Nonlinear Elastic Material Response Cauchy-Born Elasticity

2 Nonlinear Elastic Material Response Cauchy-Born Elasticity 

As yet, our reflections on the elastic properties of solids have ventured only so far as the small-strain regime. On the other hand, one of the powerful inheritances of our use of microscopic methods for computing the total energy is the ease with which we may compute the energetics of states of arbitrarily large homogeneous deformations. Indeed, this was already hinted at in fig. 4.1. 

As one of our central missions is to uncover the relation between microscopic and continuum perspectives, it is of interest to further examine the correspondence between kinematic notions such as the deformation gradient and conventional ideas from crystallography. One useful point of contact between these two sets of ideas is provided by the Cauchy-Bom rule. The idea here is that the rearrangement of a crystalline material by virtue of some deformation mapping may be interpreted via its effect on the Bravais lattice vectors themselves. In particular, the Cauchy-Bom mle asserts that if the Bravais lattice vectors before deformation are denoted by Ej, then the deformed Bravais lattice vectors are determined by e = FEj. As will become evident below, this mle can be used as the basis for determining the stored energy function W (F) associated with nonlinear deformations F. 

The Cauchy-Born strategy yields interesting insights in a number of contexts one of which is in the kinematic description of stmctural transformations. A 

Further details of this idea can be found in Tadmor et al. (1996). 

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