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1 Figure# A variant of the linear isotropic indeterminate couple-stress model with symmetric local force-stress, symmetric nonlocal force-stress, symmetric couple-stresses and orthogonal boundary conditions

Abstract

In this paper we venture a new look at the linear isotropic indeterminate couple-stress model in the general framework of second-gradient elasticity and we propose a new alternative formulation which obeys Cauchy–Boltzmann’s axiom of the symmetry of the force-stress tensor. For this model we prove the existence of solutions for the equilibrium problem. Relations with other gradient elastic theories and the possibility of switching from a fourth-order (gradient elastic) problem to a second-order micromorphic model are also discussed with the view of obtaining symmetric force-stress tensors. It is shown that the indeterminate couple-stress model can be written entirely with symmetric force-stress and symmetric couple-stress. The difference of the alternative models rests in specifying traction boundary conditions of either rotational type or strain type. If rotational-type boundary conditions are used in the integration by parts, the classical anti-symmetric nonlocal force-stress tensor formulation is obtained. Otherwise, the difference in both formulations is only a divergence-free second-order stress field such that the field equations are the same, but the traction boundary conditions are different. For these results we employ an integrability condition, connecting the infinitesimal continuum rotation and the infinitesimal continuum strain. Moreover, we provide the orthogonal boundary conditions for both models.

arXiv:1504.00868v1 [math-ph] 3 Apr 2015

A variant of the linear isotropic indeterminate couple stress model

with symmetric local force-stress, symmetric nonlocal force-stress,

symmetric couple-stresses and complete traction boundary conditions

Ionel-Dumitrel Ghiba1and Patrizio Neﬀ2and Angela Madeo3and Ingo M¨unch4

April 6, 2015

Dedicated to Richard Toupin, in deep admiration of his scientiﬁc achievements.

Abstract

In this paper we venture a new look at the linear isotropic indeterminate couple stress model in the

general framework of second gradient elasticity and we propose a new alternative formulation which obeys

Cauchy-Boltzmann’s axiom of the symmetry of the force stress tensor. For this model we prove the existence

of solutions for the equilibrium problem. Relations with other gradient elastic theories and the possibility

to switch from a 4th order (gradient elastic) problem to a 2nd order micromorphic model are also discussed

with a view of obtaining symmetric force-stress tensors. It is shown that the indeterminate couple stress

model can be written entirely with symmetric force-stress and symmetric couple-stress. The diﬀerence of

the alternative models rests in specifying traction boundary conditions of either rotational type or strain

type. If rotational type boundary conditions are used in the partial integration, the classical anti-symmetric

nonlocal force stress tensor formulation is obtained. Otherwise, the diﬀerence in both formulations is only a

divergence–free second order stress ﬁeld such that the ﬁeld equations are the same, but the traction bound-

ary conditions are diﬀerent. For these results we employ a novel integrability condition, connecting the

inﬁnitesimal continuum rotation and the inﬁnitesimal continuum strain. Moreover, we provide the com-

plete, consistent traction boundary conditions for both models.

Key words: symmetric Cauchy stresses, generalized continua, non-polar material, microstructure, size ef-

fects, microstrain model, non-smooth solutions, gradient elasticity, strain gradient elasticity, couple stresses,

polar continua, hyperstresses, Boltzman axiom, dipolar gradient model, modiﬁed couple stress model, con-

formal invariance, micro-randomness, symmetry of couple stress tensor, consistent traction boundary condi-

tions.

AMS 2010 subject classiﬁcation: 74A30, 74A35.

1Ionel-Dumitrel Ghiba, Lehrstuhl f¨ur Nichtlineare Analysis und Modellierung, Fakult¨at f¨ur Mathematik, Universit¨at Duisburg-

Essen, Thea-Leymann Str. 9, 45127 Essen, Germany; Alexandru Ioan Cuza University of Ia¸si, Department of Mathematics,

Blvd. Carol I, no. 11, 700506 Ia¸si, Romania; Octav Mayer Institute of Mathematics of the Romanian Academy, Ia¸si Branch,

700505 Ia¸si; and Institute of Solid Mechanics, Romanian Academy, 010141 Bucharest, Romania, email: dumitrel.ghiba@uni-due.de,

dumitrel.ghiba@uaic.ro

2Corresponding author: Patrizio Neﬀ, Head of Lehrstuhl f¨ur Nichtlineare Analysis und Modellierung, Fakult¨at f¨ur Mathematik,

Universit¨at Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen, Germany, email: patrizio.neﬀ@uni-due.de

3Angela Madeo, Laboratoire de G´enie Civil et Ing´enierie Environnementale, Universit´e de Lyon-INSA, Bˆatiment Coulomb,

69621 Villeurbanne Cedex, France; and International Center M&MOCS “Mathematics and Mechanics of Complex Systems”, Palazzo

Caetani, Cisterna di Latina, Italy, email: angela.madeo@insa-lyon.fr

4Ingo M¨unch, Institute for Structural Analysis, Karlsruhe Institute of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany,

email: ingo.muench@kit.edu

1

Contents

1 Introduction 3

1.1 General viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The linear indeterminate couple stress model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Our perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Notational agreements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Preliminaries 8

