## Abstract

A mechanistic model is presented for an open-cell cellular solid consisting of a three-dimensional network of elastic struts. By considering the bending and torsion as well as stretching and buckling of the struts, we allow for length-scale effects in the macroscopic response. Constitutive equations are developed for the force and couple stress tensors, accounting for finite deformations and anisotropy. The consistent tangent stiffness operators are derived and the equations are fully implemented in a nonlinear two-dimensional finite-element solution scheme for the coupled displacement/rotation problem. A boundary-value problem of a shear gap with prescribed boundary rotations is analysed, and the model is shown to predict the well-known gap–size effect. The mechanistic model allows some detailed interpretation of the micropolar behaviour, such as the effects of strut slenderness, strut length and anisotropy.

## 1. Introduction

Many cellular solids have a microstructure that is coarse compared with external dimensions. In some applications, e.g. expanding adhesives, foam is confined in a narrow gap, which may cover only a few cells across its height. As with any microstructured material, one should expect the standard continuum description (which allows for no length-scale effects) to fail in the vicinity of boundaries, within a distance from the boundary of a few times the cell size. Indeed, such effects have been proven. Here, Lakes (1986), for example, reported specimen size effects in bending and torsion experiments on closed-cell polymeric foams, where the specimen dimensions were comparable with the cell size. Andrews *et al*. (2001) found that aluminium foam loaded in simple shear appeared to become stiffer as the test specimen was scaled down. This is because the specimen was adhesively bonded to a rigid wall and therefore the boundary is more constrained than the bulk material, thus forming a stiff boundary layer whose thickness does not scale with the specimen size.

In order to capture the effect of an internal length scale, one may enrich the continuum description, by introducing additional higher gradients of displacements into the equations, or by introducing additional degrees of freedom (Forest & Sievert 2006). One of the simplest extensions of the standard continuum theory that includes an internal length scale is the micropolar theory. Micropolar continuum theory (Eringen 1998), or Cosserat theory (which is the same thing for our purposes), extends the standard kinematics by an independent rotation field. Thus, one may use the micropolar rotation field to represent some rotation in the internal structure of the material. The gradient of the extra rotation field then represents an internal curvature, which produces internal couples and an external (continuum) couple stress.

The existence of an internal length scale is conspicuous evidence of the role played by microstructural elements of the corresponding size. To understand the origins of the length-scale effects, one may directly model the internal structure itself. Onck *et al*. (2001) modelled the size effects in two-dimensional hexagonal foam structures using Euler–Bernoulli beam elements. Diebels & Steeb (2003) used the same approach, and showed that rotational degrees of freedom are necessary in order for a macroscopic continuum model to capture the observed boundary effects. They also showed that it was possible to describe the macroscopic behaviour using a linear Cosserat elastic solid. The independent rotation field of the Cosserat theory may then be interpreted as the average rotation of the strut connections or vertices. Tekoğlu & Onck (2008) reported that the length-scale effects in a two-dimensional foam structure in the linear elastic regime were best captured by a gradient theory obtained by constraining the rotational degree of freedom in the micropolar theory. This appears to contradict the findings of Diebels & Steeb (2002), but is possibly a matter of choice of model structure.

The response of a microstructural model may be averaged to yield overall stress and couple stress fields (see Onck 2002; Diebels & Steeb 2003). It is thus possible to combine the macroscopic micropolar or Cosserat theory with microstructural modelling. The idea, known as second-order computational homogenization, is to replace the constitutive equations of, for example, Cosserat elasticity by an explicit microstructural model, which is solved as an embedded boundary-value problem. During the last decade, quite a few second-order homogenization schemes have appeared, with different types of generalized continuum theory on the macroscale: Forest & Sab (1998) and Forest *et al*. (1999) used a linear Cosserat theory; Larsson & Diebels (2007) considered a nonlinear micropolar description; and Kouznetsova *et al*. (2002) used a large strain full gradient theory. Second-order homogenization applied to cellular solids was described by Dendievel *et al*. (1998) and Pradel & Sab (1998), using linear Cosserat theory. Ebinger *et al*. (2005) homogenized a two-dimensional truss structure of Timoshenko beam elements, and showed that size effects may be captured by Cosserat theory. It has thus been established that the micropolar continuum theory is able to describe the size-dependent behaviour of coarse foams, and that the micropolar rotations may be interpreted in terms of mechanisms on the microlevel.

