A constitutive model for elastic–plastic materials using scalar conjugate stress/strain base pairs

Abstract

A constitutive model for elastic–plastic materials is developed using scalar, conjugate, stress–strain, base pairs in a finite deformation setting. These conjugate base pairs arise from an alternative QR decomposition of the deformation gradient that decomposes its matrix into an orthogonal rotation and an upper-triangular matrix, called the Laplace stretch. This decomposition is particularly useful from an experimental standpoint, as it enables one to directly measure the components of Laplace stretch and its plastic contributions in a specific coordinate system. Moreover, from an experimental standpoint, it is difficult to parameterize current material models due to a covariance between their tensor invariants, traditionally used in their constructions. The use of scalar, conjugate, base pairs are also helpful from that point of view. Interestingly, the multiplicative elastic–plastic decomposition of Laplace stretch leads to an additive decomposition of the total strain attributes into their corresponding elastic and plastic components. Although an additive strain decomposition is commonly used in small-strain theory, here such a decomposition is possible even for finite deformations. An additive decomposition of the strain attributes has a deeper consequence in the construction of our constitutive model. A maximum rate of dissipation criterion has been used in deriving our constitutive equations, as this criterion is valid for a wider class of materials. Two constitutive assumptions – one for a Helmholtz potential, and one for the rate of dissipation function – are required for our constitutive construction. This model does not presuppose the existence of a yield surface. In fact, it is shown that whether a material exhibits a yielding or a creep-like behavior depends upon the differentiability of the rate of dissipation function. Two cases of plastic deformation – volume-preserving and dilatant-pressure dependent deformations – have been considered. To illustrate the proposed model, finite strain versions of classical J2 plasticity and a Drucker–Prager model are derived.