A BEM analysis of a penny-shaped interface crack using the open and the frictionless contact models: Range of validity of various asymptotic solutions

The global elastic solution for the problem of a pressurized penny-shaped crack at the interface of two dissimilar half spaces has been numerically obtained employing the boundary element method (BEM). Using the Williams’ open model (for the whole range of feasible bi-material combinations), the comparison of the global BEM solution with an existing analytical asymptotic solution has shown: (i) that the extent of the zone in which the first term is dominant is always larger than the extent of the zone in which the interpenetrations take place and (ii) that, in the former zone, a recently proposed relation between the components of the complex stress intensity factor (SIF) and the components of the energy release rate (ERR) always yields accurate results. Consequently, the appearance of negative values of the normal contribution to the ERR in certain cases has been confirmed by the BEM solution, thus questioning the significance of the asymptotic results obtained from the open model in those cases. If the Comninou's frictionless contact model is employed, the ability of the BEM formulation employed to obtain accurate elastic solutions is shown by comparisons with an existing semi-analytical solution (for a particular bi-material combination).     

A set of interface cracks with contact zones in a combined tension-shear field

Summary. A set of cracks lying along the interface of two dissimilar isotropic materials under a mixed-mode loading is considered. The interface cracks are assumed to be fully open, partially closed with frictionless contact zones and fully closed. The problem is reduced to a homogeneous combined Dirichlet-Riemann boundary value problem, which is solved in closed form. A set of transcendental equations for the determination of the contact zone lengths for an arbitrary number of cracks and the closed-form expressions for the stresses and the displacement jumps on the material interface are obtained. A single crack with one and two contact zones has been considered in details. An explicit set of two transcendental equations for the relative contact zone length and closed-form expressions for the stress intensity factors at the crack tips are obtained for both cases. The contact zone lengths and the stress intensity factors are investigated numerically for different material pairs under different values of the loading, and a comparison of the results for a crack with one and two contact zones is carried out.     

A numerical approach to subsurface crack propagation assessment in rolling contact

Subsurface crack mode II propagation parallel to the contact surface is a damage mechanism leading to dramatic failure in many components subjected to cyclic loading. A weight function (WF) was elaborated for calculating the applied mode II stress intensity factor (SIF) of a crack in a two‐dimensional half‐space in plane strain condition, for crack completely closed and frictionless contact between the crack faces. With respect to other methods, the WF allows faster SIF calculation, thus being suitable for simulation of many repeated load cycles and fatigue crack propagation. The WF was applied for simulating a case of rolling contact experiments found in the literature, and good agreement between experimental and numerical results was obtained, showing the effectiveness of the WF method in damage tolerant design.     

Elastoplastic crack analysis for pressure-sensitive dilatant materials-Part II: Interface cracks

The problem of a plane strain crack lying along an interface between a rigid substrate and an elastic-plastic material has been studied. The elastic-plastic material exhibits pressure-sensitive yielding and plastic volumetric deformation. Two-term expansions of the asymptotic solutions for both closed frictionless and open crack-tip models have been obtained. The Mises effective stress in the interfacial crack-tip fields is a decreasing function of the pressure-sensitivity in both open and closed-crack tip models. The variable-separable solution exists for most pressure-sensitive materials and the limit values for existence of the variable-separable solution vary with the strain-hardening exponents. The governing equations become singular as the pressure-sensitivity limit is approached. Strength of the leading stress singularity is equal 1/(n+1) for both crack-tip models, regardless of the pressure-sensitivity. The second-order fields have been solved as an additional eigenvalue problem and the elasticity terms do not enter the second-order solutions as n2. The second exponents for the closed crack model are negative for the weak strain hardening, whereas the negative second-order eigenvalue in the open crack model slightly grows with the pressure-sensitivity. The second-order solutions are of significance in characterising the crack-tip fields. The leading-order solution contains the dominant mode I components for both open and closed crack-tip models when the materials do not have substantial strain hardening. The second-order solutions are generally mode-mixed and depend significantly on the pressure-sensitivity.     

