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The dynamics of a “peeling front” or an elastic line is studied under creep (constant load) conditions. Our experiments show in most cases an exponential dependence of the creep velocity on the inverse force (mass) applied. In particular, the dynamical correlations of the avalanche activity are discussed here. We compare various avalanche statistics to those of a line with non-local elasticity, and study various measures of the experimental avalanche-avalanche and temporal correlations such as the autocorrelation function of the released energy and aftershock activity. From all these we conclude, that internal avalanche dynamics seems to follow “line depinning”-like behavior, in rough agreement with the depinning model. Meanwhile, the correlations reveal subtle complications not implied by depinning theory. Moreover, we also show how these results can be understood from a geophysical point of view. [ABSTRACT FROM AUTHOR]

The promising phase-field method has been intensively studied for crack approximation in brittle materials. The realistic representation of material degradation at a fully evolved crack is still one of the main challenges. Several energy split formulations have been postulated to describe the crack evolution physically. A recent approach based on the concept of representative crack elements (RCE) in Storm et al. (The concept of representative crack elements (RCE) for phase-field fracture: anisotropic elasticity and thermo-elasticity. Int J Numer Methods Eng 121:779–805, 2020) introduces a variational framework to derive the kinematically consistent material degradation. The realistic material degradation is further tested using the self-consistency condition, which is particularly compared to a discrete crack model. This work extends the brittle RCE phase-field modeling towards rate-dependent fracture evolution in a viscoelastic continuum. The novelty of this paper is taking internal variables due to viscoelasticity into account to determine the crack deformation state. Meanwhile, a transient extension from Storm et al. (The concept of representative crack elements (RCE) for phase-field fracture: anisotropic elasticity and thermo-elasticity. Int J Numer Methods Eng 121:779–805, 2020) is also considered. The model is derived thermodynamic-consistently and implemented into the FE framework. Several representative numerical examples are investigated, and consequently, the according findings and potential perspectives are discussed to close this paper. [ABSTRACT FROM AUTHOR]

Biomaterials use a hierarchical structure to optimize their self-healing behavior, for instance. However, the behavior may be constrained under different mechanical environments. In this paper, a system is suggested that the mechanical environment is modeled as a spring connected in series with the fiber bundle material. For the spring, the elastic behavior with stiffness is obeyed while, for the fiber bundle material, the nonlinear elastic constitutive relation is obeyed according to the Weibull distribution and the Daniels’ theory. Relying on the principle of total potential, the applicability condition is proposed for the system and the critical stiffness is thus derived for the spring. The applicability of hierarchical fiber bundle materials is finally investigated. The results show that the hierarchy can significantly change the critical stiffness, and hence demonstrates quite different applicability to a given mechanical environment. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of slow crack growth in heterogeneous media. The crack is subjected to arbitrary pressure distribution on the crack surface. The problem relates to construction of the so-called equilibrium crack. For such a crack, stress intensity factors are equal to the material fracture toughness at each point of the crack contour. The crack shape and size depend on spatial distributions of the elastic properties and fracture toughness of the medium, and the type of loading. In the paper, attention is focused on the case of layered elastic media when a planar crack propagates orthogonally to the layers. The problem is reduced to a system of surface integral equations for the crack opening vector and volume integral equations for stresses in the medium. For discretization of these equations, a regular node grid and Gaussian approximating functions are used. For iterative solution of the discretized equations, fast Fourier transform technique is employed. An iteration process is proposed for the construction of the crack shape in the process of crack growth. Examples of crack evolution for various properties of medium and types of loading are presented. [ABSTRACT FROM AUTHOR]

This paper examines the mechanics of mode-III defect initiation and quasi-static growth in a composite consisting of N layers separated by nonuniform nonlinear cohesive surfaces. The exact analysis is based on the elasticity solution to the problem of a single layer of finite width and thickness subjected to arbitrary, nonuniform but equilibrated shear tractions on top and bottom surfaces. The formulation leads to a system of integral equations governing rigid body layer translations and cohesive surface slip fields (tangential jump discontinuities across layer interfaces). Cohesive surfaces are modelled by traction-slip relations (exponential, modified exponential) characterized by a cohesive strength, a force length and possibly friction parameters. Symmetric center, edge and array defects are modeled by cohesive strength functions that vary with surface coordinate. Infinitesimal strain equilibrium solutions are sought by eigenfunction approximation of the solution of the governing integral equations. Solutions indicate that quasi-static defect initiation/propagation occur under increasing applied shear traction. For small values of force length, brittle behavior occurs corresponding to sharp crack growth. At larger values of force length, ductile response occurs similar to linear "spring" cohesive surfaces. Both behaviors ultimately cause abrupt failure of the surface. Results for the stiff, strong cohesive surface with a center defect under a small applied traction agree well with static crack solutions taken from the literature. Detailed results are obtained for (i) an array of symmetric collinear defects accounting for defect–defect interaction, (ii) a symmetric defect in a bilayer with nonuniform material properties/thicknesses, (iii) a center defect accounting for decohesion and friction. [ABSTRACT FROM AUTHOR]

This paper evaluates the validity of a new test methodology for measuring shear fracture toughness (mode II) of high density polyethylene (HDPE). The methodology adopts Iosipescu test for the shear loading, and determines the toughness based on the essential work of fracture (EWF) concept. The results show that even under the Iosipescu loading, tensile deformation (mode I) is still involved in the fracture process, possibly due to the significant work hardening that HDPE develops during the plastic deformation. The study found that the mode II fracture toughness can be determined through data analysis using double linear regression, i.e., by extrapolating specific work of fracture to zero ligament length and zero ligament thickness. The paper demonstrates that the new test methodology can be used to evaluate mode II fracture toughness of ductile polymers like HDPE in which significant work-hardening may be involved in the fracture process. The paper also provides quantitative comparison of the fracture toughness for HDPE in mode II with its mode I counterpart. [ABSTRACT FROM AUTHOR]

