## Abstract

The stress-invariance problem for a chiral (non-centrosymmetric) micropolar material model is explored in two different planar problems: the in-plane and the anti-plane problems. This material model grasps direct coupling between the Cauchy-type and Cosserat-type (or micropolar) effects in Hooke’s law. An identical strategy of invariance is set for both problems, leading to a remarkable similarity in their results. For both problems, the planar components of stress and couple-stress undergo strong invariance, while their out-of-plane counterparts can only attain weak invariance, which restricts all compliance moduli transformations to a linear type. As an application, when heterogeneous (composite) materials are subjected to weak invariance, their effective (volume-averaged) compliance moduli undergo the same linear shift as that of the moduli of the local phases forming the material, independently of the microstructure, geometry and phase distribution. These analytical results constitute a valuable means to validate computational procedures that handle this particular type of material model.

## 1. Introduction

The stress-invariance problem investigates the possibility of modifying the stiffness constants of a heterogeneous elastic body of arbitrary shape without altering the stress field within this deformed body, provided it is subjected to static traction boundary conditions throughout its entire boundary. The schematic layout of figure 1 illustrates the concept. This invariance was first recognized to hold in planar elasticity (Cherkaev *et al.* 1992; Thorpe & Jasiuk 1992), and subsequently explored and generalized to various situations (e.g. Dundurs & Markenscoff 1993; Moran & Gosz 1994; Norris 1999; Hu & Weng 2001; Jasiuk & Ostoja-Starzewski 2003). Reviews of this subject and related topics appeared in Milton (2002), Ostoja-Starzewski (2008) and Jasiuk (2009).

One extension of the stress-invariance problem focused on the Cosserat (micropolar) elasticity (Ostoja-Starzewski & Jasiuk 1995). As is well known, theory grants a micropolar particle six degrees of freedom (three translations *u*_{i} and three rotations *ϕ*_{i}, *i*=1,2,3), allowing it to support moment loads in addition to forces. Thus, a quasi-infinitesimal element does not reduce to zero size in contradistinction to its classical counterpart: the continuum point. As a result, the micropolar theory has an intrinsic length scale and can describe various phenomena more accurately, e.g. liquid crystals, complex mixtures, granular media and lattice beam structures. As for the kinematics, we define the strain tensor *γ*_{ij} and curvature tensor *κ*_{ij} that are related to the displacement and rotation vectors as follows (*e*_{kij} is the Levi-Civita permutation tensor):
1.1a
and
1.1b
The constitutive relations for force-stress (simply referred to as stress) *τ*_{ij} and couple-stress *μ*_{ij} are specified as
1.2a
and
1.2b
where and are generalized elastic tensors. The form of these tensors varies depending on the material isotropy/anisotropy and homogeneity. In the work of Mora & Waas (2000) and Chung & Waas (2009), the micropolar elasticity constants are determined for circular cell honeycomb structures. The equilibrium equations are derived from the generalized linear and angular momentum equations, and are given as
1.3a
and
1.3b

So far, the constitutive relations relate the stresses to the strains, on the one hand, and the couple-stresses to the curvatures, on the other hand. Special microstructures of some materials make both the stress and the couple-stress directly dependent on both strains and curvatures, representing the phenomenon of chirality (see equations (2.2)), this is most easily demonstrated in a one-dimensional helix model. Consider the helicoidal rod (of non-zero helix angle) of radius *r* and Young’s modulus *E* that is subject to an applied axial force *f* and twisting moment *m* on both ends. As a result, the rod extends and twists according to the coupled equations
1.4
For non-zero helix angles, the coupling terms *c*_{12} and *c*_{21} are non-zero. The introduction of chirality into a general three-dimensional micropolar model was conducted by Nowacki (1986) and this topic will be thoroughly discussed in the following section. Two examples of lattice structures that exhibit non-centrosymmetric micropolar behaviour are shown in figure 2.