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 mode III full-field solution in elastic materials with strain gradient effects

The near-tip asymptotic field and full-field solution are obtained for a mode III crack in an elastic material with strain gradient effects. The asymptotic analysis shows that, even though the near-tip field is governed by a single parameter B (similar to the mode III stress intensity factor), the near-tip field is very different from the classical K_III field; stresses have r ^-3/2 singularity near the crack tip, and are significantly larger than the classical K _III field within a zone of size l to the crack tip, where l is an intrinsic material length, depending on microstructures in the material. This high-order stress singularity, however, does not violate the boundness of strain energy around a crack tip. The parameter B of the near-tip asymptotic field has been determined for two anti-plane shear loadings: the remotely imposed classical K _III field, and the arbitrary shear stress tractions on crack faces. The mode III full-field solution is obtained analytically for an elastic material with strain gradient effects subjected to remotely imposed classical K _III field. It shows that the near-tip asymptotic field dominates within a zone of size 0.5 l to the crack tip, while strain gradient effects are clearly observed within 5l. It is also shown that the conventional way to evaluate the crack tip energy release rate would lead to an incorrect, infinite value. A new evaluation gives a finite crack tip energy release rate, and is identical to the J-integral. This revised version was published online in August 2006 with corrections to the Cover Date.     

On contact zone models for an electrically limited permeable interface crack in a piezoelectric bimaterial

An electrically limited permeable crack with a frictionless contact zone at the right crack tip between two semi-infinite piezoelectric spaces under the action of a remote electromechanical loading is considered. Attention is focused on the influence induced from the permittivity of the medium inside the crack gap on the contact zone length and the fracture mechanical parameters. Assuming the electric displacement constant inside the open region of the crack, the problem is reduced to a combined Dirichlet-Riemann and Hilbert boundary value problems which have been solved exactly. Stress and electric displacement intensity factors as well as the crack tip energy release rate are found in a clear analytical form. Furthermore, transcendental equations for the determination of the real contact zone length have been obtained for a general case and for a small contact zone length in an especially simple form. The dependencies of the mentioned values on the intensities of the electromechanical loading are presented in tables and associated diagrams.     

Crack kinking from an initially closed, ordinary or interface crack, in the presence of friction

One theoretically studies crack kinking from an ordinary crack (in some homogeneous solid) or an interface crack (between two dissimilar materials), in the situation where this crack is closed prior to kinking but open after it. This problem was recently considered by the authors with the simplifying, but physically quite unrealistic hypothesis of absence of friction between the crack lips. Their work is extended here to account for possible friction governed by Coulomb’s law. Problems of elastic fracture mechanics with unilateral contact and friction between the crack lips being not only non-linear, but incremental in nature, the theoretical treatment becomes notably more involved than without friction. It is still based, however, on the same basic ingredients, namely “homogeneity” properties of the type of problems considered, changes of scale and some reasonable hypotheses. It is shown that whatever the geometry of the body and the crack and whatever the loading, the asymptotic expression of the stress intensity factors (SIF) at the tip of a vanishingly small kinked crack extension depends solely upon the initial (mode II) SIF prior to kinking, the kink angle, Dundurs’s famous parameters α and β and the friction coefficient. The (history-independent) functions involved in the general formulae established are determined numerically through finite element computations. From there, using Goldstein and Salganik’s famous principle of local symmetry to predict the crack path, one derives a theoretical value for the kink angle. This value depends upon the loading only through the sign of the initial stress intensity factor; it also depends on the mismatch of elastic properties and the friction coefficient. However, its range of variation is numerically found to be rather narrow. Experiments conducted by various authors seem to confirm these theoretical predictions.     

