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TECHNICAL PAPERS

# Analytical Derivation of Cosserat Moduli via Homogenization of Heterogeneous Elastic Materials

[+] Author and Article Information
D. Bigoni1

Dipartimento di Ingegneria Meccanica e Strutturale, Università di Trento, Via Mesiano 77, 38050 Povo, Trento, Italybigoni@ing.unitn.it

W. J. Drugan

Dipartimento di Ingegneria Meccanica e Strutturale, Università di Trento, Via Mesiano 77, 38050 Povo, Trento, Italy and Department of Engineering Physics, University of Wisconsin-Madison, 1500 Engineering Drive, Madison, WI 53706-1687drugan@engr.wisc.edu

Only $νm$ is needed to determine $l$. However, the knowledge of $μm$ would allow us to determine $μ¯$ and $κ¯$ from Eqs. 1, which compared to experimental results by Lakes would permit an assessment of the quality of the estimate.

1

Corresponding author.

J. Appl. Mech 74(4), 741-753 (Apr 20, 2006) (13 pages) doi:10.1115/1.2711225 History: Received February 10, 2006; Revised April 20, 2006

## Abstract

Why do experiments detect Cosserat-elastic effects for porous, but not for stiff-particle-reinforced, materials? Does homogenization of a heterogeneous Cauchy-elastic material lead to micropolar (Cosserat) effects, and if so, is this true for every type of heterogeneity? Can homogenization determine micropolar elastic constants? If so, is the homogeneous (effective) Cosserat material determined in this way a more accurate representation of composite material response than the usual effective Cauchy material? Direct answers to these questions are provided in this paper for both two- (2D) and three-dimensional (3D) deformations, wherein we derive closed-form formulae for Cosserat moduli via homogenization of a dilute suspension of elastic spherical inclusions in 3D (and circular cylindrical inclusions in 2D) embedded in an isotropic elastic matrix. It is shown that the characteristic length for a homogeneous Cosserat material that best mimics the heterogeneous Cauchy material can be derived (resulting in surprisingly simple formulae) when the inclusions are less stiff than the matrix, but when these are equal to or stiffer than the matrix, Cosserat effects are shown to be excluded. These analytical results explain published experimental findings, correct, resolve and extend prior contradictory theoretical (mainly numerical and limited to two-dimensional deformations) investigations, and provide both a general methodology and specific results for determination of simple higher-order homogeneous effective materials that more accurately represent heterogeneous material response under general loading conditions. In particular, it is shown that no standard (Cauchy) homogenized material can accurately represent the response of a heterogeneous material subjected to a uniform plus linearly varying applied traction, while a homogenized Cosserat material can do so (when inclusions are less stiff than the matrix).

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## Figures

Figure 6

Characteristic length divided by the cell size for volume fraction of disperse phase f=0.18, for a Cosserat material deduced from homogeneization of a matrix containing a dilute distribution of parallel, infinite circular cylindrical inclusions (plane strain, Eq. 21)

Figure 5

Characteristic length divided by spherical inclusion radius and multiplied by f−1∕6 (top) and parameter η (bottom) for a homogeneous Cosserat material deduced from homogenization of a matrix containing a dilute distribution of spherical inclusions (Eq. 31)

Figure 4

The three functions gi appearing in Eq. 30, among which the minimum is selected for given values of η

Figure 3

Characteristic length divided by circular cylindrical inclusion radius for a homogeneous Cosserat material deduced from homogenization of a matrix containing a dilute distribution of parallel, infinite circular cylindrical inclusions (plane strain, Eq. 22)

Figure 2

Procedure of homogenization of a material containing a dilute distribution of circular voids and subject to a far-field bending stress distribution. Heterogeneous material (left) is an h×h prism removed from an infinite sheet that is subjected to uniaxial, linearly varying far-field stress; homogeneous Cosserat-elastic material (right) subject to the same mean moment (produced by m¯13 and σ¯11) calculated from the heterogeneous prism. For the plane strain problem (where η does not appear), level sets of σ11 are shown; note that the values of σ12, σ21, and σ22, shown parallel to the edges, are less than 1∕100 of the maximum value of σ11 at a distance from inclusion center equal to three times the radius of the inclusion (contrast this with the order of the effect in Fig. 1).

Figure 1

Procedure of homogenization of a material containing a dilute distribution of circular voids. Heterogeneous material (left) is an h×h prism removed from an infinite sheet that is subjected to uniform, uniaxial far-field stress; homogeneous material (right) is subjected to the mean stresses calculated from the heterogeneous prism. For the plane strain problem, level sets of σ11 are shown; note that the values of σ12, σ21, and σ22, shown parallel to the edges, are less than 1∕10 the maximum value of σ11 at a distance from inclusion center equal to three times the radius of the inclusion.

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