Contact and crack problems for an elastic wedge

In this paper the contact and the crack problems for an elastic wedge of arbitrary angle are considered. The problem is reduced to a singular integral equation which, in the general case, may have a generalized Cauchy kernel. The singularities under the stamp as well as at the wedge apex are studied and the relevant stress intensity factors are defined. The problem is solved for various wedge geometries and loading conditions. The results may be applicable to certain foundation problems and to crack problems in symmetrically loaded wedges in which cracks may initiate from the apex.     

Analytical solutions for two penny-shaped crack problems in thermo-piezoelectric materials and their finite element comparisons

Thermally loaded penny-shaped cracks in thermopiezoelectric materials are investigated in this paper. The analytical solutions for the penny-shaped cracks subjected to uniform temperature and steady heat flow are discussed. Comparisons are made between the stress-intensity factors derived by the analytical solutions and the numerical results using different finite element techniques.     

Stable generalized finite element methods for elasticity crack problems

Generalized or eXtended finite element methods (GFEM/XFEM) for crack problems have been studied extensively. The GFEM/XFEM are called extrinsic if additional functions are enriched at every node in certain domains, while they are called degree of freedom (DOF)-gathering if the singular enriched functions are gathered using cutoff functions. The DOF-gathering GFEM/XFEM save the additional DOFs compared with the extrinsic approach. Both extrinsic and DOF-gathering GFEM/XFEM suffer from difficulties of stabilities in a sense that their scaled condition numbers (SCN) of stiffness matrices could be much larger than those of the standard FEM. A GFEM/XFEM is referred to as stable GFEM (SGFEM) if it reaches optimal convergence orders, and its SCN is of same order as that of FEM. An extrinsic SGFEM was established in Zhang et al for the Poisson crack problems. Objective of this article is to propose the SGFEM for elasticity crack problems; both extrinsic and DOF-gathering schemes are addressed. The main idea is to modify the enriched functions by subtracting their FE interpolants, which was developed by Babuška and Banerjee. To remove local almost linear dependence introduced by multifold enrichments at one node, we propose a local principal component analysis technique to identify and analyze “contributions” of multifold enrichments at one node. Numerical studies demonstrate that the proposed SGFEM and DOF-gathering SGFEM are of optimal convergence and have the SCNs of same order as in the FEM.     

Contact problems involving forces and moments for incompressible linearly elastic materials

Aclass of rigid punch problems for an incompressible linearly elastic body involving forces and moments is considered by the theory of variational inequalities. After showing a proof of existence of the solution under a compatibility condition of the applied force and moment, extensive discussions about the reduced constraint of incompressibility in the finite element approximation are given. The reduced constraint is an explanation of the so-called reduced integration technique to resolve the incompressibility. Finally, the contact condition is controlled pointwise. Some justifications of the pointwise control are given by the idea of numerical integration.     

Plane strain crack problems for elastic materials with variable properties

Distributions of stress, strain and displacement occurring at the tip of a crack in a material with properties dependent on the type of loading are investigated for the conditions of plane strain in both far-field tensile and shear loads. The causes of the dependence of material properties on the type of external forces are the various inhomogeneities such as microcracks, pores, inclusions or reinforcing components in a material. The behaviour of these inhomogeneities depends substantially on the conditions of loading or deformation. Hence, the deformation properties of a material are not fixed intrinsic material characteristics that are invariant to the loading conditions, but rather the macroproperties of such materials are stress-state-dependent ones, and this effect becomes more noticeable as the volume content of the inhomogeneities increases. The asymptotic solutions of crack problems are obtained on the basis of proposed stress-strain relations describing not only the stress-state dependence of material properties, but the interrelation between the characteristics of volume and shear deformation as well. In a non-uniform stress state the primary macrohomogeneous material becomes an heterogeneous one. The use of the stress function is not effective for the solution of plane strain crack problems for the materials under consideration. Therefore, an approach based on the corresponding representation for the strains is used. It is shown that the commonly used suppositions of the symmetry or anti-symmetry in the stress distribution relative to the crack plane can not be accepted, since they do not allow all the boundary conditions to be satisfied. The opening of the crack surfaces in the case of far shear field is observed. The influence of stress-state sensitivity of material properties on the values of the stress intensity factor is more significant for tensile crack than for the crack in far shear field.     