This paper explores the efficacy of employing local damage models, normally applied to ductile material systems manufactured by subtractive techniques, to additively manufactured laboratory specimens. While these specimens were ductile and metallic, their additive character (i.e. porosity and surface roughness) could have had potential to activate multiple life-limiting failure paths, thus obfuscating failure prediction. Herein, two damage models are considered and compared: the micromechanical Gurson–Tvergaard–Needleman model and a Crack Band model of the strain-based, phenomenological genre. Simulations used to calibrate elastic and plastic material properties and predict damage in a novel, non-standard specimen were quasi-static, explicit. Both damage models proved capable in resolving the experimentally-observed failure path and associated loading conditions. The analyses described herein were made as part of the Third Sandia Fracture Challenge. [ABSTRACT FROM AUTHOR]

This paper studies the dynamic characteristics of the debonding mechanism in beams strengthened with FRP. The paper formulates a special finite element for the dynamic analysis of the layered beam and looks into the parameters that govern the dynamic phenomenon. The finite element is based on an extended high order theory that incorporates a rich 2D elasticity based stress and displacement fields in the adhesive layer and allows to consider debonding at the two physical interfaces of that layer through cohesive interfaces. These concepts are combined with the basic form of inertial forces to formulate a multi-layered finite element that utilizes the versatility of the FE method and avoids the need to mesh through the thickness of each layer. The computational framework is then used to study the dynamic debonding process and to examine the impact of a series of geometrical, elastic, mechanical, and physical parameters. Among other findings, the investigation reveals the potentially negative impact of stiffening the FRP layer on the dynamic failure, the sensitivity to the properties of the adhesive, and the interesting insensitivity to the loading rate (within the examined range). Finally, the sensitivity of the response to the properties of the cohesive interfaces and the idea of characterizing those interfaces based on dynamic observations are discussed. [ABSTRACT FROM AUTHOR]

This paper provides a solution for a crack stiffened by elliptic layer in antiplane elasticity. The crack is embedded in an elliptic region and stiffened by a confocally elliptic layer. The whole medium is composed of three portions, the cracked elliptic plate, the confocally elliptic layer and the infinite matrix. The remote loading is applied. The cracked elliptic plate and the infinite matrix have the same shear modulus of elasticity. The stiffening elliptic layer has a higher shear modulus of elasticity. By using the complex variable and the continuity conditions along interfaces, the problem can be solved. One numerical example with different sizes and properties of materials is given to show the effect of the stiffening layer. [ABSTRACT FROM AUTHOR]

Fatigue crack propagation is by far the most important failure mechanism. Often cracks under low-cycle fatigue conditions and, especially, short fatigue cracks cannot be treated with the conventional stress intensity range $$\Delta K$$ -concept, since linear elastic fracture mechanics is not valid. For such cases, Dowling and Begley (ASTM STP 590:82-103, ) proposed to use the experimental cyclic $$J$$ -integral $$\Delta J^{\exp }$$ for the assessment of the fatigue crack growth rate. However, severe doubts exist concerning the application of $$\Delta J^{\exp }$$ . The reason is that, like the conventional $$J$$ -integral, $$\Delta J^{\exp }$$ presumes deformation theory of plasticity and, therefore, problems appear due to the strongly non-proportional loading conditions during cyclic loading. The theory of configurational forces enables the derivation of the $$J$$ -integral independent of the constitutive relations of the material. The $$J$$ -integral for incremental theory of plasticity, $$J^{\mathrm{ep}}$$ , has the physical meaning of a true driving force term and is potentially applicable for the description of cyclically loaded cracks, however, it is path dependent. The current paper aims to investigate the application of $$J^{\mathrm{ep}}$$ for the assessment of the crack driving force in cyclically loaded elastic-plastic materials. The properties of $$J^{\mathrm{ep}}$$ are worked out for a stationary crack in a compact tension specimen under cyclic Mode I loading and large-scale yielding conditions. Different load ratios, between pure tension- and tension-compression loading, are considered. The results provide a new basis for the application of the $$J$$ -integral concept for cyclic loading conditions in cases where linear elastic fracture mechanics is not applicable. It is shown that the application of the experimental cyclic $$J$$ -integral $$\Delta J^{\exp }$$ is physically appropriate, if certain conditions are observed. [ABSTRACT FROM AUTHOR]

In the case of sufficiently brittle material the use of stress intensity factor as a fracture parameter alone is well justified within the Linear-Elastic Fracture Mechanics. This is because the singular stress field associated with the stress intensity factor is dominant near the crack tip. However, there are numerous experimental evidences that the critical stress intensity factor to cause fracture initiation (or fracture toughness) can be affected by the specimen geometry as well as loading conditions. To address this issue a number of twoparameter criteria have been proposed in the past, which often utilise non-singular terms of the classical asymptotic expansion of the stress field near the crack tip. Therefore, there is a problem of the selection of an appropriate parameter in addition to the stress intensity factor, which could account for various effects induced by the specimen geometry and loading on initiation of brittle fracture. This short paper demonstrates that brittle fracture conditions can be successfully predicted with various two-parameter criteria, and there are no clear advantages in the use of T-stress as the additional parameter in fracture criterion, in comparison with the next non-singular term, A1, of the asymptotic expansion. [ABSTRACT FROM AUTHOR]

The crack propagation simulation is still an open problem in the mechanical simulation field. In the present work this problem is analyzed using a version of truss-like Discrete Element Method, that here we called DEM. This method has been used with success in several applications in solid mechanical problems where the simulation of fracture and fragmentation is relevant. The formulation of DEM explaining the way the process of rupture could be simulated in consistent form is showed. Also are described details about how the dynamical fracto-mechanical stress intensity factors are computed. The main aim of this paper is to show the ability of this method in simulating fracture and crack propagation in solids, for this, three examples with different levels of complexity are analyzed. The obtained results are presented in terms of the variation of dynamic stress intensity factor in the fracture process, the stress map and geometric configuration on different steps in the simulation of the fracture process, the crack speed and the energetic balance during all the process. These results are compared with experimental and numerical results obtained by other researchers and published in recognized scientific papers. Final commentaries about the performance of the version of lattice model considered are carried out. [ABSTRACT FROM AUTHOR]

