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A new analytical procedure is developed for the deduction of the asymptotic series of the singular solutions in displacements and stresses near the vertex of the linear elastic isotropic corners with the Dirichlet–Robin (fixed-spring) and Neumann–Robin (free-spring) boundary conditions. Under the assumption of antiplane shear loading, the corresponding elastic problem reduces to the Laplace equation for the out-of-plane displacement. In the deduction of such singular solution, the complex variable is used to propose a harmonic function in the form of an asymptotic series including both power and logarithmic terms. This original procedure is suitable for its implementation in a computer algebra software which makes all the necessary symbolic computing, simplifications and rearrangements. This is a key issue due to the fact that the complexity of terms in these series may increase with increasing order of terms. These series are composed by the main terms (also called main singularities), solutions of the corresponding Dirichlet–Neumann or Neumann–Neumann problems, and the associated finite or infinite series of the so-called shadow terms (also called shadow singularities). These terms are determined by solving systems of recursive inhomogeneous Dirichlet–Neumann or Neumann–Neumann problems, respectively. A general classification of the behaviours of the asymptotic series covering all the considered corner problems is introduced. A few examples of the asymptotic series for corners with Dirichlet–Robin and Neumann–Robin boundary conditions are presented to illustrate the capabilities of this procedure.

We have studied cracks traveling along weak interfaces. We model them using harmonic and anharmonic forces between particles in a lattice, both in tension (Mode I) and antiplane shear (Mode III). One of our main objects has been to determine when supersonic cracks traveling faster than the shear wave speed can occur. In contrast to subsonic cracks, the speed of supersonic cracks is best expressed as a function of strain, not stress intensity factor. Nevertheless, we find that supersonic cracks are more common than has previously been realized. They occur both in Mode I and Mode III, with or without anharmonic changes of interparticle forces prior to breaking, and with or without dissipation. The extent and shape of the supersonic branch of solutions depends strongly on details such as lattice geometry, force law anharmonicity, and amount of dissipation. Particle forces that stiffen prior to breaking lead to larger supersonic branches. Increasing dissipation also tends to promote the existence of supersonic states. We include a number of other results, including analytical expressions for crack speeds in the high-strain limit, and numerical results for the spatial extent of regions where particles interact anharmonically. Finally, we note a curious phenomenon, where for forces that weaken with increasing strain, cracks can slow down when one pulls on them harder.

In this paper, the interface cracking between a functionally graded material (FGM) and an elastic substrate is analyzed under antiplane shear loads. Two crack configurations are considered, namely a FGM bonded to an elastic substrate containing a single crack and a periodic array of interface cracks, respectively. Standard integral-transform techniques are employed to reduce the single crack problem to the solution of an integral equation with a Cauchy-type singular kernel. However, for the periodic cracks problem, application of finite Fourier transform techniques reduces the solution of the mixed-boundary value problem for a typical strip to triple series equations, then to a singular integral equation with a Hilbert-type singular kernel. The resulting singular integral equation is solved numerically. The results for the cases of single crack and periodic cracks are presented and compared. Effects of crack spacing, material properties and FGM nonhomogeneity on stress intensity factors are investigated in detail.

The paper deals with a work-hardening, elastic–plastic, stress analysis of pointed V-notches under antiplane shear deformation loading both under small and large scale yielding. Stress and strain field intensities are expressed in terms of plastic Notch Stress Intensity Factors, which are analytically linked to the corresponding linear elastic ones under small scale yielding. The near tip stress and strain fields are then used to give closed-form expressions for the Strain Energy Density over a circular sector surrounding the notch tip, and for the J-integral parameter, both as a function of the relevant plastic NSIFs, these expressions being valid both under small and large scale yielding.

This paper presents an analysis of the static problem of model III crack of a functionally graded coating-substrate system with an internal crack perpendicular to the interface under antiplane shear loading when the coating layer and substrate have finite thickness. After the Fourier transform method is employed, the expressions of the displacement components can be obtained. Integral transforms are employed to reduce the problem to a singular integral equation that can be solved numerically. The influences of the nonhomogeneity constant, relative crack length and thickness ratio are quantitatively studied.

