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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.

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.

A model to predict transverse cracking in cross-ply laminates in the presence of residual thermal stresses is developed here. This model is based on the coupled criterion of the finite fracture mechanics. This criterion has been successfully used for different materials, structures and scales to predict crack initiation. It is based on two main hypotheses: (i) crack initiation occurs as a finite-length crack onset and (ii) the crack onset requires that both stress and energy criteria are fulfilled simultaneously. The present model is developed under the generalized-plane-strain hypotheses combining the results obtained using the laminate theory and a boundary element code. The present analysis shows that the residual thermal stresses affect both the stress and the energy criteria in the form of adding a residual elastic-strain to the strain imposed by external mechanical loads. An explicit expression for this residual elastic-strain is provided. For certain composite materials as carbon/epoxy the value of this residual elastic-strain is shown to be relatively large in comparison with the nominal critical transverse strain of the material. The comparison with experiments shows that considering the residual thermal stresses using the strategy proposed here improves drastically the accuracy of the model predictions.

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.

Macro-scale delamination and micro-scale fiber–matrix debonding events may notably affect the mechanical performance of fibrous composite elements. This article presents a two-dimensional finite-el...

Article number 0033974 AIP Incluida en Conference Proceedings 2309 The prediction of failure mechanisms in fibre-reinforced composite materials is of great importance for the design of composite engineering applications. With the aim of providing a tool able to predict and explain the initiation and propagation of damage in unidirectional fiber reinforced composites, in this contribution we develop a micromechanical numerical model based on a novel hybrid approach coupling the virtual element method (VEM) and the boundary element method (BEM). The BEM is a popular numerical technique, efficient and accurate, which has been successfully applied to interfacial fracture mechanics problems of fibre-reinforced composite materials. The VEM has recently emerged as a powerful and robust technique, capable of dealing with very general polygonal/polyhedral meshes, including very distorted mesh elements, and just very recently it has also been applied to fracture and damage problems. In the present model, the BEM is used to model the fiber inclusions, which are not supposed to develop non-linear deformations or damage/crack in their interior, while the VEM, which generalizes the features of FEM, is used to model the surrounding matrix material, which can develop more complex behaviors. The implemented technique has been applied to a simple fracture problem and some promising preliminary results are shown and discussed. Ministerio de Educación, Universidad e Investigación (Italia) Proyecto DEVISU 22017ZX9X4K_006

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.

This paper explores the possibility of using the Gurtin-Murdoch theory of material surface for modeling mechanical processes in nanomaterials reinforced by two-dimensional flexible and extensible nanoplatelets. In accordance with the theory, reinforcement is modeled by a vanishing thickness prestressed membrane embedded in an isotropic elastic matrix material. The governing equations for the model are reviewed with a detailed discussion of the conditions at the membrane tips. Plane strain assumption is made and with the purpose of representing the displacements in the bulk material, a single layer elastic potential is employed, with the density representing the jump in tractions across the membrane. Expressions for the remaining elastic fields are provided in terms of complex integral representations. The case of a rectilinear membrane of finite length is considered in detail. Numerical solution for this case is based on the use of approximations that automatically incorporate the membrane-tip conditions into the resulting system of boundary integral equations. Numerical examples illustrate the influence of governing dimensionless parameters and present simulations of the local elastic fields in the materials under study. Additionally, it is shown that, in absence of surface tension, the problem considered here is related to that of a membrane type elastic inhomogeneity embedded into a homogeneous elastic medium.

A new quasi-static and energy based formulation of an interface damage model which provides interface traction-relative displacement laws like in traditional trilinear (with bilinear softening) or generally multilinear cohesive zone models frequently used by engineers is presented. This cohesive type response of the interface may represent the behaviour of a thin adhesive layer. The level of interface adhesion or damage is defined by several scalar variables suitably defined to obtain the required traction-relative displacement laws. The weak solution of the problem is sought numerically by a semi-implicit time-stepping procedure which uses recursive double minimization in displacements and damage variables separately. The symmetric Galerkin boundary-element method is applied for the spatial discretization. Sequential quadratic programming is implemented to resolve each partial minimization in the recursive scheme applied to compute the time-space discretized solutions. Sample 2D numerical examples demonstrate applicability of the proposed model.

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.

