On the crack-tip stress singularity of interfacial cracks in transversely isotropic piezoelectric bimaterials

In this paper, characteristics of the interface crack-tip stress and electric displacement fields in transversely isotropic piezoelectric bimaterials are studied. The authors have proven, within the framework of the generalized Stroh formalism for piezoelectric bimaterials, that there is no coexistence of the parameters (oscillating) and κ (non-oscillating) in the interface crack-tip generalized stress field for all transversely isotropic piezoelectric bimaterials. This leads to the classification of piezoelectric bimaterials into one group that exhibits the oscillating property in the interface crack-tip generalized stress field and the other that does not. Fifteen (15) pair-combinations of six (6) piezoelectric materials PZT-4, PZT-5H, PZT-6B, PZT-7A, P-7, and BaTiO₃, which are commonly used in practice, are numerically analyzed in this study, and the results backup the above theoretical conclusions. Moreover, the associated eigenvectors for such material systems (with either =0 or κ=0) are also obtained numerically, and the result show that there still exist four linear independent associate eigenvectors for each bimaterial.     

Inelastic material behavior and fracture mechanics: a variational approach

Summary A variational principle is presented, which relates the macroscopic fracture response of a mechanical component to its microscopic, inelastic material behavior. The principle allows a comparison between the crack driving force, expressed by the J-integral, and an integral expression of the fracture resistance. On this basis, the critical values of J are calculated for a Griffith crack under mixed-mode loading. The preliminary check with data available in literature shows a fairly good agreement. Received 18 July 1998; accepted for publication 9 February 1999     

Analysis of a permeable interface crack in elastic dielectric/piezoelectric bimaterials

A permeable interface crack between elastic dielectric material and piezoelectric material is studied based on the extended Stroh’s formalism. Motivated by strong engineering demands to design new composite materials, the authors perform numerical analysis of interface crack tip singularities and the crack tip energy release rates for 35 types of dissimilar bimaterials, respectively, which are constructed by five kinds of elastic dielectric materials: Epoxy, Polymer, Al₂O₃, SiC, and Si₃N₄ and seven kinds of practical piezoelectric ceramics: PZT-4, BaTiO₃, PZT-5H, PZT-6B, PZT-7A, P-7, and PZT-PIC 151, respectively. The elastic dielectric material with much smaller permittivity than commercial piezoelectric ceramics is treated as a special transversely isotropic piezoelectric material with extremely small piezoelectricity. The present investigation shows that the structure of the singular field near the permeable interface crack tip consists of three singularities: and , which is quite different from that in the impermeable interface crack. It can be concluded that different far field loading cases have significant influence on the near-tip fracture behaviors of the permeable interface crack. Based on the present theoretical treatment and numerical analysis, the electric field induced crack growth is well explained, which provides a better understanding of the failure mechanism induced from interface crack growth in elastic dielectric/piezoelectric bimaterials. The project supported by the National Natural Science Foundation of China (10572110), Doctor Foundation of the Chinese Education Ministry and Doctorate Foundation of Xi’an Jiaotong University. The English text was polished by Yunming Chen.     

Effect of residual surface stress on the fracture of nanoscale materials

The problem of a mode I crack in nanomaterials under a remote mechanical load is investigated. The effect of the residual surface stress on the crack surface is considered and the solutions to the crack opening displacement (COD) and the stress intensity factor (K_I) are obtained. The results show that the surface effect on the crack deformation and crack tip field are prominent at nanoscale. Moreover, COD and K_I are influenced by the residual surface stress not only on the surface near the crack tip region but also on the entire crack surface.     

Analytical solution for motion of an elliptical inclusion in gradient stress field

We present a rigorous analytical solution for motion of an elliptical inclusion in isotropic matrix driven by gradient stress field. The interfacial diffusion is considered as the dominant mechanism for the motion. We demonstrate that normal stress gradient on the interface is the major driven force, while the strain energy density gradient is negligible. A key prediction of the solution is that for a given inclusion the motion velocity is proportional to stress gradient only, indicating that the solution is applicable for inclusion motion in nonuniform stress field of varying stress gradient, and that the inclusion tends to move towards the region of lower stress in nonuniform stressed materials.     