## 2. Constitutive model

In formulating the chiral material model, Nowacki (1986) considered a generalized form of the free energy *g*, involving three components: strain-induced, curvature-induced and strain-curvature-induced; it is given as
2.1
Differentiating this energy expression with respect to strain and curvature tensors, the following Hooke’s law was obtained:
2.2a
and
2.2b
The stiffness tensors can be explicitly determined from the unit cells of the chiral lattice structures shown in figure 2 by modifying the methods employed for micropolar lattices (Ostoja-Starzewski 2008). In the following, we assume these tensors to be class *C*^{2} functions and take the material to be simply connected. For an isotropic continuum, these tensors are defined as follows (*δ*_{ij}≡Kronecker’s tensor):
2.3a
2.3b
and
2.3c
This material model has nine independent stiffness constants. Their physical significance is given in table 1. Consequently, an equal number of compliance constants can be formed. We introduce these compliance constants because they frequently appear in the invariance relations derived later
2.4a
2.4b
2.4c
2.4d
2.4e
2.4f
2.4g
2.4h
and
2.4i
The stiffness constants introduced in the definitions of tensors *C*_{ijkl} have their units presented in table 2. In contradistinction to classical elasticity, the micropolar theory has an inherent ‘length scale’, allowing it to study problems of mechanics of microstructures. Inherent length scales can be formed by taking appropriate ratios of stiffness constants in the material model. For the non-centrosymmetric micropolar model, we can form two independent sets of length scales defined as
2.5a
and
2.5b
In total, we have 18 length scales (nine in each set) inherent to this non-classical model. Note that the non-chiral micropolar model can only admit one set of length scales ({*L*_{1}} in this case). The stress-invariance problem is nevertheless indifferent to these length scales because it involves a macro-scale analysis conducted independently of the material’s microstructure.

## 3. First planar problem

Consider the planar problem where the displacement field has no component in the out-of-plane direction (*x*_{3}): ** u**=(

*u*

_{1},

*u*

_{2},0), while the rotation field is restricted in that direction:

**=(0,0,**

*ϕ**ϕ*

_{3}). As the problem is planar,

*u*

_{1},

*u*

_{2},

*ϕ*

_{3}depend only on

*x*

_{1}and

*x*

_{2}. The resulting strain and curvature tensors are expressed as follows: 3.1a and 3.1b The stress tensor components are then expressed in term of strains and curvatures by applying the constitutive law of equation (2.2a), 3.2a 3.2b 3.2c 3.2d 3.2e 3.2f 3.2g 3.2h and 3.2i Similarly, we obtain the couple-stress tensor components by applying equation (2.2b), 3.3a 3.3b 3.3c 3.3d 3.3e 3.3f 3.3g 3.3h and 3.3i The non-trivial compatibility relations that govern this problem are given as 3.4a 3.4b and 3.4c Various possibilities exist when it comes to substituting for the strains and curvatures in these compatibility equations leading to multiple invariance approaches. Among all these approaches, one prevails by maximizing the number of invariant components. The steps of this approach are

— consider the stress set {

*τ*_{11},*τ*_{22},*τ*_{12},*τ*_{21}} for substitution in the compatibility equations,— derive transformation relations for compliance constants that ensure invariance, and

— express the remaining stress and couple-stress components in terms of the invariant set, that is {

*τ*_{11},*τ*_{22},*τ*_{12},*τ*_{21}}, and derive the remaining transformation relations.

This approach prioritizes invariance for the planar components of the stress tensor {*τ*_{11},*τ*_{22},*τ*_{12},*τ*_{21}}. The invariance of the planar components of the couple-stress tensor {*μ*_{11},*μ*_{22},*μ*_{12},*μ*_{21}} becomes second in importance. Finally, the *x*_{3}-direction components are the least important to be rendered invariant. It becomes clear that equations (3.2a,*b*) must be used to solve for *γ*_{11} and *γ*_{22}, while equations (3.2d,*e*) should be used to solve for *γ*_{12} and *γ*_{21}. As a result, we obtain
3.5a
3.5b
3.5c
and
3.5d
For the shear curvatures *κ*_{13} and *κ*_{23}, any of these four couples {(*τ*_{13}, *τ*_{23}); (*τ*_{31}, *τ*_{32}); (*μ*_{13}, *μ*_{23}); (*μ*_{31}, *μ*_{32})} is a mathematically valid candidate for substitution. Nevertheless, we notice that, if stress components (first or second choice) are picked for substitution, the stiffness terms *ρ* and *σ* appear in the resulting equations and consequently in the transformation relations. The planar couple-stress components have *ρ* and *σ* in their definitions (equations (3.3a,*b*,*d*,*e*)), thus, their invariance generates transformations for *ρ* and *σ* that differ from those generated for the stress components’ invariance. Since the planar couple-stress components have an invariance precedence over the three-direction stress components, *ρ* and *σ* must be suppressed when executing the first step of our invariance approach. Therefore, the choice to substitute with stress components is dropped and couple-stress components are picked instead. As a result, we have
3.6a
and
3.6b
Substituting the previous seven equations into the compatibility relations, we get, after extensive algebraic work, the following field equations:
3.7a
3.7b
3.7c
3.7d
and
3.7e
The above field equations are identical to those obtained by Ostoja-Starzewski & Jasiuk (1995), where no chirality is considered. As such, the invariance relations must be no different than those of the micropolar situation discussed in that reference,
3.8a
3.8b
3.8c
and
3.8d
where *m* and *c* are two independent constants arbitrarily chosen but constrained to keep the new compliance constants positive. Equation (3.8d) implies *m*>0.

