We propose a discontinuous finite element approximation for a model of quasi-static growth of brittle fractures in linearly elastic bodies formulated by Francfort and Marigo, and based on the classical Griffith's criterion. We restrict our analysis to the case of anti-planar shear and we consider discontinuous displacements which are piecewise affine with respect to a regular triangulation.
In this work we derive by Gamma-convergence techniques a model for brittle fracture linearly elastic plates. Precisely, we start from a brittle linearly elastic thin film with positive thickness $ρ$ and study the limit as $ρ$ tends to 0. The analysis is performed with no a priori restrictions on the admissible displacements and on the geometry of the fracture set. The limit model is characterized by a Kirchhoff-Love type of structure.
A mathematical continuum limit of the interaction energy of a random particle chain is shown to yield new insight into the effect of microscopic heterogeneities on macroscopic fracture laws in brittle materials. We derive a formula which yields that either elastic behaviour or a crack is energetically preferred. The formula explicitly shows the dependence on the boundary condition and the microstructure of the chain. The mathematical analysis is based on a variational convergence $Γ$-convergence of convex-concave potentials together with ergodic theorems which are common tools in stochastic ho…
A model for the quasi-static growth of brittle fractures: existence and approximation results
本文给出了G.A. Francfort和J. - J. Marigo的裂纹扩展理论，并基于Griffith的裂纹扩展理论，对裂纹扩展进行了理论分析。在二维情形下，我们证明了准静态演化的存在性，并证明了总能量是时间的绝对连续函数，尽管我们不能排除体能和表面能可能存在跳跃间断.这个存在性的结果证明了时间离散化过程中，在每一步的全局能量最小化进行，与约束，新的裂纹包含所有的裂纹形成在以前的时间步骤。该方法提供了一种有效的逼近连续时间演化的方法。
A model for the quasi-static growth of a brittle fracture: existence and approximation results
本文给出了G.A.提出的脆性断裂不可逆准静态演化的变分模型的精确数学表达式。Francfort和J. - J. Marigo的裂纹扩展理论，并基于Griffith的裂纹扩展理论，对裂纹扩展进行了理论分析。在二维情形下，我们证明了准静态演化的存在性，并证明了总能量是时间的绝对连续函数，尽管我们不能排除体能和表面能可能存在跳跃间断.这个存在性的结果证明了时间离散化过程中，在每一步的全局能量最小化进行，与约束，新的裂纹包含所有的裂纹形成在以前的时间步骤。该方法提供了一种有效的逼近连续时间演化的方法。
The phase-field model for fracture, despite its popularity and ease of implementation comes with its set of computational challenges. They are the non-convex energy functional, variational inequality due to fracture irreversibility, the need for extremely fine meshes to resolve the fracture. In this manuscript, the focus is on the numerical treatment of variational inequality. In this context, the popular history-variable approach suffers from variationally inconsistency and non-quantifiable nature of the error introduced. A better alternative, the penalisation approach, has the potential to r…
We study a planar thin brittle beam subject to elastic deformations and cracks described in terms of a nonlinear Griffith energy functional acting on $SBV$ deformations of the beam. In particular we consider the case in which elastic bulk contributions due to finite bending of the beam are comparable to the surface energy which is necessary to completely break the beam into several large pieces. In the limit of vanishing aspect ratio we rigorously derive an effective Griffith-Euler-Bernoulli functional which acts on piecewise $W^{2,2}$ regular curves representing the midline of the beam. The e…
We formulate a nonlocal cohesive model for calculating the deformation state inside a cracking body. In this model a more complete set of physical properties including elastic and softening behavior are assigned to each point in the medium. We work within the small deformation setting and use the peridynamic formulation. Here strains are calculated as difference quotients. The constitutive relation is given by a nonlocal cohesive law relating force to strain. At each instant of the evolution we identify a process zone where strains lie above a threshold value. Perturbation analysis shows that …
A nonlocal field theory of peridynamic type is applied to model the brittle fracture problem. The elastic fields obtained from the nonlocal model are shown to converge in the limit of vanishing non-locality to solutions of classic plane elastodynamics associated with a running crack.
Phase-field approaches to fracture based on energy minimization principles have been rapidly gaining popularity in recent years, and are particularly well-suited for simulating crack initiation and growth in complex fracture networks. In the phase-field framework, the surface energy associated with crack formation is calculated by evaluating a functional defined in terms of a scalar order parameter and its gradients, which in turn describe the fractures in a diffuse sense following a prescribed regularization length scale. Imposing stationarity of the total energy leads to a coupled system of …
Starting from a particle system with short-range interactions, we derive a continuum model for the bending, torsion, and brittle fracture of inextensible rods moving in three-dimensional space. As the number of particles tends to infinity, it is assumed that the rod's thickness is of the same order as the interatomic distance. Fracture energy in the $Γ$-limit is expressed by an implicit cell formula, which covers different modes of fracture, including (complete) cracks, folds and torsional cracks. In special cases, the cell formula can be significantly simplified. Our approach applies e.g. to …