Transverse cracking in cross-ply laminates is a key failure mechanism which needs to be understood in order to predict the failure initiation in these laminates. A new 2D model has recently been presented (Garcia et al., Int J Solids Struct 51, pp. 3844–3856, 2014) for predicting the first crack onsets in cross-ply laminates under tensile loading, which is based on the coupled criterion of the Finite Fracture Mechanics (FFM). This 2D model is able to predict a strong size effect on the critical strain leading to the first crack onsets, which agrees with some experiments found in the literature. However, some earlier studies have shown that the 3D effects can be relevant in determining the origin of the size effect found in these experiments. Therefore, a 3D application of the coupled criterion, which formulates this criterion as an optimization problem, is proposed here. The objective function to be minimized, under the stress and energy criteria described previously, is the critical strain leading to the first crack onset. The finite number of variables of the optimization define the crack geometry after its onset. Since the crack geometry is parametrized, the variables in the optimization are the parameters defining the crack geometry. The objective function is evaluated by combining FRANC3D and Abaqus. As occurs in the 2D model, the results predict the strong influence of the transverse-ply thickness on the process of transverse cracking. In particular, a slightly larger critical strain is predicted for thinner laminates. In addition, the optimal crack position also varies with the thickness: a free-edge crack onset is predicted for thick laminates, whereas for thin ones, the inside and free-edge crack positions are essentially equivalent.

Certain flows denominated as amplifiers are characterized by their global linear stability while showing large linear amplifications to sustained perturbations. As the forcing amplitude increases, a strong saturation of the response appears when compared to the linear prediction. However, a predictive model that describes the saturation of the response to higher amplitudes of forcing in stable laminar flows is still missing. While an asymptotic analysis based on the weakly nonlinear theory shows qualitative agreement only for very small forcing amplitudes, the linear response to harmonic forcing around the mean flow computed by direct numerical simulations presents a good prediction of the saturation also at higher forcing amplitudes. These results suggest that the saturation process is governed by the Reynolds stress and thus motivate the introduction of a simple self-consistent model. The model consists of a decomposition of the full nonlinear Navier–Stokes equations in a mean flow equation together with a linear perturbation equation around the mean flow, which are coupled through the Reynolds stress. The full fluctuating response and the resulting Reynolds stress are approximated by the first harmonic calculated from the linear response to the forcing around the aforementioned mean flow. This closed set of coupled equations is solved in an iterative manner as partial nonlinearity is still preserved in the mean flow equation despite the assumed simplifications. The results show an accurate prediction of the response energy when compared to direct numerical simulations. The approximated coupling is strong enough to retain the main nonlinear effects of the saturation process. Hence, a simple physical picture is formalized, wherein the response modifies the mean flow through the Reynolds stress in such a way that the correct response energy is attained.

The determination of the tensile and shear strengths of homogeneous materials can be easily performed by standard tensile and shear (e.g. Iosipescu) tests. Nevertheless, when the determination of these strengths involves a bimaterial interface, the standard samples present bimaterial corner configurations at their free-edges which generate singular stress fields. In the presence of these singular stress fields, the tensile and shear stress distributions are strongly non-uniform at these edges, where failure initiates and propagates along the bimaterial interface. The apparent strength obtained from these tests is not representative of the regularized strength of the bimaterial interface. To eliminate the stress singularities, a small notch is made on one of the materials along the interface perimeter, in this study. This idea, originally proposed by Lauke and Barroso (Compos. Interface, 18:661-669, 2011) for ascertaining tensile strength, is now adapted to ascertain shear strength, using a modified geometry of the Iosipescu sample, and it has also been generalized to configurations involving composite materials. Both proposals, for the tensile and shear tests, are performed using the bimaterial configuration of a composite and an adhesive; a bimaterial interface which typically appears in adhesive joints with composites. The local notch geometry is defined using semi-analytical tools developed by the authors and numerically verified by Finite Element models. The modified bimaterial geometries, tested under tension, demonstrated a higher tensile strength. However, the modified bimaterial geometries tested in shear did not show any clear influence over the failure load with or without the notch in the particular bimaterial configuration tested in this study.

