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An analytical procedure predicting a debond (interface crack) onset and growth in an adhesive joint between two beams or plates is developed and applied to a specific configuration often used in reinforcement tests in civil engineering. The procedure is based on Timoshenko beam theory and Linear Elastic-(Perfectly) Brittle Interface Model (LEBIM) combined with the Coupled Criterion of the Finite Fracture Mechanics (CC-FFM) for mixed-mode fracture. First, a sixth order differential equation in the shear stresses along the adhesive layer is deduced and solved, leading to closed form expressions for both shear and normal stresses in the adhesive. Then, the critical value of the applied load necessary to produce debonding is predicted by coupling a stress and an energy condition based on: (i) the stress distribution produced in the interface before the debond onset and (ii) the energy released during the debonding process along the interface. Although the developed procedure can be applied to several types of joints with different geometries, materials and loads (e.g., double lap joint tests including adherents made of steel or composites), herein it is applied to the double pull-push shear test where the debond onset and growth between a Carbon Fibre Reinforced Polymer (CFRP) laminate and a concrete block occurs. For such a case, the debond is produced under predominant fracture mode II; nevertheless, it is shown that relevant normal (peeling) stresses associated to mode I may appear as well. A comparison of the present solution with a previous one by the shear-lag model is provided as well.

In the present work the Coupled Criterion of Finite Fracture Mechanics (CCFFM) is used to predict crack onset or growth by finite increments in a linear elastic interface. The predictions of the interface failure by two alternative approaches of the CCFFM applied to the Linear Elastic Brittle Interface Model (LEBIM): the widely used method based on looking for an intersection of stress and energy criteria curves and the novel Principle of Minimum Total Energy subjected to a Stress Condition (PMTE-SC), are studied and compared. For this purpose, two analytical studies, based on the stress and energy criteria curves and the PMTE-SC, are carefully explained, providing appropriate graphical representations, by considering the widely used Double Cantilever Beam (DCB) test as benchmark problem. For the sake of simplicity, the Euler–Bernoulli beam model including an elastic interface (the Winkler interface) to model the adhesive layer joint is used in this study. To the authors’ best knowledge, this is the first study showing, for both load and displacement control, that the PMTE-SC is equivalent to the classical formulation of the CCFFM, providing exactly the same analytical predictions for the crack onset and propagation. The fact that onset of a finite crack-advance typically predicted by CCFFM is associated to tunnelling the total energy barrier is also illustrated on the DCB test. The main advantages of the PMTE-SC are its versatility and possibility of applying it to complex configurations including multiple cracks and fracture mixed-mode behaviour. Spanish Ministry of Science, Innovation and Universities and European Regional Development Fund Project PGC2018- 099197-B-I00 Consejería de Economía y Conocimiento de la Junta de Andalucía (Spain) Project P18-FR-1928 Universidad de Sevilla (España) and Programa Operativo FEDER Andalucía 2014–2020 contract US-1266016 Junta de Andalucía (Spain) and European Social Fund Acciones de transferencia del conocimiento AT17-5908-USE

The antiplane problem of an infinite isotropic elastic medium subjected to a far-field load and containing a zero thickness layer of arbitrary shape described by the Gurtin-Murdoch model is considered. It is shown that, under the antiplane assumptions, the governing equations of the complete Gurtin-Murdoch model are inconsistent for non-zero surface tension. For the case of vanishing surface tension, the analytical integral representations for the elastic fields and the dimensionless parameter that governs the problem are introduced. The solution of the problem is reduced to the solution of the hypersingular integral equation written in terms of elastic stress of the layer. For the case of a layer along a straight segment, theoretical analysis of the hypersingular equation is performed and asymptotic behavior of the elastic fields near the tips is studied. The appropriate numerical solution techniques are discussed and several numerical results are presented. Additionally, it is demonstrated that the problem under study is closely related to the specific case of the well-known problem of a thin and stiff elastic inhomogeneity embedded into a homogeneous elastic medium.