An eigenfunction expansion method is presented to obtain three-dimensional asymptotic stress fields in the vicinity of the front of a semi-infinite infinitely rigid inclusion, subjected to the far-field antiplane shear and extension/bending loadings. Four different rigid inclusion side-surface boundary conditions are considered: (i) anticrack or perfectly bonded rigid inclusion, (ii) transversely rigid inclusion (longitudinal slip permitted), (iii) rigid inclusion in part perfectly bonded, the remainder with slip, (iv) rigid inclusion located alongside a crack. The computed stress singularity for an anticrack is the same as its plane strain counterpart, while the expression for the stress intensity factor varies in the thickness direction.

Effects of electric boundary conditions on electroelastic field in a cracked piezoelectric strip are examined. Attention is focussed on an antiplane shear central crack normal to the strip surfaces. By decoupling equations and using the conformal mapping technique, expressions for electroelastic field in the piezoelectric strip are determined under the assumptions of an impermeable, permeable, or conducting crack, respectively. Comparison for the singularity near the crack tips among the obtained electroelastic fields is made.

We consider the role of small scale geometric and material features at geometric discontinuities other than a crack and study their relevance to fracture analysis. We are motivated by relatively recent experiments that show that under certain circumstances fracture initiation from geometric (sharp notches) and material (bimaterial interface corners) discontinuities can be successfully correlated with critical values of stress intensities that arise from a linear elastic analysis of the corresponding singular stress state. Implicit to such an approach is the idea that perturbations of the elastic fields near the discontinuity, which of course destroy the singular stresses, occur over a scale that is sufficiently small so that the complex behavior in this region is correlated by the elastic stress intensity. While the fracture mechanician will recognize these ideas as extensions of classical linear elastic fracture mechanics, significant differences exist and these are discussed in detail. We motivate the ideas through the use of a series of model problems in antiplane shear that are mostly amenable to exact analyses. We expect that the ideas carry through, albeit at the expense of far more complicated analysis, to planar and even three-dimensional situations.

A crack in a viscoelastic functionally graded material (FGM) layer sandwiched between two dissimilar homogeneous viscoelastic layers is studied under antiplane shear conditions. The shear relaxation modulus of the FGM layer follows the power law of viscoelasticity, i.e., μ = μ0exp (βy/h) [t0exp (δy/h) /t]q, where h is a scale length, and μ0,t0,β,δ and q are material constants. Note that the FGM layer has position-dependent modulus and relaxation time. The shear relaxation functions of the two homogeneous viscoelastic layers are μ=μ1(t1/t)q for the bottom layer and μ=μ2(t2/t)q for the top layer, where μ1 and μ2 are material constants, and t1 and t2 are relaxation times. An elastic crack problem of the composite structure is first solved and the `correspondence principle' is used to obtain stress intensity factors (SIFs) for the viscoelastic system. Formulae for SIFs and crack displacement profiles are derived. Several examples are given which include interface cracking between a viscoelastic functionally graded interlayer and a viscoelastic homogeneous material coating. Moreover, a parametric study is conducted considering various material and geometric parameters and loading conditions.

We present a general method for analyzing dynamically accelerating multiple co-linear cracks that can be applied to the contexts of plane strain or antiplane shear in an elastic material. The difficulty in solving such problems lies in the fact that the space-time regions containing known data evolve as the crack propagates in an a priori unknown manner. Using an analog to a Dirichlet-to-Neumann map, we can find complete knowledge of the stress and displacement along the fracture plane, facilitating the application of fracture criteria that require these values away from the crack tip. The method is demonstrated for a semi-infinite or finite mode III crack as well as for a pair of cracks in elastic material, using a stress intensity factor fracture criterion for simplicity.

