Abstract

Nonlocal damage models are typically used to model failure of quasi-brittle materials. Due to brittleness, the choice of a particular model or set of parameters can have a crucial influence on the structural response. To assess this influence, it is essential to keep finite element discretization errors under control. If not, the effect of these errors on the result of a computation could be erroneously interpreted from a constitutive viewpoint. To ensure the quality of the FE solution, an adaptive strategy based on error estimation is proposed here. It is based on the combination of a residual-type error estimator and quadrilateral h-remeshing. Another important consequence of brittleness is that it leads to structural responses of the snap-through or snap-back type. This requires the use of arc-length control, with a definition of the arc parameter that accounts for the localized nature of quasi-brittle failure. All these aspects are discussed for two particular nonlocal damage models (Mazars and modified von Mises) and for two tests: the Brazilian tensile splitting test and the single-edge notched beam test. For the latter test, the capability of the Mazars model to capture the curved crack pattern observed in experiments – a topic of debate in the literature – is confirmed.

Abstract

An adaptive finite element strategy for nonlocal damage computations is presented. The proposed approach is based on the combination of a residual-type error estimator and quadrilateral h-remeshing. A distinctive feature of the error estimator is that it consists in solving simple local problems over elements and so-called patches. The paper focuses on how the nonlocality of the constitutive model should be accounted for when solving these local problems. It is shown that the nonlocal damage models must be slightly modified. The resulting adaptive strategy is illustrated by means of some numerical examples involving the single-edge notched beam test.

Abstract

We take a heterodimer multiscale reaction-diffusion model as a starting point and modify it to account for the nonlocality of the underlying processes. The resulting model is a coupled system of integro-differential equations. The effect of nonlocal interactions is quantified based on the comparisons of the newly developed model with its local counterpart. Our primary focus is given to the estimation of conversion times from healthy to toxic proteins. This is done based on the stability analysis of a simplified model, as well as on the fully coupled numerical implementation. Several clinically relevant patient proteopathies are highlighted. They include the primary tauopathy, as well as the secondary tauopathy where the sustained presence of toxic tau protein requires also to be accompanied by toxic amyloid-beta peptides. We demonstrate the dynamics of concentrations of toxic proteins under several clinically realistic scenarios. Finally, we carry out a detailed parametric study, providing further insight into the role of nonlinear effects controlled via the Holling type II functional response.