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An example of a shrinking set where infinitely many exclusions during an " infinitesimal " time-interval have occured.
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In this work, we introduce a degenerating PDE system with a time-depending
domain for complete damage processes under time-varying Dirichlet boundary
conditions. The evolution of the system is described by a doubly nonlinear
differential inclusion for the damage process and a quasi-static balance
equation for the displacement field which are strong...
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Given a set of inelastic material models, a microstructure, a macroscopic structural geometry, and a set of boundary conditions, one can in principle always solve the governing equations to determine the system's mechanical response. However, for large systems this procedure can quickly become computationally overwhelming, especially in three-dimen...
Citations
... Here C is independent of α but the relevant mixed term including the damage variable degenerates. Restricting to elasticity-damage models, in [8,45,41,33] some cases of complete damage are addressed, and also in the dynamic plasticity-damage model [22] Hooke's tensor is only positive semidefinite. Moreover, in plasticity-damage models the L 2 regularisation in the damage gradient is at the moment not enough, unless having the presence of strain gradient [13] or further assumptions on elastic strain [16]: an L p regularisation is required for p > n [12,22] or for p = n [15]. ...
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.
... Here C is independent of α but the relevant mixed term including the damage variable degenerates. Restricting to elasticity-damage models, in [8,45,41,33] some cases of complete damage are addressed, and also in the dynamic plasticity-damage model [22] Hooke's tensor is only positive semidefinite. Moreover, in plasticity-damage models the L 2 regularisation in the damage gradient is at the moment not enough, unless having the presence of strain gradient [13] or further assumptions on elastic strain [16]: an L p regularisation is required for p > n [12,22] or for p = n [15]. ...
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 present coupling is such that 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.
... This type of local existence result is what is proved in several related papers (see, e.g., [5,12,13]) and will also be the object of the present note. Indeed, it seems that the description of complete damaging of the material, i.e., of what happens after the onset of some macroscopic fracture, requires a different modeling approach, see, e.g., [7,16,21]. ...
... For these reasons, at least up to our knowledge, local existence has been obtained so far only in presence of additional smoothing terms. Actually, common regularizations considered in the literature are: viscoelastic (rather than purely elastic) behavior for u [3,6,15,18], presence of inertial effects in (1.1) [6,15,17,18], and replacement of the Laplacian in (1.2) by a more regularizing operator like the fractional Laplacian (−∆) s with suitable s > 1 [19] or the p-Laplacian −∆ p with suitable p > 2 [16,17,18]. In this work, we will consider the "original" system (1.1)-(1.2) with no occurrence of any regularizing term. ...
We consider a model for the evolution of damage in elastic materials originally proposed by Michel Fr\'emond. For the corresponding PDE system we prove existence and uniqueness of a local in time strong solution. The main novelty of our result stands in the fact that, differently from previous contributions, we assume no occurrence of any type of regularizing terms.
... It was the basis for many subsequent publications concerning phase separation coupled with damage, see [22,27] for quasi-static setting and [20] for dynamic one. Also in the just linear elastic case (quasistatic or dynamic) without considering phase separation the outlined notions were used to investigate solvability in [19,18]. ...
We discuss some principle concepts of damage mechanics and outline a possibility to address the open question of the damage-to-deformation relation by suggesting a parameter identification setting. To this end, we introduce a variable motivated by the physical damage phenomenon and comment on its accessibility through measurements. We give an extensive survey on analytic results and present an isotropic irreversible partial damage model in a dynamic mechanical setting in form of a second-order hyperbolic equation coupled with an ordinary differential equation for the damage evolution. We end with a note on a possible parameter identification setting.
... For different studies on rate-dependent damage we refer to e.g. [8,9,22] in the isothermal case and [3,28,62,63] for temperature-dependent systems. ...
We present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is independent of temperature.
