modeling of a material aging
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Introduction
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We seek to develop a model of inelastic processes in the aging materials by employing a 4-D inner material metric tensor $G$ as the aging (damage) parameter of a material continuum. Aging here implies any variation in the chemical make-up, i.e., chemical degradation, phase transformation, phase coarsening, nucleation, and growth of micro-defects such as dislocations and voids, shear bands, crazes, micro-cracks, etc. Material engineering and failure analysis indicate that, in addition to the stress and strain tensors, a parameter of state (the “aging” parameter) is needed to represent on a continuum level the sub-micro and micro-structural changes of material. A kinetic equation for the evolution of the aging parameter will represent the aging process of a material. The equations of evolution for the material metric G are the Euler-Lagrange equations resulting from a Variational Principle. The conjugate force of the evolution of metric G (and of the related quantities characterizing the properties of the material) is the Energy-Momentum Tensor of Elasticity introduced by J. Eshelby (see Sec.7 below).

A 3-D material metric $g$ has long been employed as an internal variable in continuum mechanics. For example, it was used for studying the 