Proceedings of the Royal Society of Edinburgh, 136A, 997–1012, 2006

Existence of minimizers for a finite-strainmicromorphic elastic solid

Patrizio NeffDepartment of Mathematics, Darmstadt University of Technology,Schlossgartenstrasse 7, 64289 Darmstadt, Germany(neff@mathematik.tu-darmstadt.de)

(MS received 13 April 2005; accepted 30 November 2005)

We investigate geometrically exact generalized continua of micromorphic type in thesense of Eringen. The two-field problem for the macrodeformation ϕ and the affinemicrodeformation P ∈ GL+(3, R) in the quasistatic, conservative load case isinvestigated in a variational form. Depending on material constants, two existencetheorems in Sobolev spaces are given for the resulting nonlinear boundary-valueproblems. These results comprise existence results for the micro-incompressible caseP ∈ SL(3, R) and the Cosserat micropolar case P ∈ SO(3, R). In order to treatexternal loads, a new condition, called bounded external work, has to be included,which overcomes the conditional coercivity of the formulation. The possible lack ofcoercivity is related to fracture of the micromorphic solid. The mathematical analysisuses an extended Korn first inequality. The methods of choice are the direct methodsof the calculus of variations.

1. Introduction

This article addresses the mathematical analysis of geometrically exact (fully frameindifferent, i.e. form invariant under superposed rotations) generalized continua ofmicromorphic type in the sense of Eringen in the elastic case. General continuummodels involving independent rotations were introduced by the Cosserat brothers [9]at the beginning of the last century. Their development was largely forgotten fordecades, only to be rediscovered in the early 1960s [1,11,14,17,18,27,35,38,40–42].At that time, theoretical investigations on non-classical continuum theories werethe main motivation [25]. Since then, the Cosserat concept has been generalized invarious directions (for an overview of these so-called microcontinuum theories werefer the reader to [4–6,12,13,19,26]). Recently, in [7,8], the micromorphic balanceequations derived by Eringen were formally justified as a more realistic continuummodel based on molecular dynamics and ensemble averaging. The micromorphicmodel includes in a natural way size effects, i.e. the behaviour of small samples iscomparatively stiffer than that of large samples. These effects have recently receivedmuch attention in conjunction with nano-devices. From a computational point ofview, theories with size effects are increasingly used to regularize non-well-posedsituations, e.g. shear banding in elastoplasticity without hardening. It has alreadybeen shown that infinitesimal elastoplasticity augmented by purely elastic Cosserateffects indeed leads to a well-posed problem, for both the quasistatic and dynamiccase [31,32].

997c© 2006 The Royal Society of Edinburgh

998 P. Neff

The mathematical analysis of general micromorphic solids is at present restrictedto the infinitesimal, linear elastic models, see, for example, [10, 15, 16, 20, 21] forlinear micropolar models and [22–24] for linear microstretch models. The majordifficulty of the mathematical treatment in the finite strain case is related to thegeometrically exact formulation of the theory and the appearance of nonlinearmanifolds necessary for the description of the microstructure. In addition, coercivityturns out to be a delicate problem related to the possible fracture of the material.An existence result for the simpler, geometrically exact nonlinear micropolar casehas been given in [30].

This paper is organized as follows: first, we briefly review the basic conceptsof the geometrically exact elastic micromorphic theories in a variational context,i.e. we formulate the quasistatic conservative load case as a two-field minimizationproblem. The existence proof is given in § 3. There, the complete problem state-ment of the geometrically exact elastic micromorphic case in a variational contextis repeated. Since the two-field variational problem is only conditionally coercive,we need to introduce a modification for the applied loads in order to ensure firstlythat the functional to be minimized is bounded below and secondly that the cur-vature contribution can be controlled. This modification of the loads, herein calledthe ‘principle of bounded external work’, expresses merely the physical fact that,by moving the solid arbitrarily in a force field, only a finite amount of work canbe gained. Such a condition is, however, unnecessary in classical finite elasticity.With this preparation, the existence of minimizers in Sobolev spaces is then estab-lished using the direct methods of the calculus of variations and an extended Kornfirst inequality. The relevant notation is introduced in the appendix. Readers inter-ested in the application of this micromorphic model and constitutive issues shouldconsult [33].

2. The finite-strain elastic micromorphic model

Let us now motivate a finite-strain micromorphic approach.1 For our developmentwe choose a strictly Lagrangian description. We first introduce an independentkinematical field of microdeformations P ∈ GL+(3, R), together with its right polardecomposition

P = RpUp = polar(P )Up = Rpeαp/3Up, det P = eαp ,

Up =Up

det Up1/3 ∈ SL(3, R), P =

P

det P 1/3 ∈ SL(3, R),

⎫⎪⎬

⎪⎭(2.1)

with Rp ∈ SO(3, R) and Up ∈ PSym(3, R) ∩ SL(3, R). The microdeformations P aremeant to describe the substructure of the material which can rotate, stretch, shearand shrink. We refer to Rp as the microrotations.

The micromorphic theory we deal with can formally be obtained by introduc-ing the multiplicative decomposition of the macroscopic deformation gradient F

1Following Eringen [12, p. 13], we distinguish the general micromorphic case: P ∈ GL+(3, R) =R

+ ·SL(3, R) with nine additional degrees of freedom (DOF); the micro-incompressible micromor-phic case: P ∈ SL(3, R) with 8 DOF; the microstretch case: P ∈ R

+ · SO(3, R) with 4 DOF andthe micropolar case: P ∈ SO(3, R) with only three additional DOF.