The paper provides statistical analysis on the effect of mutual positions of individual contacts on the overall resistance and incremental elastic stiffness of a cluster of contacts. We consider an example of a cluster with 76 contact spots of the same radii. The regular lattice of the contacts is compared with the perturbed ones (random clusters). The constriction resistance is determined as a sum of self resistance of the individual contacts and interactions between them. It is shown that the effect of perturbations is very small and, therefore, the mean distance between the centers of individual contacts can be used to estimate the overall cluster resistance. Using elasticity-conductivity cross-property connections, this result is transferred to the incremental elastic stiffness of the cluster. Thus, we established a correspondence between cluster of regular structure and cluster of randomly located contacts. The geometrical parameters governing elastic compliance and spreading resistance of the clusters are number of contact spots and the average distance between individual contacts. [ABSTRACT FROM AUTHOR]

Cohesive zone models are explored in order to study cleavage fracture in adhesive bonded joints. A mode I cohesive model is defined which correlates the tensile traction and the displacement jump (crack faces opening) along the fracture process zone. In order to determine the traction-separation relation, the main fracture parameters, namely the cohesive strength and the fracture energy, as well as its shape, must be specified. However, owing to the difficulties associated to the direct measurement of the fracture parameters, very often they are obtained by comparing a measured fracture property with numerical predictions based on an idealized traction separation relation. Likewise in this paper the cohesive strength of an adhesive layer sandwiched between elastic substrates is adjusted to achieve a match between simulations and experiments. To this aim, the fracture energy and the load-displacement curve are adopted as input in the simulations. In order to assess whether or not the shape of the cohesive model may have an influence on the results, three representative cohesive zone models have been investigated, i.e. exponential, bilinear and trapezoidal. A good agreement between experiments and simulations has been generally observed. However, a slight difference in predicting the loads for damage onset has been found using the different cohesive models. [ABSTRACT FROM AUTHOR]

On the physical nature, most crack tips are not ideally sharp but have a small curvature radius. Both surface energy and crack-root curvature affect the stress and displacement fields in the vicinity of the crack tip. In the present paper, a numerical method, which incorporates the effect of surface elasticity into the finite element method, is employed to study the surface effects on the mode-II crack tip fields. It is found that when the curvature radius of the crack root decreases to micro-/nanometers, surface elasticity has a significant influence on the stresses near the crack tip. For a mode-II crack, surface effects alter both the magnitude and position of the maximum stresses, as is different from a mode-I crack, in which case only the stress magnitude is influence by surface stresses. [ABSTRACT FROM AUTHOR]

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

This paper describes a theoretical model and related computational methods for examining the influence of inhomogeneous material properties on the crack driving force in elastic and elastic-plastic materials. Following the configurational forces approach, the crack tip shielding or anti-shielding due to smooth (e.g. graded layer) and discontinuous (e.g. bimaterial interface) distributions in material properties are derived. Computational post-processing methods are described to evaluate these inhomogeneity effects. The utility of the theoretical model and computational methods is demonstrated by examining a bimaterial interface perpendicular to a crack in elastic and elastic-plastic compact tension specimens. [ABSTRACT FROM AUTHOR]

The paper gives some closed form expressions for the strain energy averaged in a finite size volume surrounding the root of blunt V-shaped notches under Mode I loading. The control volume, reminiscent of Neuber’s concept of elementary structural volumes, is thought of as dependent on the ultimate tensile strength and the fracture toughness K IC in the case of brittle or quasi-brittle materials subjected to static loads. Expressions for strain energy density under plane strain conditions and Mode I loading have been derived from an analytical frame recently reported in the literature, which matches Williams and Creager-Paris’ solutions in the particular cases of plates weakened by sharp V-notches or blunt cracks (U-notches), respectively. In order to validate a local-strain-energy based approach, a well-documented set of experimental data recently reported in this journal by Gómez and Elices has been used. Data refer to blunt and sharp V-specimens of PMMA subjected to static tension loads and characterised by a large variability of notch root radius (from 0 to 4.0 mm) and notch angle (from 0° to 150°). Critical loads obtained experimentally have been compared with the theoretical ones, estimated here by keeping constant the mean value of the strain energy in a well-defined small size volume. [ABSTRACT FROM AUTHOR]

This paper considers the non-axisymmetric three-dimensional problem of a penny-shaped crack with permeable electric conditions imposed on the crack surfaces, subjected to a pair of point normal forces applied symmetrically with respect to the crack plane. The crack is embedded in an infinite transversely isotropic piezoelectric body with the crack face perpendicular to the axis of material symmetry. Applying the symmetry of the problem under consideration then leads to a mixed-mixed boundary value problem of a half-space, for which potential theory method is employed for the purpose of analysis. The cases of equal eigenvalues are also discussed. Although the treatment differs from that for an impermeable crack reported in literature, the resulting governing equation still has a familiar structure. For the case of a point force, exact expressions for the full-space electro-elastic field are derived in terms of elementary functions with explicit stress and electric displacement intensity factors presented. The exact solution for a uniform loading is also given. [ABSTRACT FROM AUTHOR]