Interface models for triggering damage events have been extensively used in many engineering applications due to their inherent versatility and relatively simple numerical implementations. Usually, the non-linear behavior of interfaces and thin adhesive joints between solids has been modeled using Cohesive Zone Models (CZMs) although other alternative models have been proposed as well. This is the case of the recently proposed Linear Elastic-Brittle Interface Model (LEBIM). The LEBIM is characterized by a linear constitutive law (traction-relative displacement relationship) in the undamaged state up to failure. Damage occurs when a material point reaches a critical energy release rate leading to an abrupt failure representation. Currently, from the numerical point of view, a great deal of research dealing with interface debonds and adhesive failure is still mainly focused on 2D models. The aim of the present investigation is to extend the LEBIM (originally proposed for 2D problems) for its applications to 3D problems under general fracture conditions and to examine its reliability against experimental data and cohesive-based interfaces pinpointing the main theoretical and numerical differences with respect to alternative methods. Thus, we focus our attention on addressing the main advantages of the proposed LEBIM over the nonlinear CZMs: (i) a simple interface law without the need of triggering a gradual stiffness degradation, (ii) a reliable accuracy in comparison with experimental results, endowing a clear identification of the crack front due to the preclusion of the fracture process zone concept, and (iii) a very high efficiency in terms of mesh requirements stemming from the absence of nonlinear cohesive crack representation. To show that 3D LEBIM is feasible it is implemented into the general purpose Finite Element Method (FEM) package ABAQUS through the user subroutine UMAT . Finally, special attention is given to the adequate treatment of the mixed mode fracture that may appear in complex 3D crack propagation cases.

Crack onset in PMMA holed plates subjected to tensile stresses is studied experimentally and by the coupled stress and energy criterion of the Finite Fracture Mechanics (CCFFM). The elastic, strength and fracture properties of PMMA are determined by the standard tests, a clearly nonlinear stress-strain relation being identified in the tensile tests. Thus, a novel numerical implementation of the CCFFM considering a non-linear elastic (NLE) material model, using the Ramberg-Osgood approximation, in addition to the usually used linear elastic (LE) model, is developed. Testing of plates with different hole sizes shows a hole size effect in the nominal failure load as expected. For a better fitting of the experimental results, higher strength values obtained by three point bending (TPB) flatwise and edgewise coupons (without any notch), for these material models, are used, apparently for the first time, in the CCFFM predictions. This approach reflects the observation that the strength values associated to smaller but highly stressed volumes, like those located at stress maxima in the holed plates and TPB specimens, are higher. For finite-width holed plates and both material behaviours, suitable FEM models are developed to implement the CCFFM for both LE and NLE models, considering plane stress state. Moreover, an inverse procedure is devised, using the experimental data for holed plates and predictions by CCFFM, to estimate the strength and fracture properties to be used in both material models, providing very good correlations of the CCFFM predictions with the experimental results.

Transverse cracking of off-axis plies is typically the first step of some failure mechanisms in long fiber reinforced laminated composites. It is well known that the off-axis ply thickness affects strongly the strain level which this ply is able to withstand without cracking. Experimental evidences show that this critical strain level increases strongly with decreasing ply thickness for the thinnest plies, confirming a clear size effect. This fact could be used to inhibit transverse cracking with the use of the recently developed ultra-thin plies. However the classical failure criteria for composites are not able to predict this size effect. From the first tests in the 1970s, several models with very different hypotheses were proposed to predict this size effect. Surprisingly there is still not a clear consensus in the scientific community about which is the adequate model, though their fundamentals are radically different. In this work an extensive test campaign is carried out on self-similar [ 0 / 90 ] s cross-ply laminates of AS4/8552 carbon/epoxy. The size effect is evaluated spanning a broader spectrum of thicknesses than in the tests presented in the literature until now, in order to explore the range of thicknesses where the differences between predictions of the main theoretical models are larger. In spite of the inherent dispersion in the experimental results, the comparison shows that the purely stress-based criteria underestimates the apparent strength of the off-axis ply. The other models predict very similar results, particularly for the thinnest laminates, with slight underestimations or overestimations, depending on the model.

A procedure based on the Linear Elastic Brittle Interface Model (LEBIM) combined with the Coupled Criterion of Finite Fracture Mechanics (CCFFM) is successfully implemented in a 2D Boundary Element Method (BEM) code. In the original LEBIM formulation, the values of the interface strength, fracture toughness and stiffness are dependent on each other. Therefore, for a large interface stiffness, when the elastic interface tends to a perfect (infinitely stiff) interface, LEBIM is not able to properly characterize the crack propagation. The use of the CCFFM applied to LEBIM, with both the stress and energy criteria imposed as independent fracture conditions, allows to uncouple the interface fracture toughness and strength, obtaining realistic predictions for crack propagation even for stiff interfaces. This code is successfully applied to the problem of debond onset and growth in the pull push test. A benchmark problem is solved, focusing on the convergence of the load-displacement curve and crack-tip solution for h-refinements of BE meshes.

