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The peridynamic (PD) theory is a nonlocal reformulation of mechanics with various advantages over common approaches, mainly local continuum mechanics and molecular dynamics (MD). PD theory can capture phenomena at different dimensions, including nanoscale. However, limited studies have been performed by this theory in nanoscale, which have generally focused on the feasibility and accuracy of using PD in atomic-scale modeling. In the present study, based on the ordinary state-based peridynamic method, we investigate the fracture of pre-cracked single layer graphene sheets (SLGSs) under uniaxial tension. By simulating the exact atomic model of graphene, the failure strain and crack growth pattern in the zigzag and armchair directions in PD are compared with MD. We show that by considering some restrictions, these two methods have a good consistency with each other. Afterward, we study two different coarse-grained PD models and demonstrate this method can simulate the failure of graphene with acceptable accuracy. A significant reduction in simulation cost is an excellent point of the PD compared to the MD simulation model. Under these conditions, a massive atomic model with several million atoms can be easily simulated. [ABSTRACT FROM AUTHOR]

A new approach of multi-scale simulation via coupling of boundary element and finite element is proposed to predict fracture toughness and crack propagation of a single layer graphene sheet. In this simulation a molecular dynamics-based nonlinear beam element is developed for the atomistic model near the crack tip, whereas a special two-dimensional boundary element is employed for the continuum model in the remaining field of the cracked specimen. The material and section properties required in the nonlinear beam element are estimated through the equivalence between the potential energy of molecular dynamics and the elastic strain energy of continuum mechanics. With the estimated properties of beam element, the material properties of boundary element are further estimated by applying loads on the specimen of graphene, which is formed by a hexagonal lattice of carbon atoms. Coupling the nonlinear beam elements by boundary elements with the local-global concept of multi-scale modeling, a vast of computational time can be saved. The accuracy of near tip stresses obtained in our simulation remedies the inaccuracy of linear elastic fracture mechanics, and can be used for the prediction of atomic bond-breaking, which leads to crack propagation. The associated critical load can then be applied in the continuum model for the cracked specimen to predict mode I and mode II fracture toughness of graphene. The obtained values are then verified by the published results measured or predicted by the other methods. By varying the size of cracks and orientation of applied loads, several interesting phenomena have been observed and discussed in this paper. [ABSTRACT FROM AUTHOR]

Peridynamics is a nonlocal formulation of solid mechanics capable of unguided modelling of crack initiation, propagation and fracture. Peridynamics is based upon integral equations, thereby avoiding spatial derivatives, which are not defined at discontinuities, such as crack surfaces. Rice's J-contour integral is a firmly established expression in classic continuum solid mechanics, used as a fracture characterizing parameter for both linear and nonlinear elastic materials. A corresponding nonlocal J-integral has previously been derived for peridynamic modelling, which is based on the calculation of a set of displacement derivatives and force interactions associated with the contour of the integral. In this paper, we present an alternative calculation of the classical linear elastic J-integral for use in peridynamics, by writing Rice's J-integral as a function entirely of displacement derivatives. The accuracy of the proposed J-integral on displacement formulation is investigated by applying it to the exact analytical displacement solution of an infinite specimen with a central crack and comparing the exact analytical expression of its J-integral K I 2 / E . Further comparison with a well-known peridynamic crack problem shows very good agreement. The suggested method is computationally efficient and further allows testing of the accuracy of a peridynamic model as such. [ABSTRACT FROM AUTHOR]

A continuum mechanics theory of deformable solids is formulated to account for large deformations, nonlinear elasticity, inelastic deformation mechanisms, microstructure changes, and time dependent fields, i.e., dynamics. The theory incorporates notions from Finsler differential geometry, and it provides a diffuse interface description of surfaces associated with microstructure. Mechanisms include phase transitions and inelastic shearing, with phase boundaries and shear planes the associated surfaces. A director or internal state vector of pseudo-Finsler space is viewed as an order parameter. Newly derived in the present work are the governing equations for dynamics, including kinematic relations, balances of momentum and energy, and evolution law(s) for the internal state. Also derived are jump conditions pertinent to shock loading. Metric tensors and volume can vary isotropically with internal state via a conformal transformation. The dynamic theory is applied to describe shock loading of ceramic crystals of boron carbide, accounting for inelastic mechanisms of shear accommodation and densification upon amorphization under high pressure loading. Analytical predictions incorporating the pseudo-Finsler metric demonstrate remarkable agreement with experimental data, without parameter fitting. Additional solutions suggest that dynamic shear strength could be improved significantly in boron-based ceramics by increasing surface energy, decreasing inelastic shear accommodation in softened amorphous bands, and to a lesser extent, by increasing the energy barrier for phase transformation. [ABSTRACT FROM AUTHOR]