For efficient computations, it is desirable to represent the homogenized response by a constitutive equation. Wang & Stronge (1999) derived a constitutive equation for a two-dimensional regular structure (honeycomb), based on beam theory on the microlevel. Their model is linear and restricted to small deformations.

The objective of the present work is to provide a set of nonlinear micropolar constitutive equations for large deformations, based on simple relevant micromechanisms in a cellular solid. This is accomplished by modelling a three-dimensional affine network of elastic lines. A statistical mechanical approach will be adopted from a standard continuum model by Hård af Segerstad & Toll (2008), which was based on the strut forces caused by stretching and bending of a strut and compared well with experimental data. Micropolar character will be introduced by also including the couples caused by bending and twisting of struts.

## 2. Micropolar continuum

We shall begin by briefly developing the theory of a micropolar continuum for large deformations and large rotations in three dimensions. The theory is largely based on the work of Eringen (1998). The micropolar nonlinear kinematics is mostly due to Steinmann (1993).

A micropolar (or Cosserat) continuum involves two vectorial degrees of freedom: a velocity field and an angular velocity field,(2.1)respectively (figure 1). The overbars will be used to denote field quantities associated with the macroscale. To these fields correspond two conjugate surface forces, a force stress vector ** t** and a couple stress vector

**, such that the work rate done on a body enclosed by a boundary**

*m**Γ*is(2.2)The stress vectors are specified for any surface element with normal

**as(2.3)where is the (Cauchy) force stress tensor and is the couple stress tensor. Requiring balance of linear and angular momentum for**

*ν**Ω*leads to the field equations(2.4)(2.5)where the spatial gradient vector is defined as (

*e*_{k}being Cartesian basis vectors).

To establish conjugate deformation rates for and , we introduce (2.3) into (2.2), use equations (2.4) and (2.5) and take the limit as *Ω*→0,(2.6)where is the third-order permutation symbol. Equation (2.6) shows that the deformation rates conjugated to the force and couple stress are and , respectively.

In order to establish proper conjugate measures of stresses and strains, one introduces a nonlinear deformation map (figure 1)(2.7)such that(2.8)Using typical notation, we introduce the standard deformation gradient and its determinant as(2.9)noting that the gradient operator ∇=∂/∂** X** is defined in the reference configuration. The deformation gradient then gives the differential motion,(2.10)

We may now introduce a rotation map, represented by an orthogonal rotation tensor , such that (cf. Steinmann 1993)(2.11)In analogy with (2.9) and (2.10) a material third-order curvature tensor is defined as(2.12)which gives the differential rotation(2.13)It is noted that is skew symmetric, since(2.14)The product serves to left conjugate the term in square brackets. The rate of is related to by time differentiation of (2.12) using (2.11),(2.15)with . In the last step, we used the identity(2.16)

The work rate (2.6) may now be rewritten, using (2.8), (2.9) and (2.11),(2.17)This may be rearranged to identify the relevant measures of stress and strain(2.18)where(2.19)(2.20)are the Lagrangian stress and couple stress tensors, respectively, and(2.21)(2.22)are the conjugate strain rates. The strain measure follows immediately from (2.21) and the curvature measure follows from the comparison of (2.22) and (2.15)(2.23)(2.24)The second-order tensor is usually interpreted as a polar stretch tensor. Note, however, that, unlike the standard stretch tensor, is *not* symmetric since is the independent *micropolar* rotation and *not* the standard rotational part of . Nevertheless, and clearly provide the deformation gradient as(2.25)The second-order curvature tensor , equation (2.24), is said to be the *axial tensor* of .

For later purposes, we note that the rotation field may also be represented by a rotation vector , which is an eigenvector of ,(2.26)The relation between and is known as the Euler–Rodrigues representation(2.27)where(2.28)and is the rotation unit vector.

## 3. Micro-model

Having established the macroscopic formulation of the micropolar continuum, we now wish to model a cellular solid consisting of a network of interconnected struts of finite length. The idea is simple: struts will be considered as elastic lines, connecting at vertices. The vertices are assumed to follow the deformation mapping affinely and the rotation of the vertices is similarly linked to the rotation mapping . The deformation of the elastic lines gives rise to reaction forces and couples acting at the vertices. The force stress will represent the statistical (ensemble) average sum of forces per unit surface carried by struts crossing an infinitesimal surface element. Similarly, couple stress is taken to represent the corresponding average of couples transferred by struts across a surface element.