A 2-dimensional, elastic-plastic finite element model has been developed to simulate plasticity induced crack closure in slanted and deflected cracks growing outside the small scale yielding (SSY) regime. The finite element model allows for contact between deformable surfaces to capture the complex contact interaction between the crack faces. Coulomb's friction law has been used to model friction between the crack faces and has been incorporated in the finite element model. This paper examines the mode I and mode II behavior of slanted cracks subjected to remote mode I, constant amplitude cyclic loading. Two possible types of mode II crack face interaction have been identified: (a) complete slip in mode II before mode I opening and, (b) mode I crack opening before the crack faces undergo mode II displacements. Both types of interactions were observed in slanted cracks. The finite element study also reveals a clear dependence of mode I and mode II crack opening levels for a slanted crack on R ratio and maximum stress, Smax/δ0. The crack opening levels for a slanted crack are found to be significantly higher than the stable opening values for a straight crack growing in pure mode I. The mode I and mode II crack opening levels are also found to depend on the friction between the crack faces. A four-fold increase in friction coefficient resulted in almost 50% increase in normalized mode I and mode II opening values. This paper also describes the effect of crack deflection on closure. Deflection of a fatigue crack from 45° inclination to pure mode I caused a decrease in mode I opening level, but, an increase in mode II opening level. This difference in opening behavior is attributed to the transition of the nature of crack interaction from `complete slip before opening' to `opening in mode I before mode II shear offset'. Final stable opening levels for a deflected crack are found to be close to the stable value for straight cracks. [ABSTRACT FROM AUTHOR]

A criterion to predict crack onset at a sharp notch in homogeneous brittle materials has been presented in a previous paper of one of the authors. It is reviewed and improved herein. It fulfils both the energy and the strength criteria and takes an Irwin-like form involving the generalized intensity factor of the singularity governing the elastic behaviour in the vicinity of a notch tip. The prediction agrees fairly well with the experiments although it slightly underestimates the experimental measures. A cause of this discrepancy can be that a small notch tip radius blunts the sharp corner. It is analysed in this paper by means of matched asymptotics involving 2 small parameters: a micro-crack increment length and the notch tip radius. A correction is brought to the initial prediction and a better agreement is obtained with experiments on PMMA notched specimens. Experiments performed on a stiffer material (Alumina/Zirconia) show that it is less sensitive to small notch tip radii. A remaining small discrepancy between experiments and predictions can be due to some non linear behaviour of the materials near the notch tip. In addition, without new developments, the method allows to determine the stress intensity factor at the tip of a short crack emanating from a sharp or a rounded v-notch. [ABSTRACT FROM AUTHOR]

This paper presents an efficient numerical procedure for analysis of arbitrary shaped, planar cracks in a three-dimensional, linear elastic infinite medium by taking the surface elasticity into account. The concept of surface stresses via Gurtin-Murdoch surface theory is integrated with the classical theory of linear elasticity to form a mathematical model capable of simulating cracks under general loading conditions and nano-scale influences. The key governing equations for the bulk material are established in terms of weakly singular displacement and traction boundary integral equations involving only unknowns on the crack face, whereas that for a zero-thickness material layer on each crack face is established in a weak form following the weighted residual technique. The final governing equations for the bulk material and the two crack-face layers are fully coupled via the continuity condition along the interface. The solution of the resulting coupled system is determined numerically by the coupling of a standard finite element procedure and a symmetric Galerkin boundary element method. Owing to the weakly singular feature of all involved boundary integrals, C $$^{0}$$ basis functions are adopted everywhere in the approximation of crack-face data and only the special quadrature for evaluating nearly singular and weakly singular double surface integrals is required. Once the implemented numerical scheme is fully tested with existing reference solutions, it is then applied to investigate the role and influence of surface parameters such as the residual surface tension and the in-plane modulus on the near-front field and the size dependency of predicted solutions. Results of several examples under various scenarios are reported not only to demonstrate the capability and robustness of the proposed method but also to indicate the significance of the surface stresses in enhancing the near-surface stiffness and reducing stresses in a local region ahead of the crack front. [ABSTRACT FROM AUTHOR]

The in-plane classical dislocation-based linear elastic fracture mechanics analysis is extended to the case of strain gradient elasticity. Nonsingular stress and smooth-closure crack profiles are derived. As in the classical treatment, the crack is represented by a distribution of climb edge dislocations (for Mode I) or glide edge dislocations (for mode II). These distributions are determined through the solution of corresponding integral equations based on variationally consistent boundary conditions. An incompatible framework is used and the nonsingular full-field plastic distortion tensor components are calculated. Numerical results and related graphs are provided illustrating the nonsingular behaviour of the stress/strain components and the smooth cusp-like closure of the crack faces at the crack tip. The work provides an alternative approach to celebrated 'Barenblatt's treatment' of cracks, without the introduction of a cohesive zone and related to intermolecular forces ahead of the physical crack tip. It also supplements a recent paper by the authors in which the mode III crack, represented by an array of screw dislocations, was solved within the present gradient elasticity framework. [ABSTRACT FROM AUTHOR]

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

In this paper a comprehensive investigation is carried out with regard to the state of the stress and strain in the neighbourhood of notches in bodies subjected to an anti-plane state of shear stress, within the context of a strain limiting theory of elasticity. Taking advantage of a unified analytical framework, the strain-limiting theory of elasticity is used to determine the full stress and strain field close to a pointed or radiused notch with any notch opening angle. An extensive discussion is provided that highlights the main features of stress and strain distributions, and the implications of the new theory for fracture assessments. In particular, it is proved that the obtained stress and strain solution predicts finite strains at the notch tip and allows the intensity of the stress field to be written as a function of the elastic Notch Stress Intensity Factor $$K_{3}$$ , as in the case of conventional linearized elasticity theory. This makes the strain limiting elasticity an excellent vehicle for justifying theoretically a K based-approach to the fracture of brittle elastic solids, within the context of a self consistent theory, unlike the classical linearized theory that predicts singularities for the strain at crack tips. [ABSTRACT FROM AUTHOR]

