Fracture analysis of a penny-shaped magnetically dielectric crack in a magnetoelectroelastic material

In this paper we investigate the magnetoelectroelastic behavior induced by a penny-shaped crack in a magnetoelectroelastic material. The crack is assumed to be magnetically dielectric. A closed-form solution is derived by virtue of Hankel transform technique with the introduction of certain auxiliary functions. Field intensity factors are obtained and analyzed. The results indicate that the stress intensity factor depends only on the mechanical loads. However, all the other field intensity factors depend directly on both the magnetic and dielectric permeabilities inside the crack as well as on the applied magnetoelectromechanical loads and the material properties of the magnetoelectroelastic material. Several special cases are further discussed, with the reduced results being in agreement with those from literature. Finally, according to the maximum crack opening displacement (COD) criterion, the effects of the magnetoelectromechanical loads and the crack surface conditions on the crack propagation and growth are evaluated.     

Application of an orthotropy rescaling method to edge cracks and kinked cracks in orthotropic two-dimensional solids

Oblique edge cracks and kinked cracks in orthotropic materials with inclined principal material directions under inplane loadings are investigated. The Stroh formalism is modified by introducing new complex functions, which recovers a classical solution for a degenerate orthotropic material with multiple characteristic roots. An orthotropy rescaling technique is presented based on the modified Stroh formalism. Stress intensity factors for edge cracks as well as kinked cracks are obtained in terms of solutions for a material with cubic symmetry by applying the orthotropy rescaling method. Explicit expressions of the stress intensity factors for a degenerate orthotropic material are obtained in terms of solutions for an isotropic material. The effects of orthotropic parameter, material orientation, and crack angle on the stress intensity factors for the degenerate orthotropic material are discussed. The stress intensity factors for cubic symmetry materials are calculated from finite element analyses, which can be used to evaluate the stress intensity factors for orthotropic materials. The energy release rate for the kinked crack in an orthotropic material is also obtained.     

The crack problem in a specially orthotropic shell with double curvature

In this paper the crack problem of a shallow shell with two nonzero curvatures is considered. It is assumed that the crack lies in one of the principal planes of curvature and the shell is under Mode I loading condition. The material is assumed to be specially orthotropic. After giving the general formulation of the problem the asymptotic behavior of the stress state around the crack tip is examined. The analysis is based on Reissner's transverse shear theory. Thus, as in the bending of cracked plates, the asymptotic results are shown to be consistent with that obtained from the plane elasticity solution of crack problems. Rather extensive numerical results are obtained which show the effect of material orthotropy on the stress intensity factors in cylindrical and spherical shells and in shells with double curvature. Other results include the stress intensity factors in isotropic toroidal shells with positive or negative curvature ratio, the distribution of the membrane stress resultant outside the crack and the influence of the material orthotropy on the angular distribution of the stresses around the crack tip.     

Effective spring stiffness for a planar periodic array of collinear cracks at an interface between two dissimilar isotropic materials

Explicit analytical expressions are obtained for the longitudinal and transverse effective spring stiffnesses of a planar periodic array of collinear cracks at the interface between two dissimilar isotropic materials; they are shown to be identical in a general case of elastic dissimilarity (the well-known open interface crack model is employed for the solution). Since the interfacial spring stiffness can be experimentally determined from ultrasound reflection and transmission analysis, the proposed expressions can be useful in estimating the percentage of disbond area between two dissimilar materials, which is directly related to the residual strength of the interface. The effects of elastic dissimilarity, crack density and crack interaction on the effective spring stiffness are clearly represented in the solution. It is shown that in general the crack interaction weakly depends on material dissimilarity and, for most practical cases, the crack interaction is nearly the same as that for crack arrays between identical solids. This allows approximate factorization of the effective spring stiffness for an array of cracks between dissimilar materials in terms of an elastic dissimilarity factor and two factors obtained for cracks in a homogeneous material: the effective spring stiffness for non-interacting (independent) cracks and the crack interaction factor. In order to avoid the effect of the crack surface interpenetration zones on the effective spring stiffness, the range of the tensile to transverse load ratios is obtained under the assumption of small-scale contact conditions. Since real cracks are often slightly open (due to prior loading history and plastic deformation), it is demonstrated that for ultrasound applications the results obtained are valid for most practical cases of small interfacial cracks as long as the mid-crack opening normalized by the crack length is at least in the order of 10⁻⁵.     