The first-kind and the second-kind boundary integral equation systems for solution of frictionless contact problems

An original approach to the numerical solution of displacement boundary integral equation (BIE) and traction hypersingular boundary integral equation (HBIE) by the boundary element method (BEM) for contact problems is given. The main point is to show, how the contact conditions are used to formulate the first-kind and the second-kind BIE systems in the case of frictionless two-body elastic contact. The solution of the first-kind BIE is performed by symmetric Galerkin BEM; the second-kind BIE is solved by an appropriate collocation BEM. The contact problem in itself is solved by the method of subsequent approximations of contact region. Both forms of BIE system are compared in several numerical examples. This comparison is made for different kinds of contact problem. The major emphasis is put on the evaluation of contact pressure. The obtained results are compared with referenced numerical and with the analytical ones.     

Investigation of an interface crack with a contact zone in a piezoelectric bimaterial under limited permeable electric boundary conditions

Summary The plane strain problem for an interface crack between two bonded piezoelectric semi-infinite planes under remote electromechanical loading is considered. Mechanically frictionless and electrically permeable contact zones are assumed at the crack tips and the remaining part of the crack is considered as electrically limited permeable with a certain permeability of the crack medium. Patron’s way of modelling limited permeable conditions is used. By means of integral transforms the problem is reduced to a nonlinear system of singular integral equations. An iterative scheme together with discretization and utilization of Gauss-Chebishev quadrature rule is applied for the solution of this system. Distributions of the electric displacement along the crack region as well as the stress and electric intensity factors and the energy release rate are found for different electromechanical loads and crack permeabilities. Calculations are performed for an artificial contact zone length, however the way of an easier determination of the associated values for the real contact zone length is shown. As a particular case of the obtained solution the crack in a homogeneous piezoelectric media is considered. The results of the calculations are compared to the corresponding results obtained earlier by means of Hao and Shen’s way of modelling the crack permeability. Even though the electric displacements obtained in the respective framework of these models differ essentially, it appears that the fracture mechanical parameters are in good agreement with each other.     

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.     

A generalized comninou contact model for interface cracks in anisotropic elastic solids

An open question of interest to the mechanics of interface fracture is how to generalize the Comninou contact model for interface cracks in isotropic solids to the general anisotropic case. Part of the difficulty lies in that the peculiar oscillatory behavior can not be fully eliminated by Comninou's original assumption of pure pressure contact between the crack surfaces. In this paper, we propose a model that strictly enforces the non-oscillatory condition by allowing the crack face contact force to have a shear component normal to the direction of slip, which is somewhat reminiscent of frictionless slip between a pair of grooved surfaces. Based on that model, complex variable representations are adopted to determine the complete series expansion for the crack-tip field. The solutions are incorporated into a hybrid finite element procedure to develop a special element for closed interfacial crack tips obeying the generalized contact model. Numerical examples involving a partially closed crack between a pair of misoriented cubic crystals are given to illustrate how the special crack-tip element helps in determining the stress intensity factors as well as the contact zone geometry.     

A hybrid finite element analysis of interface cracks

A versatile hybrid finite element scheme consisting of special crack-tip elements and crack face contact elements is developed to analyse a partially closed interface crack between two dissimilar anisotropic elastic materials. The crack-tip element incorporates higher-order asymptotic solutions for an interfacial crack tip. These solutions are obtained from complex variable methods in Stroh formalism. For a closed interfacial crack tip, a generalized contact model in which the crack-tip oscillation is eliminated is adopted in the calculation. The hybrid finite element modelling allows the stress singularity at an open and closed crack tip to be accurately treated. The accuracy and convergence of the developed scheme are tested with respect to the known interface crack solutions. Utilizing this numerical scheme, the stress intensity factors and contact zone are calculated for a finite interface crack between a laminated composite material.     

