Caveats concerning conjugate stress and strain measures for frame indifferent anisotropic elasticity

Abstract

Small-distortion constitutive laws are often extemporaneously generalized to large deformations by merely applying them in the unrotated material frame or, equivalently, by using polar rates. This approach does render the resulting constitutive model indifferent to large superimposed rigid rotations, but it may lead to incorrect predictions for both the magnitude and direction of stress whenever there is significant material distortion. To demonstrate this claim, an exact large-deformation solution is derived for the stress in an idealized fiber-reinforced composite. This example shows that the Cauchy tangent stiffness tensor (corresponding to the conjugate pair of Cauchy stress and the symmetric part of the velocity gradient) must evolve in both magnitude and direction whenever the material distorts. Volume changes necessarily lead to a loss of major-symmetry of this Cauchy tangent stiffness tensor, which can be rectified by instead using specific or Kirchhoff stress. A previous work that correctly pointed out the need for the Cauchy stiffness tensor to distort is shown to have overlooked an additional contribution from the rate of distortion. Some of the anomalous properties of the Cauchy stiffness tensor are eradicated by instead using the second Piola-Kirchhoff stress or, equivalently, convected coordinates. Such an approach, however, demands accurate measurements of large-distortion material response, not only to obtain physically realistic results, but also to avoid potential instabilities in numerical computations.