Minimizers for a finite-strain micromorphic elastic solid 999

into independent microdeformation P and the micromorphic, non-symmetric rightstretch tensor U (first Cosserat deformation tensor) with

F = PU, U = P−1F, U ∈ GL+(3, R), (2.2)

leading altogether to a micro-compressible, micromorphic formulation.2

The notion micromorphic is nevertheless prone to misunderstandings: the micro-deformation P must be considered as a macroscopic (average) quantity as the defor-mation gradient and the resulting model is still phenomenological. However, geo-metrical features of the real substructure to be modelled determine the choice ofgeometric manifolds for P . Since the substructure can in principle be crushed, thechoice P ∈ GL+(3, R) is mandatory.

In the quasistatic case, the micromorphic theory is derived from a two-field vari-ational principle by postulating the following action euclidienne [9, p. 156] I forthe finite macroscopic deformation ϕ : [0, T ] × Ω → R

3 and the independent micro-deformation P : [0, T ] × Ω → GL+(3, R):

I(ϕ, P ) =∫

Ω

W (F, P,DxP ) − Πf(ϕ) − ΠM (P ) dV

−∫

ΓS

ΠN (ϕ) dS −∫

ΓC

ΠMc(P ) dS → min with respect to (ϕ, P ),

P |Γ = Pd, ϕ|Γ = gd(t).

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(2.3)The elastically stored energy density W depends not only on the macroscopic defor-mation gradient F = ∇ϕ as usual but additionally on the microdeformation Ptogether with their first-order space derivatives, represented through the third-order tensor DxP . Here Ω ⊂ R

3 is a domain with boundary ∂Ω and Γ ⊂ ∂Ωis that part of the boundary, where Dirichlet conditions gd and Pd for displace-ments and microdeformations, respectively, can be prescribed, while ΓS ⊂ ∂Ω isa part of the boundary, where traction boundary conditions in the form of thepotential of applied surface forces ΠN are given with Γ ∩ ΓS = ∅. The potential ofexternal applied volume force is Πf and ΠM takes on the role of the potential ofapplied external volume couples (appearing in a non-mechanical context, for exam-ple, as the influence of a magnetic field on the polarization of a substructure of thebulk). In addition, ΓC ⊂ ∂Ω is the part of the boundary where the potential ofapplied surface couples ΠMc are applied with Γ ∩ ΓC = ∅. On the free boundary∂Ω \ Γ ∪ΓS ∪ΓC corresponding natural boundary conditions for ϕ and P apply;these are obtained automatically in the variational process.

Variation of the action I with respect to ϕ yields the traditional equation for bal-ance of linear momentum, and variation of I with respect to P yields the additionalgeneralized balance of moment of momentum.

The standard conclusion from frame indifference (here, invariance of the freeenergy under superposed rigid body motions not merely observer invariance of themodel [3, 28,39]) is as follows: for all

Q ∈ SO(3, R) : W (F, P,DxP ) = W (QF, QP,Dx[QP ])2The strain measure U induced by this definition corresponds to CT

KL presented in [12, (1.5.11)1,p. 15].

1000 P. Neff

leads to the reduced representation of the energy (specify Q = RTp ):

W (F, P , DxP ) = W (RTp F, RT

p P, RTp DxP )

= W (UpU , Up, RTp DxP )

= W (U , Up,Kp,∇αp), (2.4)

where for P = RpUp ∈ SL(3, R) we set

Kp := RTp DxP = (RT

p ∇(P · e1), RTp ∇(P · e2), RT

p ∇(P · e3)) ∈ M3×3 × M

3×3 × M3×3.(2.5)

For a geometrically exact (macroscopically isotropic) theory we assume in thefollowing an additive split of the total free energy density into micromorphic relativelocal stretch (macroscopic), stretch of the substructure itself (microscopic) andmicromorphic curvature part according to

W = Wmp(U)︸ ︷︷ ︸

relativemacroscopic

energy

+ Wfoam(Up, αp)︸ ︷︷ ︸

microscopiclocal

energy

+ Wcurv(Kp,∇αp)︸ ︷︷ ︸

microscopicinteraction

energy

, (2.6)

since a possible coupling between U and Kp for centrosymmetric bodies can be ruledout [34, p. 14].

2.1. The elastic macroscopic micromorphic strain energy density

For a macroscopic theory which is relevant mainly for small elastic strain werequire that Wmp(U) is a non-negative isotropic quadratic form (leading to a phys-ically linear problem). This should already cover many cases of physical interest.We assume moreover the relative macroscopic stretch energy density normalized to

Wmp(1l) = 0, DUWmp(U)|U=1l = 0. (2.7)

For the local energy contribution elastically stored in the substructure we assumethe nonlinear expression

Wfoam(Up) = µm∥∥∥∥

Up

det Up(1/3) − 1l

∥∥∥∥

2

︸ ︷︷ ︸isochoric substructure energy

+λm

4

(

(det Up − 1)2 +(

1det Up

− 1)2)

︸ ︷︷ ︸volumetric energy

= µm‖Up − 1l‖2 + 14λm((eαp − 1)2 + (e−αp − 1)2)

=: Wfoam(Up, αp), (2.8)

avoiding self-interpenetration in a variational setting, since Wfoam → ∞ as det P =det Up → 0 if λm > 0.

