Numerical simulation of plasticity and failure of solids with defects under dynamic loading

The developed statistical model of solid with mesoscopic defects allowed the formulation of phenomenological model in terms of two independent variables – the defect density tensor and structural scaling parameter and the simulation of shock wave propagation in the linkage with structural relaxation phenomena [1-5]. It was established the link of non-linearity of plasticity and failure and the kinetics of structural scaling transition in the metastability area of out-of-equilibrium thermodynamic potential of solid with mesodefects with application to the analysis of spall failure. The purpose of the present work consists of development of three-dimensional mathematical model of solid with defects and the further verification of model. According the decomposition of stress and strain tensors on isotropic and deviatoric parts the kinetic equations for respective parts of defect density tensor were introduced. The deviatoric part evolution corresponds to mechanisms of plastic relaxation and isotropic part evolution corresponds to microcrack growth and failure. Developed constitutive equations of solids with mesoscopic defects were implemented into commercial numerical codes (Abaqus/Explicit). Introduction Deformation and failure under intensive loading are linking to the defect induced relaxation process that has qualitative different scenarios depending on the initial structure and strain rates. Structure induced relaxation properties and failure are related to the multiscale dislocation evolution that include the stages of new dislocation substructure nucleation and growth in the presence of external stress and due to interaction with other dislocation and dislocation substructures (mesodefects). The statistical model of solid with mesodefects [1-7] that is operated with internal variables, the defect density tensor, characterizing the mesodefect induced strain and an additional variable – the structural-scaling parameter, which reflects the spatial scale evolution in mesodefects ensembles – the scale of corresponding dislocation substructures and the spacing between them. The influence of mesodefects on the relaxation properties and damage-failure transition was described in the framework of statistically based phenomenological approach that reflects specific types of nonlinear behavior of the out-of-equilibrium systems “ solid with mesodefects” called as structural-scaling transition [1]. The purpose of this work is the adaptation of mentioned model to the impact loading and the following verification of material parameters using the data of plate impact test for the vanadium (pull-back velocity measurement after VISAR). Mathematical statement Mathematical statement of moderate shock wave front propagation in sold with defects is analyzed. Constitutive modeling follows to results given by the statistical theory of collective behavior of defects and for the conditions of the plate impact test is formulated for the component of the strain rate 0 xx   (others components of the strain tensor equal zero). Plasticity mechanisms are linked to the structural variables (defect density tensor and structural scaling parameter) and corresponding collective modes that reflect the collective behavior of defects. Defect density tensor p n s  is associated with the defect induced strain in the specific volume, where s is microscopic tensor related to the geometrical strain induced by microcrack or microshear; n is the microcrack (microshear) density. Defect evolution is characterized by the defect nucleation, defect growth and the motion of defects in the condition of the interactions of defects in the presence of structural and applied stresses. These mechanisms give the corresponding part in the total strain (or strain rate) that represents the visco-elastic plastic flow: e p p       (1) where e  is the elastic strain rate, p is the geometrical defect induced strain rate, p  is the viscous plastic strain rate. Statistical model allowed the formulation of thermodynamic potential (nonequilibrium free energy F ) and the definition of the part of the dissipation function related to defects: 0 F F p TS p p                        (2) where S is the entropy, T is the temperature. To follow the Onsager principle the constitutive relations of visco-elastic plastic solid can be written as the link of thermodynamic forces and thermodynamic fluxes: