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Discrete & Continuous Dynamical Systems - S

2013 , Volume 6 , Issue 1

Issue on Rate-Independent Evolutions

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Preface: Rate-independent evolutions
Gianni Dal Maso , Alexander Mielke and  Ulisse Stefanelli
2013, 6(1): i-ii doi: 10.3934/dcdss.2013.6.1i +[Abstract](30) +[PDF](115.3KB)
The term rate-independent is usually referred to time-dependent processes which are invariant under time rescaling. In other words, the output of the process is independent of the rate at which the input assumes its values. This feature is common to many physical and engineering systems which do not possess, or possess a very small, internal relaxation time and hence react immediately to the change of external conditions.

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Relaxation and microstructure in a model for finite crystal plasticity with one slip system in three dimensions
Sergio Conti , Georg Dolzmann and  Carolin Kreisbeck
2013, 6(1): 1-16 doi: 10.3934/dcdss.2013.6.1 +[Abstract](58) +[PDF](745.0KB)
Modern theories in crystal plasticity are based on a multiplicative decomposition of the deformation gradient into an elastic and a plastic part. The free energy of the associated variational problems is given by the sum of an elastic and a plastic energy. For a model with one slip system in a three-dimensional setting it is shown that the relaxation of the model with rigid elasticity can be approximated in the sense of $\Gamma$-convergence by models with finite elastic energy and diverging elastic constants.
Young-measure quasi-static damage evolution: The nonconvex and the brittle cases
Alice Fiaschi
2013, 6(1): 17-42 doi: 10.3934/dcdss.2013.6.17 +[Abstract](30) +[PDF](483.1KB)
A rate-independent model for incomplete damage is considered, with nonconvex energy density, mixed boundary condition, and nonzero external load, both for non-brittle and brittle materials. An existence result for a Young measure quasi-static evolution is proved.
On the Fleck and Willis homogenization procedure in strain gradient plasticity
Gilles A. Francfort , Alessandro Giacomini and  Alessandro Musesti
2013, 6(1): 43-62 doi: 10.3934/dcdss.2013.6.43 +[Abstract](35) +[PDF](428.6KB)
We revisit the homogenization process for a heterogeneous small strain gradient plasticity model considered in [5]. We derive a precise homogenized behavior, independently of any kind of periodicity assumption and demonstrate that it reduces to a model studied in [8] when periodicity is re-introduced.
Computational aspects of quasi-static crack propagation
Dorothee Knees and  Andreas Schröder
2013, 6(1): 63-99 doi: 10.3934/dcdss.2013.6.63 +[Abstract](35) +[PDF](876.2KB)
The focus of this note lies on the numerical analysis of models describing the propagation of a single crack in a linearly elastic material. The evolution of the crack is modeled as a rate-independent process based on the Griffith criterion. We follow two different approaches for setting up mathematically well defined models: the global energetic approach and an approach based on a viscous regularization.
    We prove the convergence of solutions of fully discretized models (i.e. with respect to time and space) and derive relations between the discretization parameters (mesh size, time step size, viscosity parameter, crack increment) which guarantee the convergence of the schemes. Further, convergence rates are provided for the approximation of energy release rates by certain discrete energy release rates. Thereby we discuss both, models with self-contact conditions on the crack faces as well as models with pure Neumann conditions on the crack faces. The convergence proofs rely on regularity estimates for the elastic fields close to the crack tip and local and global finite element error estimates. Finally the theoretical results are illustrated with some numerical calculations.
The Preisach hysteresis model: Error bounds for numerical identification and inversion
Pavel Krejčí
2013, 6(1): 101-119 doi: 10.3934/dcdss.2013.6.101 +[Abstract](39) +[PDF](277.3KB)
A structure analysis of the Preisach model in a variational setting is carried out by means of an auxiliary hyperbolic equation with memory variable playing the role of time, and amplitude of cycles as spatial variable. Using this representation, we propose an algorithm and derive error estimates for the identification of the Preisach density function and for an approximate inversion of the Preisach operator.
Local minimality and crack prediction in quasi-static Griffith fracture evolution
Christopher J. Larsen
2013, 6(1): 121-129 doi: 10.3934/dcdss.2013.6.121 +[Abstract](30) +[PDF](328.2KB)
The mathematical analysis developed for energy minimizing fracture evolutions has been difficult to extend to locally minimizing evolutions. The reasons for this difficulty are not obvious, and our goal in this paper is to describe in some detail what precisely the issues are and why the previous analysis in fact cannot be extended to the most natural models based on local minimality. We also indicate how the previous methods can be modified for the analysis of models based on a recent definition of stability that is a bit stronger than local minimality.
Some remarks on the viscous approximation of crack growth
Giuliano Lazzaroni and  Rodica Toader
2013, 6(1): 131-146 doi: 10.3934/dcdss.2013.6.131 +[Abstract](41) +[PDF](390.3KB)
