Closed form solution for two unequal collinear semi-permeable straight cracks in a piezoelectric media

The problem of two unequal collinear straight cracks weakening a poled transversely isotropic piezoelectric ceramic is addressed under semi-permeable electric boundary conditions on the crack faces. The plate has been subjected to combined in-plane normal(to the faces of the cracks) mechanical and electric loads. Problem is formulated employing Stroh formalism and solved using complex variable technique. The elastic field, electric field and energy release rate are obtained in closed analytic form. A case study is presented for poled PZT-5H cracked plate to study the effect of prescribed mechanical load, electric load, inter-crack distance and crack lengths on crack arrest parameters stress intensity factor (SIF), electric displacement intensity factor (EDIF) and mechanical and total energy release rates (ERR). Moreover a comparative study is done of impermeable and semi-permeable crack face boundary conditions on SIF, EDIF and ERR, and results obtained is presented graphically. It is observed that the effect of dielectric medium in the crack gap cannot be ignored.     

A Zener–Stroh crack interacting with a coated inclusion with generalized Irwin plastic zone correction

The elastic–plastic fracture behavior of a Zener–Stroh crack interacting with a coated inclusion in composite materials has been investigated with crack tip plastic zone corrections. With the distributed dislocation method, the crack problem is formulated into a set of singular integral equations which are solved numerically. The plastic zone sizes at the both crack tips are determined by a generalized Irwin model where Von Mises stress yielding criterion is used. The stress intensity factor (SIF), the plastic zone size (PZS), the crack tip opening displacement (CTOD) and the effective stress intensity factor have been evaluated. In the numerical examples, the influence of the inclusion shear modulus, the coating-layer thickness and shear modulus, as well as the distance between the crack and inclusion, on the SIF, the PZS and the CTOD are discussed in detail. Numerical examples show that increasing the shear modulus or the thickness of the coating phase, the influence of the inclusion on the normalized SIF and the normalized PZS will be shielded.     

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e. , a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.     

梯度弹性基础上正交异性薄板的弯曲

基于能量法和变分原理,采用双参数弹性基础模型,研究了梯度弹性基础上正交异性薄板在分布载荷作用下的弯曲问题。首先,根据能量法与变分原理,给出了梯度弹性基础上正交异性薄板的弯曲微分平衡方程,并得到了梯度弹性基础刚度系数 与 的计算表达式;进而,假设 向正应力在厚度方向上均匀分布,推导了弹性基础 向位移衰减函数 的计算式。在算例中,通过将梯度弹性基础退化为均质基础,并与Vlazov模型对比,证明了本文理论的正确性;最后,求解了弹性模量呈幂律分布的梯度基础上薄板的挠度分布,分析了基础上下表层材料弹性模量比 与体积分数指数 对薄板挠度分布的影响。     

On the stress state of a piezoceramic body with a flat crack under symmetric loads

A static-equilibrium problem is solved for an electroelastic transversely isotropic medium with a flat crack of arbitrary shape located in the plane of isotropy. The medium is subjected to symmetric mechanical and electric loads. A relationship is established between the stress intensity factor (SIF) and electric-displacement intensity factor (EDIF) for an infinite piezoceramic body and the SIF for a purely elastic material with a crack of the same shape. This allows us to find the SIF and EDIF for an electroelastic material directly from the corresponding elastic problem, not solving electroelastic problems. As an example, the SIF and EDIF are determined for an elliptical crack in a piezoceramic body assuming linear behavior of the stresses and the normal electric displacement on the crack surface __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 67–77, November 2005.     

有限厚度板穿透裂纹前缘附近三维弹性应力场分析

通过三维有限元计算来研究有限宽度、有限厚度含有穿透裂纹板的裂纹前缘应力场，从中找出应力强度因子与板的厚度、裂纹长度之间的关系，同时还分析了裂尖的三维约束程度和三维约束区的大小。分析结果表明：应力强度因子沿厚度的分布是不均匀的，应力强度因子的最大值及其位置与厚度有关；有限厚度板中面应力强度因子(KI)m-p及最大应力强度因子(KI)max均大于平面应力或平面应变的应力强度因子。对有限厚度裂纹问题，按平面应力或平面应变来考虑是不安全的；板中面的应力强度因子(KI)m-p及最大应力强度因子(KI)max是厚度B／a的函数；板的中面离面约束系数Tx最大，自由面(z=B)Tx=0。沿厚度方向裂尖附近的离面约束系数Tx也是z／B和B／a的函数，随着厚度的增加离面约束系数Tx增大，离中面越近离面约束系数Tx越大。Tx随着x的增大急剧减小，三维约束影响区域大小大约为板厚的一半，且裂纹长度a／W对应力强度因子沿厚度变化规律及Tx影响区域大小影响较小。     

Higher-order crack tip fields for functionally graded material plate with transverse shear deformation