This paper discusses dynamic crack growth and arrest in an elastic double cantilever beam (DCB) specimen, simulated using the Bernoulli-Euler beam theory. The specimen is made from two different materials. The section of interest, where the dynamic crack growth takes place, is made from a material, the fracture energy of which will be denoted $$2\Gamma _1 $$ . The initial crack grows slowly in a starter material with a fracture energy $$2\Gamma _{0} \;(\Gamma _{0} >\Gamma _1 )$$ , while opposed displacements on both arms of the specimen are continuously increased. As the crack reaches the material interface at $$\ell =\ell _c $$ , the loading displacement is instantly suspended, and the crack suddenly propagates through the test zone, until it stops at $$\ell =\ell _A $$ . During this process, the energy $$2\Gamma _1 (\ell _A -\ell _c )$$ is dissipated. The beam motion and the fracture process during the fast crack growth stage are investigated, based on the balance energy associated to the Griffith criterion. The motion equations are approximated using a modal decomposition up to order $$N$$ of the beam deflection (the analysis has been performed up to $$N=10$$ but in most cases $$N=5$$ is sufficient to obtain an accurate solution). This process leads to a set of N second order differential equations whose unknowns are the mode amplitudes and their derivatives, and another equation the unknowns of which are the current crack length $$\ell (t)$$ , velocity $$\dot{\ell }(t)$$ and acceleration $$\ddot{\ell }(t)$$ . To demonstrate the accuracy of this method, it is first tested on a one dimensional peeling stretched film problem, with an insignificant bending energy. An exact solution exists, accurately approximated by the modal solution. The method is then applied to the DCB specimen described above. Despite the rather crude nature of the Bernoulli-Euler model, the results crack kinematics, and specially the arrest length, correspond well to those obtained by the combined use of finite elements and cohesive zone models, even for a few modes. Moreover, for the basic mode $$N=0$$ (also referred to as Mott solution), even if the crack kinematics is not accurately reproduced, the prediction of the crack arrest length remains correct for moderate ratios. Some parametric studies about the beam geometry and the initial crack velocity are performed. The relative crack arrest $$\ell _A /\ell _c $$ appears to be almost insensitive to these parameters, and is mainly governed by the ratio $$R=\Gamma _{0} /\Gamma _1 $$ which is the key parameter to predict the crack arrest. [ABSTRACT FROM AUTHOR]

In the present paper, the roles of a linear spring interface and a membrane shell interface on the mechanical behaviour of a bimaterial are investigated by using a simplified matrix-inhomogeneity elastic system. The system consists of a circular inhomogeneity embedded in an unbounded matrix and is in shear deformation, either coming from a remote uniform stress applied in the matrix, or from a uniform shear transformation in the inhomogeneity. It was shown that, if a perfect interface of the system is transformed to a spring-type one, while the system is stressed by the above loadings, the change in the total potential energy is negative. The opposite occurs, if a membrane shell interface is formed in the stressed system. The elastic energies stored in the system constituents are determined by considering the interface as a separate phase of the system. [ABSTRACT FROM AUTHOR]

This paper focuses on the characterization of mixed-mode fracture parameters through use of two formalisms based on Crack Relative Displacement Factors and Stress Intensity Factors, respectively. The evaluation of Crack Relative Displacement Factors is based on a kinematic approach that integrates the experimental displacement field measured by a digital image correlation method. In parallel with this step, the stress intensity factor is calculated from a finite element analysis. The coupling between these two approaches allows for the identification of fracture parameters in terms of an energy release rate without any prior knowledge of material elastic properties. Depending on the mixed-mode configuration, the proportion of the energy release rate corresponding to opening and shear modes can be calculated. Moreover, the proposed formalism allows determining, in addition to fracture parameters, the local elastic properties in terms of reduced elastic compliance directly from the test sample. Experimental protocols are carried out using a Single-Edge notched specimen made from a rigid Polyvinyl Chloride polymer loaded at various mixed-mode ratio values. [ABSTRACT FROM AUTHOR]

The main purpose of this study is to investigate the efficacy and usefulness of a class of recently proposed models that could be reasonable candidates for describing the response of brittle elastic materials. The class of models that are considered allows for a non-linear relationship between the linearized elastic strain and the Cauchy stress, and this allows one to describe situations wherein the stress increases while the strain yet remains small. Thus one would be in a position to model the response of brittle elastic bodies in the neighborhood of the tips of cracks and notches. In this paper we study the behavior of such models in a plate with a V-notch subject to a state of anti-plane stress. This geometrical simplification enables us to characterize the governing equation for the problem by means of the Airy stress function, though the constitutive relation is a non-linear relation between the linearized strain and the stress. We study the problem numerically by appealing to the finite element method. We find that the numerical solutions are stable. We are able to provide some information regarding the nature of the solution near the tip of the V-notch. In particular we find stress concentration in the vicinity of the singularity. [ABSTRACT FROM AUTHOR]

In this paper using polynomial approximation of the dispersion relations near the center of the first Brillouin zone, we construct a continuum model with internal length scale starting from the discrete valence force field model. As a consequence, and in agreement with a recent result by Maranganti and Sharma, we obtain an explicit estimate of the internal lengths which is ≃0.1Å. [ABSTRACT FROM AUTHOR]

The virtual crack closure technique (VCCT) is a well-established method for computing the energy release rate when analysing fracture problems via the finite element method. For mixed-mode fracture problems, the VCCT is also commonly used to partition the fracture modes, i.e. to determine the energy release rate contributions related to the three classical fracture modes. A perhaps little known fact, however, is that in some circumstances the standard VCCT predicts physically inconsistent, negative values for the modal contributions to the energy release rate. Focusing on I/II mixed-mode problems, this paper presents a revised VCCT which furnishes a physically consistent partitioning of fracture modes by associating the mode I and II contributions to the amounts of work done in a suitably defined two-step process of closure of the virtually extended crack. Deeper investigation pinpoints the origins of the physically inconsistent predictions of the standard VCCT in the lack of energetic orthogonality between the crack-tip force components used to compute the modal contributions. Further insight into the problem is offered by a geometric construction, which introduces the 'ellipse of crack-tip flexibility'. In closing, the phenomena of contact, interpenetration, and friction between the crack surfaces are briefly touched upon. [ABSTRACT FROM AUTHOR]