Effect of T-stress on the mode-I fracture toughness of concrete

T-stress expressions are provided for three-point bending (TPB) beams and compact tension (CT) specimens and then its influence on mode I fracture toughness of concrete is investigated. The study shows that T-stress is dependent on the specimen's geometry and the material's property as well, and for TPB and CT specimens of regular size, T-stress is so small that its consequences can be neglected. The study also indicates that concrete specimen size should be carefully chosen to make sure the existence of K-dominance ahead of the crack tip, thus fracture toughness extracted from these specimen configurations can be reliable.     

Effect of thickness and boundary conditions on the behavior of viscoelastic layers in sliding contact with wavy profiles

In this work, the sliding contact of viscoelastic layers of finite thickness on rigid sinusoidal substrates is investigated within the framework of Green's functions approach. The periodic Green's functions are determined by means of a novel formalism, which can be applied, in general, to either 2D and 3D viscoelastic periodic contacts, regardless of the contact geometry and boundary conditions.Specifically, two different configurations are considered here: a free layer with a uniform pressure applied on the top, and a layer rigidly confined on the upper boundary. It is shown that the thickness affects the contact behavior differently, depending on the boundary conditions. In particular, the confined layer exhibits increasing contact stiffness when the thickness is reduced, leading to higher loads for complete contact to occur. The free layer, instead, becomes more and more compliant as thickness is reduced.We find that, in partial contact, the layer thickness and the boundary conditions significantly affect the frictional behavior. In fact, at low contact penetrations, the confined layer shows higher friction coefficients compared to the free layer case; whereas, the scenario is reversed at large contact penetrations. Furthermore, for confined layers, the sliding speed related to the friction coefficient peak is shifted as the contact penetration increases. However, once full contact is established, the friction coefficient shows a unique behavior regardless of the layer thickness and boundary conditions.     

Solution of Eshelby's inclusion problem with a bounded domain and Eshelby's tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory

A solution for Eshelby's inclusion problem of a finite homogeneous isotropic elastic body containing an inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). An extended Betti's reciprocal theorem and an extended Somigliana's identity based on the SSGET are proposed and utilized to solve the finite-domain inclusion problem. The solution for the disturbed displacement field is expressed in terms of the Green's function for an infinite three-dimensional elastic body in the SSGET. It contains a volume integral term and a surface integral term. The former is the same as that for the infinite-domain inclusion problem based on the SSGET, while the latter represents the boundary effect. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is not considered. The problem of a spherical inclusion embedded concentrically in a finite spherical elastic body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. This Eshelby tensor depends on the position, inclusion size, matrix size, and material length scale parameter, and, as a result, can capture the inclusion size and boundary effects, unlike existing Eshelby tensors. It reduces to the classical Eshelby tensor for the spherical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing as the inclusion becomes large enough, and the boundary effect is vanishing as the inclusion volume fraction gets sufficiently low.     

Analysis of a cracked concrete containing an inclusion with inhomogeneously imperfect interface

The distributed dislocation technique is applied to determine the behavior of a cracked concrete matrix containing an inclusion. The analysis of cracked concrete in the presence of inclusions such as steel expansions is a practical problem that needs special attention. The solution to the problem of interaction of an edge dislocation with a circular inclusion having circumferentially inhomogeneously imperfect interface is available in the literature. This analytical solution is used in the distributed dislocation technique to obtain the stress intensity factor for the cracked concrete in the presence of inclusion. The interface of the matrix and the inclusion is assumed inhomogeneously imperfect and the stress intensity factor is determined for the cracked concrete for a case of two identical cracks on diametrically opposite sides of the inclusion. Consideration of this general inhomogeneously imperfect interface is the contribution of this paper. The variation of the inhomogeneity parameters is studied and presented. Additionally, the general assumption for the interface is simplified to the special case of perfectly bonded interface. The observations for the perfect interface are coincident with the previously reported results.     