A numerical study of the onset and growth of debonds at fibre–matrix interfaces, for a fibre bundle under far field biaxial transverse loads, is presented. The behaviour of the fibre–matrix interface cracks was studied by means of a Linear Elastic–Brittle Interface Model (LEBIM). The simplified, but representative, model of an actual unidirectional lamina considered included ten fibres embedded in a matrix cell whose external dimensions were much larger than the fibre radius. The results presented extend the results obtained by the authors in previous studies using a single fibre model. The aim was, firstly to predict the failure loads (critical loads) originating the first debond onset in a small bundle of fibres and, secondly, to verify the unstable character of subsequent debond growth under transverse loads. The failure curves obtained in this simplified problem help to elucidate some aspects of the failure mechanisms of an actual composite material subjected to these kinds of loads.

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.

Keywords: Interface crack, LEBIM, FFM, BEM Abstract : The purpose of this work is to study the interaction between two fibres embedded in a large matrix and its influence on the fibre-matrix interface failure. In order to model the fibre-matrix interfaces the Linear Elastic Brittle Interface Model (LEBIM) [1] is used. In this way, an undamaged point of interface behaves as a linear-elastic spring, which has a linear relation between its tractions and relative displacements. The LEBIM follows a law which takes into account the variation of the fracture toughness with the fracture mode mixity of a crack growing along the interface between bonded solids. The failure criterion used for this model is based on a Finite Fracture Mechanics (FFM) approach [2, 3], where the stress and energy criteria are appropriately coupled. The FFM approach applied to the LEBIM is able to make the values of interface strength and fracture toughness to be independent, in opposite to the original LEBIM, where these values are somewhat restrictively tied. The present method uses a 2D Boundary Element Method (BEM) code to carry out the analysis of interface failure in the present problem [4]. The numerical results show that the distance between the fibres affects the position of the debond (interface crack) onset and the subsequent debond growth along their interfaces. Nevertheless, for increasing distances the problem tends to the single-fibre problem solution, as could be expected. References: Mantic V., Tavara L., Blazquez A., Graciani E., Paris F., A linear elastic-brittle interface model: application for the onset and propagation of a fibre-matrix interface crack under biaxial transverse loads, International Journal of Fracture 195 (2015) 15-38. Mantic V., Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion, International Journal of Solids Structures 46 (2009) 1287–304. Cornetti P., Mantic V., Carpinteri A., Finite Fracture Mechanics at elastic interfaces, International Journal of Solids and Structures 49 (2012) 1022–1032. Munoz-Reja M., Tavara L., Mantic V., Cornetti P., Crack onset and propagation at fibre–matrix elastic interfaces under biaxial loading using Finite Fracture Mechanics, Composites Part A: Applied Science and Manufacturing (in press).

In the present work, the coupled stress and energy criterion of Finite Fracture Mechanics (FFM) is applied to investigate negative T-stress effects related to mixed-mode brittle fracture of cracked elements. Only two material parameters are involved in the analysis, the tensile strength and the fracture toughness, which are independent of the mode-mixity. Below a critical T-value, the shear contribution to the strain energy release rate (ERR) starts to prevail in the mode II-dominated zone. This affects FFM results in terms of: (1) the fracture loci, with the critical mode II-stress intensity factor (SIF) never exceeding the fracture toughness; (2) the critical kinking angle and the actual crack advance (which results to be a structural parameter), both decreasing to infinitesimal quantities as mode II-loading conditions are approached. These predictions can be revised by considering a large amount of energy dissipated under mode II loading conditions and by assuming a mode-mixity dependent ERR. A discussion on experimental data for brittle and quasi-brittle materials available in the literature is included.

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.