This paper shows a new approach to estimate the critical traction for Mode I crack opening before crack growth by numerical simulation. For quasi-static loading, Linear Elastic Fracture Mechanics predicts the critical traction before crack growth. To simulate the crack growth, we used bond-based peridynamics, a non-local generalization of continuum mechanics. We discretize the peridynamics equation of motion with a collocation by space approach, the so-called EMU nodal discretization. As the constitutive law, we employ the improved prototype micro brittle material model. This bond-based material model is verified by the Young's modulus from classical theory for a homogeneous deformation for different quadrature rules. For the EMU-ND we studied the behavior for different ratios of the horizon and nodal spacing to gain a robust value for a large variety of materials. To access this wide range of materials, we applied sparse grids, a technique to build high-dimensional surrogate models. Sparse grids significantly reduce the number of simulation runs compared to a full grid approach and keep up a similar approximation accuracy. For the validation of the quasi-static loading process, we show that the critical traction is independent of the material density for most material parameters. The bond-based IPMB model with EMU nodal discretization seems very robust for the ratio $${\delta }/{\varDelta X}=3$$ for a wide range of materials, if an error of 5 % is acceptable. [ABSTRACT FROM AUTHOR]

A continuum model of ductile fracture initiation is presented which is based on consideration of the work done on the system sufficient to cause release of the binding energy necessary for crack formation. The fracture criterion is posed for plastic materials with no pre-existing crack as a critical state reached when the energy release rate of the bulk system is balanced by the energy release rate associated with the fractured medium. Two fracture modes are considered which are commonly addressed in fracture mechanics: mode I crack opening perpendicular to the fracture plane and mode II shear rupture tangential to the fracture plane. A study of the use of the criterion for a plastic material with power law hardening is examined and compared with published experimental data and empirical criteria. [ABSTRACT FROM AUTHOR]

In the cohesive surface model cohesive tractions are transmitted across a two-dimensional surface, which is embedded in a three-dimensional continuum. The relevant kinematic quantities are the local crack opening displacement and the crack sliding displacement, but there is no kinematic quantity that represents the stretching of the fracture plane. As a consequence, in-plane stresses are absent, and fracture phenomena as splitting cracks in concrete and masonry, or crazing in polymers, which are governed by stress triaxiality, cannot be represented properly. In this paper we extend the cohesive surface model to include in-plane kinematic quantities. Since the full strain tensor is now available, a three-dimensional stress state can be computed in a straightforward manner. The cohesive band model is regarded as a subgrid scale fracture model, which has a small, yet finite thickness at the subgrid scale, but can be considered as having a zero thickness in the discretisation method that is used at the macroscopic scale. The standard cohesive surface formulation is obtained when the cohesive band width goes to zero. In principle, any discretisation method that can capture a discontinuity can be used, but partition-of-unity based finite element methods and isogeometric finite element analysis seem to have an advantage since they can naturally incorporate the continuum mechanics. When using interface finite elements, traction oscillations that can occur prior to the opening of a cohesive crack, persist for the cohesive band model. Example calculations show that Poisson contraction influences the results, since there is a coupling between the crack opening and the in-plane normal strain in the cohesive band. This coupling holds promise for capturing a variety of fracture phenomena, such as delamination buckling and splitting cracks, that are difficult, if not impossible, to describe within a conventional cohesive surface model. [ABSTRACT FROM AUTHOR]

A recently formulated triaxiality dependent cohesive model for plane strain is implemented and its versatility is tested in simulation of ductile fracture of mild steel at different states of stress. The triaxiality dependent model was implemented as linear displacement formulation based elements whose constitutive behaviour was dependent on the stress-state of the neighbouring continuum element. By comparing the experimental data and predictions of corresponding plane strain simulations, the model parameters are estimated. The model is shown to be effective in reproducing characteristic features of the macroscopic response of both pre-cracked as well as geometries without a preexisting nominal defect. Since the model parameters are held constant for simulations at different stress-states, they are effectively material constants. [ABSTRACT FROM AUTHOR]

The modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations of complex crack topologies including branching. This drawback can be overcome by a diffusive crack modeling based on the introduction of a crack phase field as proposed in Miehe et al. (Comput Methods Appl Mech Eng 19:2765-2778, ; Int J Numer Meth Eng 83:1273-1311, ), Hofacker and Miehe (Int J Numer Meth Eng, ). In this work, we summarize basic ingredients of a thermodynamically consistent, variational-based model of diffusive crack propagation under quasi-static and dynamic conditions. It is shown that all coupled field equations, in particular the balance of momentum and the gradient-type evolution equation for the crack phase field, follow as the Euler equations of a mixed rate-type variational principle that includes the fracture driving force as the mixed field variable. This principle makes the proposed formulation extremely compact and provides a perfect basis for the finite element implementation. We then introduce a local history field that contains a maximum energetic crack source obtained in the deformation history. It drives the evolution of the crack phase field. This allows for the construction of an extremely robust operator split scheme that updates in a typical time step the history field, the crack phase field and finally the displacement field. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples, which show the evolution of complex crack patterns under dynamic loading. [ABSTRACT FROM AUTHOR]