Transverse cracking, i.e. matrix cracking in the off-axis plies of the laminate, is widely recognized as the first damage mode to appear in continuous fibre-reinforced composite laminates subjected to in-plane loading. Since transverse cracking has a great influence on the subsequent damage steps such as delaminations or oblique cracks, it is important to be able to predict its onset and growth accurately. In this paper, it is proposed to use a combination of the Coupled Criterion of Finite Fracture Mechanics (FFM) and the Equivalent Constraint Model (ECM) to predict the evolution of crack density with increasing applied load. Two formulations – a discrete formulation and a continuous formulation – are developed for the energy criterion within the Coupled Criterion. Some dependences between the two formulations are proved, which justifies the good agreement found by the models based on continuous formulations presented by other authors despite the inherent discrete nature of the phenomenon. Dependence of the failure load predicted by the Coupled Criterion on the layer thickness ratio and brittleness number (a structural parameter that characterizes a combination of the stiffness, strength, fracture toughness and the thickness of the cracked ply of the laminate) is examined and discussed for carbon/epoxy and glass/epoxy laminates. Finally, a comparison against experimental results shows a good agreement.

A comparison of the interface fracture energy in the adhesive joints PMMA/PMMA bonded by epoxy and PMMA/epoxy is presented. The work focuses on the question of how similar is the fracture energy in these joints, including its dependence on the fracture mode mixity. Series of tests for Brazilian disk specimens subjected to compressive loading were carried out to obtain the fracture energy as a function of the fracture mode mixity. A careful observation with a high-speed camera of the steps of crack growth allowed to determine the validity of each sample and explore the different forms of failure. The results show a high level of similarity in the fracture energy prediction for the two configurations, having the largest differences for the largest fracture mode-mixity angle. The similarity between the results for the two configurations is useful to simplify some analyses in practical applications. A nonlinear finite element model, including plastic effects, showed that different process zones in the two configurations could be the source of the small discrepancy found between the two configurations in the experimental results. These results show the relevance of the constraining effect of the thin adhesive layer on the process zone.

In this paper we focus on the prediction of mode-I debonding for a double cantilever beam (DCB). Among the various modeling approaches available, the Cohesive Crack Model (CCM) and Finite Fracture Mechanics (FFM) are selected for the analytical investigation, due to their ability to reconcile the stress- and energy-based approaches. The specimen is considered as an assemblage of two identical beams partly bonded together by an initially elastic interface. After the elastic stage, according to the CCM approach, it is assumed that, ahead of the physical crack tip, there exists a cohesive zone where the interface behavior is described by a stress-separation law. The interfacial stresses and length of the process zone are determined in closed form, along with the global load-displacement response. The method is first compared to the simple beam theory (SBT) and the enhanced beam theory (EBT) approaches, which are found to provide larger values of the debonding load; the difference between predictions of CCM and SBT/EBT is more pronounced for less brittle interfaces, i.e. for larger process zones. Then the analytical solution obtained by means of FFM is presented, which, despite being simply based on the elastic foundation model, closely matches the CCM results. Finally a numerical solution is achieved by a finite element analysis where generalized zero-thickness contact interface elements are adopted. An excellent agreement with these results confirms the good performance of the proposed CCM and FFM approaches.

Onset and growth of debonds at fibre-matrix interfaces in a bundle of fibres subjected to transverse loads are studied numerically. In particular, the crack path formed by debonded neighbour fibres is analysed. The Linear Elastic–Brittle Interface Model (LEBIM) is used to model the fibre-matrix interface behaviour. This simple model of a Long Fibre Reinfoced Polymer (LFRP) composite includes ten parallel fibres embedded in a matrix cell whose external dimensions are much larger than the fibre radius. The advantage of the present LEBIM formulation of the so-called matrix cracking lies in its ability to make quantitative predictions about the concurrent fibre-matrix debond onset and mixed-mode interface crack growth in a fibre bundle. The numerical analysis predicts failure loads producing the first and subsequent debond onsets, leading to a crack path. A discussion on the position where debond occurs is also included. Finally, the effect of the load biaxiality on the crack path is studied in detail.

A new numerical procedure for predicting interface failures between solids is developed. The procedure is based on the Linear Elastic-(Perfectly) Brittle Interface Model (LEBIM) combined with the Coupled Criterion of the Finite Fracture Mechanics (CCFFM). Although in the present investigation this procedure is implemented in a 2D BEM code, a general pseudocode is devised allowing its implementation in any BEM or FEM code. The pull-push shear test is used as a benchmark problem, where the fracture mode II is dominant. Nevertheless, the present procedure can tackle a debond growth occurring under any fracture mode mixity. The pull-push problem is chosen since it allows us to check the obtained numerical results against an available analytical solution based on a beam model. Additionally, the numerical results are compared with some experimental data from literature. Furthermore, an inverse analysis is applied to obtain the interface strength and fracture parameters that the model needs.