The primary objective of this paper is to study the influence of the electroelastic interactions on the stress intensity factor in bonded layers of piezoelectric and orthotropic materials containing a crack along the interface under antiplane shear. Attention is given to a two-layer hybrid laminate formed by adding a layer of piezoelectric ceramic to a unidirectional graphite/epoxy composite or an aluminum layer. Electric displacement or electric field is prescribed on the surfaces of the piezoelectric layer. The problem is formulated in terms of a singular integral equation which is solved by using a relatively simple and efficient technique. A number of examples are given for various material combinations. The results show that the effect of the electroelastic interactions on the stress intensity factor and the energy release rate can be highly significant.

The higher order solutions of stress and deformation fields near the tip of a sharp V-notch in a power-law hardening material with continuous damage formation are analytically investigated under antiplane shear loading condition. The interaction between a macroscopic sharp notch and distributed microscopic damage is considered by describing the effect of damage in terms of a damage variable in the framework of damage mechanics. A deformation plasticity theory coupled with damage and a damage evolution law are formulated. A hodograph transformation is employed to determine the solution of damaged nonlinear notch problem in the stress plane. Then, inversion of the stress plane solution to the physical plane is performed. Consequently, higher order terms in the asymptotic solutions of the notch tip fields are obtained. Analytical expressions of the dominant and second order singularity exponents and associated angular distribution functions of notch tip stress and strain are presented. Effects of damage and strain hardening exponents and notch angle on the singular behavior of the notch tip quantities are discussed detailly. It is found that damage can lead to a weaker singularity of the dominant term of stress on one hand, but to stronger singularities of the second order term of stress and the dominant and second order terms of strain compared to that for undamaged case on the other. Also, both hardening exponent and notch angle have important effects on the notch tip quantities. Moreover, reduction of the notch tip solutions to a damaged nonlinear crack problem is carried out, and higher order solutions of the crack tip fields are obtained. Effects of damage and hardening exponents on the dominant and second order terms in the crack tip solutions are detailly discussed. Discussions on some other special cases are also presented, which shows that if damage exponent equals to zero, then the present solutions can be easily reduced to the solutions for undamaged cases.

In this paper, the scattering of harmonic antiplane shear waves by a finite crack is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt method instead of the first or the second integral equation method. Contrary to the classical elasticity solution, it is found that no stress singularity is presented at the crack tip. The non- local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length.

In this article, we examine the dynamic interaction between two cracks in a piezoelectric medium under incident antiplane shear wave loading. The theoretical formulations governing the steady-state problem are based upon the use of integral transform techniques and a self-consistent iterative method. The resulting dynamic stress intensity factors at the interacting cracks are obtained by solving the appropriate singular integral equations using Chebyshev Polynomials at different loading frequencies. Numerical examples are provided to show the effect of the geometry of the cracks, the piezoelectric constants of the material and the frequency of the incident wave upon the dynamic stress intensity factor of the cracks.

A semi-infinite crack with a nonlinear zone around the crack tip is studied in detail in the following four cases: (i) antiplane shear deformation, (ii) plane deformation, (iii) plane anisotropic deformation with purely imaginary characteristic roots, and (iv) interface between dissimilar solids, respectively. The complete Williams eigenfunction expansion forms including both positive and negative powers of a distance from the crack tip in each of the four cases are considered, which could be used to describe the elastic state in an annulus around the nonlinear zone. Explicit formulations of the path independent J-integral are presented by utilizing the differential property and the so-called pseudo-orthogonal property of the complete Williams expansion forms in each of the four cases. It is shown that the complete Williams expansion forms in every case mentioned above have two kinds of contributions to the J-integral. The first one is similar to the traditional one arising from the well-known r−1/2 singularity (or r−1/2+ie singularity for the interface crack). The second one is a summation induced from the interaction of higher order singular terms and nonsingular terms of the expansion forms. Although the coefficients of the complete Williams expansion forms in each of the four cases should be determined not only by the prescribed outer boundary conditions but by some specific material model for the nonlinear zone surrounding the crack tip, once they are determined by whatever method, the J-integral could be calculated by using the formulations derived in this paper without any difficulties. It is found also that the elastic T-term acting parallel to the crack plane has no direct interaction with the higher order singular terms such that has no direct effect to the J-integral, although the presence of the T-term will dramatically affect the size of the nonlinear zone and in this way affect the coefficients of the higher order singular terms and in turn the values of the J-integral. The present results in the four cases support the conclusion derived by Hui and Ruina (1995) that the nonsingular terms and the higher order singular terms in the complete Williams expansion forms are of equal importance. Thus, in order to improve and to confirm the small scale yielding description, not only the nonsingular terms, but also the higher order singular terms should be determined for a given crack configuration, body geometry, loading conditions and the prescribed material model in the nonlinear zone.

