A criterion for the growth of cracks with bonds in the end zone

A fracture criterion which takes account of the work done in the deformation of bonds in the end zone of a crack is proposed for analysing the quasistatic growth of a crack with bonds in the end zone. The energy condition that the deformation energy release rate at the crack tip is equal to the rate of deformation energy consumption by the bonds in the end zone of the crack (the first fracture condition) corresponds to the state of limit equilibrium of the crack tip. The rupture of bonds at the trailing edge of the end zone is determined by the condition for their limiting traction (the second fracture condition). Starting from these two conditions, the processes of subcritical and quasistatic crack growth are considered for the case of a rectilinear crack at interface of materials and the two basic fracture parameters, the critical external load and the size of the end zone of the crack in the state of limit equilibrium, are determined. Analytical expressions are obtained for the deformation energy release rate at the crack tip and the rate of deformation energy consumption by the bonds and, also, the dependences of the critical external load and size of the end zone of the crack on the crack length in the case of a rectilinear crack in a homogeneous body with bond tractions which are constant and independent of the external load. The limit cases of a crack which is filled with bonds and a crack with a short end zone are considered.     

Cracks in piezoelectric and electroconductive bodies

Singularities are studied of the elastic and electric fields near a tip of a crack on the interface of two piezoelectric bodies. An analog of the Griffith formula is obtained for the increment of the potential energy of deformation due to development of a rectilinear crack. The external electrical forces result in the decrease of the energy release rate which explains an experimentally-known possibility of controlling the fracture process by some additional electric fields.     

Revisiting Energy Release Rates in Brittle Fracture

We revisit in a 2d setting the notion of energy release rate, which plays a pivotal role in brittle fracture. Through a blow-up method, we extend that notion to crack patterns which are merely closed sets connected to the crack tip. As an application, we demonstrate that, modulo a simple meta-stability principle, a moving crack cannot generically kink while growing continuously in time. This last result potentially renders obsolete in our opinion a longstanding debate in fracture mechanics on the correct criterion for kinking.     

Transient response of dissimilar piezoelectric layers with multiple interacting interface cracks

In this study, we examine the dynamic behavior of two bonded dissimilar piezoelectric layers containing multiple interfacial cracks subjected to electro-mechanical impact loading. The problem was formulated through Fourier transformation into singular integral equations in which the unknown variables are the jumps of displacement and electric potential across the crack surface in the Laplace transform domain. The resulting integral equations together with the corresponding single-valued conditions are solved numerically for the densities of electro-elastic dislocations on a crack surface. The dynamic field intensity factors and dynamic energy release rate (DERR) history are obtained for both permeable and impermeable crack. The stress field is also obtained for the interface crack under impact loads. The results show that the field intensity factors at the crack tips and dynamic energy release rate depend on the interfacial crack geometry, electromechanical coupling and the electric boundary conditions on the crack surface.     

Energy release rate for cracks in finite‐strain elasticity

Griffith's fracture criterion describes in a quasistatic setting whether or not a pre‐existing crack in an elastic body is stationary for given external forces. In terms of the energy release rate (ERR), which is the derivative of the deformation energy of the body with respect to a virtual crack extension, this criterion reads: if the ERR is less than a specific constant, then the crack is stationary, otherwise it will grow. In this paper, we consider geometrically nonlinear elastic models with polyconvex energy densities and prove that the ERR is well defined. Moreover, without making any assumption on the smoothness of minimizers, we rigorously derive the well‐known Griffith formula and the J‐integral, from which the ERR can be calculated. The proofs are based on a weak convergence result for Eshelby tensors. Copyright © 2007 John Wiley & Sons, Ltd.     

Deformation Driven Homogenization of Fracturing Solids

The paper discusses numerical formulations of the homogenization for solids with discrete crack development. We focus on multi–phase microstructures of heterogeneous materials, where fracture occurs in the form of debonding mechanisms as well as matrix cracking. The definition of overall properties critically depends on the developing discontinuities. To this end, we extend continuous formulations [1] to microstructures with discontinuities [2]. The basic underlying structure is a canonical variational formulation in the fully nonlinear range based on incremental energy minimization. We develop algorithms for numerical homogenization of fracturing solids in a deformation–driven context with non–trivial formulations of boundary conditions for (i) linear deformation and (ii) uniform tractions. The overall response of composite materials with fracturing microstructures are investigated. As a key result, we show the significance of the proposed non–trivial formulation of a traction–type boundary condition in the deformation–driven context. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Toughening effect of ferroelectric ceramics induced by domain switching and dislocations