A new linear elastic and perfectly brittle interface model for mixed mode is presented and analysed. In this model, the interface is represented by a continuous distribution of springs which simulates the presence of a thin elastic layer. The constitutive law for the continuous distribution of normal and tangential initially-linear-elastic springs takes into account possible frictionless elastic contact between adherents once a portion of the interface is broken. A perfectly brittle failure criterion is employed for the springs, which enables the study of crack onset and propagation. This interface failure criterion takes into account the variation of the interface fracture toughness with the fracture mode mixity. A unified way to represent several phenomenological both energy and stress based failure criteria is introduced. A proof relating the energy release rate and tractions at an interface point (not necessarily a crack tip point) is introduced for this interface model by adapting Irwin’s crack closure technique for the first time. The main advantages of the present interface model are its simplicity, robustness and computational efficiency, even in the presence of snap-back and snap-through instabilities, when the so-called sequentially linear (elastic) analysis is applied. This model is applied here in order to study crack onset and propagation at the fibre-matrix interface in a composite under tensile/compressive remote biaxial transverse loads. Firstly, this model is used to obtain analytical predictions about interface crack onset, while investigating a single fibre embedded in a matrix which is subjected to uniform remote transverse loads. Then, numerical results provided by a 2D boundary element analysis show that a fibre-matrix interface failure is initiated by the onset of a finite debond in the neighbourhood of the interface point where the failure criterion is first reached (under increasing proportional load); this debond further propagates along the interface in mixed mode or even, in some configurations, with the crack tip under compression. The analytical predictions of the debond onset position and associated critical load are used for several parametric studies of the influence of load biaxiality, fracture-mode sensitivity and brittleness number, and for checking the computational procedure implemented.

Composite to metal adhesively bonded joints generates critical points where geometry and material properties change abruptly. These points (multimaterial corners) are potential locations for failure initiation. There exist proposals predicting the initiation of failure at these multimaterial corners using a Fracture Mechanics approach, in which the Generalized Stress Intensity Factors (GSIFs) at the multimaterial corner control the failure initiation. The calculation of these GSIFs at anisotropic multimaterial corners involves not-straightforward calculations of the stress singularities and characteristic angular functions, numerical modeling of the joint and a careful postprocessing of the results. In this study a parametric Finite Element Analysis has been carried out allowing the generation of plots to calculate the GSIFs for unit values of the axial force, shear force and bending moment at one end of the overlap length. These results allow calculating the GSIFs at the multimaterial corners by a simple...

The model of quasistatic rate-independent evolution of a delamination at small strains in the so-called mixed mode, i.e. distinguishing opening (Mode I) from shearing (Mode II), devised in [Delamination and adhesive contact models and their mathematical analysis and numerical treatment, Chap. 9, in Mathematical Methods and Models in Composites, ed. V. Mantič (Imperial College Press, 2014), pp. 349–400; and in Quasistatic mixed-mode delamination model, Discrete Contin. Dynam. Syst. Ser. S 6 (2013) 591–610], is rigorously analyzed in the context of a concept of stress-driven local solutions. The model has separately convex stored energy and is associative, namely the one-homogeneous potential of dissipative forces driving the delamination depends only on rates of internal parameters. An efficient fractional-step-type semi-implicit discretization in time is shown to converge to (specific, stress-driven like) local solutions that may approximately obey the maximum-dissipation principle. Making still a spatial discretization, this convergence as well as relevancy of such solution concept are demonstrated on a nontrivial two-dimensional example.

One of the reasons for the application of composites to mobile structures is their capacity to absorb energy under impact. As their components have a very reduced interval of deformation till breakage, the mechanism of absorption of energy cannot be based on plasticity but on the capacity of generating free surfaces inside the material, the debonding between fibers and matrix being the most direct materialization of the mechanism of absortion of energy in composites. In this chapter the onset and propagation of damage in between fiber and matrix in a fibrous composite, particularly under transverse loading is studied and characterized. The tools to apply are interfacial fracture mechanics (IFM), linear elastic brittle interface model, finite fracture mechanics (FFM), and cohesive zone model (CZM), combined with a numerical tool, in this case the boundary element method (BEM). Different loading situations are analyzed, single tension and compression, bi-dimensional loading as well as curing effects. Finally single fiber, two fibers, and multi-fiber configurations are considered in this chapter, the key being the evolution of the energy released by the debonding crack representing damage in between fiber and matrix. Numerical predictions are compared with experimental evidence to support the adequacy of the approaches followed.