This paper is concerned with the development of a model for the treatment of a circular arc-crack in piezoelectric materials under antiplane shear and inplane electric field using the complex variable method. The problem is formulated in terms of the analytical continuation principle and complex series expansion, and as a result is reduced to two Riemann–Hilbert problems. The resulting closed form solution was then used to obtain the electroelastic field intensity factor and the energy release rate. Unlike the electric enthalpy release rate, the internal energy release rate is always positive and can thus be used as a more reliable fracture parameter in piezoelectric materials. It is also observed that the crack configuration has a significant influence upon the energy release rate.

An analytical study of the higher-order asymptotic solutions of the stress and strain fields near the traction-free crack tip under antiplane shear in a linear hardening material is investigated. The results show that every term of the asymptotic fields is controlled by both elasticity and plasticity and all the higher-order asymptotic fields are governed by linear nonhomogeneous equations. The first four term solutions are presented analytically and the first four terms are described by two independent parameters J and K2. The amplitude of the second order term solution is only dependent on the material properties, but independent of loading and geometry. This paper focuses on the case with traction-free crack surface boundary conditions. The effects of different crack surface boundary conditions, such as clamped and mixed surfaces, on the crack-tip fields are also presented. Comparison of multi-term solution with leading term solution, and finite element solution in an infinite strip with semi-infinite crack under constant displacements along the edges is provided.

The stress, strain, displacement and damage fields near the tip of a crack in a power-law hardening material with continuous damage formation under antiplane longitudinal shear loading are investigated analytically. The interaction between a major crack and distributed microscopic damage is considered by describing the effect of damage in terms of a damage variable D. A deformation plasticity theory coupled with damage and a damage evolution law are formulated. A hodograph transformation is employed to determine the singularity and angular distribution of the crack-tip quantities. Consequently, analytical solutions for the antiplane shear crack-tip fields are obtained. Effects of the hardening exponent n and the damage exponent m on the crack-tip fields are discussed. It is found that the present crack-tip stress and strain solutions for damaged nonlinear material are similar to the well-known HRR fields for virgin materials. However, damage leads to a weaker singularity of stress, and to a stronger singularity of strain compared to that for virgin materials, respectively. The stress associated with damage always falls below the HRR field for virgin material; but the distribution of strain associated with damage lies slightly above the HRR field for r/(J/τ0) > 1.5 while the difference becomes negligible when r/(J/τ0) > 2. The limiting distributions of stress and strain may indeed be given by the HRR field.

A hodograph transformation in conjunction with an appropriate affine transformation are both used to investigate the strain and stress fields near the crack tip in an anisotropic power law material under antiplane shear. Stress and strain exponents as well as angular distributions for the asymptotic stress and strain fields are obtained analytically. All the stress strain exponents are independent of material anisotropy, and the effect of material anisotropy on the asymptotic stress and strain field is discussed including higher order terms.

Analytical solutions of higher order fields in a fully plastic power-law hardening material are presented. By the use of hodograph transformation and asymptotic analysis the stress and strain exponents, angular distributions of shear stresses and strains are analytically determined. Special cases, such as linearly elastic, perfectly plastic materials are discussed. Similar characteristics between mode III and mode I plane strain, and mode II plane stress are examined. Comparison of four-term asymptotic solutions with exact and leading term solutions in an infinite strip with a semi-infinite crack under constant displacements along its edges is provided.