This note discusses the peridynamic horizon (the nonlocal region around a material point), its role, and practical use in modelling. The objective is to eliminate some misunderstandings and misconceptions regarding the peridynamic horizon. An example of crack branching in a nominally brittle material (homalite) is addressed and we show that crack branching takes place without wave interaction. We explain under what conditions the crack propagation speed depends on the horizon size and the role of incident stress waves on this speed. [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]

We apply the HotQC method of Kulkarni et al. (J Mech Phys Solids 56:1417-1449, ) to the study of quasistatic void growth in copper single crystals at finite temperature under triaxial expansion. The void is strained to 30% deformation at initial temperatures and nominal strain rates ranging from 150 to 600 K and from 2.5 × 10 to 2.5 × 10s, respectively. The interatomic potential used in the calculations is Johnson's Embedded-Atom Method potential Johnson (Phys Rev B 37:3924-3931, ). The computed pressure versus volumetric strain is in close agreement with that obtained using molecular dynamics, which suggests that inertia effects are not dominant for the void size and conditions considered. Upon the attainment of a critical or cavitation strain of the order of 20%, dislocations are abruptly and profusely emitted from the void and the rate of growth of the void increases precipitously. Prior to cavitation, the crystal cools down due to the thermoelastic effect. Following cavitation dislocation emission causes rapid local heating in the vicinity of the void, which in turn sets up a temperature gradient and results in the conduction of heat away from the void. The cavitation pressure is found to be relatively temperature-insensitive at low temperatures and decreases markedly beyond a transition temperature of the order of 250 K. [ABSTRACT FROM AUTHOR]

A new technique is presented to study fracture in nanomaterials by coupling quantum mechanics (QM) and continuum mechanics (CM). A key new feature of this method is that broken bonds are identified by a sharp decrease in electron density at the bond midpoint in the QM model. As fracture occurs, the crack tip position and crack path are updated from the broken bonds in the QM model. At each step in the simulation, the QM model is centered on the crack tip to adaptively follow the path. This adaptivity makes it possible to trace paths with complicated geometries. The method is applied to study the propagation of cracks in graphene which are initially perpendicular to zigzag and armchair edges. The simulations demonstrate that the growth of zigzag cracks is self-similar whereas armchair cracks advance in an irregular manner. The critical stress intensity factors for graphene were found to be 4.21 MPa $${\sqrt {\rm m}}$$ for zigzag cracks and 3.71 MPa $${\sqrt{\rm m}}$$ for armchair cracks, which is about 10% of that for steel. [ABSTRACT FROM AUTHOR]

The fidelity of the peridynamic theory in predicting fracture is investigated through a comparative study. Peridynamic predictions for fracture propagation paths and speeds are compared against various experimental observations. Furthermore, these predictions are compared to the previous predictions from extended finite elements (XFEM) and the cohesive zone model (CZM). Three different fracture experiments are modeled using peridynamics: two experimental benchmark dynamic fracture problems and one experimental crack growth study involving the impact of a matrix plate with a stiff embedded inclusion. In all cases, it is found that the peridynamic simulations capture fracture paths, including branching and microbranching that are in agreement with experimental observations. Crack speeds computed from the peridynamic simulation are on the same order as those of XFEM and CZM simulations. It is concluded that the peridynamic theory is a suitable analysis method for dynamic fracture problems involving multiple cracks with complex branching patterns. [ABSTRACT FROM AUTHOR]

perturbative approach applied to prestressed elastic solids exhibiting strain softening (of the true-stress/true-strain response) provides an explanation to the common observation that shear bands formed within ductile and quasi-brittle materials tend to grow rectilinearly under Mode II and that slip surfaces tend to nucleate and grow inside them. The effects of an ‘obstacle’ (a rigid line inclusion) within a shear band are also analyzed and quantified in terms of inhibition of shear band growth. [ABSTRACT FROM AUTHOR]

The method of simultaneous dual integral equations is used for obtaining the exact analytical solution for the weight function for an elliptical crack embedded in an infinite elastic solid. We show that the solution is unique and can be reduced to the known solutions for a number of particular cases. [ABSTRACT FROM AUTHOR]