The crack tip fields are investigated for a cracked functionally graded material (FGM) plate by Reissner’s linear plate theory with the consideration of the transverse shear deformation generated by bending. The elastic modulus and Poisson’s ratio of the functionally graded plates are assumed to vary continuously through the coordinate y, according to a linear law and a constant, respectively. The governing equations, i.e., the 6th-order partial differential equations with variable coefficients, are derived in the polar coordinate system based on Reissner’s plate theory. Furthermore, the generalized displacements are treated in a separation-of-variable form, and the higher-order crack tip fields of the cracked FGM plate are obtained by the eigen-expansion method. It is found that the analytic solutions degenerate to the corresponding fields of the isotropic homogeneous plate with Reissner’s effect when the in-homogeneity parameter approaches zero.     

Fracture in plates of finite thickness

Application of the plane theory of elasticity to planar crack or angular corner geometries leads to the concept of stress singularity and stress intensity factor, which are the cornerstone of contemporary fracture mechanics. However, the stress state near an actual crack tip or corner vertex is always three-dimensional, and the meaning of the results obtained within the plane theory of elasticity and their relation to the actual 3D problems is still not fully understood. In particular, it is not clear whether the same stress field as found from the well-known 2D solutions of the theory of elasticity do describe the corresponding stress components in a plate made of a sufficiently brittle material and subjected to in-plane loading, and what effect the plate thickness has. In the present study we adopt, so called, first order plate theory to attempt to answer these questions. New features of the elastic solutions obtained within this theory are discussed and compared with 2D analytical results and experimental studies as well as with 3D numerical simulations.     

Fracture of Edge Cracked Layered Plates Subjected to In-plane Bending

Layered structures are used in protection systems such as personal and heavy armor, windshields and also in functionally graded thermal barriers. The focus of this study is to understand the behavior of cracks in such systems, especially when the crack orientation is such that there are property changes along the crack front. Layered plates were prepared by bonding together sheets of epoxy and Polymethylmethacrylate (PMMA) using an epoxy adhesive. Among the two, epoxy has higher elastic modulus and lower fracture toughness compared to PMMA. Two different layer configurations; a single sheet of epoxy bonded to a single PMMA sheet (two layer) and a single sheet of epoxy sandwiched between two PMMA sheets (three layer) were considered. Single edge notched specimens were loaded in three point bending and the thickness averaged stress intensity factor (SIF) was estimated through photoelasticity. Subsequently, the behavior of crack-extension in these plates was also investigated. In both configurations, crack growth initiated in the epoxy layer first and extended stably before the start of crack extension in the PMMA layer. Once the crack extension started in the PMMA layer, the plate looses its structural integrity. It was observed that the onset of crack extension in the epoxy layer can be predicted using the thickness averaged SIF. A method using analysis of patched cracks is presented for estimating the load at which the plate completely looses its load carrying capacity. The estimates from this method match well with experimental results.     

功能梯度压电材料反平面裂纹问题

基于三维弹性理论和压电理论，导出了材料系数在横观各向同性平面内梯度分布的压电体的状态方程，进而对材料系数指数函数规律分布的半无限大压电体中的反平面裂纹问题进行了求解，利用Fourier变换给出了半无限大压电体中位移，应力，电势及电位移的解析表达式，并求得了裂纹尖端的应力强度因子和电位移强度因子，分析了不同的非均匀材料系数及几何尺寸对它们的影响。     

Stress state of a piezoceramic body with a plane crack opened by a rigid inclusion

The paper establishes a relationship between the solutions for cracks located in the isotropy plane of a transversely isotropic piezoceramic medium and opened (without friction) by rigid inclusions and the solutions for cracks in a purely elastic medium. This makes it possible to calculate the stress intensity factor (SIF) for cracks in an electroelastic medium from the SIF for an elastic isotropic material, without the need to solve the electroelastic problem. The use of the approach is exemplified by a penny-shaped crack opened by either a disk-shaped rigid inclusion of constant thickness or a rigid oblate spheroidal inclusion in an electroelastic medium __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 47–60, July 2008.     

A study on caustic method in cracked plate under symmetric bending

Based on Chun-Tu Liu's result for Reissner's plate, an equation of caustics in cracked plate under symmetric bending was obtained. The shape and the distribution of light intensity of the caustic curve were discussed in detail. The relation between the stress intensity factor (SIF) and the size of the caustic curve was found by numerical method, and a simple and accurate experimental method for evaluating SIF in plate under bending was proposed.     

Nonlocal theory solution of two collinear cracks in the functionally graded materials

In this paper, the interaction of two collinear cracks in functionally graded materials subjected to a uniform anti-plane shear loading is investigated by means of nonlocal theory. The traditional concepts of the nonlocal theory are extended to solve the fracture problem of functionally graded materials. To make the analysis tractable, it is assumed that the shear modulus varies exponentially with the coordinate vertical to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near the crack tips. The nonlocal elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials. The magnitude of the finite stress field depends on the crack length, the distance between two cracks, the parameter describing the functionally graded materials and the lattice parameter of the materials.     

