Integral-equation representations of flow elastoplasticity derived from rate-equation models

Summary Two integral-equation representations are presented in this paper, based on the exact integrations of the conventional rate-equation model of associativeJ₂ flow elastoplasticity with combined-isotropic-kinematic hardening-softening. Among them the strain-controlled integral-equation representation has two new naturally defined material functionsY(Z) andU(Z) of the normalized active workZ, which plays the role of intrinsic time. One of the immediate benefits derivable from the new representations is, owing to the explicit unfolding of the highly nonlinear path-dependence between stress and strain without a detour to the evolutions of internal state variables, their adaptability for direct calculations without any iteration. Indeed, it is itself a constructive algorithm. It is shown that at a realistic level of precision, the strain-controlled integral-equation representation saves 99% or more of the CPU time compared with the widely used elastic predictor-radial return algorithm of the rate-equation representation.List of symbols e_ij,e_ij^e,e_ij^p strain deviator, elastic strain deviator, plastic strain deviator - effective strain - ^p effective plastic strain - e₁,e₂,e₃ principal strain deviator,e₃=–e₁–e₂ - e_tan,e_rad tangential strain increment, radial strain increment - E Young's modulus, assumed to be constant - f yield function in stress space - F yield function in strain space - G shear modulus, assumed to be constant - G(Z₁,Z₂) shear relaxation function of elastoplasticity - h(^p),k(^p) material functions of plasticity for the stress-space rate-equation representation - material functions of plasticity for the strain-space rate-equation representation - I₂ second invariant of the deviatoric strain tensor - J₂ second invariant of the deviatoric stress tensor - J(z₁,z₂) shear creep function of elastoplasticity - K bulk modulus, assumed to be constant - p dummy variable of integration in place of the effective plastic strain - r_ij active stress - R_ij active strain - effective active-stress, i.e. times Euclidean length of active stress - effective active-strain, i.e. times Euclidean length of active strain - S_ij,S_ij^e,S_ij^r stress deviator, elastic stress deviator, stress relaxation - effective stress - effective stress relaxation - S₁,S₂,S₃ principal stress deviator,S₃=–S₁–S₂ - t, , , time - t₀ zero-value time - t_u latest unloading time - y(z), u(z) material functions of plasticity for the stress-controlled integral-equation representation - Y(Z), U(Z) material functions of plasticity for the strain-controlled integral-equation representation - z normalized active complementary-work - material functions defined for use in convertingh(^p) andk(^p) toy(z) andu(z) - Z normalized active work - material functions defined for use in convertingh(^p) andk(^p) toY(Z) andU(Z) - _ij back stress - A_ij back strain - _ij, _ij^e, _ij^p strain, elastic strain, plastic strain - _y (initial) yield strain, _y=h(0)/2G - Poisson's ratio assumed to be constant - _ij, _ij^e, _ij^r stress, elastic stress, stress relaxation - _y (initial) yield stress, yield strength, _y=h(0)