In fracture and fragmentation research the technique of electromagnetic forming, which uses electromagnetic (Lorentz) body forces to shape metallic parts, is finding significant use due to the high velocity, high strain rate loading it can impart without contact on workpieces. The same process is also becoming increasingly relevant for manufacturing processes in sheet metal forming, where this technique offers several advantages: speed, repeatability, non-contact loading, reduced springback and considerable ductility increase in several metals. Current modeling techniques for these coupled electromagnetic and thermomechanical processes are not based on coupled variational principles that can simultaneously account for electromagnetic and mechanical effects. Typically, separate solutions to the electromagnetic (Maxwell) and motion (Newton) equations are combined in staggered or lock-step methods, sequentially solving the mechanical and electromagnetic problems. To address this issue, Thomas and Triantafyllidis (J Mech Phys Solids 57:1391–1416, 2009) have recently introduced a fully coupled Lagrangian (reference configuration) variational principle, involving the magnetic field potential and the displacement field as independent variables. The corresponding Euler-Lagrange equations are Maxwell’s and Newton’s equations in the reference configuration under the eddy current approximation. This novel approach is used here to simulate free expansion experiments of AA6063-T6 aluminum tubes. A viscoplastic constitutive model, developed independently by the authors (Thomas et al. Acta Mater 55:2863–2873, 2007) for necking experiments in tubes of the same aluminum alloy, is used in the simulations. The measured electric currents and tube deformation—the latter obtained by Photon Doppler Velocimetry—show reasonably good agreement with the corresponding simulations, which are obtained using a variational integration numerical scheme that results in an efficient staggered solution algorithm. [ABSTRACT FROM AUTHOR]

In the paper, the results of numerical failure analysis of plain concrete beams loaded by impact three-point bending load are presented and discussed. The theoretical framework for the numerical analysis is continuum mechanics and irreversible thermodynamics. The spatial discretization is performed by the finite element method using update Lagrange formulation. Green–Lagrange stain tensor is used as a strain measure. To account for cracking and damage of concrete, the beam is modeled by the rate sensitive microplane model with the use of the so-called co-rotational stress tensor. Damage and cracking phenomena are modeled within the concept of smeared cracks. To assure objectivity of the analysis with respect to the size of the finite elements, crack band method is used. The contact-impact analysis is based on the mechanical interaction between two bodies—concrete beam (master) and dropping hammer (slave) falling on the mid span of the beam. The contact constrains are satisfied by Lagrange multiplier method, which is adapted for the explicit time integration scheme. To investigate the influence of loading rate on the failure mode of the beam parametric study is carried out. The numerical results are evaluated, discussed and compared with test results known from the literature. It is shown that the beam resistance and failure mode strongly depend on loading rate. For lower loading rates beam fails in bending (mode-I fracture). However, with increasing loading rate there is a transition of the failure mechanism from bending to shear. The results are in good agreement with theoretical and experimental results known from the literature. [ABSTRACT FROM AUTHOR]

In this study we first obtain the explicit expressions for the 15 effective reduced elastic compliances of an elastically anisotropic solid containing multiple microcracks with an arbitrary degree of alignment under two-dimensional deformations within the framework of the non-interaction approximation (NIA). Under special situations, our results can reduce to the classical ones derived by Bristow (J Appl Phys 11: 81–85, 1960), and Mauge and Kachanov (J Mech Phys Solids 42(4):561–584, 1994). Some interesting phenomena are also observed. For example, when the undamaged solid is orthotropic, the effective in-plane shear modulus is dependent on the degree of the crack alignment. The NIA method is then extended to obtain the effective electroelastic properties of an anisotropic piezoelectric solid containing two-dimensional insulat- ing, permeable or conducting microcracks with an arbitrary degree of alignment. We also derive a set of fifteen coupled nonlinear equations for the unknown effective reduced elastic compliances of a microcrac- ked, anisotropic, elastic solid by using the generalized self-consistent method (GSCM). The set of coupled nonlinear equations can be solved through iteration. [ABSTRACT FROM AUTHOR]

A continuum model is proposed to study the effects of deformation twinning on interface crack kinking in metal/ceramics layered materials. At the final stage of material failure, plastic work hardening exhausts and lattice rotation becomes main mechanism after competing with dislocation gliding. The crack-tip plasticity is established in terms of the second gradient of microrotation due to the coupling effect of the twins. The formed twinning structures not only shield the crack tip, but inhibit further dislocation emission by increasing the near-tip stress levels. A Dislocation-Free Zone (DFZ) can exist in the immediate vicinity of the tip. The model is based on the equivalence of the stresses derived from twin-based crack-tip plasticity, macroscopic plasticity and elasticity on the boundary. The two-parameter characterization of near-tip stress fields is used for the outer plastic zone to account for constraint effects. Crack kinking out of the interface follows the direction of the maximum flow stress from the crack-tip plasticity. The DFZ size and the crack-tip shielding ratio, as well as the kink angle, are obtained for various values of low hardening exponents and crack-tip constraints. [ABSTRACT FROM AUTHOR]

For evolutionary processes of material remodelling and growth, a comparison is drawn between a conventional formulation and one that postulates the existence of additional balance laws for the configurational forces. [ABSTRACT FROM AUTHOR]