The Notch Stress Intensity Factors (NSIFs) quantify the intensities of the asymptotic linear elastic stress distributions of sharp (zero radius) V-shaped notches. When the notch tip radius is different from zero, the singular sharp-notch field diverges from the rounded-notch solution in the close neighborhood of the notch tip. Nevertheless the NSIFs might continue to be parameters governing fracture if the notch root radius is small enough. Otherwise they can be seen simply as stress field parameters useful in quantifying the stress distributions ahead of the specific notch. Taking advantage of some analytical formulations which are able to describe stress distributions ahead of parabolic, hyperbolic and V-shaped notches with end holes, the paper discusses the form and the significance of the NSIFs with reference to in-plane shear loading, considering explicitly the role played by the notch opening angle and the notch tip radius. These parameters quantify the stress redistribution due to the root radius with respect to the sharp notch case to which they should naturally tend for decreasing values of the notch radius. [ABSTRACT FROM AUTHOR]

This paper is the first part of an extended program to develop a theory of fracture in the context of strain-limiting theories of elasticity. This program exploits a novel approach to modeling the mechanical response of elastic, that is non-dissipative, materials through implicit constitutive relations. The particular class of models studied here can also be viewed as arising from an explicit theory in which the displacement gradient is specified to be a nonlinear function of stress. This modeling construct generalizes the classical Cauchy and Green theories of elasticity which are included as special cases. It was conjectured that special forms of these implicit theories that limit strains to physically realistic maximum levels even for arbitrarily large stresses would be ideal for modeling fracture by offering a modeling paradigm that avoids the crack-tip strain singularities characteristic of classical fracture theories. The simplest fracture setting in which to explore this conjecture is anti-plane shear. It is demonstrated herein that for a specific choice of strain-limiting elasticity theory, crack-tip strains do indeed remain bounded. Moreover, the theory predicts a bounded stress field in the neighborhood of a crack-tip and a cusp-shaped opening displacement. The results confirm the conjecture that use of a strain limiting explicit theory in which the displacement gradient is given as a function of stress for modeling the bulk constitutive behavior obviates the necessity of introducing ad hoc modeling constructs such as crack-tip cohesive or process zones in order to correct the unphysical stress and strain singularities predicted by classical linear elastic fracture mechanics. [ABSTRACT FROM AUTHOR]

This paper studies the importance of the cohesive zone in the modelling of a fluid driven fracture under plain strain conditions. The fracture is driven by pumping of an incompressible viscous fluid at the fracture inlet. Rock deformation is modeled for linear elastic and poroelastic solids. Fluid flow in the fracture is modeled by lubrication theory. The cohesive zone approach is used as the fracture propagation criterion. Finite element analysis was used to compute the solution for the crack length, the fracture opening and propagation pressure as a function of the time and distance from the wellbore. It is demonstrated that the crack profiles and the propagation pressures are larger in the case of elastic-softening cohesive model compared to the results of the rigid-softening cohesive model for both elastic and poroelastic cohesive solids. It is found that the results are affected by the slope of the loading branch of the cohesive model and they are nearly unaffected from the exact form of the softening branch. Furthermore, the size of the process zone, the fracture geometry and the propagation pressure increase with increasing confining stresses. These results may explain partially the discrepancies in net-pressures between field measurements and conventional model predictions. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the asymptotic problem of a body of infinite extent with a notch (re-entrant corner) under remotely applied plane-strain or anti-plane shear loadings. The problem is formulated within the framework of the Toupin-Mindlin theory of dipolar gradient elasticity. This generalized continuum theory is appropriate to model the response of materials with microstructure. A linear version of the theory results by considering a linear isotropic expression for the strain-energy density that depends on strain- gradient terms, in addition to the standard strain terms appearing in classical elasticity. Through this formulation, a microstructural material constant c is introduced, in addition to the standard Lamé constants ( λ, μ). The faces of the notch are considered to be traction-free and a boundary-layer approach is followed. The boundary value problem is attacked with the asymptotic Knein-Williams technique. Our analysis leads to an eigenvalue problem, which, along with the restriction of a bounded strain energy, provides the asymptotic fields. The cases of a crack and a half-space are analyzed in detail as limit cases of the general notch (infinite wedge) problem. The results show significant departure from the predictions of the standard fracture mechanics. [ABSTRACT FROM AUTHOR]

It is often suggested in literature to use change in the effective elastic properties (stiffness loss) as an indicator of reduction of strength due to defects such as cracks and pores. We argue that the key parameter is not the reduction of the average (over the specimen) stiffness but its local minimal values caused by formation of defect clusters. These defect clusters can be identified by the emergence of spatial gradients of elastic stiffness on a smaller scale. A convenient tool of detecting these gradients is provided by the elasticity-conductivity connection: the electric conductance gradient is usually easier to measure than the stiffness gradient. This concept is supported by computational and experimental results reported in two accompanying papers published in the present issue (by Sevostianov et al and Caiulo and Kachanov). [ABSTRACT FROM AUTHOR]

In this paper the integral equation approach is developed to describe elastic-damaging materials. An isotropic damage model is implemented to study nonlinear structural problems involving localisation phenomena. Especially for the cases that exhibit stress or strain concentrations, an integral approach can be recommended. Besides, the technique is able to represent well high gradients of stress/strain. The governing integral equations are discretised by using quadratic isoparametric elements on the boundary and quadratic continuous/discontinuous cells in the zone where the nonlinear phenomenon occurs. Two numerical examples are presented to show the physical correctness and efficiency of the proposed procedure. The results are compared with the local theory and they turn out to be free of the spurious sensitivity to cell mesh refinement. [ABSTRACT FROM AUTHOR]

Bi-material interfaces are studied with cracks that end perpendicular to the interface. As is well-known, singularities in the stresses appear when classical elasticity is used. Moreover, the nature of the singularity depends on the difference in elastic constants of the two materials. In this paper, the gradient elasticity theory of Aifantis is used to remove these singularities. This is demonstrated for a range of ratios between the two Young’s moduli. [ABSTRACT FROM AUTHOR]

