Averaging theorems in finite deformation plasticity

The transition from micro- to macro-variables of a representative volume element (RVE) of a finitely deformed aggregate (e.g., a composite or a polycrystal) is explored. A number of exact fundamental results on averaging techniques, valid at finite deformations and rotations of any arbitrary heterogeneous continuum, are obtained. These results depend on the choice of suitable kinematical and dynamical variables. For finite deformations, the deformation gradient and its rate, and the nominal stress and its rate, are optimally suited for the averaging purposes. A set of exact identities is presented in terms of these variables. An exact method for homogenization of an ellipsoidal inclusion in an unbounded finitely deformed homogeneous solid is presented, generalizing Eshelby's method for application to finite deformation problems. In terms of the nominal stress rate and the rate of change of the deformation gradient, measured relative to any arbitrary state, a general phase-transformation problem is considered, and the concepts of eigenvelocity gradient and eigenstress rate are introduced. It is shown that the velocity gradient (and hence the nominal stress rate for rate-independent models) in an ellipsoidal region within an unbounded uniform and uniformly deformed solid, remains uniform when this region undergoes a uniform phase transformation corresponding to a constant eigenvelocity gradient. The generalized Eshelby tensor and its conjugate are defined and used to obtain the field quantities in an ellipsoidal inclusion which is embedded in an unbounded, uniformly deformed medium, leading to exact expressions for the concentration tensors and a set of identities relating these quantities. The Green functions for the rate quantities measured from a finitely deformed state, are formulated, and their properties are discussed. The details for calculating the Green function, Eshelby's tensor and its conjugate, and the concentration tensors, are presented. The relation between strong ellipticity and Green's function is examined in connection with the problem of the possible loss of stability of a uniformly stressed homogeneous solid. Exact values of the average nominal stress rate and the average velocity gradient, taken over an ellipsoidal region in a finitely deformed unbounded homogeneous solid, are obtained when arbitrary (variable) eigenvelocity gradients or eigenstress rates are prescribed in any region within the ellipsoid. It is shown that many results for single- and double-inclusion problems in linear elasticity, also apply to the finite-deformation rate problems, provided suitable kinematical and dynamical variables are used. The problem of the double inclusion is considered and exact expressions are given for the average field quantities, taken over the region between the two ellipsoidal domains, one containing the other, when arbitrary eigenvelocity gradients are prescribed within an arbitrary region contained in the inner ellipsoid. These results are further generalized to obtain exact expressions for the average field quantities taken over each annulus of a nested sequence of ellipsoidal regions within a finitely (uniformly) deformed infinite solid, where a constant distinct eigenvelocity gradient is prescribed on each annulus. These results are new, and should provide effective tools for homogenization of multi-component composites at finite deformations and rotations.