A wedge crack in an anisotropic material under antiplane shear

A crack emanating from the apex of an infinite wedge in an anisotropic material under antiplane shear is investigated. An isotropic wedge crack subjected to concentrated forces is first solved by using the conformal mapping technique. The solution of an anisotropic wedge crack is obtained from that of the transformed isotropic wedge crack based on a linear transformation method. Expressions for the stress intensity factor for the anisotropic wedge crack with both concentrated and distributed loads are derived. The stress intensity factors are numerically calculated for generally orthotropic wedge cracks with various crack and wedge angles as well as anisotropic parameters.     

The elastostatic approach to compression of a cracked medium with dry friction

Summary In the frame of a linear elastic material compression of a cracked plane is considered. Friction during a mutual sliding of the crack surfaces can be responsible for some non-linear effects, in particular, for hysteresis. A rigorous solution to the problem of non-axisymmetric compression of a space weakened by a circular crack with dry friction is given. This solution is obtained in displacements and the field of displacements is represented in elementary functions.     

Elastoplastic crack analysis by boundary integral method and surface spectrum measurement

The paper describes a hybrid experimental-numerical technique for elastoplastic crack analysis. It consists of the experimental surface spectrum measurement of plastic strains ahead the crack tip and the boundary element method (BEM). The light scattering method is used to measure the power density spectrum from which the values of plastic strains are obtained by comparison with a calibration experiment on the same material. Plastic strains obtained experimentally are conveniently used for the calculation of unknown boundary displacement or traction vectors by the boundary element method. Instead of an iterative solution of the boundary integral equations in pure numerical solution, the boundary unknowns are computed once for a required loading level. Also asymptotic distribution of strains or stresses is not needed in the evaluation of the domain integral for the BEM formulation in the vicinity of the crack tip. Significant CPU time saving is achieved in comparison with the pure BEM solution. The method presented is illustrated by the example for a three point bending specimen with an edge crack.     

Stress intensity factors around a crack in a nonhomogeneous interfacial layer between two dissimilar elastic half-planes

The stresses around a crack in an interfacial layer between two dissimilar elastic half-planes are obtained. The crack is parallel to the interfaces. The material constants of the layer vary continuously within a range from those of the upper half-plane to those of the lower half-plane. An internal gas pressure is applied to the surfaces of the crack. To derive the solution, the nonhomogeneous interfacial layer is divided into several homogeneous layers with different material properties. The boundary conditions are reduced to dual integral equations, which are solved by expanding the differences of the crack face displacements into a series. The unknown coefficients in the series are determined using the Schmidt method, and a stress intensity factor is calculated numerically for epoxy-aluminum composites.     

Higher-order solution for near-tip fields of cracks growing in elastic perfectly-plastic compressible materials

The higher-order asymptotic solution of a quasi-static steadily propagating mode-I crack under the plane strain condition in an elastic perfectly-plastic compressible material is studied. In order to statisfy the higher-order compatibility equation for the rate of deformation in the centered fan sector, the stress near the crack tip is expanded asymptotically as an irregular logarithmic power series. The higher order terms near the crack tip were successfully derived. These higher order solutions are distinctly different from those for a stationary crack. The present solution for a growing crack is a one-parameter near-tip field based on a characteristic length A, through which the influence of loading and crack geometry enter into the near-tip field. This feature is substantiated by the numerical solution obtained by A.G. Varias and C.F. Shih. Comparisons between the analytic solution and the numerical results are presented.Presented at the Far East Fracture Group (FEFG) International Symposium of Fracture and Strength of Solids, 4–7 July 1994 in Xi'an, China.     