Since the deformation of the struts is directly linked to the macro-kinematics, there is no need to specify explicitly the topology of the network. It is sufficient to specify a statistical ensemble of strut properties.

### (a) Strut deformations

Each strut has a centroid ** X** and a

*vertex-to-vertex*vector

*r*_{0}in the reference configuration, and

**and**

*x***in the current configuration as shown in figure 2. It is assumed to have 12 degrees of freedom: three placements,**

*r***′ and three rotations**

*Χ***′ at one end and**

*θ***″ and**

*Χ***″ at the other. The axis of the vertex-to-vertex vector is given by a director (unit vector)**

*θ***in the reference configuration and**

*N***in the current configuration (figure 2).**

*n*Following Hård af Segerstad & Toll (2008), the placements ** Χ**′,

**″ are assumed to be affine so that, by Taylor expansion of the deformation map about the strut centre,(3.1)(3.2)Equations (3.1) and (3.2) give the vertex-to-vertex vector of a strut(3.3)This implies the orientation unit vector**

*Χ***in the current configuration and stretch**

*n**λ*of the vertex-to-vertex vector as(3.4)To model the deflection due to bending of the strut, we introduce the transverse displacement vector (cf. Hård af Segerstad & Toll 2008)(3.5)This is an objective measure of the deflection of one end of the strut with respect to the opposite vertex. The meaning of the vector

**is illustrated in figure 2.**

*w*The rotations of the strut ends are similarly linked affinely to the macro-rotation field at the vertices and ,(3.6)(3.7)The *relative rotation* between the two ends, defined by(3.8)is obtained by substitution of equations (3.6) and (3.7) into (3.8)(3.9)Further terms in the expansions (3.1), (3.2), (3.6) and (3.7) would require strain and curvature measures of higher order than provided by the micropolar theory. Thus, we are restricted to the expressions (3.3) and (3.9), approximating and ** r** to first order. To ensure that these representations are sufficient, we shall impose the following restriction:(3.10)which implies that the relative rotation between the two ends of a strut is small. Now introducing a vector(3.11)for the relative rotation, and using the smallness of , the Euler–Rodrigues representation (2.27) provides the relation(3.12)Also, from (2.24)(3.13)We thus obtain, for the relative rotation vector,(3.14)Applying once again the identity (2.16), we find that , whence(3.15)We now split into a torsional component and a bending component :(3.16)(3.17)so that(3.18)Using equation (3.4)

_{1}to pull back the directors, we obtain(3.19)

(3.20)

### (b) Forces and couples

Each strut is taken to carry end forces and end couples ** f**′ and

**′ at**

*c***−(1/2)**

*x***and**

*r***″ and**

*f***″ at**

*c***+(1/2)**

*x***. The strut forces are modelled following the approach of Hård af Segerstad & Toll (2008). We formulate an uncoupled response due to elongation and bending,(3.21)where**

*r**λ*is the stretch of the vertex-to-vertex vector, equation (3.4)

_{1}. The longitudinal response

*f*_{n}(

*λ*,

**) is assumed to be bilinear, with a high stiffness in the small strain regime, i.e. in the absence of buckling, and a low stiffness in the post-buckling regime. The bending force**

*n*

*f*_{t}(

**) is assumed linear in the bending deflection. Thus,(3.22)(3.23)where**

*w**k*

_{1}is the tensile stiffness of a straight strut;

*k*

_{2}is the corresponding stiffness of a strut compressed beyond the buckling limit

*λ*<

*λ*

_{c}; and

*k*

_{3}is the bending stiffness. The function

*H*is Heaviside's step function, defined as

*H*=1 when its argument is negative and

*H*=0 otherwise. Applying equations (3.4)

_{1}and (3.5), we obtain for the total strut force(3.24)

The strut is in moment equilibrium, i.e.(3.25)Since each strut is loaded by a couple, ** c**′ and

**″, and a force,**

*c***′ and**

*f***″, respectively, at the vertices, the couple varies linearly between the ends, so that the**

*f**mean couple*along the strut is(3.26)The mean couple is independent of the strut force, which is seen by comparing equation (3.25) and (3.26). The end couples

**′ and**

*c***″ are determined by (3.25), but are not needed for our purposes. We split the mean couple into an axial and transverse component and assume a simple linear response,(3.27)where**

*c**k*

_{n}and

*k*

_{b}are the torsional and bending stiffness, respectively. Now using equations (3.19), (3.20) and (3.4)

_{1}in equation (3.27),(3.28)Note that the forms (3.21) and (3.27) neglect any coupling between rotation and stretch.