... More precisely, we shall consider the function spaces with constants 0 < a < b is assumed to be monotonically increasing. By this assumption on f the model will capture partial damage only, i.e., since f (0) ≥ a, even in the state of maximal damage the solid has the ability to counteract external loadings with suitable stresses and displacements; for models allowing for complete damage, where this property is lost, we refer, e.g., to [36,124,101]. The compactness information needed to handle the product of f (z) and quadratic terms in e is provided by the total variation |Dz|(Ω) of z in Ω, weigthed with a constant κ > 0, which is referred to as the factor of influence of damage. ...
The total variation regularization of optimization problems plays an important role in various applications like image processing and continuum mechanics. In contrast to Sobolev functions, functions with bounded total variation may jump across lower-dimensional subsets. This feature is particularly interesting in applications where solutions that develop sharp transitions between different ``phases'' are desired. For example, image processing models should be designed in such a way that sharp edges in the image are preserved. In continuum mechanics it is in some cases desirable to suggest models which provide solutions that are allowed to exhibit spatial jumps, e.g., in the modelling of the evolution of damage in concrete. However, while the usage of total variation regularization is appealing from the viewpoint of mathematical modelling, the development of efficient and reliable numerical methods for minimization problems including a total variation term is challenging due to the non-differentiability of the total variation.
An overview of existing numerical methods for a finite element discretization of a popular (bilaterally constrained) total variation minimization problem from image processing - namely the Rudin-Osher-Fatemi (ROF) problem - is given. The stability and convergence of three classes of algorithms is addressed and numerical comparisons are presented to identify the most efficient iterative scheme for our purposes.
Out of all approaches for total variation minimization, the alternating direction method of multipliers (ADMM), which is a flexible method to solve a large class of convex minimization problems, is analyzed in more detail. Advantages of the ADMM are its unconditional convergence with respect to the involved step size and its direct apllicability. However, the speed of convergence critically depends on the step size. In this thesis, variable step sizes are considered for the ADMM and an adjustment rule for the step size, which relies on the monotonicity of the residual, is proposed to accelerate the algorithm. Numerical experiments reveal significant improvements over established variants of the ADMM.
When dealing with functions with bounded variation, which may exhibit spatial jumps, the appropriate resolution of the jump sets is important to improve approximation properties of finite element functions and to reduce computational costs. Therefore, adaptive mesh refinement techniques which generate meshes that are refined locally in a neighborhood of jump sets of functions with bounded variation are promising. The primal-dual gap is a natural upper bound for the energy error for convex minimization problems and therefore naturally defines a reliable error estimator for any functions that are feasible for the primal convex minimization problem and the associated dual problem. Particularly, for the derivation of the reliability of the primal-dual gap error estimator the involved functionals need not be differentiable. An abstract a posteriori error estimate is established and is applied to the nonlinear Laplace problem and the ROF problem. The primal-dual gap error estimator is used to define adaptive finite element schemes and numerical experiments illustrate the accurate, local mesh refinement, the reliability and the moderate overestimation of the error estimator.
The observations made with regards to the approximation of the (bilaterally constrained) ROF problem are finally used for the numerical simulation of a rate-independent, partial, isotropic damage evolution with a total variation regularized damage variable. Discrete solutions are obtained using an alternate time-discrete scheme in which the nonsmooth optimization problem corresponding to the damage variable is solved in each time step using the ADMM with variable step sizes. A discrete version of a semistable energetic formulation of the rate-independent system is proven and numerical results for two benchmark problems from engineering are presented.
... While some theoretical work consider complete damage ϕ el (ε ε ε, z = 0, c, θ) = 0 (see e.g. [13,31]), for numerical simulation this approach is not applicable. Instead the degradation is regularized by a small residual stiffness η > 0: ...
... As a crack propagates stored elastic energy is dissipated, transferred to surface energy or to microstructural changes. We assume a pseudo-dissipation functional as in Ref. [31]: ...
