A constitutive model for strainrate dependent ductile-to brittle-transition

Most materials exhibit rate-dependent inelastic behaviour. Increasing strain-rate usually in-creases the yield stress thus enlarging the elastic range. However, the ductility is gradually lost and for some materials there exist a rather sharp transition strain-rate zone after which the material behaviour is completely brittle. A phenomenological approach to model ductile to brittle transition of rate-dependent solids is presented. It is an extension to the model presented in [1] using vectorial damage variable [3]. The constitutive model is derived using a thermodynamic formulation, in which the material behaviour is described completely through the Helmholz free energy and the dissipation poten-tial in terms of the variables of state and dissipation and considering that the Clausius-Duhem inequality is satisfied [2]. The Helmholtz free energy, ψ = ψ(ǫe,d), is assumed to be a function of the elastic strains, ǫe, and the damage vector d. Assuming small strains, the total strain can be additively decom-posed into elastic and inelastic strains ǫi as ǫ = ǫe + ǫi. The dissipation potential is additively split into brittle damage, ϕd, and visco-plastic, ϕvp, parts as ϕ(σ,y)ϕ = ϕ(σ,y) = ϕd(y)ϕtr(σ) + ϕvp(σ), (1) and is expressed in terms of the thermodynamic forces σ and y dual to the fluxes ǫ̇i and ḋ, respectively. The transition function, ϕtr, deals with the change in the mode of deformation when the strain-rate ǫ̇i increases. In the present formulation the Helmholtz free energy, ψ, is a function depending on the sym-metric second order strain tensor ǫe and the damage vector d, the integrity basis thus consists of the following six invariants I1 = tr ǫe, I2 =