## 4. Constitutive equations

### (a) Force stress

Using concepts well established in statistical mechanics (cf. Doi & Edwards 1989), we define the macrostress as follows. Equating the total force acting across a unit surface element with normal ** ν**, due to the Cauchy stress on the macroscale, to the ensemble average of the strut forces

**transferred by struts crossing the surface element on the microscale (see figure 3), one obtains(4.1)where**

*f**n*is the number of struts per unit (current) volume;

*r*is the strut length; and

**(**

*f***) is the force in a strut at orientation**

*n***. The notation denotes averaging over all possible**

*n***and**

*n**ψ*(

**) is the orientation distribution function, subject to the normalization condition(4.2)Denoting the ensemble average by angle brackets, we write(4.3)This is easily converted into the material stress , using the relations**

*n**r*=

*λr*

_{0}, and (3.4)

_{1},(4.4)noting that is the second Piola–Kirchhoff stress.

Introducing (3.24) into (4.4) and assuming that *r*_{0} is uniform, we obtain(4.5)where(4.6)

### (b) Couple stress

Similarly, the spatial couple stress tensor is the ensemble average of couples transferred by struts crossing a unit surface element having normal ** ν** (figure 3), such that(4.7)where

**(**

*c***) is the mean couple transferred by a strut crossing the surface element at orientation**

*n***. Thus,(4.8)The material couple stress, , becomes(4.9)Using the expression (3.28) for the couple, along with and**

*n**r*=

*λr*

_{0}, we obtain the constitutive equation for the couple stress,(4.10)where, again,

*r*

_{0}is assumed to be uniform.

## 5. Nonlinear solution technique

In this section, we establish the weak form of the balance equations (2.4) and (2.5), and a geometrically nonlinear finite-element solution scheme. We make the restriction to macroscopically two-dimensional plane strain problems. The restriction to two dimensions is a substantial simplification of the field problem, because the rotation reduces to a scalar field. A general three-dimensional treatment is considerably more complicated, as discussed by Steinmann (1993). Note, however, that the strut directors, and therefore the constitutive equations (4.5) and (4.10), are fully three dimensional.

A triad of basis vectors {*E*_{1}, *E*_{2}, *E*_{3}} is introduced, where the vector *E*_{3} is associated with the out-of-plane direction. In accordance with the restriction to plane strain, the rotation vector is *a priori* taken to be parallel with the *E*_{3} basis vector, so that and (cf. equation (2.28))(5.1)The Euler–Rodrigues representation (2.27) now reduces to(5.2)Introducing equation (5.2) in equation (2.11) yields, after lengthy manipulations,(5.3)Then introducing equations (5.1) and (5.3) in (2.22) and applying equation (2.26), one obtains(5.4)We now introduce the variations and and, again due to the two-dimensional restriction, it is noted that(5.5)The variables and will be our primary unknowns.

### (a) Weak balance equations

The momentum balance relations (2.4) and (2.5) are recast into weak form through multiplication by arbitrary test functions and , respectively, and integration over the reference domain *Ω*_{0} with bounding surface *Γ*_{0}(5.6)(5.7)Here, is the non-symmetric Kirchhoff stress, defined as , , , and denote the internal and external virtual work of linear and angular momentum and ** T** and

*T*_{m}are material traction and couple traction vectors.