This paper presents an explicit solution to the antiplane problem of a partially debonded elliptical inhomogeneity embedded in an isotropic elastic medium. The boundary value problem is formulated through the complex variable method and is reduced to the solution of a Riemann-Hilbert problem with the aid of the conformal mapping technique and the analytical continuation principle. The complex potentials governing the problem are derived explicitly in both the elliptical inhomogeneity and the surrounding matrix and the corresponding formulae for the stress intensity factors of the interface crack are provided. Several particular solutions, resulting from the current general formulation, are considered in detail and verified by comparison with those existing in the literature. In addition, the effects of material and geometrical parameters upon the change in the stress intensity factors are discussed.

Erdogan and Gupta [1] solved the problem of bonded wedges with an internal interface crack under anti-plane shear. The problem studied here is the inclined edge crack in two bonded elastic quarter planes as shown in Fig. 1. Although there have been many works [1,2] with similar geometry, this work is distinguished from the previous by either the existence of free boundary or inclination of crack with respect to the interface. The case of crack inclination angle c0 (see Fig. 1) equal to zero is the problem of the interfacial edge crack and to our knowledge, this problem has not as yet been solved. The solution method we employed is the Wiener-Hopf technique through the Mellin transform.

New aspects for the generalization of the Sokhotski-Plemelj formulae are investigated, in order to show the behaviour of the limiting values of the finite-part singular integrals, defined over a smooth closed or open contour. The new formulae are more complicated when some corner points are further included in the contour. Beyond the above, when the contour is infinite, then the limiting values of the finite-part singular integrals are calculated by using an additional method. An application of two-dimensional fracture mechanics is finally given, to the determination of the stress intensity factors near a straight crack in a bimaterial infinite and isotropic solid under antiplane shear.

The antiplane shear problem for two bonded dissimilar half planes containing a semi-infinite crack or two arbitrarily located collinear cracks is considered. For the semi-infinite crack the problem is solved for a concentrated wedge load and the stress intensity factor and the angular distribution of stresses are calculated. For finite cracks the problem is reduced to a pair of integral equations. Numerical results are obtained for cracks fully imbedded in a homogeneous medium, one crack tip touching the interface, and a crack crossing the interface for various crack angles.

A modification of the Lobatto-Chebyshev method for the numerical solution of Cauchy type singular integral equations appearing in plane or antiplane elasticity crack problems and the determination of stress intensity factors at crack tips is presented. This modification, based on a variable transformation, permits the selection of abscissas and collocation points used to be modified so as to obtain rapid convergence of the numerical results for the stress intensity factors to their correct values. The proposed technique is seen to be particularly effective for crack problems with crack tips near boundaries, interfaces or other geometrical discontinuities. Two applications of the method, to a periodic array of cracks in plane elasticity and to an antiplane shear crack near a boundary, show the effectiveness of the method.

The problem of steady growth of an antiplane shear crack in a strain rate sensitive elastic-plastic material is considered. It is assumed that the material is elastic-perfectly plastic under slow loading conditions, but that the inelastic strain rate is proportional to the difference between the stress and the yield stress raised to some power. In earlier work on this problem in which the possibility of an elastic region in stress space was not considered, it was concluded that the asymptotic crack tip field is completely autonomous, but the local field could not be shown to reduce to a generally accepted rate independent limit as the rate sensitivity of the material vanished. It is shown here that, if the possibility of elastic unloading is admitted in the formulation, the asymptotic crack tip solution does indeed approach the correct rate independent limit as the rate sensitivity of the material vanishes. Furthermore, the crack tip field loses the feature of complete autonomy under these conditions. That is, the crack tip field involves an undetermined parameter that can be determined only from the remote fields. In the analysis, both dynamic and quasistatic growth are considered.

The paper deals with the problem of finding the distribution of stress in the neighbourhood of a Griffith crack located at the interface of two bonded dissimilar elastic half-spaces. The crack is excited by a normally incident antiplane shear wave. The problem is reduced to that of solving a Fredholm integral equation. Both the iterative and numerical solutions have been obtained. The solutions are used for calculating numerical values of quantities of physical interest such as dynamic stress intensity factor and the amplitude of distance between the edges of the crack.