A unified methodology to solve problems of frictionless unilateral contact as well as adhesive contact between linear elastic solids is presented. This methodology is based on energetic principles and is casted to a minimization problem of the total potential energy. Appropriate boundary integral forms of the energy are defined and the quadratic problem form of the contact problem is proposed. The problem is solved by the collocation boundary element method (BEM). To solve the quadratic problem two algorithms are developed, both being variants of the well-known conjugate gradient algorithm. The difference between them is given by the explicit construction or not of the quadratic-problem matrix. This matrix has the same physical meaning as the stiffness matrix commonly used in the context of the finite element method (FEM). Both symmetric and non-symmetric formulations of this matrix are presented and discussed, showing that the non-symmetric one provides more accurate results. The present procedure, in addition to its interest by itself, can also be extended to problems where dissipative phenomena take place such as friction, damage and plasticity. Essential steps of the numerical implementation are briefly presented and the numerical solutions of some standard problemsof frictionless contact are given and compared to those obtained by other well-known BEM and FEM procedures for contact problems. Junta de Andalucía TEP-4051 European Social Fund TEP-4051 Spanish Ministry of Science and Innovation MAT2009-14022 Spanish Ministry of Economy and Competitiveness MAT2012-37387 MAT2015-71036-P European Regional Development Fund MAT2012-37387 MAT2015-71036-P Spanish Ministry of Education AP2009-3968 Stavros Niarchos Foundation

A theoretical model is developed to predict debond onset in spherical-particle-reinforced composites subjected to uniaxial tension. The coupled stress and incremental energy criterion of the finite fracture mechanics is applied employing an axisymmetric boundary element method code, which is highly accurate in the analysis of interface cracks in presence of contact. The model enables to predict the value of the critical remote load and the spherical angle of the debond after the onset as a function of the particle, matrix and interface properties and particle size. Results predicted are compared with experimental observations found in the literature.

This work presents three examples of standard test configurations, involving composite materials, where the presence of stress singularities induced by adhesive joints causes premature failures leading to underestimated strength values. Slight modifications of the local geometry, removing or reducing, in these critical points, the order of the stress singularities, have shown to give higher experimental failure loads, with almost double failure loads in the most striking cases. The three examples analyzed in the present work are: a) the tensile and shear strength determination of a bimaterial interface, b) the off-axis tension test, for the intralaminar shear strength of unidirectional long fibre composite materials and c) the compression test of thick composite laminates. In two of the three cases, the premature failure occurs at the corner where the tab (which is necessary for the grip jaw faces of the testing machine) is bonded to the composite laminate, while in the third one the failure occurs at the bimaterial interface of the two materials, all of them involving an adhesively bonded joint and corner configurations with stress singularities. With the use of a semi-analytical tool, developed by the authors, to calculate the order of stress singularities, slight and very local geometrical modifications have been successfully carried out to eliminate the stress singularity configuration. After the modifications, higher failure loads have been obtained in the tests carried out. The major outcome of the present work is the role that the singular stresses play in the premature failure of standard tests involving adhesive bonding. A correlation between the removal of the singularity stress configuration in critical parts of the samples and a higher experimental failure load, in all studied cases, has been clearly shown.

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.

At present, the progress in the failure criteria of unidirectional reinforced laminates requires the study of the evolution of the mechanisms of damage at micromechanical level. The objective of this paper is the study of the influence (at micromechanical scale) of a secondary transverse load (tension or compression), perpendicular to the transverse compression nominally responsible for the failure, on the inter-fibre failure under compression. The Boundary Elements Method and Interfacial Fracture Mechanics concepts are employed. It is shown that a secondary tension accelerates the generation of failure whereas a secondary compression inhibits it.

A quasi-static rate-independent model of delamination of linearly elastic bodies at small strains, sensitive to mode of delamination, using interfacial damage and interfacial plasticity as two internal parameters, is further developed with the aim to extract representations typically employed in engineering interface-models, i.e. fracture envelope and fracture energy dependence on the mode mixity, which are suitable for the model fitting to experimental data. Moreover, two concepts of solutions are implemented: globally stable energy-conserving solutions or stress-driven maximally-dissipative local solutions, arising by the fully implicit or by a semi-implicit time-stepping procedures, respectively, both yielding numerically stable and convergent time-discretizations. Spatial discretization is performed by the symmetric Galerkin boundary-element method (SGBEM). Alternating quadratic programming is implemented to cope with, respectively, global or local, energy-minimizations in the computation of the time-discretized solutions. Sample 2D numerical examples document applicability of the model as well as efficiency of the SGBEM numerical implementation and facilitate comparison of the two mentioned solution concepts.

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.