The multi-scale analysis of fracture toughness of ferroelectric ceramics under complicate mechanical–electrical coupling effect is carried out in this paper. The generalized stress intensity factor (SIF) arising from spontaneous strains and polarization transformation in switching domain zones is accurately obtained by using an extended Eshelby theory. Taking BaTiO₃ ferroelectric ceramic for example, it is discovered that the crack propagation can be induced by domain switching arising from negative electrical field when the crack surface is parallel to the isotropic plane, and the obtained critical electric displacement intensity factor (EDIF) approximates closely to that obtained by the Green’s function method. Additionally, as pinning dislocations and slip dislocations can strongly influence properties of ferroelectric devices and induce the property degradation, it is necessary to investigate the dislocation toughening effects on fatigue and fracture mechanisms. The results show that the dislocation shielding and anti-shielding effects on mode II SIF, mode I SIF and EDIF are obviously different when a dislocation locates at a position near the crack tip. Through the calculation of the critical applied EDIF for crack propagation by using mechanical energy release rate (MERR) theory, it is discovered that the slip angles obviously influence fracture toughness, and the mode II SIF arising from dislocation has little influence on fracture toughness, however, the mode I SIF and EDIF arising from dislocation have great influences on fracture toughness.     

Simulation of domain structures in cracked ferroelectric materials

The domain structure around a crack tip plays a significant role in the fracture behavior of ferroelectrics. A continuum phase field model is used to investigate the microstructure at the crack front. The concept of the Eshelby momentum tensor and configurational forces is then generalized to account for the contributions of the polarization term. Implementation of the generalized configurational force in the Finite Element code enables us to numerically obtain the driving force at the crack tip, which corresponds to the crack-tip energy release rate. Calculations show that additional positive electric fields tend to prohibit crack growth, whereas additional negative electric fields tend to promote crack growth. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

畴极化转动对多晶铁电材料断裂特性的影响

主要基于细观力学方法揭示了畴极化转动对多晶铁电陶瓷的各向异性断裂特性的平均影响。首先,用Eshelby-Mori-Tanaka理论和统计模型分析了无穷大铁电材料体中一椭球夹杂的内、外电弹性场,得到畴极化转动对电弹性场的平均影响;其次,推导了等效多晶铁电陶瓷中含一钱币状裂纹的裂纹扩展力(能量释放率)G_ext,并用它估计了畴极化转动对多晶铁电陶瓷断裂特性的影响。对BaTiO₃陶瓷中裂纹扩展力的计算结果表明,对多晶铁电材料断裂特性分析必须考虑畴极化转动的影响。计算结果得出了与实验相一致的结论:在受较小的力时,外加电场对裂纹扩展产生较大的影响,而且在某种程度上能促进了裂纹扩展。     

压电介质内裂纹问题的精确解

在压电介质断裂力学分析中,人们常假定裂纹面上的电位移法向分量为零,可是实验表明,这一假设将导致错误的结果。本文基于精确的电边界条件,并应用Stroh公式的方法,导出了含裂纹压电介质在无限远处均匀外载作用下二维问题的精确解。结果表明:(ⅰ)应力强度因子与各向同性材料相同,而电位移强度因子取决于材料常数和机械载荷,但与电载荷无关;(ⅱ)能量释放率大于纯弹性各向异性材料内的值,即总是正的,且与电载荷无关;(ⅲ)裂纹内所含空气的介电常数对介质内的场强无影响。     

Delamination growth of laminated circular plates under in-plane loads and movable boundary conditions

An investigation is made on interlaminar delamination growth of composite laminated circular plates under in-plane loads and movable delamination boundary conditions. A four-dissociated-region model is developed on the basis of von-Karman plate theory. The model is geometrically nonlinear and the laminated circular plate considered is subjected to axisymmetrical delamination. The effects of transverse shear deformation and contact effect of the delamination on the laminated plates are taking into account in the development of the governing equations of the laminated circular pates with random axisymmetrical delamination. The formulas for describing the total energy release rate and its individual mode components along the delamination front are also derived with considerations of Griffith criterion for fracture. Based on the model established, the delamination growth is numerically studied; and the influences of the parameters such as delamination radii and depths, together with material properties of the plates on the energy release rate are analyzed in detail.     

Two-dimensional analysis of a crack in quasicrystal materials

The two-dimensional problem of a crack in three-dimensional quasicrystals subject to far field loadings is studied. The analysis is based on the generalized Lekhnitskii's formalism. The analytical expressions for both the entire fields and the asymptotic fields near the crack tip are determined. The fracture quantities of quasicrystals, i.e., field intensity factors, energy release rates and so on, is a prerequisite. Numerical results for a Griffith crack under phason loading Mode I and II conditions are poltted. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experimental study of fracture toughness and energy in composite materials

The paper presents an experimental investigation of fracture characteristics of composite materials. The post-peak response of the load-crack opening displacement of notched specimens is used to evaluate the fracture energy associated with progressive matrix damage and crack growth. Effects of fiber orientation and other geometric characteristics on fracture parameters are studied. The load versus crack opening displacement as well as crack length, fracture toughness, and energy versus the number of loading cycles are obtained for different specimens. Based on the experimental results of this study, concepts of the fracture mechanics are applied to evaluate the evolution of fracture toughness and energy.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Department of Mechanical & Industrial Engineering, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2. Published in Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 323–332, May–June, 1998.     

