To characterize the response of progressively damaged glassy polymers due to the presence and evolution of voids, yield functions and flow rules were developed systematically for a pressure-dependent matrix following the modified von Mises criterion. A rigid-perfectly plastic material was first assumed. The upper bound method was used with a velocity field which has volume preserving and shape changing portions. Macroscopic yield criterion in analytical closed form was first obtained for spherical voids which is valid for all possible macroscopic strain rate fields. Macroscopic yield criteria in analytical closed form were then obtained for cylindrical voids for the special cases of axisymmetric and plane-strain modes of deformation. The upper-bound solutions were subsequently improved to better match analytical solutions for pure hydrostatic loading. Characteristics of the yield function as a function of pressure dependency and void fraction were studied in detail. Generalization of the model for spherical voids to include elasticity as well as strain hardening of the matrix was then obtained. An example for the uniaxial response of a progressively damaged material was then used to illustrate one possible application of the full set of constitutive equations. [S0021-8936(00)02902-0]

SUMMARYRadial return algorithms, for both three-dimensional and plane stress situations, are developed for a class of pressure-dependent plasticity models (formulated in state variables) with mixed hardening. The consistent tangent matrix has been developed which, among other advantages, does not require numerical inversion. The algorithms, for Gurson's mixed hardening model, are incorporated in a finite element program to solve several simple uniaxial tension problems. When compared with numerically integrated solutions, the results from the finite element analysis show excellent agreement. Finally, results of the axisymmetric necking problem, using Gurson's mixed hardening model, are presented as an example of application.

Gurson’s mixed hardening plasticity model (which takes into account the progressive damage due to void nucleation and growth of an initially dense material), with strain and stress-controlled nucleations, was used in a large deformation finite element program to study the plastic flow and damage in the uniaxial compression of cylinders under sticking friction. Effects of strain hardening, nucleation models, yield surface curvature, and geometry on the distributions and evolutions of stresses, strains, mean stress, void fractions, and coalescence are studied in detail. Using Gurson’s isotropic hardening model, positive mean and axial stresses developed at the bulge of the cylinder with growth of voids at latter stages of deformation. Due low stress triaxiality (Σm/σe<0.6) at the bulge, the process is nucleation rather than growth dominated for the majority of the cases studied. At failure, the maximum void fraction at the bulge among all cases studied is 0.085 and is far less than the critical void fraction (≈0.15) for coalescence.

The effects of Strength Differential (SD) and plastic compressibility for materials obeying the modified von Mises yield criterion were exemplified by solving two boundary-value problems. The assumptions of associated plasticity (leading to maximum plastic volume increase) and nonassociated plasticity (leading to zero plastic volume increase) were used for comparative studies on the effects of plastic compressibility. The solutions for compression processes showed that SD effects increased the pressure at initial yielding and at failure, as well as increased the capacity of the materials to withstand plastic deformations. The opposite was true for tension processes. For associated and nonassociated plasticity, upper and lower bounds for stresses and strains for load and stroke-controlled situations were indicated. The results also showed unrealistic restrictions on the Poisson’s ratio and C/T for nonassociated plasticity under certain conditions. Hence, plastic volume increase, although small, should be incorporated into a more realistic plasticity model.

In Part I [1] of this paper, Gurson’s mixed hardening plasticity model with strain and stress-controlled nucleations, was used in a large deformation finite element program to study the plastic flow and damage in the uniaxial compression of cylinders under sticking friction. Due to low stress triaxiality at the bulge of the cylinders, it was found that localization may occur before void coalescence. In this paper, necessary conditions of localizations are analyzed for the axial compression of porous cylinders under sticking friction. Shear band type of localization with a normal mode of fracture has been predicted for the majority of the cases studied. Various existing localization conditions and fracture criteria are assessed using the results from the simulation. The maximum shear stress at failure is approximately constant and a constant critical damage can not be found.