A new model for porous nonlinear viscous solids incorporating void shape effects - I: Theory

The aim of this work is to present a new, improved model for porous ductile solids at high temperatures, for which viscous effects are important. The sound matrix is assumed to obey a simple Norton law without threshold. The voids are assumed to be and remain spheroidal so that their shape is characterized by a single parameter. The model makes an essential use of the notions of gauge surface and gauge function, which extend those of yield surface and yield function of classical plasticity to the nonlinearly viscous case. It is obtained by looking for a good heuristic expression of the gauge function, using specific models pertaining to special cases as references. These reference models include: (i) that of Gologanu, Leblond and Devaux for the case of an ideal-plastic material (i.e. with an infinite Norton exponent) containing voids of arbitrary shape; (ii) that of Leblond, Perrin and Suquet for arbitrary nonlinear viscous materials with spherical or cylindrical voids; (iii) that of Ponte Castañeda and Zaidman for viscous materials with arbitrary voids, but in the linear case only (i.e. for a Norton exponent of unity), although this model was developed for arbitrary nonlinear viscous materials. The nonlinear Hashin-Shtrikman bound is used as a further reference in the case of a zero macroscopic mean strain rate. The model also includes a suitable evolution equation for the void shape parameter, which is again inspired from those proposed in some special cases by Gologanu, Leblond and Devaux, and Ponte Castañeda and Zaidman.