Gianni Dal Maso, Gilles A. Francfort, Rodica Toader
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.
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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。
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.
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…
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…
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.
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。
This paper is devoted to the characterization of the energy release rate of a crack which is merely closed, connected, and with density $1/2$ at the tip. First, the blow-up limit of the displacement is analyzed, and the convergence to the corresponding positively $1/2$-homogenous function in the cracked plane is established. Then, the energy release rate is obtained as the derivative of the elastic energy with respect to an infinitesimal additional crack increment.
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.
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。
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…
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。
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…
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.
On the crack-driving force of phase-field models in linearized and finite elasticity
本文研究了相场模型在线性和有限弹性材料中的裂纹驱动力问题，通过不同变分公式和基于断裂力学的启发式驱动力，验证了通用方法的优越性。
Linearization of finite-strain poro-visco-elasticity with degenerate mobility
本文在拉格朗日框架下研究有限应变多孔粘弹性非线性模型，利用 Moser 迭代证明浓度有界性，并在小载荷假设下严格证明非线性有限应变系统解收敛于线性小应变系统解。
Strong existence for free discontinuity problems in linear elasticity
本文通过证明 Griffith 泛函一大类拟极小元的 Ahlfors 正则性，确立线性弹性中多类自由间断问题极小元跳跃集本质闭合并具有强存在性。
This work is devoted to the variational derivation of a reduced model for brittle membranes in finite elasticity. The main mathematical tools we develop for our analysis are: (i) a new density result in $GSBV^{p}$ of functions satisfying a maximal-rank constraint on the subgradients, which can be approximated by $C^{1}$-local immersions on regular subdomains of the cracked set, and (ii) the construction of recovery sequences by means of suitable $W^{1,\infty}$ diffeomorphisms mapping the regular subdomains onto the fractured configuration.
We consider the elasticity problem in a %heterogeneous domain with contact on multiple periodic open cracks. The contact is described by the Signorini and Coulomb-friction conditions. Problem is non-linear, the dissipative functional depends on the un-known solution and the existence of the solution for fixed period of the structure is usually proven by the fix-point argument in the Sobolev spaces with a little higher regularity, $H^{1+α}$. We rescaled norms, trace, jump and Korn inequalities in fractional Sobolev spaces with positive and negative exponent, using the unfolding technique, intro…
Uniform rectifiability of brittle fractures in linear elasticity
本文研究了线弹性中脆性断裂的均匀可 rectifiability，通过证明裂缝具有大量大投影，提供了一种新的方法来克服现有方法在Griffith设定中的限制。
By a combination of geometrical and configurational analysis we study the properties of absolute minimal and equilibrium states of general Mumford-Shah functionals, with applications to models of quasistatic brittle fracture propagation. The main results concern the mathematical relations between physical quantities as energy release rate and energy concentration for 3D cracks with complex shapes, seen as outer measures living on the crack edge.
Finite-strain poro-visco-elasticity with degenerate mobility
本文在Lagrangian框架下使用二阶非简单材料概念，研究了有限应变下具有退化迁移率的孔隙-粘弹性固体拟静态非线性模型，通过交错时间增量格式和能量耗散不等式证明了弱解的存在性。
We consider the elasticity operator with zero Poisson's ratio on an infinite strip and an infinite plate with a horizontal crack. We prove an asymptotic formula for the distance of the embedded eigenvalues to some spectral threshold of the operator as the crack becomes small.
The interaction of a thin rigid inclusion with a finite crack is studied. Two plane problems of elasticity are considered. The first one concerns the case when the upper side of the inclusion is completely debonded from the matrix, and the crack penetrates into the medium. In the second model, the upper side of the inclusion is partly separated from the matrix, that is the crack length $2a$ is less than $2b$, the inclusion length. It is shown that both problems are governed by a singular integral equation of the same structure. Derivation of the closed-form solution of this integral equation i…
Existence Results for the Time-incremental Elastic Contact Problem with Coulomb Friction in 2D
本文研究了二维弹性接触问题中Coulomb摩擦的存在性，通过变分不等式和Leray-Lions算子理论，证明了各向同性和各向异性弹性下的解存在性。
Given the flexibility of choosing negative elastic parameters, we construct material structures that can induce two resonance phenomena, referred to as the elastodynamical resonances. They mimic the emerging plasmon/polariton resonance and anomalous localized resonance in optics for subwavelength particles. However, we study the peculiar resonance phenomena for linear elasticity beyond the subwavelength regime. It is shown that the resonance behaviours possess distinct characters, with some similar to the subwavelength resonances, but some sharply different due to the frequency effect. It is p…
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