A coupled dislocation-hydrogen based model of inelastic deformation

An inelastic model is developed within the thermodynamic framework of Gurtin to describe the hydrogen effect on material deformation. Internal state variables associated with statistically stored dislocations, the concentration of hydrogen, and associated gradients are introduced into the Helmholtz free energy, in addition to the standard variables of elastic strain and temperature. The power expended by microforces related to the hydrogen solutes and concentration gradients is included in the thermodynamics, and these forces are required to satisfy equilibrium conditions at the micro level. The theory is developed for small strains, whereby the total strain is decomposed into the sum of contributions form elasticity, plasticity (dislocation) and hydrogen effects. Thermodynamic restrictions from the dissipation inequality result in the restriction that the stress and microforces are defined as derivatives of the free energy with respect to elastic strain, and hydrogen concentration and its gradient, respectively. Furthermore, a general form for the chemical potential is constructed based upon thermodynamic restrictions. It is a function of the hydrogen concentration and its gradient, and includes the effects of hydrogen on lattice dilatation and dislocation density. Mass balance for the hydrogen results in Fick's law (hydrogen rate is proportional to the flux of hydrogen). The flux is then expressed as the divergence of a function proportional to the gradient of the chemical potential. The energy balance yields a heat conduction equation that includes plastic working as a source term (dissipation) and also the effects of dislocation storage and hydrogen. The plastic flow rule is based upon the thermally activated motion of dislocations over local barriers, and therefore depends upon the von Mises stress. The strength of the activation barrier depends upon the internal strength of the material through the statistically stored dislocations. These dislocations are stored inversely with their mean free path (which depends upon dislocation spacing) and recover proportionally to their density. In accordance with experimental observations, hydrogen is modeled to reduce the activation energy for dislocation slip and also reduce the dislocation spacing thereby affecting the macroscopically observed flow properties of the material.