Dynamic crack propagation in a 2D elastic body: The out-of-plane case

Already in 1920 Griffith has formulated an energy balance criterion for quasistatic crack propagation in brittle elastic materials. Nowadays, a generalized energy balance law is used in mechanics [F. Erdogan, Crack propagation theories, in: H. Liebowitz (Ed.), Fracture, vol. 2, Academic Press, New York, 1968, pp. 498-586; L.B. Freund, Dynamic Fracture Mechanics, Cambridge Univ. Press, Cambridge, 1990; D. Gross, Bruchmechanik, Springer-Verlag, Berlin, 1996] in order to predict how a running crack will grow. We discuss this situation in a rigorous mathematical way for the out-of-plane state. This model is described by two coupled equations in the reference configuration: a two-dimensional scalar wave equation for the displacement fields in a cracked bounded domain and an ordinary differential equation for the crack position derived from the energy balance law. We handle both equations separately, assuming at first that the crack position is known. Then the weak and strong solvability of the wave equation will be studied and the crack tip singularities will be derived under the assumption that the crack is straight and moves tangentially. Using the energy balance law and the crack tip behavior of the displacement fields we finally arrive at an ordinary differential equation for the motion of the crack tip.     

Phase-Field Modeling of Fracture in Anisotropic Media

In the recent years phase-field modeling of fracture has become a promising tool to describe complex crack patterns in all kinds of solid materials. Many of the models assume an isotropic material behavior, which of course is not a meaningful assumption for e.g. biological tissues such as arterial walls. Since the phase-field approach introduces an additional (smeared) phase describing the evolution of the crack, this method is well suited to be extended to anisotropic materials without thinking about an adaption of the discretization technique. Anisotropy can be incorporated in several ways, like by an extension of the surface energy, i.e. by making the energy release rate orientation dependent, as considered in [1]. Our ansatz is based on a pure geometrical approach, namely on an anisotropic formulation of the crack surface itself. Here, we will focus on transversely isotropic and cubically anisotropic solids, where the latter one makes the incorporation of the second gradient of the crack phase field necessary. At the end one numerical example is shown, which conceptually shows the influence of the anisotropy on the crack path. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New three-dimensional finite elements with embedded strong discontinuities to model solids at failure

This work focuses on the development of new finite elements which can capture strong discontinuities in three-dimensional failure problems. The displacement jumps in the solid are approximated by a linear interpolation obtained by enforcing a new class of enhanced separation modes to exactly be satisfied by the formulation. Efforts are also put towards the development of a proper crack propagation tracking algorithm needed for the complicated crack surfaces appearing in realistic 3D failure simulations, based on a combination of the global tracking algorithm and the marching cubes algorithm. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Out-of-plane displacement measurement near the crack tip during a crack propagation: Validation of a 3D formulation for a specimen in PMMA loaded in mode I.

The fundamental aim of this study is the determination zone of the 3D effects and the transient one at the vicinity of the crack tip during a crack propagation in brittle materials ( PMMA ) using an optical method (Michelson interferometer). With the obtained interferograms, we can extract the phase (thus the relief) by using a new numerical approach based on the principle of images correlation between real fringes and virtual fringes. Different dynamic tests are realized by a plate loaded in mode I under a constant loading. We compare the obtained data with the two-dimensional theory of Westergaard (plane stress hypothesis) [1]. With the divergence is established, we propose a new 3D formulation, based on a formulation employed for static crack, which takes into account 3D and transient effects. For the static cracks, the 3D effects relate to a presence of the state of three-dimensional stresses. However in dynamics, the transient effects appear and are related to the crack propagation velocity. The 3D effects and transient effects lead to results equivalent to experimental ones in terms of displacement but are completely different to results given by the two-dimensional theory near the crack tip. It is possible to quantify the zone when the plane stress hypothesis is not valid according to the crack propagation speed V. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unsteady propagation of longitudinal shear cracks

In all problems of unsteady crack propagation which have been solved to date [1 to 3], it has been assumed that the crack propagates at a constant speed. This assumption was not prompted by physical considerations of the problem, but by the methods of solution, therefore, the applicability of the results is limited. It would be more realistic to consider the speed of crack propagation as a function of time based on explicit physical hypotheses. Unfortunately, the general case of the resultant problem cannot be solved by existing methods. However, the problem of longitudinal shear cracks i.e. the plane problem in which the displacement is parallel to the crack boundary, may be solved for an arbitrary given variation in crack propagation speed, utilizing the method developed in connection with the theory of supersonic flows [4 and 5].

Note that equilibrium problems of longitudinal shear cracks have been studied in [6 and 7].     

