The evolutionary constitutive elastic-inelastic relation with its compatible objective derivative is derived in general form using the kinematics of superposition of small elastic and inelastic strains on finite elastic-inelastic strains. The equation is rendered concrete using the elastic law for a slightly compressible material.Key words: elastic-inelastic behavior, finite strains, slight compressibility, evolutionary constitutive equations.1. Preliminary Information. Using three configurations: an initial configuration ae 0 , a current configuration ae, and an intermediate configuration ae * close to the current configuration and employing the kinematics of superposition of small strains (position gradients) on finite strains, Novokshanov and Rogovoi [1] derived constitutive equations for finite elastic strains of a simple material relative to the intermediate configuration.According to the Celerier-Richter theorem or the Noll reduction theorem, the constitutive equation for a simple material that satisfies the objectivity principle is written as (see [2])( 1.1) where T is the true stress tensor; R and U are the orthogonal tensor and the symmetric positive definite pure-strain tensor in the polar decomposition of the position gradient F = R · U ;g 1 (U ) is the material response to pure strain. Relation (1.1) can be written in several equivalent forms [1], in particular,where J = I 3 (F ) is the third basic invariant F , which defines the relative volume change; andg 6 is the material response function. In [1], the functiong 6 is linked tog 1 by the relationg 1 = J −1 U ·g 6 · U . For the intermediate configuration ae * close to the current configuration, the constitutive equation (1.2) is written asHere T * is the stress reached in the configuration ae * (the initial stress for this configuration), h = ( ae * ∇ u) t is the gradient (relative to the configuration ae * u) of the vector of small displacements that transform the intermediate configuration to the current configuration, e = (h + h t )/2 is the small-strain tensor relative to the configuration ae * , d = (h − h t )/2 is the small-rotation tensor,L IV 6 is the fourth-rank tensor (generally, anisotropic) which defines the elastic response of the material to small strains relative to the intermediate configuration.The approximate relation (1.3) can be made exact by dividing by the increment in the time of transition from the intermediate to the current configuration and passing to the limit, i.e., by letting the intermediate configuration to the current configuration. As a result, we have the evolutionary equation T Tr =L IV 6 ··ė (1.4)with the Truesdell objective derivative.
The kinematic relations describing elastic-inelastic deformation that coincide in shape with the wellknown Lie representation but are free from the drawback of the latter are extended to the case of thermo-elastic-inelastic deformation with finite strains. The limitations imposed on the kinematics by the principle of objectivity are considered. Relations for the stresses and entropy are derived from the laws of thermodynamics, and a heat-conduction equation is constructed.
Polar decomposition tensors are constructed for slightly disturbed kinematic elastic, inelastic, and thermal strain tensors. Provided that the inelastic and thermal site gradients are pure deformations without rotations, relations are obtained between inelastic small strains and small rotations and between thermal small strains and small rotations which transform an intermediate configuration to a close current configuration.
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