Complex variable method for problems of a laminate composed of multiple isotropic layers

A new method of the analysis of a plate composed of thin layers of isotropic linear elastic material is developed. A general solution for displacement, resultant stress and resultant moment fields is obtained by using the complex function theory. It is proved that the complete solutions of the laminated plate subjected to tractions prescribed on its boundary can be obtained from the sum of solutions for uncoupled plates. Particular attention is devoted to the crack tip field and energy release rate for the laminated plate. A closed form solution for singular fields near the crack tip and the relation between the J-integral and the intensity factors are derived through the complex potential formula. Complete forms of the complex potentials for a crack in an infinite laminate as well as for the singularities such as a point force, a point moment and a dislocation are also obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fracture of a finite piezoelectric layer with a penny-shaped crack

This paper studies a penny-shaped crack in a finite thickness piezoelectric material layer. The piezoelectric medium is subjected to a thermal flux on its top and bottom surfaces. Both thermally insulated crack and heated crack are considered. Numerical solution for the finite layer thickness is obtained through the solution of a pair of dual integral equations. The result reduces to the closed form solution when the thickness of the piezoelectric layer becomes infinite. Exact expressions for the stress and electric displacement at the crack border are given as a function of the stress intensity factor, which is determined by the applied thermal flux. This paper is useful for the reliability design of piezoelectric materials in thermal environments.     

Crack kinking of a delamination at an inclined core junction interface in a sandwich beam

The paper considers a general interface delamination and crack kinking from an inclined core junction in a sandwich beam. This particular problem is relevant for a newly developed peel-stopper component for sandwich structures.A finite element model (FEM) was developed and calibrated against a known model by He and Hutchinson. The numerically and analytically determined solution coefficients were in a perfect agreement with each other, so the necessary generalisation of results can be obtained through the application of FEM-analyses. The FE-model was used to determine solution coefficients for a number of interface compositions of practical interest. As expected some of the coefficients were quite sensitive to the specific material combination, which confirms that accurate solution strategies are important.The solution coefficients obtained were further applied to the analysis of the crack propagation and kinking process in three different sandwich beam configurations, each of which contained an inclined junction of 20°, 30°, or 40°. The objective was to examine how the core junction angle and the fracture mechanical properties of the sandwich components influenced the crack kinking tendency. The latter is vital for the design and functionality of a newly developed peel-stopper. It was shown that smaller core junction angles will lead to longer crack propagation (delamination) along the core-core interface prior to a possible kinking. The physical insight obtained is essential for optimal design of peel-stoppers.     

Green's functions for the stress intensity factor evolution in finite cracks in orthotropic materials

The elastodynamic response of an infinite orthotropic material with finite crack under concentrated loads is examined. Solution for the stress intensity factor history around the crack tips is found. Laplace and Fourier transforms are employed to solve the equations of motion leading to a Fredholm integral equation on the Laplace transform domain. The dynamic stress intensity factor history can be computed by numerical Laplace transform inversion of the solution of the Fredholm equation. Numerical values of the dynamic stress intensity factor history for some example materials are obtained. This solution can be used as a Green's function to solve dynamic problems involving fini te cracks.     

不同裂纹分布的孔隙材料渗透系数

含裂纹孔隙材料渗透性由裂纹的微观结构决定,其研究对工程实践意义重大。该文假设含裂纹孔隙材料是由孔隙基体和裂纹组成的二相复合材料,基于细观均匀化理论给出了四种不同裂纹分布的渗透张量稀疏解、相互作用直推（IDD）解和修正的IDD解。基于单元嵌入技术和弹性比拟的数值模拟方法,采用不连通的离散裂纹模型,研究了裂纹数目对有效渗透系数数值解收敛性的影响及不同裂纹分布的孔隙材料渗透性,并将得到的数值解和理论解对比分析,结果表明:随着裂纹数目的增加,有效渗透系数的变化范围逐渐减小,并最终趋于稳定,而且选择合适的裂纹数目,能同时保证计算的随机收敛性和合理的计算效率;对于所研究的四种分布的裂纹,相比稀疏解,IDD解更接近数值解,但随着裂纹密度的增加,裂纹间的相互作用增强,IDD解会逐渐偏离数值解;修正的IDD解充分考虑了裂纹间的相互作用和边界效应,能更好地估计含裂纹孔隙材料的渗透性。     