T-stress effects on crack deflection in notched brittle materials are investigated by means of an average stress approach. It is shown that the crack advance always reveals to be straight if the T contribution is negligible. On the contrary, if T is different from zero, the trajectory tends to curve: the higher is |T|, the larger is the curvature parameter, its value being either positive or negative depending on the sign of T. Analogies and differences with respect to straight crack propagation are quantified in terms of the fracture loci and of the critical kinking angle. The investigation on the stability of crack deflection under mode I loading conditions concludes the paper, together with a discussion on experimental observations.

Understanding matrix failure in LFRP composites is one of the main challenges when developing failure criteria for these materials. This work aims to study the influence of the secondary transverse load on the crack initiation at micro-scale. Four non-classical approaches of fracture mechanics are used to model the onset of fibre-matrix interface debonds: Linear Elastic Brittle Interface Model (LEBIM), an Energetic Approach for the Linear Elastic Brittle Interface Model (EA-LEBIM), an Energetic Approach for the bilinear Cohesive Zone Model (EA-CZM) and the Coupled Criterion of the Finite Fracture Mechanics (CC-FFM). Results obtained by these approaches predict that, for brittle fibre-matrix configurations, a secondary transverse compression reduces the critical value of the main transverse tension leading to the debond onset. This fact is not taken into account by the currently used failure criteria

The first damage mode to appear in continuous fibre-reinforced composite laminates subjected to in-plane loading is usually transverse cracking, i.e. matrix cracking in the off-axis plies of the laminate. Since the density of transverse cracks has a great influence on the subsequent failure steps like delaminations, it is important to be able to predict it accurately. In this paper, the evolution of crack density with increasing external load is predicted using a combination of the Coupled Criterion of Finite Fracture Mechanics and the Equivalent Constraint Model.

The contact problem with Coulomb friction together with a simple Kelvin-Voigt viscoelastic model is studied. The numerical solution is obtained using a time discretization by a semi-implicit formula, the visco-elastic solids in contact being discretized by Symmetric Galerkin Boundary Element Method (SGBEM). The resulting minimization problem with a nonsmooth cost functional is suitably transformed in several ways. Firstly, a transformation is performed to apply SGBEM without any viscoelastic fundamental solution. Secondly, a transformation of contact quantities leads to a minimimization with a quadratic programming structure. Numerical examples show the applicability of the proposed approach to solve rather intricate frictional contact problems.

The controversy about the symmetry of the debond onset at the fibre–matrix interface in single-fibre specimens under transverse tension is studied here applying the coupled stress and energy criterion of the finite fracture mechanics. This criterion enables to compare the two different post-failure configurations found in the literature studying this problem: an asymmetric configuration with a single debond and a symmetric one with two debonds. The coupled criterion applied here predicts that an asymmetric post-failure configuration is originated by a lower critical remote tension than the symmetric one, the difference being above 10% in some cases. Thus, the asymmetric debond onset is the preferred solution to the initially symmetric problem, which agrees with the experimental evidences found in the literature. This result is a consequence of the shielding effect between the two debonds in the symmetric solution. Numerical results are obtained for two common composites, glass/epoxy and carbon/epoxy, and three virtual bimaterials corresponding to the extreme values of the Dundurs elastic parameters. The stress–strain curves predicted for the two post-failure configurations are also compared. Eventually, simplified models using springs and breakable elements, which capture the essence of the problem studied and explain the energetic origin of the loss of symmetry, are introduced.

Crack initiation in the inner ply of symmetric [ 0 / 90 ] s laminates under tension is studied applying the coupled stress and energy criterion proposed by Leguillon (Eur. J. Mech. A/Solids 21, 2002) in the framework of the finite fracture mechanics. This criterion assumes that a crack of a finite extension appears abruptly when the stress criterion is fulfilled and this crack onset is energetically admissible. The stress state is calculated by applying the laminate theory to the undamaged laminate which provides explicit expressions to be used in the stress criterion. Assuming generalised plane strain, the magnitude involved in the (incremental) energy criterion is evaluated numerically by means of a boundary element code and a dimensional analysis. The two criteria lead to a theoretical model providing explicit expressions of the critical parameters for the crack onset, which depend only on the computational results through a scalar value. This model predicts that the crack grows unstably after the onset in the inner ply and is arrested close to the interface between the plies for a length larger than a certain threshold length independent of the fracture and strength properties. This threshold length depends only on the elastic and geometric properties of the laminate. The model also predicts the existence of a size effect which agrees with the experimental results found in the literature. In addition, the model provides a physical interpretation of this size effect.