Even well-established non-linear FEM codes may present convergence issues when solving highly unstable problems, e.g. fracture mechanics problems, especially in the presence of a pronounced snap-back behaviour. In composites, at lamina level, the debonds occurring between the fibres and matrix, lead to multiple-crack configurations. Each individual debond growth is an unstable process. Then, when multiple debonds occur the problem becomes highly unstable. Moreover, experimental evidence shows that a sequence of multiple debonds originates a macro-crack that extends over the whole lamina thickness. In this article a new Python based numerical tool implementing a Sequentially Linear Analysis (SLA) procedure, which is able to call the FEM software Abaqus , is described and tested. The Linear Elastic Brittle Interface Model (LEBIM) is also included in Abaqus by means of a UMAT subroutine. Numerical results for a representative 100-fibre model show that the tool is able to adequately model simultaneous onset and propagation of multiple debonds resulting in a completely unstable process composed by a series of instability events. Eventually, these numerical results together with some previous experimental results for transverse failure loads for unidirectional carbon fibre-epoxy matrix plies allow us to estimate the tensile strength and critical fracture energy of the fibre-matrix interface by a simple inverse analysis .

This work studies how the direction of crack propagation affects the translaminar fracture toughness in composites. In particular, this work focuses on propagation mode-I toughness. The two extreme cases are analyzed: crack propagating in the direction either parallel or perpendicular to the fibers (always without breaking fibers). To evaluate the effect of the direction of crack propagation, an experimental campaign was carried out by two new cracked Three Point Bending (TPB) specimens specially developed for this purpose. Using these specimens, the fracture toughness was measured for different directions of the crack propagation. In addition, a set of standard Double Cantilever Beam (DCB) specimens (corresponding to the longitudinal interlaminar fracture toughness) were tested for the sake of comparison. The results show that the translaminar fracture toughness is higher when the crack grows parallel to the fiber direction than when it grows perpendicular. In both cases, the fracture toughness is higher than that measured for the longitudinal interlaminar growth using DCB specimens. The physical reasons for these experimental results are discussed.

A method for the prediction of crack onset and propagation along an interface in a fibre-reinforced composite system is devel- oped. The method is based on the Linear Elastic-(Perfectly) Brittle Interface Model (LEBIM) and a coupled stress and energy criterion of the Finite Fracture Mechanics (FFM). Models similar to LEBIM, also called weak or imperfect interface models, are frequently applied to describe the behaviour of adhesively bonded joints. In the present paper the interface is modelled by a con- tinuous distribution of linear elastic-brittle springs where the normal and tangential stresses across the undamaged interface are, respectively, proportional to the relative normal and tangential displacements. The interface failure criterion applied takes into account the variation of the fracture toughness with the fracture mode mixity. In the present work a 2D collocational boundary element method (BEM) code is used to analyse debond onset and propagation at a fibre embedded in a matrix subjected to remote uniaxial transverse tension. One of the advantages of the method proposed here by applying the FFM criterion to the LEBIM is that we are able to make independent the two governing parameters of the model (critical stress and fracture toughness of the interface), which are related by an equation in the original LEBIM formulation. It seems that the predictions provided by FFM with LEBIM are quite similar to those obtained by the well-known Cohesive Crack Model (CCM). The advantage of the present method with respect to the CCM is evident, since a troublesome nonlinear analysis due to the softening can be avoided.

The present authors and coworkers have recently developed a new physically and mathematically well justified and efficient approach for interface crack onset and propagation, implemented in finite and boundary element method (FEM and BEM) codes and applied to several problems of engineering interest. This approach borrows concepts from damage mechanics, such as the damage variable, from plasticity, as kinematic hardening, and from interface fracture mechanics, as fracture energy dependent on the fracture mode mixity. The computational implementation is based on recursive minimizations of a total energy functional, which can be computed by FEM and BEM. Global or specific local minimizations lead to different solution types, the energetic and stress driven solutions, respectively. In opposite to the associative models, where interface plasticity is explicitly taken into account by a plastic slip variable, applied to mixed-mode crack propagation problems by the present authors so far, it seems that non-associative models have the advantage of ending up at easier (e.g. smooth instead of non-smooth) and reduced (e.g. elimination of plasticity variable) minimization problems. An implementation of such a non-associative model in a collocation BEM code is presented and applied to an engineering problem of delamination.