2.1 Related models in isotropic second gradient elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Discussion of invariance properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Conformal invariance of the curvature energy and group theoretic arguments in favour of the

modiﬁed couple stress theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 The classical indeterminate couple stress model based on

k∇[axl(skew ∇u)]k2with skew-symmetric nonlocal force-stress . . . . . . . . . . . . . . . . . . . . 14

3 The new isotropic gradient elasticity model with symmetric nonlocal force stress and sym-

metric hyperstresses 16

3.1 Formulation of the complete boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Existence and uniqueness of the solution in the Curl(sym ∇u)-formulation . . . . . . . . . . . . . 22

3.3 Traction b oundary condition in the Curl (sym ∇u)-formulation

versus the ∇[axl(skew∇u)]-formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Principle of virtual work in the indeterminate couple stress model . . . . . . . . . . . . . . . . . . 27

4 Relation to the Cosserat-micropolar and micromorphic model 31

5 Conclusion 35

6 Epilogue: Much ado about nothing 35

References 39

A The traction boundary conditions in the Curl (sym ∇u)-formulation and in the ∇[axl(skew∇u)]-

formulation are diﬀerent 42

B From second order couple stress tensors to third order moment stress tensors and back 44

C The name of the indeterminate couple stress model 45

2

1 Introduction

1.1 General viewpoint

The Cosserat model is an extended continuum model which features independent degrees of rotation in addition

to the standard translational degrees of particles, see [28, 83, 81, 66, 64] for a detailed exposition. The prize,

which has to be paid for this extension are non-symmetric force stress tensors together with so-called couple

stress tensors which then represent the response of the model due to spatially diﬀering Cosserat rotations. The

couple stress model is the Cosserat model [48] with restricted rotations, i.e. in which the Cosserat rotations

coincide with the continuum rotations. As such it belongs also to a certain subclass of gradient elasticity

models1, where the higher derivatives only act on the continuum rotations. This constitutes a big conceptual

advantage since the interpretation of the Cosserat rotations as new physical degrees of freedom is in general a

diﬃcult task. Such a model is also called a model with “latent microstructure” [11, 12].

Let F=R U be the polar decomposition of the deformation gradient F=∇ϕinto rotation R∈SO(3)

and positive deﬁnite symmetric right stretch tensor U=√FTF, where ϕ: Ω ⊂R3→R3characterizes the

deformation of the material ﬁlling the domain Ω ⊆R3. We write R= polar(F). In a variational context, the

energy density Wto be minimized in the geometrically nonlinear constrained Cosserat model is given by

W=W(U−

|{z}

strain

,polar(F)T∇xpolar(F)

|{z }

curvature

),(1.1)

which reduced form follows from left-invariance of the Lagrangian Wunder superposed rotations. In this paper,

our objectives are much more modest. We will only be concerned with the linearized variant of (1.1), which

can be written as

W=W( sym ∇u

|{z }

inﬁnitesimal

strain

,∇[axl(skew∇u)]

|{z }

inﬁnitesimal

curvature

) = Wlin(sym ∇u) + Wcurv (∇[axl(skew∇u)]),(1.2)

where u: Ω ⊂R3→R3is the displacement and

∇axl(skew∇u) = 2 curl u. (1.3)

The energy density (1.2) is the classical Lagrangian for the indeterminate couple stress formulation. As will be

seen later, this formulation leads naturally to totally skew symmetric nonlocal force stress contributions.

Toupin already remarked on an alternative representation of the energy (1.1) [102, Section 6] which leads,

in its linearized variant given by Mindlin [72, eq. (2.4)] to a dependence on

W=W(sym ∇u, Curl (sym ∇u)) = Wlin(sym ∇u) + Wcurv(Curl (sym ∇u)) ,(1.4)

due to the equivalence

∇(axl skew∇u) = 1

2∇curl u= (Curl (sym ∇u))T

instead of (1.2). The representation 1

2∇curl uis directly derived from the original Cosserat model [15, 98]. Both

authors, Toupin and Mindlin, noted that now, comparing (1.2) and (1.4) the force stress tensors and the couple

stress tensors are changed while the balance of linear momentum equation remains unchanged such that these

concepts are not uniquely deﬁned (see also Truesdell and Toupin’s remark on null-tensors [105, p. 547]. However,

they apparently did not realize that it is possible to use this ambiguity to obtain completely symmetric force

stress tensors also in the couple stress model which is otherwise the paragon for a model having non-symmetric

force stress tensors. We also need to remark that in a purely mechanical context, the observation of size-eﬀects

does not necessitate to introduce skew-symmetric stress-tensors [5].

In this paper we do not discuss in detail the ﬁeld of applications of such a special format of gradient elasticity

model. Suﬃce it to say that much attention is directed to nano-scaled material in which size-eﬀects may become

important, which may make the present model applicable at strong stress gradients in the vicinity of cracks,

or more generally, in highly heterogeneous media. We must also warn the reader: the indeterminate couple

stress model is, in our view, a certain singular limit of the Cosserat model with independent displacements and

micro-rotation and therefore some degenerate behaviour is to be expected throughout.

1Le Roux [96] seems to give for the ﬁrst time a second gradient theory in linear elasticity using a variational formulation [65, 67].

3