The most general form of Wmp consistent with the requirement (2.7) is

Wmp(U) = µe‖ sym(U − 1l)‖2 + µc‖ skew(U − 1l)‖2 + 12λe tr[sym(U − 1l)]2, (2.9)

with material constants µe, µc, λe such that µe, 3λe + 2µe, µc 0 from the non-negativity [12] of (2.9).

Minimizers for a finite-strain micromorphic elastic solid 1001

Remark 2.1. It is important to realize that µe, λe are effective elastic constantswhich in general do not coincide with the classical Lame constants.

The so-called Cosserat couple modulus µc (rotational couple modulus) remainsfor the moment unspecified, but we note that µc = 0 is physically possible, even inthe micropolar case, since the micromorphic reaction stress DUWmp(U) · UT is notsymmetric in general, i.e. the problem does not decouple.3

By formal similarity with the classical formulation, we may call µm and λm themicroscopic Lame moduli of the affine substructure.

2.2. The nonlinear elastic curvature energy density

The curvature energy is responsible for the size-dependent resistance of the sub-structure against local twisting and inhomogeneous volume change. Thus, inhomo-geneous microstructural rearrangements are penalized. For the curvature term, tobe specific, we assume that

Wcurv(Kp,∇αp)

= 112µL1+p

c (1 + α4Lqc‖Kp‖q)(α5‖ sym Kp‖2 + α6‖ skew Kp‖2 + α7 tr[Kp]2)(1+p)/2

+ 112µL1+p

c (α8‖∇αp‖1+p + α8Lc‖∇αp‖2+p),(2.10)

where Lc > 0 is setting an internal length-scale with units of length. It is to be notedthat we have decoupled the curvature coming from inhomogeneous volume changesand from pure twisting. The values α4 0, p > 0 and q 0 are additional materialconstants. The factor 1

12 appears only for convenience and α5 > 0, α6, α7 0,α8 > 0 should be satisfied as a minimal requirement. We mean tr[Kp]2 = ‖ tr [Kp]‖2

by abuse of notation. This choice for Wcurv does not presuppose any knowledge ofthe magnitude of the micromorphic curvature in the material and is non-degeneratein the origin ‖Kp‖ = 0, ‖∇αp‖ = 0.

Some care has to be exerted in the finite-strain regime: Wcurv should preferablybe coercive in the sense that we impose pointwise

∃c+ > 0, ∃r > 1 : for all Kp ∈ T(3) and all ξ ∈ R3 : Wcurv(Kp, ξ) c+‖(Kp, ξ)‖r,

(2.11)or the less demanding

∃r > 1 :Wcurv(Kp, ξ)‖(Kp, ξ)‖r

→ ∞ as ‖(Kp, ξ)‖ → ∞, (2.12)

which implies necessarily that α6, α8 > 0 in (2.10). Observe that our formulation ofthe micromorphic curvature tensor is mathematically convenient in the sense that‖Kp‖ = ‖RT

p DxP‖ = ‖DxP‖ provides pointwise control of all first derivatives of Pindependent of the values of P itself.4

3In a linearized isotropic micropolar model, the balance of forces and balance of rotationalmomentum are independent of each other if µc = 0.

4This is not true for other possible basic invariant curvature expressions like P −1DxP orPTDxP or FTDxP (see [12, §§ 1.5.4, 1.5.11]).

1002 P. Neff

Note that the presented formulation includes a finite-strain Cosserat micropolarmodel as a special case, if we set P = R ∈ SO(3, R). In this fashion, we have thefollowing correspondence of limit problems:

λm → ∞ ⇒ micro-incompressible model: manifold SL(3, R),

µm → ∞ ⇒ microstretch model: manifold R+ · SO(3, R),

µm, λm → ∞ ⇒ micropolar model: manifold SO(3, R),µm, λm, µc → ∞ ⇒ higher (second) gradient continua of Mindlin type.

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(2.13)

3. Analysis

3.1. Statement of the micromorphic problem in variational form

Let us gather the obtained three-field problem posed in a variational form. Thetask is to find a triple (ϕ, P , αp) : Ω ⊂ R

3 → R3 × SL(3, R) × R of macroscopic

deformation ϕ and independent microdeformation P = eαp/3P , minimizing theenergy functional I with

I(ϕ, P , αp) =∫

Ω

Wmp(P−1∇ϕ) + Wfoam(Up, αp) + Wcurv(RTp DxP ,∇αp)

− Πf(ϕ) − ΠM (P ) dV −∫

ΓS

ΠN (ϕ) dS

−∫

ΓC

ΠMc(P ) dS → min with respect to (ϕ, P , αp), (3.1)

under the constraints

Up = RTp P , Rp = polar(P ), U = P−1∇ϕ, P = eαp/3P , (3.2)

and the Dirichlet boundary conditions

ϕ|Γ = gd, Rp|Γ = Rpd , Up|Γ = Upd ⇒ P |Γ = RpdUpd , αp|Γ = αpd .(3.3)

Here, the constitutive assumptions on the densities are taken to be

Wmp(U) = µe‖ sym(U − 1l)‖2 + µc‖ skew(U)‖2 + 12λe tr[sym(U − 1l)]2,

Wfoam(Up, αp) = µm‖Up − 1l‖2 + 14λm((eαp − 1)2 + (e−αp − 1)2),

Wcurv(Kp,∇αp) = 112µL1+p

c (1 + α4Lqc‖Kp‖q)

× (α5‖ sym Kp‖2 + α6‖ skew Kp‖2 + α7 tr [Kp]2)(1+p)/2

+ 112µL1+p

c (α8‖∇αp‖1+p + α8Lc‖∇αp‖2+p),

Kp = RTp DxP = (RT

p ∇(P · e1), RTp ∇(P · e2), RT

p ∇(P · e3)), (3.4)

where Kp denotes the third-order curvature tensor. The total elastically storedenergy W = Wmp + Wfoam + Wcurv depends on the deformation gradient F = ∇ϕ,and the microdeformations P together with their spatial derivatives.