Relations between different solutions of an interface crack in a neighborhood of the crack tip given by the open model, frictionless and frictional contact models of interface cracks are analyzed numerically for a penny-shaped interface crack subjected to remote tension. A new analytic expression for the size of the near-tip contact zone in presence of Coulomb friction between crack faces is proposed in the so-called case of the contact zone field embedded in the oscillatory field.

A simple yet efficient procedure to solve quasistatic problems of special linear visco-elastic solids at small strains with equal rheological response in all tensorial components, utilizing boundary element method (BEM), is introduced. This procedure is based on the implicit discretisation in time (the so-called Rothe method) combined with a simple “algebraic” transformation of variables, leading to a numerically stable procedure (proved explicitly by discrete energy estimates), which can be easily implemented in a BEM code to solve initial-boundary value visco-elastic problems by using the Kelvin elastostatic fundamental solution only. It is worth mentioning that no inverse Laplace transform is required here. The formulation is straightforward for both 2D and 3D problems involving unilateral frictionless contact. Although the focus is to the simplest Kelvin–Voigt rheology, a generalization to Maxwell, Boltzmann, Jeffreys, and Burgers rheologies is proposed, discussed, and implemented in the BEM code too. A few 2D and 3D initial-boundary value problems, one of them with unilateral frictionless contact, are solved numerically.

The problem of fiber–matrix debonding due to transverse loading is revisited. Predictions of the critical load for the debond onset obtained by a Cohesive Zone Model combined with contact mechanics and by a Finite Fracture Mechanics model based on a coupled stress and energy criterion are compared. Both models predict a strong nonlinear dependence of the critical load on the fiber size. A good agreement between the predictions provided by these models is found for large and medium fiber radii. However, different scaling laws for small fiber radii are noticed. A discussion of the asymptotic trends for very small and very large fiber radii is presented. Limitations of both models are also discussed. For very small fibers, it is shown that matrix plasticity can prevail over fiber–matrix debonding, leading to an upper bound for the critical load. When fiber–matrix debonding prevails over plasticity for large enough fibers, the predictions provided by the two models are still in fair good agreement.

The objective of this paper is the study of the inter-fibre/matrix failure under biaxial loading. In particular the influence at micromechanical scale of the presence of a secondary transverse load (tension or compression) perpendicular to the transverse tension nominally responsible for the failure is studied for all phases of the mechanism of damage. The Boundary Element Method is employed and Interfacial Fracture Mechanics concepts are used for the analysis of the results. The results lead to the conclusion that the presence of a compressive secondary load would promote failure whereas a tensile one would inhibit the mechanism of damage only slightly.

In this paper the experimental part of the procedure for the Generalized-Fracture-Toughness determination in multimaterial closed corners with two stress singularities is presented. The so-called Generalized-Fracture-Toughness (the critical values of the Generalized Stress Intensity Factors (GSIFs) at failure) can only be determined if each singular term can be isolated, after which the value of GSIF at the instant of failure can be evaluated. The loading configuration which allows the isolation of the stress singularities was analyzed in Part I, while the details of the testing procedure and postprocessing of the experimental results are covered in the present paper. The proposed test is a modified configuration of the Brazilian test in which the multimaterial corner tip is located at the center of the disk. A failure envelope based on the GSIFs and Generalized-Fracture-Toughness values at the corner tip is defined and proposed as a failure criterion for joints including this type of corners. Results for real adhesively bonded double-lap joints are presented showing a satisfactory agreement with the proposed failure criterion.

In the present work a general procedure for the experimental evaluation of Generalized-Fracture-Toughness values at multimaterial closed corners is defined. The lack of symmetries in the stress fields at multimaterial corners with non-isotropic materials makes it extremely difficult to define standard test procedures for Generalized-Fracture-Toughness determination. The proposed procedure is suitable for closed corners (all wedge faces being bonded) having two singular terms with real singularity exponents. The procedure begins by finding the load configurations at which one of the singular terms in the asymptotic series expansion vanishes, allowing the failure to be controlled by the other, non-vanishing, singular term. An example of a typical bimaterial corner in an adhesively bonded joint with composite materials is extensively analyzed and suitable test configurations for the experimental evaluation of the Generalized-Fracture-Toughness values K1C and K2C associated to the singular terms are defined. The experimental part of this work using a novel modified configuration of the Brazilian test will be described in a follow up paper.