To linearize equations (5.6) and (5.7), we establish the (material) rates of the deformation gradient and micropolar rotation and rewrite the relations (2.21) and (2.22), for the micropolar stretch and the curvature, respectively,(5.8)(5.9)(5.10)(5.11)with the spatial velocity and angular velocity gradients, and , defined as(5.12)(5.13)Time derivation of the left-hand side of equation (5.6) yields(5.14)where the stress rate is defined as(5.15)Evaluating with use of equations (5.8)–(5.10) one finds(5.16)with(5.17)Thus, equation (5.15) may equivalently be expressed as(5.18)and the expression for turns into(5.19)

To linearize the angular momentum balance, equation (5.7), we note that(5.20)and the linearization becomes(5.21)Using equations (5.10) and (5.11), we may write the couple stress rate as(5.22)with(5.23)(5.24)Substitution of equations (5.16) and (5.22) into equation (5.21) finally yields(5.25)

### (b) Linearization of the constitutive equations

In order to establish the fourth-order operator ^{Σ}, equation (5.17), we differentiate the constitutive equation for the force stress , equation (4.5), with respect to the right micropolar stretch tensor and introduce the result into equation (5.17),(5.26)with(5.27)

Similarly, the operators and , are obtained by differentiation of the constitutive equation for the couple stress equation (4.10), and substitution into equations (5.23) and (5.24),(5.28)

(5.29)

### (c) Discrete averaging

The ensemble average will be approximated by the average over a discrete ensemble of struts. Since the strut parameters *r*_{0}, *k*_{1}, *k*_{2}, *k*_{3}, *k*_{n} and *k*_{b} are here taken to be uniform, the discretization involves only the strut director ** N**. Thus, we form a set of

*N*directors, {

**}={**

*N*

*N*_{1}, …,

*N*_{N−1},

*N*_{N}}, typically but not necessarily isotropically distributed. This amounts to(5.30)(5.31)(5.32)(5.33)where the functions and are given by equations (4.6) and (3.4)

_{2}. Since

*N*_{i}are independent of deformation, equation (5.30) needs to be computed only once. However, equations (5.31)–(5.33) depend on the deformation and must be evaluated repeatedly.

We note that the model automatically accounts for any induced anisotropy, under the assumption of affine evolution of the spatial strut vectors ** n** (cf. equation (3.4)

_{1}). The initial anisotropy, or isotropy, is determined by {

**}.**

*N*### (d) Finite-element equations

Finally, the deformation and rotation maps are interpolated on the basis of a finite-element subdivision of the region *Ω*_{0} into elements , *e*=1, …, nel, where nel is the number of elements. Each element has the interpolation(5.34)(5.35)(5.36)(5.37)with(5.38)where nod is the number of nodes on the element. In the above equation, *N*_{Χ} and *N*_{θ} are the element basis functions with respect to displacements and rotations, respectively. The material gradient of the basis functions becomes(5.39)(5.40)Inserting the discretization into the weak form of the transposed balance equations, equations (5.6) and (5.7), we obtain the discretized formulation(5.41)where(5.42)

(5.43)

(5.44)

(5.45)

The FE equations (5.41) are solved using Newton's method. Thus, the incrementally linear format of equation (5.41) is obtained by means of equations (5.19) and (5.29)(5.46)where Δ• denotes an increment in •. The submatrices in the stiffness matrix from equation (5.46) are evaluated with use of equations (5.26), (5.28) and (5.29) as(5.47)(5.48)

(5.49)

(5.50)

## 6. Analysis of length-scale effects

It can be said immediately from dimensional considerations that the force stress must be independent of the internal length scale and that the couple stress goes with the square of the internal length scale. The natural choice of internal length parameter is *r*_{0} (any other microstructural dimensions may be related to *r*_{0} by suitable dimensionless numbers). Hence(6.1)where *G* is the shear modulus of the solid constituent.

The various stiffness parameters have dimensions of force (*k*_{1}, *k*_{2} and *k*_{3}) and couple (*k*_{n} and *k*_{b}). Those will naturally depend on the geometry and topological arrangement of the struts, which is not completely known. Some insight may be gained by considering the scalings of simple theory of beams and columns. If a strut is idealized as a slender linear elastic rod being subjected to longitudinal stretching, buckling, bending and torsion, we are led to the following scaling relations:(6.2)(6.3)(6.4)(6.5)where *α* is the strut aspect ratio (length-to-diameter ratio) and *ϕ*_{0} is the solid volume fraction in the reference (undeformed) state. These scalings are simplistic in several ways: the expression (6.3) for *k*_{1} is based on the tensile response of a rod, but in most foams the axial strut force is in fact governed by the bending of connecting struts. Therefore, *k*_{1} may well obey the same scaling as *k*_{2} and *k*_{3}. The critical stretch *λ*_{c} would also be expected to depend on *α* but is kept constant here.

