压电复合材料的共线周期性裂纹平面问题的电弹性场分析

利用复变函数知识、半逆解法及待定系数法, 研究了压电复合材料的共线周期性裂纹问题, 给出了在电不可渗透边界条件下的应力、电位移、应力强度因子、电位移强度因子和机械应变能释放率的解析解。当裂纹间距趋于无穷时, 共线周期性裂纹退化为一条单裂纹, 得到了压电复合材料一条单裂纹的结果。通过数值算例讨论了共线周期性裂纹的裂纹长度、裂纹间距和机电载荷对机械应变能释放率的影响规律。结果表明, 机械应变能释放率随着共线周期性裂纹的裂纹长度、共线周期性裂纹的裂纹间距、机械载荷和正电场的增大而增大, 随着负电场的增大而减小。     

On a generalised plane strain crack problem for inhomogeneous anisotropic elastic materials

A generalised plane strain crack problem is considered for a class of inhomogeneous anisotropic elastic materials. The problem is reduced to a boundary integral equation involving hypersingular integrals. The boundary integral equation may be solved numerically using standard procedures. Some crack problems for a particular inhomogeneous material are considered in detail and the stress intensity factors are obtained in order to assess the effect of the anisotropy and inhomogeneity on the stress field near the crack tips.     

A hypersingular boundary integral equation for antiplane crack problems for a class of inhomogeneous anisotropic elastic materials

An antiplane multiple crack problem is considered for a class of inhomogeneous anisotropic elastic materials. The problem is reduced to a boundary integral equation involving hypersingular integrals. The boundary integral equation may be solved numerically using standard procedures. Some crack problems for a particular inhomogeneous material is considered in detail and the stress intensity factors are obtained in order to assess the effect of the anisotropy and inhomogeneity on the stress field near the crack tips.     

Collinear Zener-Stroh crack problem in plane elasticity

This paper investigates the collinear Zener-Stroh crack problem in plane elasticity. Two cracks in series are chosen as an example in analysis. Different to the Griffith crack problem, an initial displacement discontinuity exists in the Zener-Stroh crack problem, which in turn is the increment of displacement when a moving point goes around the crack in a closed loop. From the traction free condition along the cracks, the dislocation distribution function as well as the complex potential with undetermined coefficients is suggested. The involved undetermined coefficients can be evaluated from the condition of the assumed initial displacement discontinuity. Finally, a closed form solution for the problem is obtained and the calculated stress intensity factors at crack tips are presented. A problem for two collinear Zener-Stroh cracks with different lengths is also studied.     

The inclusion problem with a crack crossing the boundary

The problem of an elastic plane containing an elastic inclusion is considered. It is assumed that both the plane and the inclusion contain a radial crack and the two cracks are collinear. The problem is formulated in terms of a system of singular integral equations. In the interesting limiting cases in which the crack tips approach the interface from either one or both sides, the dominant parts of the kernels become generalized Cauchy kernels giving rise to stress singularities of other than 13-1 power. For these unusual cases of a crack terminating at or crossing the interface stress intensity factors are defined and some detailed results are given for various crack-inclusion geometries and material combinations.     

Non-local theory solution for the anti-plane shear of two collinear permeable cracks in functionally graded piezoelectric materials

In this paper, the non-local theory of elasticity is firstly applied to obtain the behavior of two collinear cracks in functionally graded piezoelectric materials under anti-plane shear loading for permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near the crack tips. The non-local elastic solutions yield finite stresses at the crack tips, thus allows us to use the maximum stress as a fracture criterion. The finite stresses at the crack tips depend on the distance between two collinear cracks, the functionally graded parameter and the lattice parameter of the materials, respectively.     

Fracture of a piezoelectric material layer bonded by two elastic layers

A piezoelectric material layer bonded by two elastic layers under mechanical and electrical loads is studied. The piezoelectric material layer contains a central crack or two collinear cracks. Both mixed-mode crack and anti-plane crack are considered for the impermeable crack assumption and the permeable crack assumption. The effect of electric boundary conditions on electrical and mechanical field intensity factors are discussed. Some new observations are found.     