We describe an existence result for quasistatic evolutions of cracks in antiplane elasticity obtained in [16] by a vanishing viscosity approach, with free (but regular enough) crack path. We underline in particular the motivations for the choice of the class of admissible cracks and of the dissipation potential. Moreover, we extend the result to a model with applied forces depending on time.
Crack propagation by a regularization of the principle of local symmetry
Matteo Negri
2013, 6(1): 147-165 doi: 10.3934/dcdss.2013.6.147 +[Abstract](24) +[PDF](479.0KB)
For planar mixed mode crack propagation in brittle materials many similar criteria have been proposed. In this work the Principle of Local Symmetry together with Griffith Criterion will be the governing equations for the evolution. The Stress Intensity Factors, a crucial ingredient in the theory, will be employed in a 'non-local' (regularized) fashion. We prove existence of a Lipschitz path that satisfies the Principle of Local Symmetry (for the approximated stress intensity factors) and then existence of a $BV$-parametrization that satisfies Griffith Criterion (again for the approximated stress intensity factors).
A characterization of energetic and $BV$ solutions to one-dimensional rate-independent systems
Riccarda Rossi and  Giuseppe Savaré
2013, 6(1): 167-191 doi: 10.3934/dcdss.2013.6.167 +[Abstract](57) +[PDF](538.1KB)
The notion of BV solution to a rate-independent system was introduced in [8] to describe the vanishing viscosity limit (in the dissipation term) of doubly nonlinear evolution equations. Like energetic solutions [5] in the case of convex energies, BV solutions provide a careful description of rate-independent evolution driven by nonconvex energies, and in particular of the energetic behavior of the system at jumps.
    In this paper we study both notions in the one-dimensional setting and we obtain a full characterization of BV and energetic solutions for a broad family of energy functionals. In the case of monotone loadings we provide a simple and explicit characterization of such solutions, which allows for a direct comparison of the two concepts.
Thermodynamics of perfect plasticity
Tomáš Roubíček
2013, 6(1): 193-214 doi: 10.3934/dcdss.2013.6.193 +[Abstract](35) +[PDF](581.8KB)
Viscoelastic solids in Kelvin-Voigt rheology at small strains exhibiting also stress-driven Prandtl-Reuss perfect plasticity are considered quasistatic (i.e. inertia neglected) and coupled with heat-transfer equation through dissipative heat production by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Enthalpy transformation is used and existence of a weak solution is proved by an implicit suitably regularized time discretisation.
Thermalization of rate-independent processes by entropic regularization
T. J. Sullivan , M. Koslowski , F. Theil and  Michael Ortiz
2013, 6(1): 215-233 doi: 10.3934/dcdss.2013.6.215 +[Abstract](27) +[PDF](533.1KB)
We consider the effective behaviour of a rate-independent process when it is placed in contact with a heat bath. The method used to ``thermalize'' the process is an interior-point entropic regularization of the Moreau--Yosida incremental formulation of the unperturbed process. It is shown that the heat bath destroys the rate independence in a controlled and deterministic way, and that the effective dynamics are those of a non-linear gradient descent in the original energetic potential with respect to a different and non-trivial effective dissipation potential.
Quasistatic damage evolution with spatial $\mathrm{BV}$-regularization
Marita Thomas
2013, 6(1): 235-255 doi: 10.3934/dcdss.2013.6.235 +[Abstract](46) +[PDF](538.8KB)
An existence result for energetic solutions of rate-independent damage processes is established. We consider a body consisting of a physically linearly elastic material undergoing infinitesimally small deformations and partial damage. In [23] an existence result in the small strain setting was obtained under the assumption that the damage variable $z$ satisfies $z\in W^{1,r}(\Omega)$ with $r\in(1,\infty)$ for $\Omega⊂ \mathbb{R}^d.$ We now cover the case $r=1$. The lack of compactness in $W^{1,1}(\Omega)$ requires to do the analysis in $\mathrm{BV}(\Omega)$. This setting allows it to consider damage variables with values in {0,1}. We show that such a brittle damage model is obtained as the $\Gamma$-limit of functionals of Modica-Mortola type.
Structural stability of rate-independent nonpotential flows
Augusto Visintin
2013, 6(1): 257-275 doi: 10.3934/dcdss.2013.6.257 +[Abstract](30) +[PDF](430.4KB)
Several phenomena may be represented by doubly-nonlinear equations of the form $$ \alpha(D_tu) - \nabla\cdot \gamma(\nabla u)\ni h, $$ with $\alpha$ and $\gamma$ (possibly multivalued) maximal monotone mappings. Hysteresis effects are characterized by rate-independence, which corresponds to $\alpha$ positively homogeneous of zero degree.
    Fitzpatrick showed that any maximal monotone relation may be represented variationally. On this basis, an initial- and boundary-value problem associated to the equation above is here formulated as a null-minimization problem, without assuming $\gamma$ to be cyclically monotone. Existence of a solution $u\in H^1(0,T; H^1(\Omega))$ is proved, as well as its stability with respect to variations of the data, of the mapping $\gamma$, and of the domain $\Omega$.

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