A Generalized Backward Euler algorithm for the numerical integration of a viscous breakage model

Abstract

Summary This paper discusses the formulation and the numerical performance of a fully implicit algorithm used to integrate a rate‐dependent model defined within a breakage mechanics framework. For this purpose, a Generalized Backward Euler (GBE) algorithm has been implemented according to two different linearization strategies: The former is derived by a direct linearization of the constitutive equations, while the latter introduces rate effects through a consistency parameter. The accuracy and efficiency of the GBE algorithm have been investigated by (1) performing material point analyses and (2) solving initial boundary value problems. In both cases, the overall performance of the underlying algorithm is inspected for a range of loading rates, thus simulating comminution from slow to fast dynamic problems. As the viscous response of the breakage model can be recast through a viscous nucleus function, the presented algorithm can be considered as a general framework to integrate constitutive equations relying on the overstress approach typical of Perzyna‐like viscoplastic models.

A Cosserat Breakage Mechanics model for brittle granular media

“…one that includes the second gradient as well, or by using a viscous regularisation. Recent work has related rate-dependence at the continuum scale to crack growth within the individual grains (Marinelli and Buscarnera, 2019;Zhang and Buscarnera, 2017). A more refined constitutive model including one or both of these regularisation methods, combined with the additional physics discussed above, would seem to be the natural next step in model development.…”

Consistent Implicit Time Integration for Viscoplastic Modelingof Subsidence above Hydrocarbon Reservoirs

“…De Borst and Heeres [22] provided a unified scheme for standard and viscous plasticity models in geomechanics. Grimstad and Degago in [23] implemented an implicit backward Euler integration scheme for a similar (n-SAC) model (see also, e.g., [24,25] and the references therein).…”