The interface crack under combined loading: an eigenvalue problem for the gap

To avoid the difficulty of overlapping material near the crack tips, the tips of the crack are assumed to be closed (the Comninou model). The exact series solution to the problem of the interface crack under combined loading is complicated, and it is difficult to take advantage of the smallness of the left contact zone to obtain simple asymptotic approximations to the quantities of physical interest. Here, the gap is shown to satisfy an eigenvalue problem. The method of matched inner and outer expansions is used to obtain a simple uniform approximation to this quantity. Similar results are obtained for the tangential shift, which satisfies the same differential equation as the gap. The tractions at the interface also satisfy a second order differential equation and simple uniform approximations are obtained for these quantities.     

3-D boundary element method for cracks in a plate

New fundamental solutions which automatically satisfy boundary conditions at the interfaces of an elastic plate perfectly bonded to two elastic halfspaces are implemented in a 3-D boundary element method (BEM) for crack problems. The BEM features a new integration scheme for highly singular kernels. The capability is achieved through a part analytic and part numerical integration procedure, such that the analytic part of the integration is similar for all slip/opening variations, ‘Part-through’ elliptic cracks in an elastic plate with traction-free surfaces are analysed and the stress intensity factor (SIF) values along the crack front are found to compare favourably with widely accepted numerically obtained SIF results by Raju and Newman.¹     

A finite element analysis of plane strain dynamic crack growth at a ductile-brittle interface

In this work, steady, dynamic crack growth under plane strain, small-scale yielding conditions along a ductile-brittle interface is analysed using a finite element procedure. The ductile solid is taken to obey the J ₂ flow theory of plasticity with linear isotropic strain hardening, while the substrate is assumed to exhibit linear elastic behaviour. The objectives of this work are to establish the validity of an asymptotic solution for this problem which has been derived recently [12], and to examine the effect of changing the remote (elastic) mode-mixity on the near-tip fields. Also, the influence of crack speed on the stress fields and crack opening profiles near the propagating interface crack tip is assessed for various bi-material combinations. Finally, theoretical predictions are made for the variation of the dynamic fracture toughness with crack speed for crack growth under a predominantly tensile mode along ductile-brittle interfaces. Attention is focused on the effect of mismatch in stiffness and density of the constituent phases on the above aspects.     

Dual boundary element analysis of cracked plates: singularity subtraction technique

The present paper is concerned with the formulation of the singularity subtraction technique in the dual boundary element analysis of the mixed-mode deformation of general homogeneous cracked plates.The equations of the dual boundary element method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation is applied on the other, general mixed-mode crack problems can be solved in a single region boundary element formulation, with both crack surfaces discretized with discontinuous quadratic boundary elements.The singularity subtraction technique is a regularization procedure that uses a singular particular solution of the crack problem to introduce the stress intensity factors as additional problem unknowns. The single-region boundary element analysis of a general crack problem restricts the availability of singular particular solutions, valid in the global domain of the problem. A modelling strategy, that considers an automatic partition of the problem domain in near-tip and far-tip field regions, is proposed to overcome this difficulty. After the application of the singularity subtraction technique in the near-tip field regions, regularized locally with the singular term of the Williams' eigenexpansion, continuity is restored with equilibrium and compatibility conditions imposed along the interface boundaries. The accuracy and efficiency of the singularity subtraction technique make this formulation ideal for the study of crack growth problems under mixed-mode conditions.     

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

A receding contact problem between a functionally graded layer and two homogeneous quarter planes

In this paper, the plane problem of a frictionless receding contact between an elastic functionally graded layer and two homogeneous quarter planes is considered when the graded layer is pressed against the quarter planes. The top of the layer is subjected to normal tractions over a finite segment. The graded layer is modeled as a non-homogeneous medium with a constant Poisson’s ratio and exponentially varying shear modules. The problem is converted into the solution of a Cauchy-type singular integral equation in which the contact pressure and the receding contact half-length are the unknowns using integral transforms. The singular integral equation is solved numerically using Gauss–Jacobi integration. The corresponding receding contact half-length that satisfies the global equilibrium condition is obtained using an iterative procedure. The effect of the material non-homogeneity parameter on the contact pressure and on the length of the receding contact is investigated.