Substitution of the above expressions for the stiffness parameters in equations (4.5), (4.6) and (4.10) verifies the *G*- and *r*_{0}-dependence in (6.1), but also yields the more specific scalings (supposing that *α*≳1)(6.6)These scalings are based on a strut aspect ratio that is constant under deformation. This is consistent with our assumption that large deformations of the struts occur by buckling and deflection only, so that the stretching of the arc length of a strut is always small. It should be carefully noted, however, that the above interpretation of the stiffness parameters is a simplistic one, being based on elastic line models.

A way of applying this model would be to use the above scalings with adjustable numerical factors in front of each one. Here, we are only interested in the general behaviour of this type of model, so we shall simply use equations (6.2)–(6.5) as they stand and study the equations(6.7)

(6.8)

## 7. Simulation of shear layer

We now perform simulations using the reduced constitutive equations (6.7) and (6.8). The material parameters that can be varied are the aspect ratio *α*, the strut length *r*_{0} and orientation distribution {** N**} of the struts. We consider a shear gap with rotations as well as displacements prescribed on the top and bottom boundaries and completely free edges. A pure shear displacement is imposed on the top boundary. When nothing else is mentioned

*u*/

*H*=1/20. The width-to-height ratio of the domain is chosen as

*L*/

*H*=3, which is sufficient to avoid edge disturbances at the midsection AB, yet keeps the computational load reasonable. The boundary-value problem is defined in figure 4.

All computations reported here were performed with a sufficient FE discretization, using triangular elements with quadratic basis functions for the displacements and linear for the rotations. The choice of a quadratic/linear approximation was suggested by Diebels & Steeb (2002).

When nothing else is mentioned the orientation distribution of struts is isotropic, and the rotations are set to zero on the top and bottom boundaries. The critical stretch is set to *λ*_{c}=0.99 throughout. The strut aspect ratio is set to *α*=1 in most of the examples; this is to obtain a clear couple stress effect. The strut aspect ratio should, however, not be interpreted too literally in this reduced form of the constitutive equations. It is mainly a consistent way of controlling the relations between the various stiffnesses of the struts via equations (6.2)–(6.5).

Figure 5 shows the solution fields and for a strut length *r*_{0}=*H*/6, a strut aspect ratio of *α*=1 and a total shear deformation of *u*/*H*=1/2. It is clearly seen that the no-rotation condition on the top and bottom boundaries leads to boundary layers with reduced rotations and reduced shear strains. This general behaviour is characteristic of the micropolar theory and quite the same as that described by, for example, Diebels & Steeb (2002). If the boundary condition is changed to zero couple stress, the boundary layers disappear, the rotations become constant and displacements linear along the vertical section AB. Figure 6 shows the response, according to equations (6.7) and (6.8).

### (a) Aspect ratio effect

The strut aspect ratio *α* controls the relative magnitude of the different strut stiffnesses according to the elastic line approximations. A short and ‘chubby’ strut, as compared with a slender one, will be more responsive to curvature relative to displacement gradients. Figure 7 shows that the gapwise displacement distribution (section AB) is linear for slender struts while it becomes S-shaped when *α* is small. The shear strain is thus suppressed in the boundary layers and concentrates towards the centre of the gap. This results in an augmented shear stress: figure 6 shows that both shear stress and couple stress increase dramatically when *α* decreases. Note that the displacement field only is plotted for 0.5≤*x*_{2}≤1.0.

On the other hand, when the struts are sufficiently slender the couple stresses vanish and the displacement field becomes linear, as in a standard continuum. The micropolar rotation field still exhibits the boundary layers, due to the prescribed boundary rotation, but it has little influence on either the displacement field or the shear stress response. Hence, the case of slender struts is adequately described by a standard continuum.

### (b) Strut length effect

Figure 8 shows how the displacement and rotation fields vary across the gap (along AB) for different strut lengths. These plots clearly show that the boundary-layer thickness scales with the strut length (being≈3*r*_{0}). Thus, when the strut length is less than approximately 1/6 of the gap, the rotation reaches the rotation of the displacement field () at the centre of the gap. (Note that a standard continuum in this problem would produce a *constant* rotation of −*u*/2*H* across the gap.) Increasing the strut length makes the boundary layers meet, and suppresses the rotations throughout the gap.