We present a continuum model that incorporates rate-dependent damage and fracture, a material order-parameter field and temperature within a phase-field approach. The models covers partial damage as well as the formation of macro-cracks. For the material order parameter we assume a Cahn Larché-type dynamics, which makes the model in particular applicable to binary alloys. We give thermodynamically consistent evolution equations resulting from a unified variational approach. Diverse coupling mechanisms can be covered within the model, such as heat dissipation, thermal-expansion-induced failure and crack deflection due to inhomogeneities. With help of an adaptive finite element code we conduct numerical experiments of different complexity in order to study the possibilities and limitations of the presented model. We furthermore include anisotropic linear elasticity in our model and investigate the effect on the crack pattern.
... [14,15] for a first local existence result in 1D) in the spirit of phase transition models. Hence, we refer to [5,6,16,17,23], and references therein, for some analytical results on this type of model. ...
In this paper, we consider a model describing evolution of damage in elastic materials, in which stiffness completely degenerates once the material is fully damaged. The model is written by using a phase transition approach, with respect to the damage parameter. In particular, a source of damage is represented by a quadratic form involving deformations, which vanishes in the case of complete damage. Hence, an internal constraint is ensured by a maximal monotone operator. The evolution of damage is considered “reversible”, in the sense that the material may repair itself. We can prove an existence result for a suitable weak formulation of the problem, rewritten in terms of a new variable (an internal stress). Some numerical simulations are presented in agreement with the mathematical analysis of the system.
... e.g. [BMR09,MRZ10,Mie11a,HK15]), and the case in which some residual stiffness is guaranteed even for χ = 0 (i.e., K(χ) is positive definite for every χ). In fact, in what follows we shall confine our analysis to the latter case. ...
This contribution deals with a class of models combining isotropic damage with plasticity. It has been inspired by a work by Freddi and Royer-Carfagni [FRC10], including the case where the inelastic part of the strain only evolves in regions where the material is damaged. The evolution both of the damage and of the plastic variable is assumed to be rate-independent. Existence of solutions is established in the abstract energetic framework elaborated by Mielke and coworkers (cf., e.g., [Mie05, Mie11b]).
... Therefore, global-in-time existence results for complete damage models are rare. Modeling and existence of weak solutions for purely mechanical complete damage systems with quasi-static force balances are studied in [BMR09,Mie11,HK12] and with visco-elasticity in [MRZ10,RR12]. ...
... The elasticity is considered to be linear and the system is assumed to be in quasi-static mechanical equilibrium since diffusion processes take place on a much slower time scale. The main modeling idea in [HK12] has been to formulate such degenerating system on a time-dependent domain which consists of the not completely damaged regions that are connected to the Dirichlet boundary. In this context, the concept of maximal admissible subsets is introduced to specify the domain of interest. ...
... In the following, a weak formulation of the system above combining the ideas in [HK11] and [HK12] is given. ...
In this work, we analytically investigate a degenerating PDE system for phase separation and complete damage processes considered on a nonsmooth time-dependent domain with mixed boundary conditions. The evolution of the system is described by a degenerating Cahn–Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a degenerating quasi-static balance equation for the displacement field. All these equations are highly nonlinearly coupled. Because of the doubly degenerating character of the system, the doubly nonlinear differential inclusion and the nonsmooth domain, the structure of the model is very complex from an analytical point of view.A novel approach is introduced for proving existence of weak solutions for such degenerating coupled system. To this end, we first establish a suitable notion of weak solutions, which consists of weak formulations of the diffusion and the momentum balance equation, a variational inequality for the damage process and a total energy inequality. To show existence of weak solutions, several new ideas come into play. Various results on shrinking sets and its corresponding local Sobolev spaces are used. It turns out that, for instance, on open sets which shrink in time a quite satisfying analysis in Sobolev spaces is possible. The presented analysis can handle highly nonsmooth regions where complete damage takes place. To mention only one difficulty, infinitely many completely damaged regions which are not connected with the Dirichlet boundary may occur in arbitrary small time intervals.