Mode-I crack problem for functionally graded layered structures

This paper deals with two bonded functionally graded finite strips with two collinear cracks. Different layers may have different nonhomogeneity properties in the structure. A bi-parameter exponential function was introduced to simulate the continuous variation of material properties. The problem is solved by using the integral transform, singular integral equation methods and the theory of residues. Various internal cracks and edge crack and crack crossing the interface configurations are investigated, respectively. The asymptotic stress field near the tip of a crack crossing the interface is examined and it is shown that, unlike the corresponding stress field in piecewise homogeneous materials, in this case the “kink” in material property at the interface does not introduce any singularity. Numerical calculations are carried out, and the influences of nonhomogeneity constants, geometric parameters and crack interactions on the stress intensity factors are investigated.     

Multiple cracks in transformation toughened ceramics

The problem of a collinear array of cracks in a transformation toughened ceramic under monotonically increasing load is analyzed by the theory of complex potentials and dislocation formalism. The analysis is applicable to both a homogeneous material containing multiple cracks and a brittle matrix ceramic reinforced by distributed transformable particulates. The detrimental effect of transformation at the onset of crack growth is generally found to be amplified by crack tip interactions. For the particulate reinforced ceramic the maximum relative strengthening is achieved for moderate volume fractions of large particles.     

Collinear cracks in a layered half – plane with a graded nonhomogeneous interfacial zone – Part II: Thermal shock response

In Part I of the paper, the problem of collinear cracks in a layered half-plane with a graded nonhomogeneous interfacial zone was investigated under mechanical loading and the cracking behavior was addressed by evaluating the stress intensity factors as functions of various geometric and material parameters. In Part II, the solution framework is extended to the problem of thermal shock on the basis of uncoupled, quasi-static thermoelasticity. The interfacial zone, in this case, is assumed to have the graded thermoelastic properties. Using the principle of superposition, a system of singular integral equations is solved subjected to equivalent crack surface tractions obtained form the transient thermoelasticity solution for a uncracked medium. Main results presented are the transient thermal stress intensity factors of collinear cracks to illustrate the parametric effects of geometric and material combinations of the layered medium with the thermoelastically graded interfacial zone. This revised version was published online in July 2006 with corrections to the Cover Date.     

磁电弹性功能梯度板共线Griffith裂纹断裂行为

该文考察了磁电弹性功能梯度板的反平面问题。该板具有多个垂直于边界的共线裂纹。裂纹表面采用磁电不穿透或可穿透假设。应用积分变换和位错密度函数将问题化为柯西奇异积分方程求解。导出和分析了场强度因子和能量释放率。数值结果表明了载荷组合参数、材料梯度指数及裂纹构形对裂尖断裂行为的影响。     

Anti-plane strain around two equal collinear cracks and a crack containing dislocations in a non-work hardening elastic-plastic material loaded uniformly at infinity

The stress distribution is determined for uniform loading at infinity of a non work hardening elastic-plastic material containing either two equal collinear cracks or a single crack containing dislocations. Alternative assumptions are made, either that plasticity is confined to the plane of the crack or that it occurs in a centred fan. The usual criteria for failure, the plastic zone size and the crack opening displacement at the tip are plotted as a function of applied stress to yield stress ratio, and compared with results previously derived from the assumption that slip only occurs in the plane of the crack. There is little significant difference between the results, as had previously been shown for a single crack and for an infinite array of equal, equally spaced parallel or collinear cracks.     

The J and M Integrals and Energy of Two Unequal Collinear Cracks

The energetics of two unequal-length collinear cracks for modes- I, II, III is considered. The analysis is performed for remote uniform load normal to the plane of cracks by applying the J and M path-independent integrals and the relationship between them. The material forces for each crack as well as the total crack energy of the system are evaluated. The exact closed form solution to this interaction problem is expressed as a function of cracks’ dimensions and spacing between them. The expression derived for the energy of interaction between a main crack and a nearby microcrack can be used for an estimate of the toughness degradation due to microcracks. Another application of the two-crack energy solution is related to the elastic compliance of cracked solids relevant for the cases when crack faces are in contact along certain areas.     

Perturbation method of the piezoelectric fracture mechanics considering the permittivity of the medium in the crack gap

In this paper, the problem of the cracks with arbitrary forms in piezoelectric material is studied. The permittivity of the medium in the crack gap is considered. Except the collinear cracks, this boundary condition is too difficult to deal with; therefore, a perturbation method is recommended. By the way, the electric boundary conditions of electric fracture mechanics are discussed. For example, a small parameter solution of a crack is given and compared with the known `exact' (it will be discussed later) solution. This result shows that the impermeable or permeable conditions are only the boundary conditions for the first approximations of the perturbation solutions.     

