Implicit–Explicit Integration of Gradient-Enhanced Damage Models

Abstract

Quasi-brittle materials exhibit strain softening. Their modeling requires regularized constitutive formulations to avoid instabilities on the material level. A commonly used model is the implicit gradient enhanced damage model. For complex geometries, it still shows structural instabilities when integrated with classical backward Euler schemes. An alternative is the implicit-explicit (IMPL-EX) integration scheme. It consists of the extrapolation of internal variables followed by an implicit calculation of the solution fields. The solution procedure for the nonlinear gradient enhanced damage model is thus transformed into a sequence of problems that are algorithmically linear in every time step. Therefore, they require one single Newton-Raphson iteration per time step to converge. This provides both additional robustness and computational speedup. The introduced extrapolation error is controlled by adaptive time stepping schemes. Two novel classes of error control schemes that provide further performance improvements are introduced and assessed. In a three dimensional com-1 Titscher, January 30, 2019 pression test for a mesoscale model of concrete, the presented scheme provides a speedup of about 40 compared to an adaptive backward Euler time integration.

A stable mesh-independent approach for numerical modelling of structured soils at large strains

“…For the rest of constitutive models or in cases where the governing equations are formulated using the concept of large strains, the number of iterations required to converge the global problem drastically decrease with respect to usual algorithms. This benefit, due to the extrapolative nature of the technique, comes with a price: each IMPLEX iteration introduces some error to the solution that depends on the size of the time step (Titscher et al, 2019).…”

“…In the original proposal of the method, the equations for the evaluation of the constitutive response in the extrapolation step (with an assumed increment of the plastic multiplier) and the correction step are discretized employing an implicit approach (Oliver et al, 2008;Titscher et al, 2019). This is not the only possibility, for instance, Prazeres et al (2016) employed an explicit approach for the extrapolation step.…”

“…To address these issues, Oliver et al (2008) proposed a numerical technique, the so-called IMPLEX, for the integration of constitutive models that provides enhanced robustness and computability with respect to usual methods. This scheme, or variations of it, has been successfully applied to thermomechanical problems (Rodríguez et al, 2016(Rodríguez et al, , 2017, singlephase geomechanical problems with simple elasto-plastic constitutive responses (Prazeres et al, 2016) and even the integration of gradient-enhanced damage models (Titscher et al, 2019).…”

Incorporation of gradient-enhanced microplane damage model into isogeometric analysis

“…Thus, some sorts of regularization techniques are necessarily required to overcome this issue, which originated non-local integral algorithms (Pijaudier-Cabot and Bažant, 1987) and gradient-enhanced approximations (Peerlings et al, 1996). These regularization approaches have provoked a great deal of controversy about how to account for isotropic damage (Triantafyllidis and Aifantis, 1986;Vree et al, 1995;Poh and Sun, 2017;Nguyen et al, 2018;Titscher et al, 2019) and anisotropic damage (Desmorat et al, 2007;Germain et al, 2007;…”