It is assumed that µe, λe > 0, µc 0 and µm, λm, Lc > 0. The parameters αi,i = 1, . . . , 8, are dimensionless weighting factors. If not stated otherwise, we assumethat α5 > 0, α6 > 0, α8 > 0 and α7 0.

Minimizers for a finite-strain micromorphic elastic solid 1003

A finite Cosserat micropolar theory is included in the formulation (3.1), (3.2),(3.4) by restricting it to P ∈ SO(3, R) or setting µm, λm = ∞, formally. Similarly,for µm = ∞ only we recover the micro-stretch formulation with P ∈ R

+ · SO(3, R),and for λm = ∞ we recover the micro-incompressible formulation case P ∈ SL(3, R).

3.2. The external potentials

Traditionally, in the conservative, dead-load case one would have

Πf(ϕ) = 〈f, ϕ〉, ΠM (P ) = 〈M, P 〉, ΠN (ϕ) = 〈N, ϕ〉, ΠMc(P ) = 〈Mc, P 〉(3.5)

for the potentials of applied loads with given functions

f ∈ L2(Ω, R3), M ∈ L2(Ω, M3×3), N ∈ L2(ΓS , R3), Mc ∈ L2(ΓC , M3×3).

For our treatment, we need to assume, however, that the external potentials,describing the configuration dependent applied loads, are continuous with respectto the topology of L1(Ω), L1(ΓS) and L1(ΓC), respectively, and in addition satisfythe conditions

∃C+ > 0 for all ϕ ∈ L1(Ω, R3), P ∈ L1(Ω, GL+(3, R)) :∫

Ω

Πf(ϕ) + ΠM (P ) dV,

∫

ΓS

ΠN (ϕ) dS,

∫

ΓC

ΠMc(P ) dS C+. (3.6)

While continuity is satisfied, for example, for the dead-load case Πf(ϕ) = 〈f, ϕ〉 andf ∈ L∞(Ω), the second condition (3.6) restricts our attention to ‘bounded externalwork’. If we want to describe a situation corresponding to the classical dead-loadcase, we could take

Πf(ϕ) =1

1 + [‖ϕ(x)‖ − K+]+〈f(x), ϕ(x)〉, (3.7)

for some large positive constant K+ and [·]+, the positive part of a scalar argument.It now suffices that f ∈ L1(Ω). Then

∫

ΩΠf(ϕ) dV C+, independent of ϕ ∈

L1(Ω).The new condition (3.6) can be rephrased as saying that only a finite amount of

work can be performed against the external loads, regardless of the magnitude oftranslation and microdeformation. This is certainly true for any real field of appliedloads.5

5In classical finite elasticity, such a condition is not necessary, since the elastic energy density isassumed a priori to verify an unqualified coercivity condition [36] of the type W (F ) c+‖F‖q −C, q > 1, which, together with Dirichlet conditions and Poincare’s inequality, controls the Lq(Ω)part of the deformation.

Some examples of fields satisfying (3.6) are the gravity field of a finite mass, the electric fieldof a finite charge, etc. Note also that (3.6) does not exclude local, integrable singularities. Thetraditional dead-load case in (3.5) must rather be interpreted as a linearization of the finiteexternal potential. We write ϕ(x) = x + u(x), and then Π(x, ϕ(x)) = Π(x, x + u(x)) = Π(x, x) +〈DϕΠ(x, x), u〉+ · · · = const.+〈f, u〉+ · · · with f(x) = DϕΠ(x, x). I am not aware of the previousintroduction of a condition similar to (3.6).

1004 P. Neff

In order to elucidate why we need this new assumption, consider the exemplarysituation with classical dead loads

I(ϕ, P ) =∫

Ω

‖P−1∇ϕ−1l‖2 +‖P −1l‖2 +‖DxP‖2 −〈f, ϕ〉 dV → min(ϕ, P ), (3.8)

for P ∈ SL(3, R). The first step in the direct methods of the calculus of variationsis to show that sequences (ϕk, Pk) with finite energy I(ϕk, Pk) K are boundedin some Sobolev spaces. In order to obtain the boundedness of ∇ϕ it is necessaryto control P ∈ L6(Ω) (see (3.14)). This holds true if we can already bound thecurvature DxP ∈ L2(Ω) from the embedding theorem (see (3.12)). However, thereis no way to infer an a priori bounded curvature from bounded energy I, essentiallybecause of the dead-load term, which can balance an unbounded curvature. If thelocal part ‖P − 1l‖2 of the substructure energy has a higher exponent (here 6), theproblem may be avoided in this simple setting. However, case 2 (see below) willalways need the bounded external work assumption.

3.3. The different cases

We distinguish three different situations:

Case 1 (µc > 0, α4 0, p 1, q 0). Elastic macro-stability, local first-ordermicromorphic. Fracture excluded.

