Constitutive equations for an elastic material with anisotropic rigid particles.

In this paper we have derived constitutive equations for an elastic material with anisotropic rigid particles. We have included a dependence on the Finger tensor B and the orientation tensor Q in the expression for the free energy of the system. With this expression for the free energy we have derived an expression for the stress tensor up to second order in both these variables. We have shown that the elastic modulus in this expression depends on Q, and this dependence leads to an effective elastic modulus that depends on the strain. We have calculated the explicit form of the equation for the stress tensor for a deformation in the xy plane with a strain equal to -gamma. For fully isotropic materials with Q=0 this expression reduces to an equation containing only odd powers of gamma. The inclusion of a non-zero value for the orientation tensor leads to an additional set of terms in the equation, all proportional to Qxy (the xy component of the tensor Q), and all proportional to even powers of gamma. We have qualitatively compared these expressions with Fourier transform (FT) rheological measurements of xanthan gels, at concentrations above and below the order-disorder transition. In FT rheometry an oscillatory deformation is applied in the nonlinear regime, and the resulting stress response is analyzed in Fourier space. In the 2% (w/w) xanthan system (disordered state) only odd harmonics were found in the stress response, whereas in the 4% (w/w) xanthan gel (ordered state) even harmonics could be detected. As predicted by our theory, the intensity of these even harmonics first increased with increasing gamma, until a maximum value was reached. Beyond this maximum the intensity decreased continuously with increasing gamma.