 +
kD2 u(x).ξk2 dξ + . . .
|B(0, Lc )| ξ∈B(0,Lc )
Z Lc Z
1
kD2 u(x).ξk2 dξ dt + . . .
= k∇u(x)k2 +
|B(0, Lc )| t=0 ξ∈∂B(0,t)
Z Lc Z
1
= k∇u(x)k2 +
kD2 u(x).ξk2 dS2t dt + . . .
|B(0, Lc )| t=0 ξ∈t S2
Z Lc Z
1
2
= k∇u(x)k +
kD2 u(x).(t h̃)k2 t2 dS2 dt + . . .
|B(0, Lc )| t=0 h̃∈S2
Z Lc Z
1
= k∇u(x)k2 +
t4
kD2 u(x).h̃k2 dS2 dt + . . .
|B(0, Lc )| t=0
h̃∈S2
Z
1 L5
= k∇u(x)k2 + 4 π L3 c
kD2 u(x).h̃k2 dS2 + . . .
c
5 h̃∈S2
3
2 Z
3
L
c
= k∇u(x)k2 +
(3.8)
kD2 u(x).h̃k2 dS2 + . . . .
20 π h̃∈S2
=
ξ∈B(0,Lc )
Exactly the same procedure for the full linear elastic energy expression is now
possible, together with a complete analytic result of the integration. We obtain to
highest order with the assumption (3.6)
Z
b
1
K
2
µ
b k dev sym ∇u(x + ξ)k2 +
Tr [∇u(x + ξ)] dξ
|B(0, Lc )| ξ∈B(0,Lc )
2
µ
¶
Z
3 L2c
=µ
b k dev sym ∇u(x)k2 +
k dev sym D2 u(x).h̃k2 dS2
20 π h̃∈S2
¶
µ
Z
i2
h
b
K
3 L2c
2
2
2
+
Tr [∇u(x)] +
Tr D u(x).h̃ dS
2
20 π h̃∈S2
b
K
2
=µ
bk dev sym ∇u(x)k2 +
Tr [∇u(x)]
2
à µ
!
¶ b
L2c
1
λ
2
2
2
2
+
µ
b kD u(x)kR27 − k∇ curl u(x)k + k∇ Div u(x)k
, (3.9)
5
2
2
where we have used the calculation in the appendix equation (5.16). As a preliminary result we see that a mean field4 u determined as a minimiser of this energy
will be smoother than the original field u but full pointwise control of all second
derivatives of the mean field is not implied.5
4 Of course, once the original integrand is replaced, u is not any longer the true averaged field
but only related to it in the obvious way.
5 Since ∆u = ∇ Div u − curl curl u we obtain from the modification (3.10) at least that ∆u ∈
Vol. 60 (2009)
Mean field modeling of isotropic random Cauchy elasticity
489
In the two-dimensional, planar case, one can show (5.17) that for smooth functions
with compact support u ∈ C0∞ (Ω ⊂ R2 , R2 ) it holds after integration
Z
Ω
kD2 u(x)k2R27 −
1
k∇ curl u(x)k2 dV ≥ 1
2
Z
k sym ∇ curl u(x)k2 dV ,
(3.10)
Ω
b = 0 in the higher order term and using
where the constant 1 is sharp. Thus, for λ
the last result (formally also in three-dimensions) as a simplification, we obtain
the classical indeterminate couple stress integrand, see (2.7)
µ
b k dev sym ∇u(x)k2 +
b
L2
K
2
Tr [∇u(x)] + µ
b c k sym ∇ curl uk2 .
2
5
(3.11)
Note, however, that this representation implies the symmetry of the moment
stress tensor m! Based on different considerations of point mechanics in [26]
the authors have also arrived at the conclusion that the moment stress tensor in
the couple stress theory should be symmetric. It violates the pointwise positivity
condition −1 < βγ < 1.
3.2. Relaxed penalty formulation of the mean field energy
Using the obtained second order corrected integrand for the mean field we would
be bound to implement a fourth order problem on the equilibrium level. To avoid
this, we formulate a penalty approach. We add a scalar pressure variable p(x)
which is supposed to take on the role of 13 Div u and we define a microrotation
vector axl(A) ∈ R3 for A ∈ so(3) taking on the role of 12 curl u. Then we define
the relaxed and penalised integrand
b
K
µ∞
2
Tr [∇u(x)] + c k curl u − 2 axl(A)k2
2
2
!
