Dorothee Knees

On the inviscid limit of a model for crack propagation

We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffith’s local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.

A VANISHING VISCOSITY APPROACH TO A RATE-INDEPENDENT DAMAGE MODEL

We analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps. Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arclength reparametrization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.

GRIFFITH-FORMULA AND J-INTEGRAL FOR A CRACK IN A POWER-LAW HARDENING MATERIAL

We consider an elastic body with pre-existing crack which is subjected to external loadings. It is assumed that the constitutive relation is of power-law type (Ramberg/Osgood model). Several fracture criteria are based on the energy release rate, which is the derivative of the potential deformation energy with respect to the crack length. The goal of this paper is to derive the Griffith-formula and the Eshelby–Cherepanov–Rice integral for the energy release rate of this nonlinear model taking into account the actual regularity of the corresponding displacement and stress fields.

ON GLOBAL SPATIAL REGULARITY AND CONVERGENCE RATES FOR TIME-DEPENDENT ELASTO-PLASTICITY

We study the global spatial regularity of solutions of generalized elasto-plastic models with linear hardening on smooth domains. Under natural smoothness assumptions on the data and the boundary we obtain u ∈ L∞((0, T); H3/2-δ(Ω)) for the displacements and z ∈ L∞((0, T); H1/2-δ(Ω)) for the internal variables. The proof relies on a reflection argument which gives the regularity result in directions normal to the boundary on the basis of tangential regularity results. Based on the regularity results we derive convergence rates for a finite element approximation of the models.

Regularity of Elastic Fields in Composites

It is well known that high stress concentrations can occur in elastic composites in particular due to the interaction of geometrical singularities like corners, edges and cracks and structural singularities like jumping material parameters. In the project C5 Stress concentrations in heterogeneous materials of the SFB 404 it was mathematically analyzed where and which kind of stress singularities in coupled linear and nonlinear elastic structures occur. In the linear case asymptotic expansions near the geometrical and structural peculiarities are derived, formulae for generalized stress intensity factors included. In the nonlinear case such expansions are unknown in general and regularity results are proved for elastic materials with power-law constitutive equations with the help of the difference quotient technique combined with a quasi-monotone covering condition for the subdomains and the energy densities. Furthermore, some applications of the regularity results to shape and structure optimization and the Griffith fracture criterion in linear and nonlinear elastic structures are discussed. Numerical examples illustrate the results.

Convergence of alternate minimization schemes for phase field

We consider time-discrete evolutions for a phase-field model (for fracture and damage) obtained by alternate minimization schemes. First, we characterize their time-continuous limit in terms of parametrized BV-evolutions, introducing a suitable family of “intrinsic energy norms”. Further, we show that the limit evolution satisfies Griffith’s criterion, for a phase-field energy release, and that the irreversibility constraint is thermodynamically consistent.

Homogenization of elliptic systems with non-periodic, state-dependent coefficients

In this paper, a homogenization problem for an elliptic system with non-periodic, state-dependent coefficients rep- resenting microstructure is investigated. The state functions defining the tensor of coefficients are assumed to have an intrinsic length scale denoted by e> 0. The aim is the derivation of an effective model by investigating the limit process e → 0o f the state functions rigorously. The effective model is independent of the parameter e> 0 but preserves the microscopic structure of the state functions ( e> 0), meaning that the effective tensor is given by a unit cell problem prescribed by a suitable microscopic tensor. Due to the non-periodic structure of the state functions and the corresponding microstructure, the effective tensor turns out to vary from point to point (in contrast to a periodic microscopic model). In a forthcoming paper, these states will be solutions of an additional evolution law describing changes of the microstructure. Such changes could be the consequences of temperature changes, phase separation or damage progression, for instance. Here, in addition to the above and as a preparation for an application to time-dependent damage models (discussed in a future paper), we provide a Γ -convergence result of sequences of functionals being related to the previous microscopic models with state- dependent coefficients. This requires a penalization term for piecewise constant state functions that allows us to extract from bounded sequences those sequences converging to a Sobolev function in some sense. The construction of the penalization term is inspired by techniques for Discontinuous Galerkin methods and is of own interest. A compactness and a density result are provided.

A phase-field damage model based on evolving microstructure

In this paper we discuss a damage model that is based on microstructure evolution. In the context of evolutionary Γ-convergence we derive a corresponding effective macroscopic model. In this model, the damage state of a given material point is related to a unit cell problem incorporating a specific microscopic defect. The size and shape of this underlying microscopic defect is determined by the evolution. According to the small intrinsic length scale inherent to the original models a numerical simulation of damage progression in a device of realistic size is hopeless. Due to the scale separation in the effective model, its numerical treatment seems promising.

On the Energy Release Rate in Finite–Strain Elasticity

Griffiths fracture criterion describes in a quasistatic setting whether or not a pre-existing crack in an elastic body is stationary for given external forces. This fracture criterion can be reformulated in terms of the the energy release rate (ERR), which is the derivative of the deformation energy of the body with respect to a virtual crack extension. In this note we consider geometrically nonlinear elastic models with polyconvex energy densities and provide a mathematical framework which guarantees that the ERR is well defined. Moreover, without making any assumptions on the asymptotic structure of the elastic fields near the crack tip, we derive rigorously two formulas for the ERR, namely a generalized Griffith formula and the J-integral. For simplicity we consider here a straight crack in a two dimensional domain. The presented techniques are also applicable to smooth interface cracks, for which we give an example in the last section.

Regularity results for transmission problems of linear elasticity on polyhedral domains

Boundary value problems for the Lame operator with piecewise constant material coefficients are investigated on polyhedral domains. Because of geometric peculiarities and non-smooth material constants, the displacement fields and especially the stress fields have a singular behavior in the neighborhood of corners, edges and those points where the material constants jump. For 3D problems it is not clear if the displacement fields are bounded. In this article we describe sufficient conditions on the distribution of the material parameters and the geometry which guarantee that weak solutions of the BVP are bounded and piecewise continuous.