In this paper, the Symmetric Galerkin Boundary Element Method for Linear Elastic Fracture Mechanics is extended to non-linear cohesive cracks propagating through homogeneous linear elastic isotropic media. The cohesive model adopted is based on the concept of free energy density per unit undeformed area. The corresponding constitutive cohesive equations present a softening branch which induces a potential instability. Thus, a suitable solution algorithm capable of following the growth of the cohesive zone is needed, and in the present work the numerical simulation is controlled by an arc-length method combined with a Newton-Raphson algorithm for the iterative solution of nonlinear equations. The Boundary ElementMethod is very attractive for modeling cohesive crack problems as all nonlinearities are located on the boundaries of linear elastic domains. Moreover a Galerkin approximation scheme, applied to a suitable symmetric boundary integral equation formulation, ensures an easy and efficient treatment of cracks in homogeneous media and an excellent convergence behavior of the numerical solution. The cohesive zone model is applied to simulate a pure mode I crack propagation in concrete. Numerical results for three-point bending test are used to check the numerical results for mode I and are compared with some numerical results obtained by FEM analysis found in the literature.

The plane strain problem of a single circular cylindrical inclusion embedded in an unbounded matrix subjected to a remote uniform uniaxial transverse tension is studied. A theoretical model for the simultaneous prediction of the initial size of a crack originated at the inclusion/matrix interface (or equivalently the initial polar angle of this crack) and of the critical remote tension required to originate this crack is developed. Isotropic and linear elastic behaviour of both materials, with the inclusion being stiffer than the matrix, is assumed. The interface is considered to be strong (providing continuity of displacements and tractions across the interface surface) and brittle. The model developed is based on the classical analytic solutions for the above-mentioned inclusion problem without and with a crack situated at the inclusion/matrix interface and a recently introduced coupled stress and energy criterion of failure by Leguillon [Eur. J. Mech. A/Solids 21 (2002) 61–72]. A new dimensionless structural parameter γ , depending on bimaterial and interface properties together with the inclusion radius a, which plays a key role in characterizing the interface crack onset, is introduced. Asymptotic behaviour of the predicted critical remote tension and the interface crack length/polar angle at the onset are characterized for small and large values of γ and a. A size effect inherent to this problem is predicted and analysed. The following asymptotic characteristics of this size effect are noteworthy: (i) for small inclusion radii a, the polar angle of the crack at onset is constant (independent of a), whereas the critical remote tension increases with decreasing a, being inversely proportional to the square root of a; (ii) for large inclusion radii a, the length of the crack at onset and the critical remote tension are approximately constant.

A numerical study of the fundamental problem of a pressurized penny-shaped crack at the interface of two dissimilar half spaces is carried out allowing for the possibility of frictionless contact between crack faces. A new, highly accurate axi-symmetric formulation of the boundary element method (BEM) for the solution of elastic contact problems is employed. The correctness and accuracy of available predictions of different kinds for several key characteristics of the solution of this problem are checked. First, comparison of the BEM results for the near-tip contact length shows a very good agreement with some existing predictions. Second, the global solution obtained by BEM is compared with existing asymptotic solutions, obtained with both the open and the frictionless contact models. BEM results show that at the closest neighborhood to the crack tip the global solution of the problem is governed by the first term of the asymptotic solution of the frictionless contact model (up to a distance of the order of a fraction of the near-tip contact length). After a small transition region, in an adjacent surrounding zone whose extent is almost independent of the near-tip contact length, the global solution of the problem is governed by the first term of the asymptotic solution of the open model. As a result of the comparison presented, the regions in which the classical fracture parameters, stress intensity factor (SIF) and energy release rate, can be accurately obtained from the global numerical solution of a crack of this kind have been determined. Third, BEM results and previous estimations show certain discrepancies with a recently published closed form solution of the near-tip contact length and the mode II SIF of the frictionless contact model. A new closed form expression of this mode II SIF, derived from the asymptotic solution of the open model, is proposed in this paper.

Composite materials reinforced by unidirectional long fibers behave macroscopically as homogeneous transversely isotropic linear elastic materials. A general, accurate and computationally efficient procedure for the evaluation of singularity exponents and singular functions characterizing singular stress fields in multimaterial corners involving this kind of material is presented in this paper. To take full advantage of the sextic Stroh formalism of anisotropic elasticity applied to this particular problem, the complete set of explicit expressions of the eigenvalues and eigenvectors of the real 6 × 6 fundamental elasticity matrix N has been deduced for all the non-degenerate and degenerate (repeated roots of the sextic Stroh equation) cases. These expressions will also facilitate further applications of the Stroh formalism to these materials. Several numerical examples of singularity analysis of multimaterial corners appearing in adhesively bonded joints and damaged cross-ply laminates of composite materials are presented.