Effect of interface stresses on the image force and stability of an edge dislocation inside a nanoscale cylindrical inclusion

Dislocation mobility and stability in inclusions can affect the mechanical behaviors of the composites. In this paper, the problem of an edge dislocation located within a nanoscale cylindrical inclusion incorporating interface stress is first considered. The explicit expression for the image force acting on the edge dislocation is obtained by means of a complex variable method. The influence of the interface effects and the size of the inclusion on the image force is evaluated. The results indicate that the impact of interface stress on the image force and the equilibrium positions of the edge dislocation inside the inclusion becomes remarkable when the radius of the inclusion is reduced to nanometer scale. The force acting on the edge dislocation produced by the interface stress will increase with the decrease of the radius of the inclusion and depends on the inclusion size which differs from the classical solution. The stability of the dislocation inside a nanoscale inclusion is also analyzed. The condition of the dislocation stability and the critical radius of the inclusion are revised for considering interface stresses.     

Asymptotic fields in the finite element analysis of electrically permeable interface cracks in piezoelectric bimaterials

Summary The interface crack problem for a piezoelectric bimaterial based on permeable conditions is studied numerically. To find the singular electromechanical field at the crack tip, an asymptotic solution is derived in connection with the conventional finite element method. For mechanical and electrical loads, the complex stress intensity factor for an interface crack is obtained. The influence of the applied loads on the electromechanical fields near the crack tip is also studied. For a particular case of a short crack with respect to the bimaterial size, the numerical results are compared with the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.One author (V.G.) gratefully acknowledges the support provided by the Alexander von Humboldt Foundation of Germany.accepted for publication 7 June 2004     

Non-axisymmetric thermal stress of a functionally graded coated circular inclusion in an infinite matrix

This paper is to study the non-axisymmetric two-dimensional problem of thermal stresses in an infinite matrix with a functionally graded coated circular inclusion based on complex variable method. With using the method of piece-wise homogeneous layers, the general solution for the functionally graded coating having radial arbitrary elastic properties is derived when the matrix is subjected to uniform heat flux at infinity, and then numerical results are presented for several special examples. It is found that the existence of the functionally graded coating can change interfacial thermal stresses, and choosing proper change ways of the radial elastic properties in the coating can obviously reduce the thermal stresses.     

Study of the tensile fracture of plates with a rigid linear inclusion

The limit equilibrium of elastoplastic body is studied under the conditions of a plane problem. The body contains a linear inclusion, which is rigid but of finite rupture strength. The plastic or prefracture zones develop near the ends of the inclusion and are modeled by slip cracks along the matrix—inclusion interface. A new interpretation of the boundary conditions is proposed to solve a model problem for such a composition, and its analytical solution is derived. Two possible mechanisms of local fracture are considered: (a) fracture of the inclusion and (b) separation of the inclusion. The critical length of the inclusion is determined. This length together with the elastic and strength parameters of the composition determines the mechanism of local fracture. The limit loads are found for each mechanism of fracture. State Academy of Water Industry, Rovno, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 123–129, July, 2000.     

Finite Element study of fracture initiation in flaws subject to internal fluid pressure and vertical stress

Hydraulic fracturing is a method used routinely in oil and gas exploitation and in engineered geothermal systems. While used frequently, there are many aspects of hydraulic fracturing, such as the direction of propagation of the newly-created fractures, which are not very well understood. Even though it is known that the local stress field plays a fundamental role in the orientation of the new fractures, there may be other factors, such as the geometry of the existing fractures and the magnitude of the hydraulic pressure applied, that may play a major role in the path that a new fracture follows when pressurized.The main goal of this study is to numerically analyze the effect of the ratio between a vertical load, or stress, and the hydraulic pressure applied in existing flaws on the stress field in the vicinity of the flaw tips. For that purpose, a double flaw geometry 2a-30-30 was modeled in the Finite Element code ABAQUS, and different vertical loads and internal flaw pressures were applied to the model. The variation of the maximum principal stresses and maximum shear stresses around the flaw tips were analyzed and related to fracture initiation.The study showed that the ratio between the water pressure applied in the flaws and the vertical load/stress (WP/VL) plays a crucial role in the magnitude and shape of the stress field around a flaw tip, and therefore in the location of tensile and shear fracture initiation. As WP/VL increases, the location of initiation of new tensile fractures shifts from the upper face of the studied flaw towards the region right ahead of the flaw tip; simultaneously, the location of initiation of new shear fractures shifts from the region ahead of the flaw tip to the upper face of the analyzed tip.     

