MICROMECHANICAL MODEL OF FRACTURE OF AN INELASTIC MATERIAL AND ITS APPLICATION TO THE INVESTIGATION OF STRAIN LOCALIZATION

We suggest a continuum-mechanics model of fracture of polycrystalline materials and apply this model to studying the processes of localization of plastic strains. The model is based on the phenomenological theory of dislocations and microdefect development. According to this model, the plastic deformation and fracture is a single process caused by the motion of dislocation and also (at a later stage) by generation and development of microdefects. The deformation process is assumed to consist of two stages. At the first stage the material is subjected to plastic deformation due to the motion of dislocations in crystal grains of the material. Part of these dislocations is accumulated at intergrain boundaries. This concentration of dislocations leads to residual microstresses. On the macrolevel, it leads to an elastoviscoplastic flow with simultaneous hardening of the material. The second stage begins after the intensity of the dislocation flux accumulated at the boundaries of grains attains a critical level, beyond which the process of annihilation of dislocations begins. This process is accompanied by disclinations of grains with the formation of voids between them. On the microlevel, this process leads to relaxation of the internal stresses and softening of the material. The flux of annihilating dislocations is characterized by the tensor of annihilation of dislocations. On the macrolevel, this flux leads to damage of the material and can be described in terms of the damage tensor. The spherical part of the damage tensor is associated with the volume of appearing voids, while the deviatoric part is associated with the fracture shear deformation of the material leading to the relaxation of the residual stresses. We assume that the voids to be nearly spherical. The effective elastoviscoplastic material with spherical voids can be described by the associated law of plastic flow depending on the content of voids in the material and on the plastic strain rate. A complete system of constitutive equations for macroparameters relates the tensors of effective and residual stresses with the tensors of viscoplastic strain rate and damage. The investigation of the system shows that it does not contradict the thermodynamic inequalities and provides well-posed formulations of boundary value problems of mechanics, including those governing the stage of softening of the material. This permits one to use the model in question for describing the process of localization of plastic strains. The quasistatic problem of a rod subjected to uniaxial tension with constant strain rate is solved. We show that, depending on the parameters of the model, the a-e diagrams of the material describe a broad range of phenomena characteristic of various materials. In particular, these phenomena include the dynamic and kinematic hardening of the material, partial softening followed by hardening, the yield spike, the delay of plasticity, softening with the tendency to an asymptotic flow under a constant stress, and hysteresis loops. Various sorts of diagrams can be obtained by varying five parameters. These parameters can be identified experimentally on the basis of these diagrams for constant rates of deformation. The potentials of the model suggested for the description of the plastic strain localization are illustrated by an example of the one-dimensional dynamic tension of a rod in the case where two loads are applied instantaneously to the ends of the rod and cause the ends of to move at a constant velocity. The structure of the bands of localization of plastic strains is investigated. It is shown that the localization of plastic strains and fracture are possible either at the center of the rod subjected to tension or at the ends of this rod, depending on the velocity of the ends and the parameters of the material.