Ã
∞
2
b
Kc
Lc
λ
2
+
Tr [∇u(x) − p11] +
µ
b k dev sym ∇(2 axl(A)k2 + k∇3p(x)k2 .
2
5
2
µ
b k dev sym ∇u(x)k2 +
(3.12)
∞
À µ are numerical
Here, Lc > 0 is the physical cut-off length and µ∞
c , Kc
penalty factors (not physical constants). Clearly, this is now a microstretch model,
having the advantage that we can precisely interpret the appearing terms. For
∞
µ∞
c , Kc → ∞ we recover, formally, the higher-order mean field integrand. In
L2 (Ω) if the energy is bounded for zero boundary conditions. This reflects that the “truly”
2 (Ω, R3 ).
averaged field u would satisfy u ∈ Hloc
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P. Neff, J. Jeong, I. Münch and H. Ramezani
ZAMP
terms of the micro-stretch formulation presented in (2.1) we can write
b
K
µ∞
2
Tr [∇u(x)] + c k curl u − 2 axl(A)k2
2
2
Ã
!
∞
2
b
Kc
Lc
λ
2
+
Tr [∇u(x) − p11] +
µ
b k dev sym ∇(2 axl(A)k2 + k∇3p(x)k2
2
5
2
³
´2
p
Kc
1 p
2
Tr [ε] +
= µ k dev sym εk2 + µc k skew εk2 +
K0 Tr [ε] + λc Tr [p 11]
2
2
(3.13)
µ
b k dev sym ∇u(x)k2 +
+
γ−β
3α + (β + γ)
a0
γ+β
2
k dev sym ∇φk2 +
k skew ∇φk2 +
Tr [∇φ] + k∇pk2 ,
2
2
6
2
∞ b
where µc = µ∞
c , Kc = Kc , K = K0 , λc = K0 and γ = β =
b L2
9λ
c
5 .
4b
µL2c
5
and 3α + (β +
γ) = 0 and a0 =
The well-posedness of this relaxed formulation, although
not pointwise positive definite, can be shown along the lines in [12].
4. Conclusion
We have shown that mean field modelling of random isotropic Cauchy material leads in a straight forward way to the well established micro-stretch model.
Thereby, the usually difficult to determine coefficients in the curvature part can
be uniquely given and are related primarily to the one cut-off length Lc used
for the averaging scale. Thus we have provided an additional argument for the
usefulness of the micro stretch-model in general, independent of more detailed
micro-structure arguments. It is interesting to note the consequences for a linear
isotropic Cosserat model which is meant to be a homogeneous substitute medium
of the random Cauchy material. In this case, in terms of the Cosserat parameters,
µc should be large and the moment stress tensor m should be symmetric and trace
free. This conclusion is consistent with experimental observations and identifications by Lakes [16] for foams. In addition we see that the mean field, obtained by
spherical averaging, gives rise to a more “symmetric” response as compared to the
full possibilities of an isotropic second gradient model.
References
[1] Z. P. Bazant and B. H. Oh, Efficient numerical integration on the surface of a sphere. Z.
Angew. Math. Mech. 66 (1986), 37–49.
[2] A. C. Eringen, Microcontinuum Field Theories. Springer, Heidelberg 1999.
[3] A.C. Eringen, Theory of thermo-microstretch elastic solids. Int. J. Engng. Sci. 28 (1990),
71291–1301.
[4] S. Forest, Homogenization methods and the mechanics of generalized continua - part 2.
Theoret. Appl. Mech. (Belgrad) 28–29 (2002), 113–143.
Vol. 60 (2009)
Mean field modeling of isotropic random Cauchy elasticity
491
[5] S. Forest and R. Sievert, Nonlinear microstrain theories. Int. J. Solids Struct. 43 (2006),
7224–7245.
[6] G. Frantziskonis and E. C. Aifantis, On the stochastic interpretation of gradient-dependent
constitutive equations. Eur. J. Mech. A/Solids 21 (2002), 589–596.
[7] R. Hill, The elastic behaviour of crystalline aggregate. Proc. Phys. Soc. (London) A65
(1952), 349–354.
[8] T. J. R. Hughes, L. Mazzei, and K. E. Jansen, Large eddy simulation and the variational
multiscale method. Comput. Visual Sci. 3 (2000), 47–59.
[9] D. Iesan and A. Pompei, On the equilibrium theory of microstretch elastic solids. Int. J.
Eng. Sci. 33 (1995), 399–410.
[10] D. Iesan and R. Quintanilla, Existence and continuous dependence results in the theory of
microstretch elastic bodies. Int. J. Eng. Sci. 32 (1994), 991–1001.
[11] D. Iesan and A. Scalia, On Saint-Venant’s principle for microstretch elastic bodies. Int. J.
Eng. Sci. 35 (1997),1277–1290.
[12] J. Jeong and P. Neff,
Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Preprint 2588, http://wwwbib.mathematik.tudarmstadt.de/Math-Net/Preprints/Listen/pp08.html, to appear in Math. Mech. Solids,
2008, doi: 10.1177/1081286508093581
[13] N. Kirchner and P. Steinmann, Mechanics of extended continua: modelling and simulation
of elastic microstretch materials. Comp. Mech. 40 (2007), 651–666.
[14] A. Kirisa and E. Inan, On the identification of microstretch elastic moduli of materials by
using vibration data of plates. Int. J. Eng. Sci. 46 (2008), 585–597.
[15] W. T. Koiter, Couple stresses in the theory of elasticity I,II. Proc. Kon. Ned. Akad.
Wetenschap, B 67 (1964), 17–44.
[16] R. S. Lakes, Experimental microelasticity of two porous solids. Int. J. Solids Struct. 22
(1985), 55–63.
[17] M. Lazar and C. Anastassiadis. Lie-point symmetries and conservation laws in microstretch
and micromorphic elasticity. Int. J. Engrg. Sci. 44 (2006), 1571–1582.
[18] J. D. Lee and Y. Chen, Constitutive relations of micromorphic thermoplasticity. Int. J.
Engrg. Sci. 41 (2002), 387–399.
[19] R. D. Mindlin and H. F. Tiersten, Effects of couple stresses in linear elasticity. Arch. Rat.
Mech. Anal. 11 (1962), 415–447.
[20] P. Neff, On material constants for micromorphic continua. In Y. Wang and K. Hutter, editors, Trends in Applications of Mathematics to Mechanics, STAMM Proceedings, Seeheim
2004, pages 337–348. Shaker Verlag, Aachen 2005.
[21] P. Neff,
The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric. Preprint 2409, http://wwwbib.mathematik.tudarmstadt.de/Math-Net/Preprints/Listen/pp04.html, Zeitschrift f. Angewandte Mathematik Mechanik (ZAMM) 86 (2006), 892–912.
[22] P. Neff, Existence of minimizers for a finite-strain micromorphic elastic solid. Preprint 2318,
http://wwwbib.mathematik.tu-darmstadt.de/Math-Net/Preprints/Listen/pp04.html, Proc.
Roy. Soc. Edinb. A 136 (2006), 997–1012.
[23] P. Neff and S. Forest, A geometrically exact micromorphic model for elastic metallic foams
accounting for affine microstructure. Modelling, existence of minimizers, identification of
moduli and computational results. J. Elasticity 87 (2007), 239–276.
[24] R. A. Toupin, Elastic materials with couple stresses. Arch. Rat. Mech. Anal. 11 (1962),
385–413.
[25] R. A. Toupin, Theory of elasticity with couple stresses. Arch. Rat. Mech. Anal. 17 (1964),
85–112.
[26] F. Yang, A. C. M. Chong, D. C. C. Lam, and P. Tong, Couple stress based strain gradient
theory for elasticity. Int. J. Solids Struct. 39 (2002), 2731–2743.
492
P. Neff, J. Jeong, I. Münch and H. Ramezani
ZAMP
5. Appendix
5.1. Notation
Let Ω ⊂ R3 be a bounded domain with Lipschitz boundary ∂Ω and let Γ be a
smooth subset of ∂Ω with non-vanishing 2-dimensional Hausdorff measure. For
a, b ∈ R3 we let ha, biR3 denote the scalar product on R3 with associated vector
norm kak2R3 = ha, aiR3 . We denote by M3×3 the set of real 3 × 3 second order
tensors, written with capital letters and Sym denotes symmetric second orders
3×3
tensors.
