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Mathematics and Mechanics of Solids
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Extreme Lamé Compliance in Anisotropic Crystals

Cliff Y. Guo

Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4006, USA

Lewis Wheeler

Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4006, USA

For a crystalline material, Poisson's ratio depends upon two orthogonal directions, one corresponding to the applied uniaxial stress and another for the resulting transverse strain. Interest in directions that yield a negative value leads to a consideration of extreme values and associated directions. Another indicator of a negative Poisson's ratio is the Lamé compliance, defined as the transverse strain response to a unit uniaxial stress. Where this quantity is positive, Poisson's ratio is negative and it is natural to associate a near minimum value of the ratio with a maximum of the Lamé compliance. Moreover, the stationary directions associated with the compliance bear a clearer relationship to the crystallographic directions than those of the ratio. Indeed, many of these directions are not dependent upon the elastic constants within a given crystal symmetry class. In the case of 1-cristobalite, the maximum value of the Lamé compliance is associated with such invariant stationary points. In the present work, we describe the invariant stationary directions and touch on a few of the simplest material-dependent stationary points.

Key Words: crystals • anisotropy • extreme elastic compliance • Poisson's ratio • auxeticity

This version was published on June 1, 2009

Mathematics and Mechanics of Solids, Vol. 14, No. 4, 403-420 (2009)
DOI: 10.1177/1081286507080807
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