Dynamic fracture: an example of convergence towards a discontinuous quasistatic solution

Considering a one-dimensional problem of debonding of a thin film in the context of Griffith’s theory, we show that the dynamical solution converges, when the speed of loading goes down to 0, to a quasistatic solution including an unstable phase of propagation. In particular, the jump of the debonding induced by this instability is governed by a principle of conservation of the total quasistatic energy, the kinetic energy being negligible.      

Effect of electric fields on fracture behavior of ferroelectric ceramics

The asymptotic problem of a semi-infinite crack perpendicular to the poling direction in a ferroelectric ceramic subjected to combined electric and mechanical loading is analyzed to investigate effect of electric fields on fracture behavior. Electromechanical coupling induced by the piezoelectric effect is neglected in this paper. The shape and size of the switching zone is shown to depend strongly on the relative magnitude between the applied electric field and stress field as well as on the ratio of the coercive electric field to the yield electric field. A universal relation between the crack tip stress intensity factor and the applied intensity factors of stress and electric field under small-scale conditions is obtained from the solution of the switching zone. It is found that the ratio of the coercive electric field to the yield electric field plays a significant role in determining the enhancement or reduction of the crack tip stress intensity factor. The fracture toughness variation of ferroelectrics under combined electric and mechanical loading is also discussed.     

A crack along the interface of a circular inclusion embedded in an infinite solid

The two-dimensional problem of an arc shaped crack lying along the interface of a circular elastic inclusion embedded in an infinite matrix with different elastic constants is considered. Based on the complex variable method of Muskhelishvili, closed-form solutions for the stresses and the displacements around the crack are obtained when general biaxial loads are applied at infinity. These solutions are then combined with A.A. Griffith's virtual work argument to give a criterion of crack extension, namely the de-bonding of the interface. The critical applied loads are expressed explicitly in terms of a function of the inclusion radius and the central angle subtended by the crack arc. In the case of simple tension the critical load is inversely proportional to the square-root of the inclusion radius. By analyzing the variation of the cleavage stress near the crack tip, the deviation of the crack into the matrix is discussed. The case of uniaxial tension is worked out in detail.     

The brittle fracture criterion based on the maximum tensile stress on the surface of blunt crack tip

The brittle fracture criterion is developed for a blunted crack. The curvature radius of the blunt crack tip is suggested as a characteristic length for brittle materials, and then the fracture toughness of the brittle materials can be determined from the cohesion strength and the characteristic length of the materials.     

硬化系数对界面端弹塑性奇异应力场的影响

本文利用弹塑性边界元分析方法,对具有不同硬化系数的线性硬化结合材料界面端进行了计算,分析结果表明,当硬化系数较大时,界面附近的弹塑怀应力与将弹塑性本构关系简化为线性后得到的理论结果相接近,而当硬化系数相对较少时,理论分析的奇异应力场的主控区变得非常小,在屈服域的绝大部分区间,应力奇异性与理论解有较大区别,本文的结果还表明,硬化系数越小,过渡区（弹塑性厅异应力场支配区到屈服边界）越大,屈服区域应力分布变得平坦,在小规模屈服条件异次数一致）,即可用弹性厅异应力场来近似地描述小规模屈服时的弹塑性界面端,但应力强度系数则比弹性时略大,且随硬化系数的减小而增大。     

Generalized stress intensity factors in the interaction within a rectangular array of rectangular inclusions

Summary To evaluate the mechanical strength of fiber-reinforced composites, it is necessary to consider singular stresses at the end of fibers because they cause crack initiation, propagation, and final failure. A square array of rectangular inclusions under longitudinal tension is considered in this paper. The body-force method is applied to a unit cell region. Then, the problem is formulated as a system of singular integral equations, where the unknown functions are the densities of body forces distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. The unknown functions are expressed as piecewise-smooth functions using power series and two types of fundamental densities which express singular stresses. Generalized stress intensity factors at the corners of inclusions are systematically calculated with varying the shape and spacing of a square array of square and rectangular inclusions.