The
is given by hX, Y iM3×3 =
£
¤ standard Euclidean scalar product on M
T
Tr XY , and thus the Frobenius tensor norm is kXk2 = hX, XiM3×3 . In the
following we omit the index R3 , M3×3 . The identity tensor on M3×3 will be denoted
by 11, so that Tr [X] = hX, 11i. We set sym(X) = 21 (X T + X) and skew(X) =
1
T
3×3
we set for the
2 (X − X ) such that X = sym(X) + skew(X). For X ∈ M
1
deviatoric part dev X = X − 3 Tr [X] 11 ∈ sl(3) where sl(3) is the Lie-algebra of
traceless matrices. The set Sym(n) denotes all symmetric n × n-matrices. The
Lie-algebra of SO(3) := {X ∈ GL(3) |X T X = 11, det[X] = 1} is given by the set
so(3) := {X ∈ M3×3 | X T = −X} of all skew symmetric tensors. The canonical
identification of so(3) and R3 is denoted by axl A ∈ R3 for A ∈ so(3). The Curl
operator on the three by three matrices acts row-wise, i.e.


 
curl(X11 , X12 , X13 )T
X11 X12 X13
(5.1)
Curl X21 X22 X23  = curl(X21 , X22 , X23 )T  .
X31 X32 X33
curl(X31 , X32 , X33 )T
Moreover, we have
∀ A ∈ C1 (R3 , so(3)) :
Div A(x) = − curl axl(A(x)) .
(5.2)
Note that (axl A) × ξ = A.ξ for all ξ ∈ R3 , such that


 
3
0
α β
−γ
X
−εijk · axl Ak ,
axl −α 0 γ  :=  β  , Aij =
−β −γ 0
−α
k=1
kAk2M3×3 = 2 k axl Ak2R3 ,
hA, BiM3×3 = 2haxl A, axl BiR3 ,
(5.3)
where εijk is the totally antisymmetric permutation tensor. Here, A.ξ denotes the
application of the matrix A to the vector ξ and a × b is the usual cross-product.
Moreover, the inverse of axl is denoted by anti and defined by


 
0
α β
−γ
−α 0 γ  := anti  β  , axl(skew(a ⊗ b)) = − 1 a × b ,
(5.4)
2
−β −γ 0
−α
and
2 skew(b ⊗ a) = anti(a × b) = anti(anti(a).b) .
(5.5)
Vol. 60 (2009)
Mean field modeling of isotropic random Cauchy elasticity
493
Moreover,
curl u = 2 axl(skew ∇u) .
(5.6)
By abuse of notation we denote the differential Dϕ of the deformation ϕ : R3 7→ R3
by ∇ϕ. This implies a transposition in certain comparisons with other literature
since here (∇ϕ)kj = ∂j ϕk is understood.
5.2. Second order expansions
Let us gather some expansions and developments which we need in the following. Note first that D2 u is interpreted as D2 u(x) ∈ Lin(R3 , M3×3 ) and therefore
[D2 u(x)]T [D2 u(x)] ∈ Lin(R3 , R3 ) = M3×3 .
[11 + ∇u(x + h)].y − [11 + ∇u(x)].y = [D2 u(x).h].y + . . . ,
£
¤
Tr [[11 + ∇u(x + h)] − [11 + ∇u(x)]] = Tr D2 u(x).h + . . . ,
Tr [[11 + ∇u(x + h)] − [11 + ∇u(x)]] = Div u(x + h) − Div u(x)
= h∇ Div u(x), hi + . . .
£
¤
= Tr D2 u(x).h + . . . ,
(5.7)
2
sym[[11 + ∇u(x + h)] − [11 + ∇u(x)]] = sym[D u(x).h] + . . . ,
dev sym[[11 + ∇u(x + h)] − [11 + ∇u(x)]] = dev sym[D2 u(x).h] + . . . ,
skew([11 + ∇u(x + h)] − [11 + ∇u(x)]) = skew(D2 u(x).h) + . . . ,
2 axl[skew([11 + ∇u(x + h)] − [11 + ∇u(x)])] = 2 axl[skew(D2 u(x).h)] + . . . ,
curl u(x + h) − curl u(x) = ∇ curl u(x).h + . . .