A variational formulation of quasi-static brittle fracture in elastic solids at small strains is proposed and an associated finite element implementation is presented. On the theoretical side, a consistent thermodynamic framework for brittle crack propagation is outlined. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip. On the numerical side, we first consider a standard finite element discretization in the two-dimensional space which yields a discrete formulation of the global dissipation in terms of configurational nodal forces. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity for two-dimensional problems is performed by the doubling of critical nodes and interface segments of the mesh. A crucial step for the success of this procedure is its embedding into a r-adaptive crack-segment re-orientation algorithm governed by configurational-force-based directional indicators. Here, successive crack propagation is performed by a staggered loading-release algorithm of energy minimization at frozen crack state followed by nodal releases at frozen deformation. We compare results obtained by the proposed formulation with other crack propagation criteria. The computational method proposed is extremely robust and shows an excellent performance for representative numerical simulations. [ABSTRACT FROM AUTHOR]

A finite element model of crack propagation along a sinusoidal interface with amplitude A and wavelength λ between identical elastic materials is presented. Interface decohesion is modeled with the Xu and Needleman (J Mech Phys Solid 42(9):1397, 1994) cohesive traction–separation law. Ancillary calculations using linear elastic fracture mechanics theory were used to explain some aspects of stable and unstable crack growth that could not be directly attained from the cohesive model. For small aspect ratios of the sinusoidal interface ( A/λ ≤ 0.25), we have used the analytical Cotterell–Rice (Intl J Fract 16:155–169, 1980) approximation leading to a closed-form expression of the effective toughness, K Ic , given by $$K_{Ic}\sqrt{(1-\nu^{2})/E\phi_n}=2/\left(1+\left[1+4\pi^{2}(A/\lambda)^{2}\right]^{-1/2}\right),$$ where $$\phi_n$$ is the work of separation, E is Young’s modulus, and ν is Poisson’s ratio. For A/λ > 0.25, both the cohesive zone model and numerical J-integral estimates of crack tip stress intensity factors suggest the following linear relationship: $$K_{Ic} \sqrt{(1-\nu^{2})/E\phi_n}=0.81+1.89(A/\lambda).$$ Parametric studies show that the length of the cohesive zone does not significantly influence K Ic , although it strongly influences the behavior of the crack between the initiation of stable crack growth and the onset of unstable fracture. [ABSTRACT FROM AUTHOR]

The application of rate-dependent cohesive elements is validated in simulation of ductile fracture in aluminum round bars under dynamic loading conditions. Smooth and notched round bars made of AA6060-T6 are tested and simulated under quasi-static and dynamic loadings. The smooth round bar is modeled using finite elements that obey Gurson–Tvergaard–Needleman (GTN) formulation as the constitutive equation. Comparing with experimental results, corresponding GTN parameters and rate-dependent plasticity of the alloy are obtained. A single strain rate-dependent GTN element with the obtained parameters is examined under different values of stress triaxiality and loading rates. The resulting stress-elongation curves represent the traction separation law (TSL) for cohesive elements and the variations of the maximum traction and the energy absorbed are investigated. The notched round bars are modeled by axisymmetric continuum and cohesive elements. The undamaged bulk material is elastic-visco plastic and the cohesive elements obey the TSL defined from the single element calculations. The experiments are simulated by these models in which the cohesive elements are rate sensitive and automatically obtain the values of the total strain rate from their adjacent continuum elements to update the values of the cohesive strength during the analysis. The results of the analysis, including maximum load, time of failure and diameter reduction are validated with the experimental results. The effects of element size, rate-dependent plasticity of the material and stress triaxiality are also discussed. [ABSTRACT FROM AUTHOR]

Rotational degrees of freedom in Cosserat continua give rise to higher fracture modes. Three new fracture modes correspond to the cracks that are surfaces of discontinuities in the corresponding components of independent Cosserat rotations. We develop a generalisation of J- integral that includes these additional degrees of freedom. The obtained path-independent integrals are used to develop a criterion of crack propagation for a special type of failure in layered materials with sliding layers. This fracture propagates as a progressive bending failure of layers – a “bending crack that is, a crack that can be represented as a distribution of discontinuities in the layer bending. This situation is analysed using a 2D Cosserat continuum model. Semi-infinite bending crack normal to layering is considered. The moment stress concentrates along the line that is a continuation of the crack and has a singularity of the power − 1/4. A model of process zone is proposed for the case when the breakage of layers in the process of bending crack propagation is caused by a crack (microcrack in our description) growing across the layer adjacent to the crack tip. This growth is unstable (in the moment-controlled loading), which results in a typical descending branch of moment stress – rotation discontinuity relationship and hence in emergence of a Barenblatt-type process zone at the tip of the bending crack. [ABSTRACT FROM AUTHOR]