Case 2 (µc = 0, α4 > 0, p 1, q > 1). Elastic pre-stability, non-local second-ordermicromorphic, macroscopic specimens, in a sense close to classical elasticity, zeroCosserat couple modulus. Fracture excluded for bounded external work.

Case 3 (µc = 0, α4 = 0, 0 < p 1, q = 0). Elastic pre-stability, non-local second-order micromorphic theory, macroscopic specimens, in a sense close to classicalelasticity, zero Cosserat couple modulus. Since possibly ϕ ∈ W 1,1(Ω, R3), due tolack of elastic coercivity, including fracture in multiaxial situations.

We refer to 0 < p < 1, q 0 as the subcritical case, to p = 1, q 0 as the criticalcase and to p 1, q > 1 as the super-critical case. We will treat the first two casesmathematically.

3.4. The coercivity inequality

The decisive analytical tool underlying the treatment of case 2 (super-critical,µc = 0) is the following inequality establishing coercivity.

Theorem 3.1 (extended three-dimensional Korn first inequality). Let Ω ⊂ R3 be

a bounded Lipschitz domain and let Γ ⊂ ∂Ω be a smooth part of the boundarywith non-vanishing two-dimensional Hausdorff measure. Define H1,2

(Ω, Γ ) := φ ∈H1,2(Ω)|φ|Γ = 0 and let Fp, F

−1p ∈ C0(Ω, GL(3, R)). Then

∃c+ > 0 ∀φ ∈ H1,2 (Ω, Γ ) : ‖∇φF−1

p (x) + F−Tp (x)∇φT‖2

L2(Ω) c+‖φ‖2H1,2(Ω).

Proof. The proof of this version of Korn’s inequality is presented in [37], whichimproves on a similar result in [29], in which the possible validity of this inequalitywas first observed.

Minimizers for a finite-strain micromorphic elastic solid 1005

3.5. Existence for the geometrically exact elastic micromorphic model

The following results extend the existence theorems for the geometrically exactmicromorphic micro-incompressible elastic solids given previously.6

Theorem 3.2 (existence for elastic micromorphic model: case 1). Let Ω ⊂ R3 be

a bounded Lipschitz domain and assume for the boundary data that gd ∈ H1(Ω, R3)and Pd ∈ W 1,1+p(Ω, GL+(3, R)). Moreover, let the applied external potentials sat-isfy (3.6). Then (3.1) with material constants conforming to case 1 and p > 1admits at least one minimizing solution triple

(ϕ, P , αp) ∈ H1(Ω, R3) × W 1,1+p(Ω, SL(3, R)) × W 1,2+p(Ω, R).

Proof. We apply the direct methods of the calculus of variations. The influence ofthe external potentials is condensed into writing Π(ϕ, P ). With the prescription of(gd, Pd) it is clear that I < ∞ for exactly this pair of functions after decomposing Pdin its rotational, isochoric stretch and volumetric stretch. Since (3.6) is assumed, it isalso clear that I is bounded below for all ϕ ∈ L2(Ω, R3) and P ∈ L2(Ω, GL+(3, R)).

We may therefore choose infimizing ‘sequences of triples’

(ϕk, P k, αkp) ∈ H1(Ω, R3) × W 1,1+p(Ω, SL(3, R)) × W 1,2+p(Ω, R), (3.9)

such thatlim

k→∞I(ϕk, P k, αk

p) = infϕ∈L1(Ω,R3),

P∈L1(Ω,SL(3,R)),αp∈L1(Ω,R)

I(ϕ, P , αp). (3.10)

The total curvature contribution Wcurv along this sequence is bounded indepen-dently of the number k, again on account of (3.6).7

We now observe that the micromorphic curvature term Kp controls

P ∈ W 1,1+p(Ω, SL(3, R)),

since ‖Kp‖ = ‖RTp DxP‖ = ‖DxP‖, pointwise, the assumption that α5, α6 > 0 and

the application of Poincares inequality with the Dirichlet conditions on P . More-over, since α8 > 0 we obtain boundedness of αk

p ∈ W 1,2+p(Ω, R), again independentof k ∈ N. This result remains true already without specification of Dirichlet bound-ary conditions for αp since the term eαp estimates any Lq-norm of αp. For p > 1Sobolev’s embedding shows that we can choose a subsequence, not relabelled, suchthat strongly

αkp → ˆαp ∈ C0(Ω, R), k → ∞. (3.11)

Now we may extract a subsequence, again denoted by P k, converging strongly inL1+p(Ω) to an element P ∈ W 1,1+p(Ω, M3×3), since p > 0 by assumption. More-over, a further subsequence can be found, such that the curvature tensor Kp,k con-verges weakly to some Kp in L1+p(Ω). For 1 < (1 + p) < 3 the embedding

W 1,1+p(Ω) ⊂ L3(1+p)/(3−(1+p))−δ(Ω), δ 0, (3.12)6The proposed finite results determine the macroscopic deformation ϕ ∈ H1(Ω, R

3) and noth-ing more. This means that discontinuous macroscopic deformations by cavities or the formation ofholes are not excluded (possible mode-I failure). If µc > 0, fracture is effectively ruled out, whichis unrealistic.

7If (3.6) does not hold, one may have infimizing sequences with unbounded curvature. Thegeometrically exact micromorphic formulation is only conditionally coercive.