Analysis of crack formation in bonded lap joints using finite fracture mechanics on the basis of linear elasticity solutions.

In this work, crack formation and the corresponding failure load of bonded lap joints is analyzed. The analysis is based on linear elasticity solutions for bonded lap joints and makes use of the finite fracture mechanics. A hybrid criterion is applied that states the spontaneous formation of a crack of finite size if a stress and an energy criterion are fulfilled simultaneously. The stress distribution of a linear elasticity solution is used for the stress criterion and for the calculation of the incremental energy release rate which is necessary for definition of the energy criterion. The resulting fracture criterion is compared to literature results and shows a good agreement. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the peculiarities of a crack growth path in anisotropic solid

One of the possibilities to increase the resistance of a structure to catastrophic fracture is to force a main line crack to deviate from its path. In this connection the influence of the elastic moduli of an anisotropic material on the possibilities of crack rotation are studied. In particular a linear elastic problem for a straight Mode I crack, located on a symmetry axis of an orthotropic plane is considered. The strength properties of the material are supposed to be isotropic. For studying a direction of a crack growth path several crack models are considered. It is shown that a thin elongated elliptical hole as a crack model leads to more plausible results concerning crack rotation conditions than an ideal cut model. The maximal tensile stresses are taken as a crack growth criterion. It is shown that for some class of orthotropic materials a crack deviates from the straight path just after it starts to grow even in the conditions of uniaxial normal tension. The problem of the stability of a straight crack path under Mode I loading is also considered. This problem is reduced to the problem of the fracture direction determination for thin elongated elliptical cavity slightly inclined to the initial direction. In the frame of the proposed approach the conditions of instability are obtained. It is shown that for some class of orthotropic materials a straight crack path is unstable in the conditions of uniaxial normal tension. This class of materials is wider than one for which a crack deviates from the straight crack path just after its start. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Phase Field Modeling of Fracture Caused by Fluid Transport in Poroelastic Solids

The numerical modeling of failure mechanisms due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by recently developed diffusive crack modeling concepts, which are based on the introduction of a crack phase field. Such an approach is conceptually in line with gradient-extended continuum damage models which include internal length scales. In this paper, we extend our recently outlined mechanical framework [1–3] towards the phase field modeling of fracture in the coupled problem of fluid transport in deforming porous media. Here, extremely complex crack patterns may occur due to drying or hydraulic induced fracture, the so called fracking. We develop new variational potentials for Biot-type fluid transport in porous media at finite deformations coupled with phase field fracture. It is shown, that this complex coupled multi-field problem is related to an intrinsic mixed variational principle for the evolution problem. This principle determines the rates of deformation, fracture phase field and fluid content along with the fluid potential. We develop a robust computational implementation of the coupled problem based on the potentials mentioned above and demonstrate its performance by the numerical simulation of complex fracture patterns. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类四元数小波包的构造

实值二维信号可以用四元数来表示,因此,四元数的尺度函数和小波的构造就成为分析二维信号的关键．引入了四元数小波包的概念,并且借助于四元数多分辨分析和四元数尺度函数和四元数小波函数的概念和若干公式,给出并构造了一类四元数正交小波包的构造方法,得到了四元数正交小波包的3个正交性公式,最后,利用四元数正交小波包给出了L^2（R...     

Asymptotic and Spectral Analysis of the Spatially Nonhomogeneous Timoshenko Beam Model

We develop spectral and asymptotic analysis for a class of nonselfadjoint operators which are the dynamics generators for the systems governed by the equations of the spatially nonhomogeneous Timoshenko beam model with a 2–parameter family of dissipative boundary conditions. Our results split into two groups. We prove asymptotic formulas for the spectra of the aforementioned operators (the spectrum of each operator consists of two branches of discrete complex eigenvalues and each branch has only two points of accumulation: +∞ and —∞), and for their generalized eigenvectors. Our second main result is the fact that these operators are Riesz spectral. To obtain this result, we prove that the systems of generalized eigenvectors form Riesz bases in the corresponding energy spaces. We also obtain the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators. The pencil asymptotics are essential for the proofs of the spectral results for the aforementioned dynamics generators.     

Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements

Motivated by models of fracture mechanics, this paper is devoted to the analysis of a unilateral gradient flow of the Ambrosio–Tortorelli functional, where unilaterality comes from an irreversibility constraint on the fracture density. Solutions of such evolution are constructed by means of an implicit Euler scheme. An asymptotic analysis in the Mumford–Shah regime is then carried out. It shows the convergence towards a generalized heat equation outside a time increasing crack set. In the spirit of gradient flows in metric spaces, a notion of curve of maximal unilateral slope is also investigated, and analogies with the unilateral slope of the Mumford–Shah functional are also discussed.