Plane strain crack problems for elastic materials with variable properties

Distributions of stress, strain and displacement occurring at the tip of a crack in a material with properties dependent on the type of loading are investigated for the conditions of plane strain in both far-field tensile and shear loads. The causes of the dependence of material properties on the type of external forces are the various inhomogeneities such as microcracks, pores, inclusions or reinforcing components in a material. The behaviour of these inhomogeneities depends substantially on the conditions of loading or deformation. Hence, the deformation properties of a material are not fixed intrinsic material characteristics that are invariant to the loading conditions, but rather the macroproperties of such materials are stress-state-dependent ones, and this effect becomes more noticeable as the volume content of the inhomogeneities increases. The asymptotic solutions of crack problems are obtained on the basis of proposed stress-strain relations describing not only the stress-state dependence of material properties, but the interrelation between the characteristics of volume and shear deformation as well. In a non-uniform stress state the primary macrohomogeneous material becomes an heterogeneous one. The use of the stress function is not effective for the solution of plane strain crack problems for the materials under consideration. Therefore, an approach based on the corresponding representation for the strains is used. It is shown that the commonly used suppositions of the symmetry or anti-symmetry in the stress distribution relative to the crack plane can not be accepted, since they do not allow all the boundary conditions to be satisfied. The opening of the crack surfaces in the case of far shear field is observed. The influence of stress-state sensitivity of material properties on the values of the stress intensity factor is more significant for tensile crack than for the crack in far shear field.     

Elastodynamic analysis of the Finite punch and finite crack problems in orthotropic materials

The transient elastodynamic response of the finite punch and finite crack problems in orthotropic materials is examined. Solution for the stress intensity factor history around the punch corner and crack tip is found. Laplace and Fourier transforms together with the Wiener–Hopf technique are employed to solve the equations of motion in terms of displacements. A detailed analysis is made in the simplified case when a flat rigid punch indents an elastic orthotropic half-plane, the punch approaches with a constant velocity normally to the boundary of the half-plane. An asymptotic expression for the singular stress near the punch corner is analyzed leading to an explicit expression for the dynamic stress intensity factor which is valid for the time the dilatational wave takes to travel twice the punch width. In the crack problem, a finite crack is considered in an infinite orthotropic plane. The crack faces are loaded by impact uniform pressure in mode I. An expression for the dynamic stress intensity factor is found which is valid while the dilatational wave travels the crack length twice. Results for orthotropic materials are shown to converge to known solutions for isotropic materials derived independently.     

A decomposition technique of generalized degrees of freedom for mixedmode crack problems

The numerical manifold method (NMM) builds up a unified framework that is used to describe continuous and discontinuous problems; it is an attractive method for simulating a cracking phenomenon. Taking into account the differences between the generalized degrees of freedom of the physical patch and nodal displacement of the element in the NMM, a decomposition technique of generalized degrees of freedom is deduced for mixed mode crack problems. An analytic expression of the energy release rate, which is caused by a virtual crack extension technique, is proposed. The necessity of using a symmetric mesh is demonstrated in detail by analysing an additional error that had previously been overlooked. Because of this necessity, the local mathematical cover refinement is further applied. Finally, four comparison tests are given to illustrate the validity and practicality of the proposed method. The aforementioned aspects are all implemented in the high‐order NMM, so this study can be regarded as the development of the virtual crack extension technique and can also be seen as a prelude to an h‐version high‐order NMM. Copyright © 2017 John Wiley & Sons, Ltd.     