Closed-form solution for an eccentric anti-plane shear crack normal to the edges of a magnetoelectroelastic strip

Summary The problem of an anti-plane shear crack embedded in a magnetoelectroelastic strip is investigated. The crack is assumed to be normal to the strip edges. By using the finite Fourier transform, the associated mixed boundary-value problem is reduced to triple series equations, then to singular integral equations. Solving the resulting equations analytically, the field intensity factors and energy release rates at the crack tips can be determined in explicit form. The influences of applied electric and magnetic loadings on the normalized energy release rate and mechanical strain energy release rate are presented graphically. Obtained results reveal that applied electric and magnetic loadings affect crack growth, depending on their directions and adopted fracture criteria. The derived solution is applicable to other cases including two collinear cracks distributed symmetrically in a magnetoelectroelastic strip, and a periodic array of collinear cracks in a magnetoelectroelastic plane.     

非均匀材料界面裂纹的Cell-Based光滑有限元法

为提高非均匀材料界面裂纹尖端断裂参数的求解精度,基于非均匀材料界面断裂力学、Cell-Based光滑有限元(Cell-SFEM)和非均匀材料的互交作用积分法,提出了求解非均匀材料界面裂纹尖端断裂参数的CellBased光滑有限元法,推导了基于Cell-Based光滑有限元法的非均匀材料的互交作用积分法,对非均匀材料间的界面裂纹尖端处正则应力强度因子进行了求解,并与参考解进行了比较,讨论了互交积分区域大小和光滑子元个数与正则应力强度因子的关系。数值算例结果表明:本方法具有很高的计算精度,对积分区域大小不敏感,可为设计、制造抗破坏非均匀材料提供依据。     

含垂直极化方向穿透的共线半无限裂纹的狭长压电体的平面问题

通过构造新的广义保角映射, 利用广义复变函数方法研究了在电不可渗透边界条件下含有沿垂直于极化方向穿透的共线半无限裂纹的狭长压电体的平面问题, 给出了裂纹尖端处的各场强度因子的解析解。此外, 当狭长体高度和裂纹尺寸按一定的趋势变化时, 还可以得到压电复合材料另外几种新构型的平面问题的解析解。通过数值算例分析得到了两裂纹之间的距离、 狭长体的高度及裂纹面上的受载长度对场强度因子的影响规律。     

The energy release rate for a quasi-static mode I crack in a nonhomogeneous linearly viscoelastic body

The Energy Release Rate (ERR) for the quasi-static problem of a semi-infinite mode I crack propagating through an inhomogeneous isotropic linearly viscoelastic body is examined. The shear modulus is assumed to have a power-law dependence on depth from the plane of the crack and a very general behavior in time. A Barenblatt type failure zone is introduced in order to cancel the singular stress and a formula for the ERR is derived which explicitly displays the combined influences of material viscoelasticity and inhomogeneity. The ERR is calculated for both power-law material and the standard linear solid and the qualitative features of the ERR are presented along with numerical illustrations.     

Collinear periodic cracks in an anisotropic medium

The problem of collinear periodic cracks in an anisotropic medium is examined in this paper. By means of Stroh formalism and the conformal mapping method, we obtain general periodic solutions for collinear cracks. The corresponding stress intensity factors, crack opening displacements and strain energy release rate are found.     

Cracks propagation and interaction in an orthotropic elastic material: Analytical and numerical methods

An elastic orthotropic material containing a crack in Mode I is considered to formulate a new analytical model. The boundary conditions for the crack existence in the material lead to the solution of the homogeneous Riemann–Hilbert problems. The mathematical model was elaborated for a single and two collinear cracks of different lengths and distance for Mode I in order to investigate cracks interaction problem. Using the theory of Cauchy’s integral and the numerical analysis, the fields in the vicinity of the crack tips were determined.Finite Element Method was applied to compare the mathematical analytical solution and to determine the fields in the vicinity of the crack tips. The critical values of applied stress which caused cracks propagation were evaluated. The interaction of cracks in an orthotropic aramid-epoxy material was studied in details. Comparison of both approaches to crack propagation leads to the conclusion that the new analytical model is correct and can be applied to more complex cracks geometries, including inclined cracks.