复合材料桥纤维拔出问题的动态裂纹模型

在一无限的正交各向异性体的弹性平面上,对具有桥纤维平行自由表面的一个内部中央裂纹,进行了弹性分析．提出了复合材料桥纤维拔出的一个动态模型．由于纤维破坏是由最大拉应力支配,纤维断裂并且裂纹扩展将以自相似的方式出现．通过复变函数的方法将所讨论的问题转化为Reimann-Hilbert混合边界值问题的动态模型,呈现一简单的和容易的解．求得了正交异性体中扩展裂纹受运动的阶梯载荷、瞬时脉冲载荷作用下问题的解析解,并利用这一解,通过迭加最终求得该模型的解．     

Crack propagation calculation using trial cracks

We present a method to perform finite element calculations for crack propagation problems with arbitrary crack directions in two dimensions. The crack direction (angle of propagation) is determined by inserting small “trial cracks” at the crack tip. For each trial crack, the domain is remeshed to allow crack propagation between elements. The trial cracks are then opened and the energy release rate is measured. The optimum crack direction (i.e., the crack direction with maximum energy release) is determined by an optimisation procedure. Although the method is computationally expensive due to the need to perform several calculations for each crack increment, it has the advantage that the energy release rate can be calculated even in cases where other methods fail. After explaining the method, it is applied to some test examples. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonsmooth domain optimization for elliptic equations with unilateral conditions

In the paper we consider elliptic boundary problems in domains having cuts (cracks). The non-penetration condition of inequality type is prescribed at the crack faces. A dependence of the derivative of the energy functional with respect to variations of crack shape is investigated. This shape derivative can be associated with the crack propagation criterion in the elasticity theory. We analyze an optimization problem of finding the crack shape which provides a minimum of the energy functional derivative with respect to a perturbation parameter and prove a solution existence to this problem.     

Numerical modeling of brittle fracture in porous CFC-materials

Both energy and stress criteria are necessary conditions for fracture but neither the one nor the other is sufficient. A combination of these criteria is proposed in [1]. This combined criterion is used for numerical simulation of crack propagation by the 4-point bending test in porous materials. Examples of such materials are carbon-carbon composites (CFC) [2, 3]. Micrographs of the cross-sections of these materials are used for FEM modeling of the crack propagation on the basis of the proposed criterion. Results of the numerical modeling are compared with experimental results. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

各向异性材料动态裂纹扩展特性和动态应力强度因子

基于各向异性材料力学,研究了无限大各向异性材料中Ⅲ型裂纹的动态扩展问题．裂纹尖端的应力和位移被表示为解析函数的形式,解析函数可以表达为幂级数的形式,幂级数的系数由边界条件确定．确定了Ⅲ型裂纹的动态应力强度因子的表达式,得到了裂纹尖端的应力分量、应变分量和位移分量．裂纹扩展特性由裂纹扩展速度M和参数alpha反映,裂纹扩展越快,裂纹尖端的应力分量和位移分量越大;参数alpha对裂纹尖端的应力分量和位移分量有重要影响．     

Analysis of extension propagation process of interface crack between belts of a radial tire using a finite element method

A radial tire is a very complex structure made from rubber elastomers and fiber–rubber composite materials. During its use, extension propagation of interface crack between belts can occur, which obviously affects its durability and life. In the present paper, a new mathematical model of extension propagation of interface crack in complex composite structures is presented. The model can reveal the extension propagation dependence of interface crack on the relative size of energy release rates at the left and right crack tips and on the interfacial material properties. The extension propagation model of interface crack, Irwin’s virtual crack close technique and the finite element analysis method are used together in simulating numerically the extension propagation process of a interface crack between belts of a radial tire. The present study numerical results show that the extension propagation model of interface crack proposed in this paper can more realistically characterize the complexity of the extension propagation process of interface crack in complex composite structures.     

Phase field simulation of thermomechanical fracture

In Francfort and Marigo's variational free-discontinuity formulation of brittle fracture [1] cracking is regarded as an energy minimization process, where the total energy is minimized with respect to any admissible crack set and displacement field. No additional criterion is needed to determine crack paths, branching of cracks and crack initiations. However, a direct discretization of the model is faced with significant technical problems, as it involves minimizations in a set of possibly discontinuous functions. A regularized version of the model has been introduced by Bourdin [2] and based on this, we use the concept of a continuum phase field model to simulate cracking processes. Cracks are indicated by the order parameter of the phase field model and cracking can be regarded as a phase transition problem. Additionally, introducing the heat equation into the model, it is capable to also take account of crack propagation due to thermal stresses. In the numerical implementation, crack parameter as well as temperature are treated as additional degrees of freedom and the coupled field equations are solved using the finite element method together with an implicit time integration scheme. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Energy driven crack propagation at finite strains based on the embedded strong discontinuity approach