In many practical applications, piezoelectric ceramics are bonded to non-piezoelectric and insulating isotropic elastic materials such as polymer. Since the conventional form of Stroh’s formulation, on which almost all of existing works on interfacial cracks in piezoelectric media have been based, breaks down or becomes complicated for isotropic elastic materials, many solutions available in the literature cannot be directly applied to interfacial cracks between a piezoelectric material and an isotropic elastic material. The present paper is devoted to a hybrid complex-variable method which combines the Stroh’s method of piezoelectric materials with the well-known Muskhelishvili’s method of isotropic elastic materials. This method is illustrated in detail for an insulating interfacial crack between a piezoelectric half-plane and an isotropic elastic half-plane, although interface cracks between piezoelectric and isotropic elastic conductor can be analyzed in a similar way. The solution obtained generally exhibits oscillatory singularity, in agreement with a previous known result based on the Stroh’s formulation. A simple explicit condition is obtained for the bimaterial constants under which the oscillatory singularity disappears. It is expected that the hybrid complex-variable method could more conveniently handle other possible complications (such as a hole or an inclusion) inside the isotropic elastic material, because it offers explicit solutions of a single complex variable rather than several different complex-variables associated with the Stroh’s formulation. [ABSTRACT FROM AUTHOR]

This paper presents a fast and accurate solution for crack interaction problems in infinite- and half- plane solids. The new solution is based on the method of complex potentials developed by Muskhelishvili for the analysis of plane linear elasticity, and it is formulated through three steps. First, the problem is decomposed into a set of basic problems, and for each sub-problem, there is only one crack in the solid. Next, after a crack-dependent conformal mapping, the modified complex potentials associated with the sub-problems are expanded into Laurent’s series with unknown coefficients, which in turn provides a mechanism to exactly implement in the form of Fourier series the boundary condition in each sub-problem. Finally, taking into account the crack interaction via a perturbation approach, an iterative algorithm based on fast Fourier transforms (FFT) is developed to solve the unknown Fourier coefficients, and the solution of the whole problem is readily obtained with the superposition of the complex potentials in each sub-problem. The performance of the proposed method is fully investigated by comparing with benchmark results in the literatures, and superb accuracy and efficiency is observed in all situations including patterns where cracks are closely spaced. Also, the new method is able to cope with interactions among a large number of cracks, and this capability is demonstrated by a calculation of effective moduli of an elastic solid with thousands of randomly-spaced cracks. [ABSTRACT FROM AUTHOR]

The paper investigates the buckling mode of failure of long interfacial cracks subjected to shear loading. A criterion which describes the transition from fracture to buckling instability failure modes is derived. [ABSTRACT FROM AUTHOR]

This paper investigates interfacial crack tip stress fields and the J-integral for bi-materials with plastic hardening mismatch via detailed elastic-plastic finite element analyses. For small scale yielding, the modified boundary layer formulation with the elastic T-stress is employed. For fully plastic yielding, plane strain single-edge- cracked specimens under pure bending are considered. Interfacial crack tip stress fields are explained by modified Prandtl slip-line fields. It is found that, for bi-materials consisting of two elastic-plastic materials, increasing plastic hardening mismatch increases both crack-tip stress constraint in the lower hardening material and the J-contribution there. The implication of asymmetric J-integral in bi-materials is also discussed. [ABSTRACT FROM AUTHOR]

The present paper proposes ductile failure criteria in terms of true fracture strain (the equivalent strain to fracture) as a function of the stress triaxiality (defined by the ratio of the hydrostatic stress to the equivalent stress) for the API X65 steel. To determine the stress-modified fracture strain, smooth and notched tensile bars with four different notch radii are tested, from which true fracture strains are determined as a function of the notch radius. Then detailed elastic–plastic, large strain finite element analyses are performed to estimate variations of stress triaxiality in the tensile bars, which leads to true fracture strains as a function of the stress triaxiality, by combining them with experimental results. Two different failure criteria are proposed, one based on local stress and strain information at the site where failure initiation is likely to take place, and the other based on averaged stress and strain information over the ligament where ductile fracture is expected. As a case study, ligament failures of API X65 pipes with a gouge are predicted and compared with experimental data. [ABSTRACT FROM AUTHOR]

The theory of photoelasticity is used to study analytically the effects of T-stress on the fringe patterns around the crack tip in mode II crack specimens. The locus of an isochromatic fringe determined by taking into account the T-stress is compared with the locus of a fringe with no T-stress. It is shown for mode II cracks that in the presence of T-stress, the fringe loops are neither symmetric nor continuous. Asymmetric and discontinuous fringe patterns predicted in this paper are consistent with the experimental results observed previously in photoelasticity tests. [ABSTRACT FROM AUTHOR]

The fracture strength of PMMA is studied using columns with rectangular cross- section that contain a central hole and are subjected to axial compression. Samples with geometries such that the width is 2, 3, 4, and 5 times the diameter of the hole are investigated. Cracks are initiated by introducing sharp notches in the axial direction at the crowns of the hole in each sample. The sample is then subjected to axial compression such that the axial cracks grow in a stable manner until a critical compression is attained, after which the cracks extend rapidly in the sample but do not reach its ends. To predict the experimental results, we have first sought to use several published models, but discovered that none would produce results in accord with our data, as they seem not to address the essential features of our beam-column experiments. We have then developed a simple model that consists of a suitably modified version of an elasticity solution based on a short-crack in an infinite plate, combined with a beam-column solution for a long-crack. This model yields results in good accord with our data and also gives reasonable values for the fracture toughness of the material. A comparative discussion of several published models is presented at the end of the paper, pointing out that linear elasticity is not an appropriate tool for solving the present beam-column problem. [ABSTRACT FROM AUTHOR]