A finite element stress analysis of a crack in a bi-material plate

A stress analysis is presented for the problem of a crack in one material of a bi-material plate located perpendicular to the material interface. A numerical solution using the finite element techniques to determine the force displacement relationships is used. Knowing this, a work integral method is used to determine the stress intensity factors for the crack. Since the work integral is independent of path, the path of integration can be chosen far enough away from the crack tip to avoid the complications of the crack tip singularity. The problem is studied for a number of cases where the crack length to plate width ratio, distance from crack tip to material interface, and the ratio of material constants were varied as parameters.     

Dynamic Stress-Intensity Factors for an Orthotropic Infinite Strip with a Semi-Infinite Crack

A clamped infinite strip of an orthotropic material and containing a semi-infinite crack is considered. The strip is loaded by time-dependent translations of the boundaries and the dynamic stress-intensity factor is obtained using a path independent integral. The solution is found to be of the same form as for the corresponding isotropic case.     

A non-linear crack model

Elliott's crack model which accounts for non-linearity of material behaviour in the vicinity of a crack tip is difficult to investigate analytically in view of the mathematical complexities associated with the governing singular integral equation. However, with a particular force law, the governing equation can be transformed into one that arises in the theory of the lift coefficient of a thin jet-flapped wing, and for which a solution has been obtained. The displacement variation ahead of the crack tip is therefore known, and the results show that the width of the crack front decreases as the force law becomes sharper, a state of affairs which is analogous to that for the dislocation width in the Peierls-Nabarro model. The fact that a solution of the governing equation has been obtained without the use of inverse methods, should act as a spur to the search for solutions for other force laws.     

Cracked Elastic Bi-Material Layer Under Compressive Loads

The boundary value problem of an elastic bi-material layer containing a finite length crack under compressive mechanical loadings has been studied. The crack is located on the bi-material interface and the contact between crack surfaces is frictionless. Based on Fourier integral transformation techniques the solution of the formulated problem is reduced to the solution of singular integral equation, then, with Chebyshev`s orthogonal polynomials, to infinite system of linear algebraic equations. The expressions for contact stresses in the elastic compound layer are presented. Based on the analytical solution it is found that in the case of frictionless contact the shear and normal stresses have inverse square root singularities at the crack tips. Numerical solutions have been obtained for a series of examples. The results of these examples are illustrated graphically, exposing some novel qualitative and quantitative knowledge about the stress field in the cracked layer and their dependence on geometric and applied loading parameters. It can be seen from this study that the crack tip stress field has a mixture of mode I and mode II type singularities. The numerical solutions show that an interfacial crack under compressive forces can become open in certain parts of the contacting crack surfaces, depending on the applied forces, material properties and geometry of the layers.     

The time-dependent interaction between inclusions and a cracked matrix subjected to antiplane shear

Summary. The time-dependent interaction between multiple circular inclusions and a cracked matrix in the antiplane viscoelastic problem is discussed in this paper. The fundamental elastic solution is obtained as a rapidly convergent series in terms of complex potentials via successive iterations of Möbius transformation in order to satisfy continuity conditions on multiple interfaces. Based on the correspondence principle, the Laplace transformed viscoelastic solution is then directly determined from the corresponding elastic one. In association with the singular integral technique, the time-dependent mode-III stress intensity factor of the crack tip can be solved numerically in a straightforward manner. Finally, some typical examples of an arbitrary crack lying in a matrix with various material properties under various loading types are also discussed. The results show that, depending on the relative locations and material properties of inclusions, the evolution of the stress intensity factor (SIF) may increase or decrease with time.