Numerical Integration Scheme for Coupled Elastoplastic–Viscoplastic Constitutive Law for Tunnels

The paper presents an efficient numerical integration scheme for coupled elastoplastic–viscoplastic constitutive behavior with internal-state variables standing for irreversible processes. In most quasi-static structural analyses, the solution to boundary value problems involving materials that exhibit time-dependent constitutive behavior proceeds from the equation integration, handled at two distinct levels. On the one hand, the first, or local, level refers to the numerical integration at each Gaussian point of the rate constitutive stress–strain relationships. For a given strain increment, the procedure of local integration is iterated for stresses and associated internal variables until convergence of the algorithm is achieved. On the other hand, the second, or global, level is related to structure equilibrium between internal and external forces achieved by the Newton–Raphson iterative scheme. A review of the elastoplastic and viscoplastic model is given, followed by the coupling between these models. Particular emphasis is given in this contribution to address the first level integration procedure, also referred to as the algorithm for stress and internal variable update, considering a general elastoplastic–viscoplastic constitutive behavior. The formulation is described for semi-implicit Euler schemes. The efficacy of the numerical formulation is assessed by comparison with analytical and numerical solutions derived for deep tunnels in coupled elastoplasticity–viscoplasticity. Finally, a parametric analysis is performed to show the importance that this model can have, in the long-term convergence profile, against other models. For the considered flow surfaces, potential functions, and properties, differences on the order of 23% to 52% are found in the long-term convergence profile.