In this paper, a cooled composite interface corner consisting of two bonded dissimilar materials is considered as a plane problem. With the complex variable method, the thermal residual stress field is studied analytically. It is found that the regular stress term possesses the singularity either of ln r or ln2 r. The exact expressions for the corresponding singular stress field are presented. [ABSTRACT FROM AUTHOR]

Due to the enormous difference in the scales involved in correlating the macroscopic properties with the micro- and nano-physical mechanisms of carbon nanotube-reinforced composites, multiscale mechanics analysis is of considerable interest. A hybrid atomistic/continuum mechanics method is established in the present paper to study the deformation and fracture behaviors of carbon nanotubes (CNTs) in composites. The unit cell containing a CNT embedded in a matrix is divided in three regions, which are simulated by the atomic-potential method, the continuum method based on the modified Cauchy–Born rule, and the classical continuum mechanics, respectively. The effect of CNT interaction is taken into account via the Mori–Tanaka effective field method of micromechanics. This method not only can predict the formation of Stone–Wales (5-7-7-5) defects, but also simulate the subsequent deformation and fracture process of CNTs. It is found that the critical strain of defect nucleation in a CNT is sensitive to its chiral angle but not to its diameter. The critical strain of Stone–Wales defect formation of zigzag CNTs is nearly twice that of armchair CNTs. Due to the constraint effect of matrix, the CNTs embedded in a composite are easier to fracture in comparison with those not embedded. With the increase in the Young’s modulus of the matrix, the critical breaking strain of CNTs decreases. [ABSTRACT FROM AUTHOR]

Weld bead geometry cannot, by its nature, be precisely defined. Parameters such as bead shape and toe radius vary from joint to joint even in well-controlled manufacturing operations. In the present paper the weld toe region is modelled as a sharp, zero radius, V-shaped notch and the intensity of asymptotic stress distributions obeying Williams’ solution are quantified by means of the Notch Stress Intensity Factors (NSIFs). When the constancy of the angle included between weld flanks and main plates is assured and the angle is large enough to make mode II contribution non-singular, mode I NSIF can be directly used to summarise the fatigue strength of welded joints having very different geometry. By using a large amount of experimental data taken from the literature and related to a V-notch angle of 135°, two NSIF-based bands are reported for steel and aluminium welded joints under a nominal load ratio about equal to zero. A third band is reported for steel welded joints with failures originated from the weld roots, where the lack of penetration zone is treated as a crack-like notch and units for NSIFs are the same as conventional SIF used in LEFM. Afterwards, in order to overcome the problem related to the variability of the V-notch opening angle, the synthesis is made by simply using a scalar quantity, i.e. the mean value of the strain energy averaged in the structural volume surrounding the notch tips. This energy is given in closed form on the basis of the relevant NSIFs for modes I and II and the radius RC of the averaging zone is carefully identified with reference to conventional arc welding processes. RC for welded joints made of steel and aluminium considered here is 0.28 mm and 0.12 mm, respectively. Different values of RC might characterise welded joints obtained from high-power processes, in particular from automated laser beam welding. The local-energy based criterion is applied to steel welded joints under prevailing mode I (with failures both at the weld root and toe) and to aluminium welded joints under mode I and mixed load modes (with mode II contribution prevailing on that ascribable to mode I). Surprising, the mean value of ΔW related to the two groups of welded materials was found practically coincident at 2 million cycles. More than 750 fatigue data have been considered in the analyses reported herein. [ABSTRACT FROM AUTHOR]

In the present work, we predict contribution of a partially debonded circular inhomogeneity into the material overall elastic compliance. Debonding at the matrix/inclusion boundary is modeled as interfacial arc cracks. The change in the elastic compliance caused by interface cracking is estimated through the accompanying energy change that is related to the mode I and mode II stress intensity factors at the crack tips. The sum of the crack compliance and the inhomogeneity compliance (with perfect bonding) gives the total compliance of the debonded inhomogeneity. The latter is obtained in terms of the material properties and crack length. Additional analysis shows that the replacement of an interface crack with a crack in a homogenized medium is an inadequate approach when seeking approximate solutions. The paper also provides guidelines how to determine properties of a fictitious perfectly bonded orthotropic inhomogeneity that has the same contribution into the material compliance as the debonded isotropic one. This problem is of practical importance when modeling damage accumulation in composite materials by means of homogenization schemes. [ABSTRACT FROM AUTHOR]

The solution for the elastic three-phase circular inclusion problem plays a fundamental role in many practical and theoretical applications. In particular, it offers the fundamental solution for the generalized self-consistent method in the mechanics of composites materials. In this paper, a general method is presented for evaluating the interaction between a pre-existing radial matrix crack and a three-phase circular inclusion. The bonding at the inclusion-interphase interface is considered to be imperfect with the assumption that the interface imperfections are constant. On the remaining boundary, that being the interphase-matrix interface, the bonding is considered to be perfect. Using complex variable techniques, we derive series representations for the corresponding stress functions inside the inclusion, in the interphase layer and the surrounding matrix. The governing boundary value problem is then formulated in such a way that these stress distributions simultaneously satisfy the traction free condition along the crack face, the imperfect interface conditions and the prescribed asymptotic loading conditions. Stress intensity factor (SIF) calculations are performed at the crack tips for different material property combinations, imperfect interface conditions and crack positions. The results illustrate convincingly the role of an interphase layer as well as the effects of an imperfect interface on crack behavior. For instance, when the interphase layer is softer than the inclusion and matrix, the results show that the radial matrix crack will propagate from the nearby crack tip regardless of the level of the imperfect (spring-layer) interface parameter. In comparison, when the interphase layer is stiffer than the inclusion and matrix, the interphase layer will shield the crack from effects of the imperfect (spring-layer) interface. Hence, these results provide a quantitative description of the interaction problem between a three-phase inclusion with interface imperfections and a radial matrix crack. [ABSTRACT FROM AUTHOR]