We now explore the invariance possibilities of the planar couple-stress components. Starting with the shear components, we exploit the fact that *τ*_{12} and *τ*_{21} are now invariant and we find, on account of equations (3.2d,*e*) and (3.3d,*e*),
3.9a
and
3.9b
As such, *μ*_{12} and *μ*_{21} can be rendered invariant if the following shifting relations apply:
3.10a
and
3.10b
Similarly, expressing *μ*_{11} and *μ*_{22} in terms of *τ*_{11} and *τ*_{22} (both already invariant) gives
3.11a
and
3.11b
Therefore, *μ*_{11} and *μ*_{22} are made invariant if the following shifting relations for *U* and *T* hold:
3.12a
and
3.12b
Note that equations (3.10a) and (3.12a) reveal identical transformations between *U* and . If this were not the case, it would become impossible to render the axial and shear components of the couple-stress tensor invariant simultaneously.

Our last step is to investigate whether some out-of-plane components can be rendered invariant without distorting the derived transformation relations necessary for the invariance of planar components. First, it is realized that *μ*_{31} and *μ*_{32} can be expressed in terms of *μ*_{13} and *μ*_{23} (already invariant) as follows:
3.13a
and
3.13b
These two components can be made invariant if *R* satisfies the transformation
3.14
For the remaining components, we express them in terms of their invariant counterparts to facilitate the detection of the invariance conditions,
3.15a
3.15b
3.15c
3.15d
3.15e
and
3.15f
As noticed, the expressions of these components involve compliance terms that were already assigned shifting relations when analysing invariance for the planar components. This however does not eliminate the possibility to grant invariance to these components. In fact, there exists a *restrained* possibility to achieve what will be referred to as *weak invariance*. We explain this concept through a brief example. Consider equation (3.15a) where invariance of *τ*_{33} requires . Equations (3.8a,*c*) tell us that this can only be true if constant *c*=0. As a matter of fact, it is shown that all components of equations (3.15) can be rendered invariant if *c*=0. In such a case, all the compliance moduli will follow a linear shift that results in weak invariance. On the other hand, if *c*≠0, none of the components defined in equations (3.15) can be made invariant, this is the situation of strong invariance. In conclusion, weak invariance is granted to all stress and couple-stress components under linear (constrained) shifting relations for all compliance moduli, while strong invariance is granted to planar components only (in addition to *μ*_{13} and *μ*_{23}) without restricting the compliance moduli shifting relations. Note that *W* (and consequently *η*) does not appear in this problem. As such, the invariance procedure is indifferent to this modulus, which can assume any physically permissible transformation without affecting the invariance of any stress component.

## 4. Second planar problem

Consider the planar problem having the non-trivial kinematic components: the out-of-plane displacement *u*_{3} and two rotations *ϕ*_{1} and *ϕ*_{2}. All three components solely depend on *x*_{1} and *x*_{2}. The resulting strain and curvature tensors are given as
4.1a
and
4.1b
The relevant compatibility equations for this problem involve the non-zero components of the above tensors. They are
4.2a
4.2b
4.2c
4.2d
and
4.2e
The stress tensor components are found by applying equation (2.2a). They are thus expressed as follows:
4.3a
4.3b
4.3c
4.3d
4.3e
4.3f
4.3g
4.3h
and
4.3i
Similarly, the couple-stress components are found using equation (2.2b),
4.4a
4.4b
4.4c
4.4d
4.4e
4.4f
4.4g
4.4h
and
4.4i
We follow the same procedure of the first problem by prioritizing invariance to planar stresses: substitute for the stresses in the compatibility equations. We first express curvatures in terms of stresses by inverting equations (4.3a,*b*,*d*,*e*) to obtain
4.5a
4.5b
4.5c
and
4.5d
For the sake of simplifying the mathematical expressions in the second planar problem, three compliance moduli *T*, *R* and *M* will be slightly modified from how they were initially defined in equations (2.4f), (2.4d) and (2.4c), respectively. The remaining moduli are unchanged. Consequently, nine compliance moduli still correlate to nine independent stiffness constants. The new definitions are specified as
4.6a
4.6b
and
4.6c
Substituting for the curvatures in the first two compatibility equations, we obtain the resulting field equations
4.7a
and
4.7b
Invariance of the planar stress components is possible providing the following shifting relations hold:
4.8a
4.8b
and
4.8c
where *m* and *c* are arbitrary constants. For the couple-stresses, we express them in terms of the planar components of the stress tensor (already invariant) to allow identification of the invariance criteria. Thus, we have
4.9a
and
4.9b
The invariance of the above two components requires
4.10a
and
4.10b
The planar shear components of the couple-stress tensor are expressed as
4.11a
and
4.11b
The invariance of these terms requires the following transformation to hold:
4.12a
and
4.12b
Note that equations (4.10a) and (4.12a) are identical. This issue was observed and interpreted in the first planar problem for modulus *U*. The same interpretation applies to this problem. We proceed to explore the invariance possibilities among the out-of-plane components. We invert the last four equations of (4.3) to obtain
4.13a
4.13b
4.13c
and
4.13d
The last three compatibility equations can be expressed in terms of the stress components as follows:
4.14a
4.14b
and
4.14c
The invariance of these four shear components is not possible unless *T* and *V* are restricted to a purely linear shift. In such a case, *S* and *P* are allowed to admit the following transformation:
4.15a
and
4.15b
As noticed, the concept of weak invariance is again invoked in this problem. The subsequent analysis will explore this type of invariance on the remaining components. We have the following relations for the out-of-plane components:
4.16a
4.16b
4.16c
4.16d
4.16e
and
4.16f
Weak invariance applies to all the above components. A similar conclusion was reached when analysing the first problem. Summarizing, the planar components of stress and couple-stress tensors undergo strong invariance while the out-of-plane components can only undertake weak invariance. Note that the compliance modulus *A* was not observable in this problem (*W* was equivalently unobservable in the first planar problem), hence it is free to be transformed without affecting the results. Finally, we remark the importance of chirality in achieving invariance in the anti-plane problem. Had the *C*^{(3)} tensor been absent in this problem, no invariance could have been realized. This corroborates the conclusion that Ostoja-Starzewski & Jasiuk (1995) had reached when investigating invariance for simple micropolar models.

