Thermomechanics of solid materials with application to the Gurson-Tvergaard material model

The elastic-plastic material model for porous material proposed by Gurson and Tvergaard is evaluated. First a general description is given of constitutive equations for solid materials by thermomechanics with internal variables. The role and definition of internal variables are briefly discussed and the following definition is given: The independent variables present (possibly hidden) in the basic laws for thermomechanics are called controllable variables. The other independent variables are called internal variables. An internal variable is shown always to be a state variable. This work shows that if the specific dissipation function is a homogeneous function of degree one in the fluxes, a description for a time-independent process is obtained. When damage to materials is evaluated, usually a scalar-valued or tensorial variable called damage is introduced in the set of internal variables. A problem arises when determining the relationship between physically observable weakening of the material and the value for damage. Here a more feasible approach is used. Instead of damage, the void volume fraction is inserted into the set of internal variables. This allows use of an analytical equation for description of the mechanical weakening of the material. An extension to the material model proposed by Gurson and modified by Tvergaard is derived. The derivation is based on results obtained by thermomechanics and damage mechanics. The main difference between the original Gurson-Tvergaard material model and the extended one lies in the definition of the internal variable 'equivalent tensile flow stress in the matrix material' denoted by M. Using classical plasticity theory, Tvergaard elegantly derived an evolution equation for M. This is not necessary in the present model, since damage mechanics gives an analytical equation between the stress tensor and M. Investigation of the Clausius-Duhem inequality shows that in compression, states occur which are not allowed.