= 2 axl[skew(D2 u(x).h)] + . . . ,
∇ curl u(x).h = 2 axl[skew(D2 u(x).h)] .
5.3. Spherical integration
We make constantly use of the following simple closed form expressions for integrals
over the unit sphere which ca be found, e.g., in [1]
Z
´
4π ³
2
2
hX.h, hi dS2 =
2 k sym Xk2 + Tr [X] ,
15
2
Zh∈S
Z
Z
4π
2
hX.h, hidS2 =
Tr [X] ,
hh, hi dS2 =
1 dS2 = 4 π ,
3
2
2
2
h∈S
h∈S
h∈S
Z
Z
4π
4π
2
2
2
Tr [v ⊗ v] =
kvk2 . (5.8)
hv, hi dS =
h(v ⊗ v).h, hidS =
3
3
h∈S2
h∈S2
494
P. Neff, J. Jeong, I. Münch and H. Ramezani
On this basis,
Z
2
4π
h∇ curl u.h̃, h̃i dS2 =
15
h̃∈S2
4π
=
15
8π
=
15
³
¡
2 k sym ∇ curl uk2 + Tr [sym ∇ curl u]
2 k sym ∇ curl uk2 + (Div curl u)2
2
´
¢
k sym ∇ curl uk2 .
On the other hand,
Z
Z
2
h∇ curl u.h̃, h̃i dS2 =
h̃∈S2
ZAMP
(5.9)
2
h2 axl[skew(D2 u(x).h̃)], h̃i dS2
h̃∈S2
Z
=2
Zh̃∈S
1
=
2
Using (5.8)3 we get as well
Z
Z
£
¤2
Tr D2 u(x).h dS2 =
h∈S2
2
2
1
hskew(D2 u(x).h̃), anti(h̃)i dS2
4
2
hD2 u(x).h̃, anti(h̃)i dS2 .
(5.10)
h̃∈S2
2
h∇ Div u(x), hi dS2 =
h∈S2
and
Z
4π
k∇ Div u(x)k2 ,
3
(5.11)
Z
k skew[D
h∈S2
Z
=
Z
h∈S2
=
h∈S2
2
u(x).h]k2M3×3
2
dS =
h∈S2
2k axl[skew[D2 u(x).h]]k2R3 dS2
1
k2 axl[skew[D2 u(x).h]]k2R3 dS2
2
Z
1
1
k∇ curl u(x).hk2R3 dS2 =
h[∇ curl u(x)]T [∇ curl u(x)].h, hi dS2
2
h∈S2 2
4π
=
k∇ curl u(x)k2 .
6
(5.12)
Moreover,
Z
h∈S2
Z
kD2 u(x).h]k2M3×3 dS2 =
=
h[D2 u(x)]T [D2 u(x)].h, hi dS2
h∈S2
4π
kD2 u(x)k2R27 .
3
(5.13)
Thus, observing that
kD2 u(x).hk2M3×3 = k dev sym[D2 u(x).h]k2M3×3 + k skew[D2 u(x).h]k2M3×3
¤2
1 £
+ Tr D2 u(x).h ,
(5.14)
3
Vol. 60 (2009)
Mean field modeling of isotropic random Cauchy elasticity
495
we obtain
Z
h∈S2
k dev sym[D2 u(x).h]k2M3×3 dS2
Z
¤2
1 £
k[D2 u(x).h]k2 − k skew[D2 u(x).h]k2 − Tr D2 u(x).h dS2
3
h∈S2
4
π
4π
4π
kD2 u(x)k2R27 −
k∇ curl u(x)k2M3×3 −
k∇ Div u(x)k2 .
(5.15)
=
3
6
9
=
Therefore
Z
b
£
¤2
K
µ
b k dev sym[D2 u(x).h]k2M3×3 +
Tr D2 u(x).h dS2
2
h∈S2
!
Ã
µ
¶
b
1
1
K
µ
b
= 4π µ
b
k∇ Div u(x)k2
kD2 u(x)k2R27 − k∇ curl u(x)k2 + 4π
−
3
6
2
3
µ
¶
b
4π
1
2π λ
2
2
2
=
µ
b kD u(x)kR27 − k∇ curl u(x)k +
k∇ Div u(x)k2
3
2
3
à µ
!