Inter-fibre failure under compression transverse to the fibres is studied at micromechanical level. Interfacial fracture mechanics concepts, associated to both the open model and the contact model, are applied. A numerical study is performed using the boundary element method aimed at explaining the origin and evolution of the damage at micromechanical level, considered as fibre–matrix interface cracks. Assuming that the damage starts as small debonds originated by shear stresses at the position where their maximum values are reached, it has been found that the crack shows different morphologies at both tips: an open one and a closed one with a large contact zone. Then the interface crack grows unstably in mixed mode only on the open tip side until this growth changes to stable, once the crack closes at this tip, with the generation of a contact zone.

A boundary element formulation of the axi-symmetric elastic problem, including contact conditions for multi-domain problems, is presented in this paper and applied to the analysis of interface cracks. A weak formulation of the equilibrium and compatibility equations has been developed to allow non-conforming discretizations to be used in the contacting boundaries. In order to obtain a high accuracy in the solution (employing continuous linear elements), the singular terms appearing in the axi-symmetric boundary integral equations have been integrated analytically. The numerical analysis of a penny-shaped crack at the interface of two dissimilar materials and of a fibre-matrix debonding crack in the single fibre fragmentation test is presented.

The question of unique solvability of the boundary integral equation of the first kind given by the single-layer potential operator is studied in the case of plane isotropic elasticity. First, a sufficient condition of the positivity, and hence invertibility, of this operator is presented. Then, considering a scale transformation of the domain boundary, the well known formula for scaling the Robin constant in potential theory is generalized to elasticity. Subsequently, an explicit equation for evaluation of critical scales for a given boundary, when the single-layer operator fails to be invertible, is deduced. It is proved that there are either two simple critical scales or one double critical scale for any domain boundary. Numerical results, obtained applying a symmetric Galerkin boundary element code, confirm the propositions of the theory developed for both single and multi-contour boundaries.

An analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane. The concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi-material corner problems. Explicit forms of eigenequations for evaluation of singularity exponent in the case of multi-material corners are derived both for all combinations of homogeneous Neumann and Dirichlet boundary conditions at faces of open corners and for multi-material planes with singular interior points. Perfect transmission conditions at wedge interfaces are considered in both cases. It is proved that singularity exponents are real for open anisotropic multi-material corners, and a sufficient condition for the singularity exponents to be real for anisotropic multi-material planes is deduced. A case of a complex singularity exponent for an anisotropic multi-material plane is reported, apparently for the first time in potential theory. Simple expressions of eigenequations are presented first for open bi-material corners and bi-material planes and second for a crack terminating at a bi-material interface, as examples of application of the theory developed here. Analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces, and also for a special case of a crack perpendicular to a bi-material interface. A numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi-material interface is presented.

Non-conforming discretizations of the surfaces involved in the contact problem are sometimes required, due to the geometry and/or the load. A strong application of the contact conditions directly relating variables (displacements and tractions) of the nodes of the discretizations, a general approach called by the authors node-to-point contact scheme, may lead to unsatisfactory results. In this paper, a weak application of the contact conditions, by means of the principle of virtual work, is developed for boundary integral equations. The formulation is presented for two-dimensional problems without or with friction, using the Coulomb model. The modelization is made using the Boundary Element Method and the problem is solved with an incremental procedure based on a displacement scaling approach. The solution scheme proposed is applicable to any contact problem (with small or large displacements) and is validated in this paper by applying it to receding, conforming and advancing contact problems, the jumps in the contact stresses that appeared in node-to-point contact schemes, not having been found in the problems tested.

Symmetrical stress representation in the Stroh formalism for anisotropic elastic bodies is introduced and the range of its applicability is analysed. By making use of this stress representation new formulae for influence functions giving stresses in an infinite anisotropic medium subjected to a straight dislocation and a straight dislocation dipole are derived. The advantage of the new formulae is that they explicitly show the symmetrical structure of these influence functions not referred to previously. Relations of these influence functions to influence functions giving stresses and Airy stress function due to a straight wedge disclination, whose explicit expressions are also introduced, are derived. Application of these results in computation of stresses by the hypersingular and regularized Somigliana stress identities is discussed.

In the present paper we provide a method to determine the load causing delamination along an interface in a composite structure. The method is based on the elastic interface model, according to which the interface is equivalent to a bed of linear elastic springs, and on Finite Fracture Mechanics, a crack propagation criterion recently proposed for homogeneous structures. The procedure outlined is general. Details are given for the pull–push shear test. For such geometry, the failure load is obtained and compared with the estimates provided by stress concentration analysis and Linear Elastic Fracture Mechanics. It is seen that Finite Fracture Mechanics provides intermediate values. Furthermore, it is shown that the predictions provided by Finite Fracture Mechanics are almost coincident with the ones provided by the Cohesive Crack Model. As far as we are concerned with the determination of the failure load, the advantage of using Finite Fracture Mechanics with respect to the Cohesive Crack Model is evident, since a troublesome analysis of the softening taking place in the fracture process zone is not necessary. A final comparison with classical fracture criteria based on critical distances, such as the average stress criterion, concludes the paper.