The antiplane shear problem of two bonded elastic half planes containing a crack perpendicular to the interface is considered. The cases of a semi-infinite crack terminating at the interface, a finite crack away from and terminating at the interface, two cracks one on each side of the interface, and a finite crack crossing the interface are separately investigated. The nature of the stress singularity for the crack terminating at and going through the interface is studied, and it is shown that at the irregular point on the interface, for the former the power of singularity is not -1/2 and for the latter the stresses are bounded. For a material pair of aluminum-epoxy some numerical results giving the stress intensity factors, the density functions, and the crack opening displacements are presented.

The problem of kinking of internal cracks in infinite plates submitted to antiplane shear at infinity was solved by using the Mellin transform. This transform allows a straightforward reduction of it to a system of second order Fredholm integral equations for a pair of functions in terms of which the stress intensity factors at the crack tips may be calculated. Of great interest and of equivocal views is the regime at the vicinity of the corner, the region where the kink is nucleated, and which reflects numerical difficulties arising in the solution of the integral equations. For this purpose an asymptotic expansion is constructed for their kernels and subsequently for the values of SIFs with respect to the length e of the kinked branch. The analysis yielded results which were physically sound, but differing fundamentally with those existing in the literature. These results were in agreement with those obtained for an in-plane loading of the cracks and for cracks in bi-phase materials whose tips are on the interface of the phases.

The quasi-static growth of a crack in an elastic-creeping material under antiplane shear or mode III loading is investigated. The creep response of the material is assumed to be governed by a power-law between the creep strain rate, creep strain, and stress. While this law is capable of describing elastic-primary, secondary, or tertiary creep, the major emphasis of this paper is on crack growth in the elastic-primary creep regime. The asymptotic crack-tip stress and strain fields for a quasi-statically extending crack in an elastic-primary creeping material are developed. This is followed by a finite element analysis to determine the complete stress and strain fields within the confines of small scale yielding. These fields are then compared with the asymptotic ones to establish the size of the zone of dominance of the crack-tip fields.

The mechanics of transient crack growth in an elastic power-law creeping material is investigated using the special case of a suddenly growing crack with constant growth rate \.a0. Small scale yielding assumption is assumed to be valid at the instant when crack growth occurs. The analysis is carried out for the case of antiplane shear mode III and plane strain mode I. The resulting transient stress field for small crack extension is analyzed using perturbation methods. For small crack extension, the HR field, the RR field and the elastic K field coexist near the crack tip, one inside another. The regions of dominance of these near tip fields are estimated and an approximate solution is provided for the near tip stresses. The effect of crack growth rate on the small scale yielding condition is also studied. For large crack extension, it is shown that creep relaxation can be neglected and results of steady state analysis can be modified to describe the near tip stress fields. Extension of these results to more general crack growth histories is also discussed

In view of the fact that non-coplaner crack growth is a common characteristic of crack propagation in fiber reinforced composite materials, an attempt is made to derive the equation of the energy release rateG for non-coplaner crack extension in orthotropic linear elastic solids. Combining the idea developed in our previous paper and the analysis by Sihet al., the equation ofG is obtained for the cases of plane symmetric loading, plane skew-symmetric loading, antiplane shear loading, and a combination of these. The result will serve as a basic tool for developing a fracture mechanics approach to composites.

The elasiostatic antiplane shear problem of two half spaces bonded through an infinite layer all having transverse fatigue cracks is considered. The cases of three imbedded cracks, cracks terminating at the interfaces, completely broken layer, and a crack in the layer spreading into the half spaces are separately treated in detail. The Fourier transform is used to formulate the problem. Use of the mixed type conditions on the crack plane reduces the formulation to a system of singular integral equations with simple or generalized Cauchy kernels. Singular behavior of the solution at the irregular points on cracks is investigated. Numerical results given in the paper include stress intensity factors and crack opening displacements for material pairs of aluminum-epoxy and steel-aluminum.