Applications of a constitutive equation for microstructural change in polymers.

Abstract

Constitutive equations currently available in non-linear elasticity and for non-linear viscoelastic solids fail to capture permanent set upon release of loading, observed experimentally in polymers under large deformation. In order to address this phenomenon and to incorporate softening and "yield" behavior also observed, the implications of a constitutive equation assuming deformation-dependent microstructural transformation are studied. The constitutive equation assumes the action of two separate micro-mechanisms for the generation of stress: the elastic distortion of cross-link networks and the simultaneous rupture of existing networks and their replacement with networks having new reference configurations. Throughout the study, materials are assumed isotropic and incompressible and the constituent networks are assumed to respond elastically. Analytical results applicable to the general statement of the constitutive equation are obtained when possible. Analytical and numerical studies assuming neo-Hookean response of the constituent networks supplement these results. The constitutive equation is studied in the contexts of homogeneous simple shear and equal biaxial extension. Numerical examples demonstrate softening, dependent on both the parameters controlling microstructural transformation and the responses of the constituent networks. It is shown that the equation may imply a local maximum in the stress-deformation relations for any constituent network materials, possibly indicating "yield." The existence of permanent set is established analytically for the general case. It is also shown that a deformation cycle is a dissipative process regardless of the constituent network responses. The constitutive equation is applied to two boundary-value problems. Results from these problems may form a basis for experimental verification of the equation. Numerical examples of the circumferential shear of a hollow cylinder show that the constitutive equation can imply the formation of a boundary layer of high shear at the inner surface. Numerical studies of a hollow sphere subjected to radial tensile tractions demonstrate that the equation can imply redistribution of normal stresses; in particular, an inner boundary layer of high residual circumferential compressive stress remains after release of external tractions. It is shown analytically that the process of microstructural transformation can affect the emergence of a local maximum in the traction-deformation relation; this result is demonstrated numerically.