Energetic solutions for the coupling of associative plasticity with damage in geomaterials

We prove existence of globally stable quasistatic evolutions, referred to as energetic solutions, for a model proposed by Marigo and Kazymyrenko in 2019. The behaviour of geomaterials under compression is studied through the coupling of Drucker–Prager plasticity model with a damage term tuning kinematical hardening. This provides a new approach to the modelling of geomaterials, for which non associative plasticity is usually employed. The kinematical hardening is null where the damage is complete, so there the behaviour is perfectly plastic. We analyse the model combining tools from the theory of capacity and from the treatment of linearly elastic materials with cracks

Density in SBD and approximation of fracture energies

We prove three density theorems, in the strong BD topology, for the three subspaces of SBD functions: SBD; SBDp∞, where the absolutely continuous part of the symmetric gradient is in Lp, with p > 1; SBDp, whose functions are in SBDp∞ and the jump set has finite Hn-1-measure. We compare them with existing results, discussing related approximation of fracture energies

Globally stable quasistatic evolution for a coupled elastoplastic-damage model

We show the existence of globally stable quasistatic evolutions for a rate-independent material model with elastoplasticity and incomplete damage, in small strain assumptions. The main feature of our model is that the scalar internal variable which describes the damage affects both the elastic tensor and the plastic yield surface. It is also possible to require that the history of plastic strain up to the current state influences the future evolution of damage

Globally stable quasistatic evolution for strain gradient plasticity coupled with damage

We consider evolutions for a material model which couples scalar damage with strain gradient plasticity, in small strain assumptions. For strain gradient plasticity, we follow the Gurtin–Anand formulation (J Mech Phys Solids 53:1624–1649, 2005). The aim of the present model is to account for different phenomena: On the one hand, the elastic stiffness reduces and the plastic yield surface shrinks due to material’s degradation, on the other hand the dislocation density affects the damage growth. The main result of this paper is the existence of a globally stable quasistatic evolution (in the so-called energetic formulation). Furthermore, we study the limit model as the strain gradient terms tend to zero. Under stronger regularity assumptions, we show that the evolutions converge to the ones for the coupled elastoplastic damage model studied in Crismale (ESAIM Control Optim Calc Var 22:883-912, 2016)

Quasistatic crack growth based on viscous approximation: a model with branching and kinking

Employing the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions of cracks in brittle materials in the setting of antiplane shear. The crack path is not prescribed a priori and is chosen in an admissible class of piecewise regular sets that allows for branching and kinking

A Reshetnyak-type lower semicontinuity result for linearised elasto-plasticity coupled with damage in W1,n

In this paper we prove a lower semicontinuity result of Reshetnyak type for a class of functionals which appear in models for small-strain elasto-plasticity coupled with damage. To do so we characterise the limit of measures αkEuk with respect to the weak convergence αk⇀ α in W1,n(Ω) and the weak∗ convergence uk⇀ ∗ u in BD(Ω) , E denoting the symmetrised gradient. A concentration compactness argument shows that the limit has the form αEu+η, with η supported on an at most countable set

Equilibrium Configurations for Epitaxially Strained Films and Material Voids in Three-Dimensional Linear Elasticity

We extend the results about the existence of minimizers, relaxation, and approximation proven by Bonnetier and Chambolle (SIAM J Appl Math 62:1093–1121, 2002), Chambolle and Solci (SIAM J Math Anal 39:77–102, 2007) for an energy related to epitaxially strained crystalline films, and by Braides et al. (ESAIM Control Optim Calc Var 13:717–734, 2007) for a class of energies defined on pairs of function-set. We study these models in the framework of three-dimensional linear elasticity, where a major obstacle to overcome is the lack of any a priori assumption on the integrability properties of displacements. As a key tool for the proofs, we introduce a new notion of convergence for (d- 1) -rectifiable sets that are jumps of GSBDp functions, called σsymp-convergence

Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model

Employing the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions for elastoplastic materials with incomplete damage affecting both the elastic tensor and the plastic yield surface, in a softening framework and in small strain assumptions

C∗ -fermi systems and detailed balance

A systematic theory of product and diagonal states is developed for tensor products of Z2-graded ∗ -algebras, as well as Z2-graded C∗-algebras. As a preliminary step to achieve this goal, we provide the construction of a fermionicC∗-tensor product of Z2-graded C∗-algebras. Twisted duals of positive linear maps between von Neumann algebras are then studied, and applied to solve a positivity problem on the infinite Fermi lattice. Lastly, these results are used to define fermionic detailed balance (which includes the definition for the usual tensor product as a particular case) in general C∗-systems with gradation of type Z2, by viewing such a system as part of a compound system and making use of a diagonal state

Fatigue Effects in Elastic Materials with Variational Damage Models: A Vanishing Viscosity Approach

We study the existence of quasistatic evolutions for a family of gradient damage models which take into account fatigue, that is the process of weakening in a material due to repeated applied loads. The main feature of these models is the fact that damage is favoured in regions where the cumulation of the elastic strain (or other relevant variables, depending on the model) is higher. To prove the existence of a quasistatic evolution, we follow a vanishing viscosity approach based on two steps: we first let the time step τ of the time discretisation and later the viscosity parameter ε go to zero. As τ→ 0 , we find ε-approximate viscous evolutions; then, as ε→ 0 , we find a rescaled approximate evolution satisfying an energy-dissipation balance