By making use of a convenient decomposition of the fundamental tractions, a new formula for the C-matrix in the Somigliana identity for a three- or two-dimensional elastic isotropic body is derived. This kind of formula is more advantageous for analytical and computational C-matrix evaluations than the currently well-known formula. A general closed analytical formula of the C-matrix for the case of any finite number of tangent planes to the boundary of the body at a non-smooth boundary point, presented in the final section of this paper, demonstrates the usefulness of the new formula.

The present work deals with an evaluation of stress intensity factors (SIFs) along straight crack fronts and edges in three-dimensional isotropic elastic solids. A new numerical approach is developed for extraction, from a solution obtained by the boundary element method (BEM), of those SIFs, which are relevant for a failure assessment of mechanical components. In particular, the generalized SIFs associated to eigensolutions characterized by unbounded stresses at a neighbourhood of the crack front or a reentrant edge and also that associated to T-stress at the crack front can be extracted. The method introduced is based on a conservation integral, called H-integral, which leads to a new domain-independent integral represented by a scalar product of the SIF times some element shape function defined along the crack front or edge. For sufficiently small element lengths these weighted averages of SIFs give reasonable pointwise estimation of the SIFs. A proof of the domain integral independency, based on the bi-orthogonality of the classical two-dimensional eigensolutions associated to a corner problem, is presented. Numerical solutions of two three-dimensional problems, a crack problem and a reentrant edge problem, are presented, the accuracy and convergence of the new approach for SIF extraction being analysed.

A theoretical model for prediction of the critical load generating a crack onset at the fibre–matrix interface under a remote biaxial transverse load is presented. In particular, this work is focused on the tension dominated failure. After an abrupt onset the crack grows unstably up to achieving an arrest length. A simple plane strain model of a single circular inclusion surrounded by an unbounded matrix allows obtaining conclusions approximately valid for a dilute fibre packing. Linear isotropic elastic behaviour is assumed for both inclusion and matrix. Two classical elastic solutions for both perfectly bonded and partially debonded circular inclusions are used together with a coupled stress and energy criterion, proposed recently in the framework of finite fracture mechanics, and a phenomenological law for fracture toughness of interface cracks growing in fracture mixed mode. The obtained analytical and semi-analytical expressions make easy studying the influence of all the dimensionless parameters governing the fibre–matrix system behaviour: Dundurs elastic bimaterial constants α and β , the interface brittleness number γ and the load biaxiality parameter η . A size effect of the inclusion radius on the critical load is predicted, smaller inclusions being stronger and less dependent on the secondary load. Finally, an experimental procedure for measurement of the fibre–matrix interface fracture and strength properties is proposed.

The presence in a laminate of plies oriented at 90° with respect to the preferred direction of load generates almost immediately the appearance in these plies of cracks transverse to the load (parallel to the fibres in the lamina). These cracks reach the interface with the neighbouring ply, which in this paper will be considered oriented 0° with respect to the direction of the load. This may cause the bifurcation of the crack, which now appears propagating as a delamination crack between the two plies. The objective of this study is to characterize the stress state at the tip of both, the transverse crack in the 90° ply reaching the interface with the 0° ply, and the delamination crack for different lengths of the debonding. The analysis is performed by means of the Boundary Element Method allowing contact, without or with friction, to take place between the faces of the crack. The plies are considered as equivalent homogeneous bodies under a generalized plane strain state. The results are compared with those predicted by the open and contact models of Interfacial Fracture Mechanics. Accurate knowledge of the stress state at the neighbourhood of the tips of the cracks studied is necessary to generate failure criteria based on Fracture Mechanics parameters to predict the appearance and growth of the type of damage described.

This paper establishes that the Stroh orthogonality relations for an anisotropic body are a direct consequence of the fact that the system of equations of equilibrium is self-adjoint and positive definite. It is demonstrated that, assuming a complex representation of displacements and boundary tractions, the Betti theorem of reciprocity implies the ‘orthogonality’, and positive definiteness of strain energy implies the full rank of the normalization matrix, in the Stroh orthogonality relations. The presented proof is applicable to both the Lekhnitskii and Eshelby theories of an anisotropic body.