The paper presents a novel physical model, based on perturbation theory, to describe localization pattern formation in a solid material as a result of system instabilities. Such kind of approach has been inspired by the theory of population dynamics. In particular, the sinergetic phenomenon of strain localization into a stressed continuum, and its subsequent evolution to cohesive cracking, is obtained through the competition of an external source of energy (e.g., strain energy) and of the internal behavior of the material. The hypothesis of mobile energy entities within material bulk is put forward. These entities, which under low strain conditions are evenly distributed throughout the body, can be considered as strain quanta. The quantization of mechanical quantities is not new in continuum and fracture mechanics, [see, e.g., Novozhilov (1969, Prik Mat Mek 33:212–222)]. With increasing strain, a certain critical point is reached when the homogeneous situation becomes unstable and the strain quanta begin to aggregate into bands, leading to periodic strain localization patterns. The model, which is only theoretical at this stage, can be applied to the particular case of dry snow avalanches. In these cases, snow avalanche triggering is due toinstability (onset of sliding onto a weak plane) and is controlled by external loading (e.g., weight of the slope, load by skiers) and by internal factors (e.g., temperature changes, snow phase transformations etc.). [ABSTRACT FROM AUTHOR]

A new quantum action-based theory, dynamic quantized fracture mechanics (DQFM), is presented that modifies continuum-based dynamic fracture mechanics (DFM). The crack propagation is assumed as quantized in both space and time. The static limit case corresponds to quantized fracture mechanics (QFM), that we have recently developed to predict the strength of nanostructures. DQFM predicts the well-known forbidden strength and crack speed bands – observed in atomistic simulations – which are unexplained by continuum-based approaches. In contrast to DFM and linear elastic fracture mechanics (LEFM), that are shown to be limiting cases of DQFM and which can treat only large (with respect to the “fracture quantum”) and sharp cracks under moderate loading speed, DQFM has no restrictions on treating defect size and shape, or loading rate. Simple examples are discussed (i) strengths predicted by DQFM for static loads are compared with experimental and numerical results on carbon nanotubes containing nanoscale defects; (ii) the dynamic fracture initiation toughness predicted by DQFM is compared with experimental results on microsecond range impact failures of 2024-T3 aircraft aluminum alloy. Since LEFM has been successfully applied also at the geophysics size-scale, it is conceivable that DQFM theory can treat objects that span at least 15 orders of magnitude in size. [ABSTRACT FROM AUTHOR]

The thermal stress induced in a spherical inclusion by the difference of the thermal expansion coefficiences of the inclusion and its embedding matrix is considered. Both the inclusion and the matrix are assumed to be of cubic symmetry. Eshelby’s equivalent inclusion method is used to solve the problem. A smple expression for the determination of thermal mismatch stress is thus derived. Some numerical examples are provided. [ABSTRACT FROM AUTHOR]

Partly in response to a communication recently published in this journal on apparent inconsistencies between certain continuum and atomistic formulations of gradient elasticity (Yang and Guo 2005), we further elaborate on this issue in view of results and works not known or not cited in the aforementioned communication. In particular, we unify the concepts and motivations of two different formats of gradient elasticity. The first format was motivated for use in statics and aims at removing strain singularities. The second format was motivated for use in dynamics and aims at describing wave dispersion. We suggest here an alternative format of gradient elasticity that is dispersive, while its static version is identical to the first format mentioned above. Also, procedures are outlined by which the higher-order coefficients can be related to micro-structural properties. Finally, solution methods are described for static and dynamic analysis. [ABSTRACT FROM AUTHOR]

A new computational damage micromodel for laminates, which takes into account classical experimental micro- and macro-observations for various stacking sequences, is described. The first computational examples are shown. [ABSTRACT FROM AUTHOR]

Maximum shear stress trajectories are obtained for the mode I and for the mode II crack in an isotropic elastic solid in plane strain. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution of an elastic half plane containing multiple edge cracks. A general procedure is proposed to formulate the multiple crack problem based on the distributed dislocation method. Three different patterns of modeling dislocation density at the crack mouth are discussed. The correct form of the weight function for the dislocation density is adopted in this analysis. The induced integral involving Cauchy kernel is evaluated by means of recursion relations. Numerical computations of problems with up to 100 edge cracks show that the current method is computationally efficient and accurate. Comparisons of 2D and 3D multiple edge cracks are also discussed. [ABSTRACT FROM AUTHOR]

A novel procedure is proposed for evaluation of stress intensity factors of planar Mode III shear cracks perpendicular to a nearby interface between two isotropic elastic solids. Shear cracks traversing a flat layer bonded to two different elastic solids are also analyzed. The method is based on superposition of singular near tip stress and displacement fields generated by both the main crack and certain image cracks. Both the main and the image cracks are loaded by self-equilibrating shear tractions of different magnitude, such that matching parts of the said fields are made to satisfy traction and displacement continuity conditions at the interface. Selected comparisons with results obtained by different methods show good agreement. Applications of the method to other crack problems are discussed. [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]

