A generalised algorithm for anelastic processes in elastoplasticity and biomechanics

Abstract

A computational algorithm for solving anelastic problems in finite deformations is introduced. The presented procedure, termed the Generalised Plasticity Algorithm (GPA) hereafter, takes inspiration from the Return Mapping Algorithm (RMA), which is typically employed to solve the Karush–Kuhn–Tucker (KKT) system arising in finite elastoplasticity, but aims to modify and extend the RMA to the case of more general flow rules and strain energy density functions as well as to non-classical formulations of elastoplasticity, in which the plastic variables are not treated as internal variables. To assess its reliability, the GPA is tested in two different contexts. First, it is used for solving two classical problems (a shear-compression test and the necking of a circular bar). In both cases, the GPA is compared with the RMA in terms of structural set-up, computational effort and flexibility, and its convergence is evaluated by solving several benchmarks. Some restrictions of the classical form of the RMA are pointed out, and it is shown how these can be overcome by adopting the proposed algorithm. Second, the GPA is applied to characterise the mechanical response of a biological tissue that undergoes large deformations and remodelling of its internal structure.

A poroplastic model of structural reorganisation in porous media of biomechanical interest

“…Equations (44a)-(44c) were solved numerically by applying a numerical procedure recently developed for monophasic continua [47], and adapted to the biphasic framework in this paper. The results of our simulations, performed with our own code, and implemented in the non-commercial software UG [96], are reported in Sect.…”

“…In this section, we demonstrate in detail the computational procedure, adapted from [47], that is used to solve numerically (50a)-(50c). We search for solutions to the problem (50a)-(50c) by means of a linearisation algorithm based on the Newton method, and articulated in two stages.…”

Finite element modeling of finite deformable, biphasic biological tissues with transversely isotropic statistically distributed fibers: toward a practical solution

“…Moreover, it would be interesting to adapt the practical solution method presented in this work to the cases of remodeling and growth [58,59]. This should be done to investigate the influence of these phenomena on the material response of biological tissues (cf., e.g., [60], where the response was studied in the isotropic case, with the aid of the computational algorithm outlined in [61]).…”

Eshelby’s inclusion theory in light of Noether’s theorem

“…Gurtin [1995;2000] reformulated and generalized Eshelby's approach with the method of the varying control volumes and considered the Eshelby stress as the appropriate stress of an independent material balance law. The Eshelby stress has been seen as the object capturing inhomogeneities and singularities (e.g., [Epstein and Maugin 1990;Gurtin 1995;2000;Epstein and Maugin 2000;Epstein and Elżanowski 2007;Verron et al 2009;Weng and Wong 2009;Maugin 2011]), or the driving force of phenomena of material evolution such as plasticity and growth-remodeling (e.g., [Maugin and Epstein 1998;Epstein and Maugin 2000;Cermelli et al 2001;Epstein 2002;Imatani and Maugin 2002;Grillo et al 2003;2005;Epstein 2009;2015;2017;Hamedzadeh et al 2019]), or phase transitions, or evolution of the interfaces among phases (e.g., [Gurtin 1986;Gurtin and Podio-Guidugli 1996;Fried and Gurtin 1994;2004]).…”