On the determination of elastodynamic crack tip stress fields

A linear elastic body in plane strain which contains a stationary crack and which is initially at rest and stress free is considered. It is shown that if the elastodynamic displacement field and stress intensity factor are known, as functions of crack length, for any symmetrical distribution of time-varying forces which acts on the body, subsequent to t=0, then the stress intensity factor due to any other symmetrical load system whatsoever which acts on the same body may be directly determined. The other load system may be of arbitrary spatial distribution and time variation. Further, that part of the elastodynamic displacement field due to the other load system, which arises from the presence of the crack, may also be directly determined. The results are obtained by extension of Rice's mode of derivation of the corresponding Bueckner-Rice elastostatic results to Laplace-transformed elastodynamic variables. Likewise, the existence of a universal elastodynamic “weight function” for any given cracked body is demonstrated. As an application, Freund's recent result for the stress intensity factor due to suddenly applied concentrated forces on the crack surfaces is derived directly by our method, from de Hoop's earlier solution for suddenly applied uniform pressures.     

功能梯度压电带中裂纹对SH波的散射

研究功能梯度压电带中裂纹对SH波的散射问题,为了便于分析,材料性质假定为指数模型,并假设裂纹面上的边界条件为电渗透型的.根据压电理论得到压电体的状态方程,利用Fourier积分变换,问题转化为对偶积分方程的求解.用Copson方法求解积分方程.求得了裂纹尖端动应力强度因子、电位移强度因子的解析表达式,最后数值结果显示了标准动应力强度因子与入射波数、材料参数、带宽、波数以及入射角之间的关系.     

均布荷载作用下功能梯度简支梁弯曲的解析解

利用应力函数半逆解法,研究了均布载荷作用下、材料属性在厚度上任意变化的功能梯度简支梁弯曲的解析解,给出了各向应力应变与位移的解析显式表达式.首先根据平面应力状态的基本方程,得出了功能梯度梁的应力函数应满足的偏微分方程,并根据应力边界条件得出了各应力分布的表达式;进而根据功能梯度材料的本构方程和位移边界条件,得出了应变和位移的分布.最后,通过将本文的解退化到均质各向同性梁并与经典弹性解比较,证明了本文理论的正确性,并求解了材料组分呈幂律分布的功能梯度梁的应力和位移分布,分析了上下表层材料的弹性模量比λ与组分材料体积分数指数n对应力和位移分布的影响.     

Torsional impact of transversely isotropic solid with functionally graded shear moduli and a penny-shaped crack

The torsional impact response of a penny-shaped crack in an unbounded transversely isotropic solid is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace transform and Hankel transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress fields are obtained. Investigated are the influence of material nonhomogeneity and orthotropy on the dynamic stress intensity factor. The peak value of the dynamic stress intensity factor can be suppressed by increasing the shear moduli's gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface.     

复合材料断裂力学参量的实验测定

本文采用三维云纹干涉仪同步测量含裂纹复合材料弯曲板的面内位移场及离面位移场，并对离面位移场云纹图进行计算机自动采集和处理，编制了云纹数字图像处理软件。文中推导了正交异性复合材料弯曲问题的离面位移场表达式，得到了弯曲板裂纹尖端位移w与应力场强度因子K1的关系式。通过实测位移，结合裂尖位移场推导出应力强度因子K1值。     

Parameters controlling fracture resistance in functionally graded materials under mode I loading

In this investigation the fracture behavior of functionally graded materials (FGMs) was studied by means of experiments carried out on model polymer-based FGMs. Model graded materials were manufactured by selective ultraviolet irradiation of ECO [poly(ethylene carbon monoxide)], a photo-sensitive ductile copolymer that becomes more brittle and stiffer under exposure to ultraviolet light. The mechanical response of the graded material was characterized using uniaxial tensile tests. Single edge notched tension graded ECO specimens possessing different spatial variations of Young’s modulus, failure stress and failure strain were tested under remote opening loading. A full-field digital image correlation technique was used to measure in real-time the displacement field around the crack tip while it propagated through the graded material. The measured displacement field was then used to extract fracture parameters such as stress intensity factor and T-stress, and thus construct resistance curves for crack growth in the FGMs. For this loading configuration it was found that the nonsingular T-stress term in the asymptotic expansion for stresses needs to be accounted for in order to accurately measure the fracture resistance in FGMs. In addition, the influence of local failure properties (i.e., failure stress and failure strain) on crack growth resistance was investigated in detail. It was found that depending on the combined effects of the spatial variation of these two failure parameters, regardless of the spatial variation of the Young’s modulus, the FGM fracture resistance can either increase, decrease or remain constant with continued crack growth.