## 5. Invariance in composite materials

We now apply the stress-invariance property to composite materials. In particular, we investigate the transformation that effective properties of a composite body must undergo to secure stress invariance. For our material model, the constitutive law can be expressed as the inverse of equations (2.2a,*b*),
5.1a
and
5.1b
It can be shown that the compliance tensors, *S*_{ijkl}’s admit the same mathematical representation as that of their stiffness counterparts *C*_{ijkl}’s. Let us consider the volume average of both sides of the above equation, and define the effective compliance tensors as explained below
5.2a
and
5.2b
where all the bracketed quantities represent volume-averaged values. Consider the composite body subjected to weak invariance (all stress and couple-stress components are invariant, and so are their volume-averaged values), so that
5.3a
and
5.3b
In the case of weak invariance, all compliance moduli are subject to a linear shift, and so is the compliance tensor *S*, thus,
5.4
We easily observe that the effective compliance tensors in both configurations are related by
5.5
As a result, the effective compliance tensors undergo the same linear shift as their microscopic counterparts, hence
5.6

The above results, in conjunction with the general transformation relations derived for stress invariance in both planar problems, constitute a practical means to validate computational procedures (e.g. a finite-element method) for simulating complex mechanics problems adopting this material model. Technically, the computational solver treats one of the discussed planar problems in two different configurations; the arbitrary domain of interest and traction boundary conditions are kept identical in both configurations, while the compliance moduli are to obey the invariance transformations. The equality of the numerically evaluated stress and couple-stress fields in both configurations would confirm the accuracy of the computational model.

## 6. Conclusions

The stress-invariance problem for the chiral micropolar material model is explored in two different planar problems (in-plane and anti-plane problems). An identical strategy of invariance is set for both problems, leading to a remarkable similarity in their results. For both problems, the planar components of stress and couple-stress undergo a strong invariance, while their out-of-plane counterparts can only attain weak invariance, which restricts all compliance moduli transformations to a linear type. As an application, when heterogeneous (composite) materials are subject to weak invariance, their effective (volume-averaged) compliance moduli undergo the same linear shift as that of the moduli of the local phases forming the material, independently from the microstructure, geometry and phase distribution. These analytical results provide a practical means to validate complex computational procedures designed to simulate the mechanical behaviour of a given body that constitutively obeys this material model. In summary, the consideration of the non-centrosymmetric micropolar material model enriched the investigation of the generalized stress-invariance problem. Further advancements featured by extending the material model (e.g. nonlinear effects) or diversifying the problem attributes for the same material model (e.g. effects of body forces or eigenstrains) can be achieved in accordance with the findings of this work.

## Acknowledgements

This research was made possible by the support from Sandia National Laboratories DTRA (grant HDTRA1-08-10-BRCWMD) and the NSF (grant CMMI-1030940).

- Received December 21, 2010.
- Accepted April 13, 2011.

- This journal is © 2011 The Royal Society