A new analytical approach for’ micromechanical modeling of the effective viscoelastic behavior of a’ composite material is presented. Fractionexponential operators are. used to describe the viscoelastic properties of the constituents. To construct the corresponding elastic solution, effective field method is used. Effective viscoelastic operators are obtained from the Volter ra’s elasticity-viscoelasticity correspondence principle. Incompatible deformation that often occurs during the manufacturiig process is taken intp account. All the formulas are obtained in explicit ready-to-use form. [ABSTRACT FROM AUTHOR]

Nonlinear viscoelastic relations to describe the mechanical behavior of polymers and polymer based composites are developed. The nominal stress - strain and creep curves for different values of the current coefficient of lateral deformation are constructed. To model some effects associated with lateral deformation, relations for a nonlinear composite formed of elastic and viscous components are formulated. They are applied to creep fracture of polymers subjected to ultraviolet radiation. [ABSTRACT FROM AUTHOR]

A new method is proposed for shape sensitivity analysis of a crack in a homogeneous, isotropic, and nonlinearly elastic body subject to mode I loading conditions. The method involves the material derivative concept of continuum mechanics, domain integral representation of theJ-integral, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is required in the proposed method. Since the governing variational equation is differentiated before the process of discretization, the resulting sensitivity equations are independent of any approximate numerical techniques. Based on the continuum sensitivities, the first-order reliability method was employed to perform probabilistic analysis. Numerical examples are presented to illustrate both the sensitivity and reliability analyses. The maximum difference between the sensitivity of stress-intensity factors calculated using the proposed method and the finite-difference method is less than four percent. Since all gradients are calculated analytically, the reliability analysis of cracks can be performed efficiently. [ABSTRACT FROM AUTHOR]

During dynamic crack propagation, several distinct changes in the morphology of the dissipative region near the crack edge occur, and they have a pronounced influence on the main propagation mechanism. There are also distinct morphological differences between mode I and mode II. For mode I crack expansion, four successive generations of localization may be observed: micro-separations coalescing with the main crack, protruding clusters of micro-separations, micro-branches, and finally (macro-)branches. The region of localizations is increasing laterally from the main crack plane with crack growth and velocity, as it appears, because of high normal stresses in planes normal to the crack direction. If sufficient space is available, an expanding mode I crack accelerates to a constant velocity, which appears to prevail even after branching and multiple-branching. This indicates an amazing self-similarity over the four generations of localization. The morphology changes during crack propagation depend both on the magnitude of the applied load and on the travelled length of the crack edge. For mode II, the energy dissipation seems generally to be much more concentrated to the crack plane than for mode I. A main reason appears to be that normal stresses in planes normal to the crack direction are comparatively small in front of the crack. Therefore, strong micro-separation localizations seem to appear mainly in shear planes parallel with the crack plane. The appearance of such localizations may be analogous to the remarkable flow velocity gradient discontinuity discovered in turbulent shear flow near a wall.As a consequence of the apparently stronger concentration of the dissipative region to the crack plane, a mode II crack can reach higher velocities than a mode I crack, and it may even reach intersonic velocities. [ABSTRACT FROM AUTHOR]

In the evaluation of accurate weight functions for the coefficients of first few terms of the linear elastic crack tip fields and the crack opening displacement (COD) using the finite element method (FEM), singularities at the crack tip and the loading point need to be properly considered. The crack tip singularity can be well captured by a hybrid crack element (HCE), which directly predicts accurate coefficients of first few terms of the linear elastic crack tip fields. A penalty function technique is introduced to handle the point load. With the use of these methods numerical results of a typical wedge splitting (WS) specimen subjected to wedge forces at arbitrary locations on the crack faces are obtained. With the help of appropriate interpolation techniques, these results can be used as weight functions. The range of validity of the so-called Paris equation, which is widely used in the evaluation of the COD from the stress intensity factors (SIFs), is established. [ABSTRACT FROM AUTHOR]

This paper presents an analytical solution, substantiated by extensive finite element calculations, for the stress field at a notch root in a plate of arbitrary thickness. The present approach builds on two recently developed analysis methods for the in-plane stresses at notch root under plane-stress or plane strain conditions, and the out-of-plane stresses at a three-dimensional notch root. The former solution (Filippi et al., 2002) considered the plane problem and gave the in-plane stress distributions in the vicinity of a V-shaped notch with a circular tip. The latter solution by Kotousov and Wang (2002a), which extended the generalized plane-strain theory by Kane and Mindlin to notches, provided an expression for the out-of-plane constraint factor based on some modified Bessel functions. By combining these two solutions, both valid under linear elastic conditions, closed form expressions are obtained for stresses and strain energy density in the neighborhood of the V-notch tip. To demonstrate the accuracy of the newly developed solutions, a significant number of fully three-dimensional finite element analyses have been performed to determine the influences of plate thickness, notch tip radius, and opening angle on the variability of stress distributions, out-of-plane stress constraint factor and strain energy density. The results of the comprehensive finite element calculations confirmed that the in-plane stress concentration factor has only a very weak variability with plate thickness, and that the present analytical solutions provide very satisfactory correlation for the out-of-plane stress concentration factor and the strain constraint factor. [ABSTRACT FROM AUTHOR]