¶ b
1
4π
λ
µ
b kD2 u(x)k2R27 − k∇ curl u(x)k2 + k∇ Div u(x)k2 .
(5.16)
=
3
2
2
5.4. Estimate for two-dimensional displacement fields
Here we show that for u(x, y, z) = (u1 (x, y), u2 (x, y), 0)T ∈ C0∞ (Ω, R3 ) it holds
that
Z
Z
1
kD2 uk2R27 − k∇ curl uk2M3×3 dx ≥
k sym ∇ curl uk2M3×3 dx .
(5.17)
2
Ω
Ω
Proof. We first observe that curl u = (0, 0, u2,x − u1,y )T from which follows


0
0
0
0
0
0 ,
∇ curl u = 
(u2,x − u1,y )x (u2,x − u1,y )y 0


0
0
(u2,x − u1,y )x
1
0
0
(u2,x − u1,y )y  ,
(5.18)
sym ∇ curl u = 
2
(u2,x − u1,y )x (u2,x − u1,y )y
0
and thus pointwise k sym ∇ curl uk2M3×3 = 12 k∇ curl uk2M3×3 . The inequality (5.17)
is therefore equivalent to
Z
Z
k∇ curl uk2M3×3 dx .
(5.19)
kD2 uk2R27 dx ≥
Ω
Ω
496
P. Neff, J. Jeong, I. Münch and H. Ramezani
The second derivatives are
µ
u1,xx
D u=
u1,yx
2
u1,xy
u1,yy
u2,xx
u2,yx
u2,xy
u2,xy
ZAMP
¶
,
kD2 uk2 = u21,xx + u21,xy + u22,xx + u22,xy
+ u21,yx + u21,yy + u22,yx + u22,xy ,
k∇ curl uk2 = u22,xx − 2u2,xx u1,yx + u21,yx
2
+ u22,xy − 2u2,xy u1,yy + u1,yy
,
kD2 uk2 − k∇ curl uk2 = u21,xx + u21,xy + u22,xy + u22,yy + 2u2,xx u1,yx
+ 2u2,xy u1,yy .
(5.20)
Partial integration for functions with compact support and the Theorem of Schwarz
for the mixed products give
Z
kD2 uk2 − k∇ curl uk2 dx
Ω
Z
=
u21,xx + u21,xy + u22,xy + u22,yy + 2u2,xx u1,yx + 2u2,xy u1,yy dx
Ω
Z
=
u21,xx + u21,xy + u22,xy + u22,yy + 2u2,xy u1,xx + 2u2,yy u1,yx dx
(5.21)
Ω
Z
= (u22,xy + 2u2,xy u1,xx + u21,xx ) + (u22,yy + 2u2,yy u1,yx + u21,xy ) dx
ZΩ
Z
= (u2,xy + u1,xx )2 + (u2,yy + u1,xy )2 dx =
k∇ Div uk2 dx ≥ 0 .
Ω
Ω
Summarising we have in the two-dimensional situation
Z
1
kD2 uk2R27 − k∇ curl uk2 dx
2
Ω
Z
=
k sym ∇ curl uk2M3×3 + k∇ Div uk2 dx + Null-Lagrangian .
Ω
Patrizio Neff
AG6, Fachbereich Mathematik
Technische Universität Darmstadt
Schlossgartenstrasse 7
64289 Darmstadt
Germany
e-mail: [email protected]
Jena Jeong
Ecole Spéciale des Travaux Publics du Bâtiment et de l’Industrie (ESTP)
28 avenue du Président Wilson
94234 Cachan Cedex
France
(5.22)
Vol. 60 (2009)
Mean field modeling of isotropic random Cauchy elasticity
e-mail: [email protected]
Ingo Münch
Institut für Baustatik
Universität Karlsruhe (TH)
Kaiserstrasse 12
76131 Karlsruhe
Germany
e-mail: [email protected]
Hamid Ramezani
CNRS-CRMD
1b Rue de la Ferollerie
45071 Orleans
France
e-mail: [email protected]
(Received: August 13, 2008; revised: November 6, 2008)
Published Online First: March 4, 2009
To access this journal online:
www.birkhauser.ch/zamp
497