1006 P. Neff

for three spatial dimensions is compact for δ > 0 and shows that the subsequenceP k can be chosen such that it indeed converges strongly in the topology of L6−δ(Ω),since, moreover, we have p 1, which implies immediately that

P ∈ W 1,1+p(Ω, SL(3, R)).

If 1 + p 3, we can use better embeddings to reach the same conclusion.Because µc > 0, we have the simple algebraic estimate

Wmp(P−1,kF k) µc‖P−1,kF k − 1l‖2

= µc(‖P−1,kF k‖2 − 2〈P−1,kF k, 1l〉 + 3)

µc(‖Uk‖2 − 2√

3‖Uk‖ + 3), (3.13)

implying the boundedness of the micromorphic stretch Uk = P−1,kF k in L2(Ω).Moreover, by Holder’s inequality, we obtain

‖F k‖s,Ω = ‖P kP−1,kF k‖s,Ω

‖P k‖r1,Ω‖P−1,kF k‖r2,Ω

= ‖eαkp/3P k‖r1,Ω‖P−1,kF k‖r2,Ω

supx∈Ω

eαkp(x)/3‖P k‖r1,Ω‖P−1,kF k‖r2,Ω ,

1s

=1r1

+1r2

. (3.14)

Since P k is bounded in L6(Ω) (see (3.12)) and P−1,kF k is bounded in L2(Ω) andαk

p is strongly converging in C0(Ω, R) (3.11), we may choose r1 = 6, r2 = 2 toobtain boundedness of F k = ∇ϕk in Ls(Ω), s = 3

2 . Using the Dirichlet boundaryconditions for ϕk and the generalized Poincare inequality, we get

‖ϕk‖W 1,s(Ω,R3) const. (3.15)

By the boundedness of ϕk in W 1,s(Ω, R3) we may extract a subsequence, not rela-belled, such that ϕk ϕ ∈ W 1,s(Ω, R3). Furthermore, we may always obtain asubsequence of (ϕk, P k) such that Uk = P−1,kF k converges weakly in L2(Ω) tosome element ˆU on account of the boundedness of the stretch energy and µc > 0.

We have already shown that for p 1 the sequence P k indeed converges stronglyin Lr(Ω) to an element ˆP ∈ W 1,1+p(Ω, SL(3, R)). Therefore,

P−1,k =1

det P kAdj P k → 1

det ˆPAdj ˆP = ˆP−1 in Lr/2(Ω, SL(3, R)),

r =3(1 + p)

(3 − (1 + p))− δ, if 1 < (1 + p) < 3,

⎫⎪⎪⎬

⎪⎪⎭

(3.16)

and for p > 1 we obtain P−1,k → ˆP−1 strongly in L3+δ(Ω, SL(3, R)), δ > 0. More-over,

P−1,k = e−αkp/3P−1,k → P−1 = e− ˆαp/3 ˆP−1,k, (3.17)

on account of the strong convergence of αkp . Thus, P−1,kF k converges certainly

weakly to P−1F in L1(Ω) on account of Holder’s inequality (sharp). The weak limitin L1(Ω) must coincide with the weak limit of Uk in L2(Ω). Hence, the identityˆU = P−1∇ϕ holds.

Minimizers for a finite-strain micromorphic elastic solid 1007

Since the mapping polar : GL+(3, R) → SO(3, R) is a bounded continuous func-tion on invertible matrices with positive determinant, it generates a nonlinear super-position operator

polar(·) : Lr(Ω, GL+(3, R)) → Lr(Ω, SO(3, R)), (3.18)

which, moreover, is continuous [2, p. 101, theorem 3.7]. Thus, Rk = polar(Pk) →ˆR = polar( ˆP ) strongly in Lr(Ω) and a similar argument as for the sequence Uk

shows that

Kp,k Kp = polar(P )TDxP (3.19)

in L1+p(Ω), weakly. Again on account of P k → ˆP in Lr(Ω, SL(3, R)) we now inferthat

Ukp =

√P k,T P k →

√ˆPT ˆP = ˆUp in Lr(Ω, SL(3, R)), (3.20)

because the map M3×3 → PSym(3), X →

√XTX is continuous and has linear

growth.Since the total energy is convex in (U , Up,Kp,∇αp) and continuous with respect

to αp, and the external potential Π is continuous with respect to strong convergencein L1(Ω) on account of (3.6), we get

I(ϕ, ˆP, ˆαp) =∫

Ω

Wmp( ˆU) + Wfoam( ˆUp, ˆαp) + Wcurv(Kp,∇ ˆαp) dV − Π(ϕ, P )

lim infk→∞

∫

Ω

Wmp(Uk) + Wfoam(Ukp )

+ Wcurv(Kp,k,∇αkp) dV − Π(ϕk, Pk)

= limk→∞

I(ϕk, P k, αkp) = inf

ϕ∈L1(Ω,R3),P∈L1(Ω,SL(3,R)),

αp∈L1(Ω,R)

I(ϕ, P , αp), (3.21)

which implies that the limit triple (ϕ, P , ˆαp) is a minimizer. Note that the limitmicrodeformations P = eαp/3RpUp may fail to be continuous, if p 2 (non-exist-ence or the limit case of Sobolev embedding). Moreover, uniqueness cannot beascertained, since SL(3, R) is a nonlinear manifold (and the problem consideredis indeed highly nonlinear), such that convex combinations in SL(3, R) may leaveSL(3, R). Since the functional I is differentiable, the minimizing pair is a stationarypoint and therefore a solution of the corresponding field equations. Note again thatthe limit microdeformations may fail to be continuously distributed in space. Thefact that a minimizing solution may nevertheless be found under these unfavourablecircumstances is entirely due to µc > 0 and p > 1. The proof simplifies considerablyin the geometrically exact Cosserat micropolar case P ∈ SO(3, R), in which casep 1 is already sufficient.