Elastic-plastic crack-tip fields and constraint levels in bending specimens under large-scale yielding (LSY) are examined. The J-A2 three-term solution is modified by introducing an additional term caused by the global bending. Three different methods, i.e. two-point matching, constant A2 and elastic stress estimation method, are proposed to determine the fourth term. It is shown that the elastic stress estimation method is the simplest, yet effective, in that the fourth term can be derived from the strength theory of materials and the concept of plastic hinge, and effectively quantifies the contribution of the global bending moment on the crack-tip field. Consequently, the modified J-A2 solution, with the inclusion of the correction for global bending, does not introduce any new parameter. The two parameters remain as the loading (J and M) and the constraint level (A2). To validate the present solution, detailed finite element analyses (FEA) were conducted for a Three Point Bend (TPB) specimen with a/W=0.59 in A285 steel, and Single Edge Notched Bend (SENB) specimen subjected to pure bending with a/W=0.5 in A533B steel at different deformation levels ranging from small-scale yielding (SSY) to LSY. Results show that the modified J-A2 solution matches fairly well with the FEA results for both TPB and SENB specimens at all deformation levels considered. In addition, the fourth stress term is (a) proportional to the global bending moment and inversely proportional to the ligament length; (b) negligibly small under SSY; and (c) significantly large under LSY or fully plastic deformation. Accordingly, the present model effectively characterizes the crack-tip constraint for bending dominated specimens with or without the large influence from the global bending stress on the crack-tip field. [ABSTRACT FROM AUTHOR]

In a previous paper, a general method was presented for constructing the solution to the problem of a semi-infinite, mode III crack propagating dynamically through an infinite, general linear viscoelastic body. The only restrictions placed upon the crack tip speed were that it have constant sign and in magnitude not exceed the glassy shear wave speed. In the present contribution, those previous analytical results are applied to a study of dynamic unsteady crack growth in a linear viscoelastic body. In particular, a numerical algorithm for computing the stress intensity factor is given along with example simulations of running cracks using the Achenbach-Chao viscoelastic model and a stress intensity factor (SIF) fracture criterion. We also compare the transient SIF with the dynamic steady state SIF, and examine the transition to constant crack speed for a dynamically accelerating crack in a viscoelastic material. [ABSTRACT FROM AUTHOR]

The singular nature of the dynamic stress fields around an interface crack located between two dissimilar isotropic linearly viscoelastic bodies is studied. A harmonic load is imposed on the surfaces of the interface crack. The dynamic stress fields around the crack are obtained by solving a set of simultaneous singular integral equations in terms of the normal and tangent crack dislocation densities. The singularity of the dynamic stress fields near the crack tips is embodied in the fundamental solutions of the singular integral equations. The investigation of the fundamental solutions indicates that the singularity and oscillation indices of the stress fields are both dependent upon the material constants and the frequency of the harmonic load. This observation is different from the well-known -1/2 oscillating singularity for elastic bi-materials. The explanation for the differences between viscoelastic and elastic bi-materials can be given by the additional viscosity mismatch in the case of viscoelastic bi-materials. As an example, the standard linear solid model of a viscoelastic material is used. The effects of the frequency and the material constants (short-term modulus, long-term modulus and relaxation time) on the singularity and the oscillation indices are studied numerically. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical studies of effective elastic properties of cracked solids. We concentrate on two dimensional media containing different patterns of parallel crack distributions. We use the Rotated Staggered Grid (RSG) which allows one to simulate elastic wave propagation very accurately in fractured media. Our aim is to compare the predictions given by several effective medium theories to the effective properties we derive from our numerical experiments. Namely, these are the ``Non-interaction approximation (NIA)'', the ``Differential scheme (DS)'' and an extension of the DS (EDS). According to our results, the DS theory and its extension perform well. Simulations of media containing very few cracks prove that for our setup the effective properties stabilize at low numbers of cracks. Finally, we studied parallel cracks clustered in stacked columns. We found that, as expected, the shielding effects dominate in such patterns. [ABSTRACT FROM AUTHOR]

A short overview of various approaches to two- and three-dimensional crack interaction problems is given. Solutions for closely spaced cracks are discussed. It is argued that such solutions are not immediately relevant for the problems of crack coalescence. [ABSTRACT FROM AUTHOR]

Eshelby's formula that relates the strain inside of an ellipsoidal inclusion in an unbounded elastic medium to the uniform strain imposed at infinity is generalized to the cases of poroelastic and thermoelastic materials. This result holds for an arbitrary anisotropy of the inclusion and of the host material. [ABSTRACT FROM AUTHOR]

A correction to the article about the determination of the effective mode-I toughness of a sinusoidal interface between two elastic solids that was published in the October 30, 2007 issue is presented.