Rodica Toader, Chiara Zanini
We introduce a new model of irreversible quasistatic crack growth in which the evolution of cracks is the limit of a suitably modified $ε$-gradient flow of the energy functional, as the "viscosity" parameter $ε$ tends to zero.
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In this paper we study the quasistatic crack growth for a cohesive zone model. We assume that the crack path is prescribed and we study the time evolution of the crack in the framework of the variational theory of rate-independent processes.
Quasistatic growth of cavities and cracks in the plane
本文针对二维非线性弹性问题，提出同时描述空腔与裂纹不可逆准静态扩展并允许空腔合并为裂纹的模型，在有限空腔数和裂纹连通分量数有先验界的条件下证明了准静态演化的存在性。
In this paper, we prove a new existence result for a variational model of crack growth in brittle materials proposed in [15]. We consider the case of $n$-dimensional finite elasticity, for an arbitrary $n\ge1$, with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads, both depending on time.
Linearization of quasistatic fracture evolution in brittle materials
本文证明非线性弹性准静态断裂演化在刚度趋于无穷时收敛到线性弹性准静态裂纹扩展解，无需裂纹几何先验假设，推广了静态线性化结果至演化情形。
Nonlocal quasistatic fracture evolution for interacting cracks is developed and supporting numerical examples are presented. The approach is implicit and is based on local stationarity and fixed point methods. It is proved that the fracture evolution decreases stored elastic energy with each load step as the cracks advance; provided the load increments are chosen sufficiently small. This is also seen in the numerical examples. The numerical examples include evolution of a straight crack, a crack propagating inside an L-shaped domain, and two offset inward propagating cracks.
In this paper we prove a two-dimensional existence result for a variational model of crack growth for brittle materials in the realm of linearized elasticity. Starting with a time-discretized version of the evolution driven by a prescribed boundary load, we derive a time-continuous quasistatic crack growth in the framework of generalized special functions of bounded deformation (GSBD). As the time-discretization step tends to $0$, the major difficulty lies in showing the stability of the static equilibrium condition, which is achieved by means of a Jump Transfer Lemma generalizing the result o…
In this paper we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e., a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approxim…
Discontinuous finite element approximation of quasistatic crack growth in finite elasticity
本文提出一种时间-空间离散化方法，用于有限弹性中准静态裂纹扩展问题，采用自适应三角剖分并证明能量及变形的收敛性。
This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution associated with the relaxed model. Firstly, a $Γ$-convergence analysis is performed with a surface energy density which does not provide weak compactness in the space of Special Functions of Bounded Variation. Then, the asymptotic analysis of the quasistatic crack evolution is presented in the case of bounded solutions that is with the simpli…
Quasistatic evolution of cohesive-type fracture
针对无预设裂纹路径且无拓扑限制的内聚断裂模型，证明了全局稳定准静态演化的存在性，通过引入新的记忆变量收敛概念并改变经典证明顺序克服了极限保持问题。
In this paper we contribute to studying the issue of quasistatic limit in the context of Griffith's theory by investigating a one-dimensional debonding model. It describes the evolution of a thin film partially glued to a rigid substrate and subjected to a vertical loading. Taking friction into account and under suitable assumptions on the toughness of the glue, we prove that, in contrast to what happens in the undamped case, dynamic solutions converge to the quasistatic one when inertia and viscosity go to zero, except for a possible discontinuity at the initial time. We then characterise the…
We deal with quasistatic evolution problems in plasticity with softening, in the framework of small strain associative elastoplasticity. The presence of a nonconvex term due to the softening phenomenon requires a nontrivial extension of the variational framework for rate-independent problems to the case of a nonconvex energy functional. We argue that, in this case, the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view. Thus, we analize a different selection criterion for the solutions of the quasistatic evolution problem, base…
Visco-energetic solutions have been recently advanced as a new solution concept for rate-independent systems, alternative to energetic solutions/quasistatic evolutions and balanced viscosity solutions. In the spirit of this novel concept, we revisit the analysis of the variational model proposed by Francfort and Marigo for the quasi-static crack growth in brittle materials, in the case of antiplane shear. In this context, visco-energetic solutions can be constructed by perturbing the time incremental scheme for quasistatic evolutions by means of a viscous correction inspired by the term introd…
We show the existence of quasistatic evolutions in a fracture model for brittle materials by a vanishing viscosity approach, in the setting of planar linearized elasticity. The crack is not prescribed a priori and is selected in a class of (unions of) regular curves. To prove the result, it is crucial to analyze the properties of the energy release rate.
Gradient-flow characterizations of one-dimensional quasistatic viscoelasticity with Bhattacharya-like viscosity
本文研究了一维准静态非线性黏弹性方程在Dirichlet边界条件下的全局存在性，通过空间离散化方法直接处理粘性相关的Riemann度量，并证明了解的梯度流表示。
We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each time interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response.
We prove the stability of a large class of unilateral minimality properties which arise naturally in the theory of crack propagation proposed by Francfort and Marigo in [Revisiting brittle fractures as an energy minimization problem. J. Mech. Phys. Solids, 46 (1998), 1319-1342]. Then we give an application to the quasistatic evolution of cracks in composite materials.
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, depend 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 ze…
In this paper we study the vanishing inertia and viscosity limit of a second order system set in an Euclidean space, driven by a possibly nonconvex time-dependent potential satisfying very general assumptions. By means of a variational approach, we show that the solutions of the singularly perturbed problem converge to a curve of stationary points of the energy and characterize the behavior of the limit evolution at jump times. At those times, the left and right limits of the evolution are connected by a finite number of heteroclinic solutions to the unscaled equation.
Balanced quasistatic evolutions of critical points in metric spaces
本文提出了一种基于分离能量演化与系统过渡时标的平衡准静态演化方法，适用于度量空间中的临界点问题，无需非退化临界点假设。
The modeling of cracks has been an intensely researched topic for decades - both from the mechanical as well as from the mathematics point of view. As far as the modeling of sharp cracks/interfaces is concerned, the resulting free boundary problem is numerically very challenging. For this reason, diffuse approximations in the sense of phase-field theories have become very popular. Within this paper, the focus is on rate-independent damage models. Since the resulting phase-field energies in general are non-convex, we are faced with a discontinuous evolution of the phase-field variable. Solution…
We consider the quasi-static evolution of a brittle layer on a stiff substrate; adhesion between layers is assumed to be elastic. Employing a phase-field approach we obtain the quasi-static evolution as the limit of time-discrete evolutions computed by an alternate minimization scheme. We study the limit evolution, providing a qualitative discussion of its behaviour and a rigorous characterization, in terms of parametrized balanced viscosity evolutions. Further, we study the transition layer of the phase-field, in a simplified setting, and show that it governs the spacing of cracks in the firs…
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