We continue with the super-critical case which is more appropriate for macro-scopic situations and closer to classical elasticity.

Theorem 3.3 (existence for elastic micromorphic model: case 2). Let Ω ⊂ R3 be

a bounded Lipschitz domain and assume for the boundary data that gd ∈ H1(Ω, R3)

1008 P. Neff

and Pd ∈ W 1,1+p+q(Ω, SL(3, R)). Moreover, let the applied external potentials sat-isfy (3.6). Then (3.1) with material constants conforming to case 2 admits at leastone minimizing solution triple:

(ϕ, P , αp) ∈ H1(Ω, R3) × W 1,1+p+q(Ω, SL(3, R)) × W 1,2+p(Ω, R).

Proof. We repeat the arguments of case 1. However, the boundedness of infimiz-ing sequences is not immediately clear. Boundedness of the microdeformationsP k holds in the space W 1,1+p+q(Ω, SL(3, R)) with 1 + p + q > N = 3. Hence, wemay extract a subsequence, not relabelled, such that P k converges strongly toˆP ∈ C0(Ω, SL(3, R)) in the topology of C0(Ω, SL(3, R)) on account of the Sobolev-embedding theorem. Since P−1,k = e−αk

p/3P−1,k, we also obtain

P−1,k → P−1 ∈ C0(Ω, GL+(3, R)), (3.22)

on account of the strong convergence of αkp .

Along such a strongly convergent sequence of microdeformations, the sequenceof deformations ϕk is also bounded in H1(Ω, R3). However, this is not due to abasically simple estimate as in case 1, but rather is only true after integration overthe domain: at face value we only control certain mixed symmetric expressions inthe deformation gradient. Let us define uk ∈ H1,2(Ω, R3) by ϕk = gd +(ϕk − gd) =gd + uk. We then have

∞ > I(gd, Pd, αpd)

>

∫

Ω

Wmp(Uk) + Wfoam(Ukp , αk

p) + Wcurv(Kp,k,∇αkp) dV − Π(ϕk, P k)

∫

Ω

Wmp(Uk) dV − Π(ϕk, P k)

∫

Ω

Wmp(Uk) dV − C

∫

Ω

14µe‖P−1,k∇ϕk + ∇ϕT

k P−T,k − 21l‖2 dV − C

=∫

Ω

14µe‖P−1,k(∇uk + ∇gd) + (∇uk + ∇gd)TP−T,k − 21l‖2 dV − C

=∫

Ω

14µe‖P−1,k∇uk + ∇uT

k P−T,k‖2

+ 142µe〈P−1,k∇uk + ∇uT

k P−T,k, P−1,k∇gd + ∇gTd P−T,k − 21l〉

+ 14µe‖P−1,k∇gd + ∇gT

d P−T,k − 21l‖2 dV − C

∫

Ω

14µe‖P−1,k∇uk + ∇uT

k P−T,k‖2

− 14µe

(

ε‖P−1,k∇uk + ∇uTk P−T,k‖2 +

1ε‖P−1,k∇gd + ∇gT

d P−T,k − 21l‖2)

+ 14µe‖P−1,k∇gd + ∇gT

d P−T,k − 21l‖2 dV − C

∫

Ω

18µe‖P−1,k∇uk + ∇uT

k P−T,k‖2 − 12µe‖P−1,k∇gd + ∇gT

d P−T,k − 21l‖2

+ 14µe‖P−1,k∇gd + ∇gT

d P−T,k − 21l‖2 dV − C

Minimizers for a finite-strain micromorphic elastic solid 1009

=∫

Ω

18µe‖P−1,k∇uk + ∇uT

k P−T,k‖2

− 14µe‖P−1,k∇gd + ∇gT

d P−T,k − 21l‖2 dV − C

∫

Ω

18µe‖P−1,k∇uk + ∇uT

k P−T,k‖2 dV − C

=∫

Ω

18µe‖(P−1,k − P−1 + P−1)∇uk + ∇uT

k (P−1,k − P−1 + P−1)T‖2 dV − C

∫

Ω

18µe ‖P−1∇uk + ∇uT

k P−T ‖2

︸ ︷︷ ︸combinations of derivatives

dV − C2‖P−1 − P−1,k‖∞‖uk‖2H1,2(Ω) − C

( 18µecK − C2‖P−1 − P−1,k‖∞)‖uk‖2

H1,2(Ω) − C, (3.23)

where we used Young’s inequality with ε = 12 , made use of the appropriate Dirich-

let boundary conditions for uk and applied the extended Korn inequality (theo-rem 3.1) yielding the positive constant cK for the continuous microdeformationP−1. Since ‖P−1 − P−1,k‖∞ → 0 for k → ∞ due to (3.22), we are able to concludethe boundedness of uk in H1(Ω). Hence, ϕk is bounded in H1(Ω). Now we find thatUk ˆU = P−1∇ϕ by construction with the notation as in case 1. The remainderproceeds as in case 1. This finishes the argument. The limit microdeformations Pare indeed found to be continuous.

Remark 3.4 (the micro-incompressible case). Both existence results can easily beadapted to cover the micromorphic micro-incompressible case αp ≡ 1.