It is also seen that, as the strut length is increased, the maximum rotation first increases, then goes through a maximum and decreases again. This is explained as follows: when the boundary layers are thinner than *H*/2, their only effect at the gap centre is to concentrate the shear strain to the centre. Since the (negative) rotation will be half the displacement gradient in this case, it too will be enhanced. When, on the other hand, the boundary layers cover the full gap the no-rotation boundary condition is felt all the way into the gap centre, thus reducing the rotation there.

### (c) Anisotropy effect

Three different sets of strut directors were generated, as illustrated in figure 9. One is approximately isotropic, denoted {** N**}

^{I}. Another one, called {

**}**

*N*^{H}, is nearly planarly oriented in the horizontal plane, obtained by vertical compression. A third one, {

**}**

*N*^{V}, is strongly orientated along the vertical axis, obtained by vertical elongation.

Figure 10 shows that the magnitude of the rotation field increases with the degree of vertical alignment. This is explained as follows: the vertical struts tend to follow vertical continuum lines, which rotate as the shear angle, while the horizontal struts tend to follow horizontal continuum lines, which do not rotate at all. The resulting rotation therefore tends to be half the shear angle (the ordinary continuum rotation) when the orientation distribution is isotropic. Then if vertical struts dominate the rotation tends towards the shear angle and if horizontal struts dominate the rotation will tend to zero.

It should be pointed out that general anisotropy implies non-uniform strut length (different *r*_{0} in different directions, which we have not considered here), leading to a directionally dependent internal length scale. However, in the extreme cases considered here, i.e. strongly aligned and strongly planar orientation (cf. figure 9), an almost monodisperse strut length is realistic.

## 8. Concluding remarks

The model presented here idealizes an open-cell cellular solid as an affine network of elastic lines. This idealization allows us to develop constitutive equations for a three-dimensional cell structure on a mechanistic basis and thus provide a clear interpretation of the micropolar degrees of freedom and stress measures. The approach strikes a balance between the efficiency of purely macroscopic, phenomenological, models and the accuracy of fully microstructural models. We have thus shown that a fairly efficient macroscopic solution can be obtained using the finite-element method, while incorporating a good deal of mechanistic interpretation on the microlevel. To our knowledge, this is the first mechanistic constitutive equation for a finite elastic generalized continuum.

In RVE-based theories, the averaged force stress is biased due to finite size of the RVE, if the microstress varies nonlinearly. To compensate for this, an additional force stress term appears in the averaged couple stress (see for example Forest *et al*. (1999) and Larsson & Diebels (2007)). Owing to its statistical mechanical nature, the present theory does not need the concept of a representative volume element. One advantage of this is that the averaging of force stress and couple stress are kept separate, see equations (4.3) and (4.8), and thereby simpler.

The general behaviour of the model in the presented shear layer problem is very similar to that reported by Diebels & Steeb (2002) for the classical Cosserat elasticity. A parameter study shows that the strut aspect ratio controls the couple stress response, and thereby the displacement field; that the strut length controls the shape of the rotation field, and the boundary-layer thickness in particular; and anisotropy in the strut orientation distribution has a strong effect on both rotation field and the displacement field.

The continuum framework is fully nonlinear and the material model is nonlinear in strain but linear in curvature. The material nonlinearities include the strain-dependent length, orientation and concentration of struts, as well as the weakening of struts due to buckling.

There is certainly scope for refinements of the model. The way that the forces and moments of a strut are linked to the rotations and displacements of its ends is nothing but a phenomenological ansatz on the microlevel, and this may of course be done differently. The models for the strut force and the strut couple, equations (3.22), (3.23) and (3.27), could, for example, be constructed to allow for the stretch–rotation coupling. The restriction to uniform strut length and aspect ratio could also be removed, at a computational cost. The simplistic scalings of the *k*-parameters in equations (6.2)–(6.5) could be developed more carefully. Some adjustable parameters will, however, inevitably remain. Any such refinements will need to be accompanied by experiments or at least numerical simulations (resolving the actual morphology). No experimental data for elastic open-cell foams within the finite strain micropolar regime seem to be available at this point.

## Acknowledgments

This work was financed by the vehicle research program (ffp) and the following participating companies: Finnveden AB, Gestamp HardTech AB, Outokumpu Stainless AB, SAAB Automobile AB, Volvo AB and Volvo Car Corporation.

## Footnotes

- Received June 27, 2008.
- Accepted October 27, 2008.

- © 2008 The Royal Society