New advances in three-dimensional finite element modeling of crack propagation at finite strains are presented. The proposed numerical model is based on the Enhanced Assumed Strain concept. The enhanced part of the deformation gradient is associated with a displacement discontinuity. In contrast to previous works, a new, energy based criterion for crack propagation is presented. The necessity for a tracking algorithm for the crack path is avoided by using more than one discontinuity within each finite element. This leads to a strictly local formulation, i.e., no information about the neighboring elements are required. Further advantages of such a formulation are a symmetric tangent stiffness matrix and the reduction of locking effects. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase Field Model for Ductile to Brittle Failure Mode Transition

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in dynamic problems with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. We outline a conceptual framework for phase field models of crack propagation in brittle elastic and ductile elastic-plastic solids under dynamic loading and investigate the ductile to brittle failure mode transition observed in the experiment performed by Kalthoff and Winkeler [3]. We develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. To this end, we define energy storage and dissipation functions for the plastic flow including the fracture phase field. The introduction of local history fields that drive the evolution of the crack phase field inspires the construction of robust operator split schemes. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the elastic symmetries of growing mixed-mode cracks

Many continuum damage mechanics models for quasi-brittle materials are based on the reduction of stiffness due to elliptical crack or penny-shaped microcracks in the material. Because of this a numerical study of growing elliptical cracks in a unit cube is undertaken with the help of an FEM simulation.The propagation of the crack is governed by the principle of maximum driving force [1]. For each propagation step the tensor of elasticity is calculated and its symmetries are analyzed. It will be shown that the elastic symmetry in each step is close to orthotropy and can be approximated by an elliptical crack. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Invariant energy integrals for the non-linear crack problem with possible contact of the crack surfaces

Generally two-dimensional and three-dimensional formulations of the non-linear crack problem when the crack surfaces do not overlap for a non-uniform anisotropic linearly elastic body are considered. The first derivative of the potential energy function with respect to the perturbation parameter and its representation in the form of an invariant integral over an arbitrary closed contour are obtained for a general form of the differentiable perturbation of a region with a cut, using the method of material derivatives. The sufficient conditions for the existence of an invariant energy integral are derived in general form, and examples of invariant integrals are constructed for different types of perturbations and a different geometry of the cut.     

Progressive thermal stress distribution around a crack under Joule heating in orthotropic materials

Stress intensity factor and stress distribution at crack tips are classical problems in solids, which are closely related to the failure and reliability of materials. A crack in a nonlinearly coupled anisotropic medium, on the other hand, is much more difficult to analyze. Using the generalized complex variable method, the thermal stress problem of a crack embedded in an orthotropic medium has been analyzed, and the progressive thermal stress distributions have been obtained in closed-forms. The analysis shows that the thermal stress intensity factors are linear functions of remote thermal flux while are nonlinear functions of remote current; the thermal stress distributions under produced by thermal flux and Joule heating are similar, but not identical; the thermal stress intensity factors are linear functions with respect to the thermal expansion coefficients; with the increase of crack length, the thermal stress intensity factor caused by Joule heat increases rapidly; the thermal stress intensity factors are directly proportional to the temperature difference between the upper and lower crack surfaces and the left and right half crack surfaces divided by the square root of the crack length, and the ratios are only determined by the material parameters. These results provide a powerful tool for the failure and reliability analysis of conductive materials, and suggested that thermal stress analysis may be localized.     

各向异性介质中周期性界面裂纹的弹塑性问题

应用富里叶积分变换方法将裂纹边值问题化为对偶积分方程组,再用定积分变换法将问题进一步化为奇异积分方程组,求得了双材料各向异性弹塑性介质中周期性界面裂纹反平面问题的封闭形式解,并作为特例讨论了各向同性双材料问题、各向异性单一材料问题及各向同性—各向异性双材料问题.结果表明:裂纹尖端前沿的塑性区尺寸、裂纹的张开位移(COD)均决定于两种材料流动极限中的较小者及裂纹的长度和相邻两裂纹的间距,此外,COD还与材料模量有关.     

含有直线状裂纹正交异性板的平面问题的应力函数

本文的解析对象为含有一与主轴呈任意角度直线状裂纹的无限大正交异性板的平面问题.采用加权积分法导出了能够表现裂纹尖端附近有限应力集中特征的应力函数.这样的计算模型消除了裂纹尖端的奇异性,可以比较真实地反映非金属材料微裂区的力学行为.     

Simulation of Hertzian cone cracks using a phase field description for fracture

The present contribution focuses on fracture caused by indentation loading on the surface of a brittle solid. Its theoretical prediction is a challenging task due to the fact that crack nucleation is not geometrically induced, but is caused by the stress concentration in the contact near-field. The application of the phase field model requires constitutive assumptions to ensure a tension-compression asymmetric material response and prevent damage in compressed regions. This is achieved at the cost of giving up the variational concept of brittle fracture. We simulate the indentation of a cylindrical flat-ended punch on brittle materials like silicate glass. In order to reduce the numerical effort, we exploit axisymmetric conditions for the finite element formulation. After crack initiation stable propagation of a cone crack can be observed in good agreement with experiments. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)