4. Final remarks

The variational micromorphic problem presented fits neatly into the frameworkof the direct methods of the calculus of variations. The coercivity part for thedeformation is, however, non-trivial and for the (uncommon) value of the Cosseratcouple modulus µc = 0 additional difficulties arise that can only be circumventedby the use of the generalized Korn first inequality. In cases 1 and 2, more realisticassumptions on the applied external loads Π are necessary to establish a lowerbound for the energy I and a control of the curvature independent of the magnitudeof deformation.

Altogether, the quasistatic finite micromorphic theory is established on firmmathematical grounds. The geometrically exact microstretch case (restricted man-ifold R

+ · SO(3, R)) can also be treated with the same method.An extension of the method to other choices of strain and curvature measures

needs to be made. However, this might be a non-trivial task due to certain deficien-cies of these measures.

The open case (case 3) allows for discontinuous macroscopic deformations andmight therefore be a model problem allowing us to describe fracture. The variationalframework presented is ideally suited for subsequent numerical treatment by thefinite-element method.

Appendix A. Notation

Let Ω ⊂ R3 be a bounded domain with Lipschitz boundary ∂Ω and let Γ be

a smooth subset of ∂Ω with non-vanishing two-dimensional Hausdorff measure.

1010 P. Neff

For a, b ∈ R3 we let 〈a, b〉

R3 denote the scalar product on R3 with associated vector

norm ‖a‖2R3 = 〈a, a〉

R3 . We denote by M3×3 the set of real 3×3 second-order tensors,

using uppercase letters, and by T(3) the set of all third-order tensors. The standardEuclidean scalar product on M

3×3 is given by 〈X, Y 〉M3×3 = tr [XY T], and thus the

Frobenius tensor norm is ‖X‖2 = 〈X, X〉M3×3 . In the following we omit the index

R3, M3×3.The identity tensor on M

3×3 will be denoted by 1l, so that trX = 〈X, 1l〉. Welet Sym and PSym denote the symmetric and positive definite symmetric tensors,respectively. We adopt the usual abbreviations of Lie-group theory, i.e.

GL(3, R) := X ∈ M3×3 | det X = 0

the general linear group, and

SL(3, R) := X ∈ GL(3, R) | det X = 1,

O(3) := X ∈ GL(3, R) | XTX = 1l,

SO(3, R) := X ∈ GL(3, R) | XTX = 1l, det X = 1

with corresponding Lie algebras so(3) := X ∈ M3×3 | XT = −X of skew

symmetric tensors and sl(3) := X ∈ M3×3 | trX = 0 of traceless tensors.

We set sym(X) = 12 (XT + X) and skew(X) = 1

2 (X − XT) such that X =sym(X) + skew(X).

For X ∈ M3×3 we set for the deviatoric part dev X = X − 1

3 tr[X] · 1l ∈ sl(3)and for vectors ξ, η ∈ R

n we have the tensor product (ξ ⊗ η)ij = ξiηj . The operatoraxl : so(3, R) → R

3 is the canonical identification. We write the polar decompositionin the form F = RU = polar(F )U with R = polar(F ) being the orthogonal partof F . For a second-order tensor X we define the third-order tensor

h = DxX(x) = (∇(X(x) · e1),∇(X(x) · e2),

∇(X(x) · e3)) = (h1, h2, h3) ∈ M3×3 × M

3×3 × M3×3.

For third-order tensors h ∈ T(3), we set

‖h‖2 =3∑

i=1

‖hi‖2

together with

sym(h) := (sym h1, sym h

2, sym h3) and tr [h] := (tr [h1], tr [h2], tr [h3]) ∈ R

3.

Moreover, for any second-order tensor X we define X · h := (Xh1, Xh2, Xh3) andh · X, correspondingly.

Quantities with a bar, e.g. the micropolar rotation Rp, represent the microp-olar replacement of the corresponding classical continuum rotation R. In generalwe work in the context of nonlinear, finite elasticity. For the total deformationϕ ∈ C1(Ω, R3) we have the deformation gradient F = ∇ϕ ∈ C(Ω, M3×3) and weuse ∇ in general only for column vectors in R

3. Furthermore, S1(F ) and S2(F )denote the first and second Piola–Kirchhoff stress tensors, respectively. The firstand second differentials of a scalar-valued function W (F ) are written DF W (F ) · H

Minimizers for a finite-strain micromorphic elastic solid 1011

and D2F W (F ) · (H, H), respectively. Sometimes we also use ∂XW (X) to denote the

first derivative of W with respect to X. We employ the standard notation of Sobolevspaces, i.e. L2(Ω), H1,2(Ω), H1,2

(Ω), which we use indifferently for scalar-valuedfunctions as well as for vector-valued and tensor-valued functions. Moreover, weset ‖X‖∞ = supx∈Ω ‖X(x)‖. We define H1,2

(Ω, Γ ) := φ ∈ H1,2(Ω) | φ|Γ = 0,where φ|Γ = 0 is to be understood in the sense of traces and by C∞

0 (Ω) we denoteinfinitely differentiable functions with compact support in Ω. We use uppercase let-ters to denote possibly large positive constants, e.g. C+, K and lower case lettersto denote possibly small positive constants, e.g. c+, d+. The smallest eigenvalue ofa positive definite symmetric tensor P is abbreviated by λmin(P ).

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(Issued 6 October 2006 )