Variational principles for generalized plane strain problems and their applications

This paper is concerned with the variational principles for the generalized plane strain problem of elasticity, which do not seem to have been documented well in the literature, hitherto. Both the total potential energy and the total complementary potential energy principles have been formulated and presented. Their counterparts in the context of generalized variational principles have also been presented. As a result, of the introduction of a discrete degree of freedom, i.e. the uniform direct strain out of the plane, which characterizes the generalized plane strain problem, a fair bit of complications arises. The minimum nature of the stationary values of the energy functionals may not be taken for granted as expected in their counterparts in conventional plane stress or strain elasticity. The expression of the total complementary potential energy obtained here has not been found before in the literature to the best of the authors' knowledge. This might be responsible for the fact that the generalized plane strain problem has been avoided in published work employing the variational principle based on the total complementary potential energy. The presently formulated total complementary potential energy functional has been applied to some classic problems in composites materials, viz. the analysis of transversely cracked laminates and the micromechanics of unidirectionally fibre-reinforced composites. Some interesting and/or new results have been obtained.     

Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials

We have studied the average magnetoelectroelastic field in a multi-inclusion or inhomogeneity embedded in an infinite matrix. The magnetoelectroelastic inclusion and inhomogeneity problems are discussed [1], and a numerical algorithm to evaluate the magnetoelectroelastic Eshelby’s tensors for the general material symmetry and ellipsoidal inclusion shape is developed. The solutions for the magnetoelectroelastic inclusion and inhomogeneity problems are applied to study the multi-inclusion and inhomogeneity problems. It is shown that the average field in an annulus surrounding an inclusion embedded in an infinite magnetoelectroelastic medium only depends on the shapes and orientations of two ellipsoids, which generalizes Tanaka and Mori's observation in elasticity [2]. The average field in a multi-inclusion is then determined exactly, from which the average field in a multi-inhomogeneity is obtained, using the equivalent-inclusion concept [3]. The solutions of multi-inclusion and inhomogeneity problems serve as basis for an averaging scheme to model the effective magnetoelectroelastic moduli of heterogeneous materials, which generalizes Nemat-Nasser and Hori's multi-inclusion model in elasticity [4]. The model is further extended to predict the effective thermal moduli of the heterogeneous magnetoelectroelastic solids, generalizing the recent work of Li on the thermal expansion coefficients of elastic composites [5]. The proposed model recovers Mori–Tanaka and self-consistent approaches as special cases. Finally, some numerical results are given to demonstrate the applicability of the model. The potential techniques to enhance the magnetoelectric effect in practical composites are also discussed.     

The potential theory method for a half-plane crack and contact problems of piezoelectric materials

The half-plane crack and contact problems for transversely isotropic piezoelectric materials are exactly analyzed. The potential theory method is employed with the resulting integro-differential (for crack problem) and integral (for contact problem) equations having identical structures with those reported earlier in the literature. Existing results in potential theory are thus utilized to obtain complete solutions of the problems under consideration. In particular, for the half-plane crack, both the permeable and impermeable electric conditions at the crack surfaces are considered. The solutions for the permeable crack and half-plane contact are entirely new to the literature.     

A generalized crack problem in plane elasticity

The plane isotropic elasticity problem of a simple curvilinear crack with non-coincident edges (contrary to the idealization usually made) is considered. The maximum opening between the edges of the crack may be as great as 0.2 of the crack length. For the solution of this problem, the model of replacing the real crack by a continuous distribution of poles (concentrated forces and edge dislocations) along a single are lying between the real crack edges is introduced. The problem is reduced to an almost singular integral equation and an approximate method for its numerical solution is proposed. An application to the case of a symmetric crack in an infinite plane medium under uniform loading at infinity is also made.     

Surface crack problems in plates

The mode I crack problem in plates under membrane loading and bending is reconsidered. The purpose is to examine certain analytical features of the problem further and to provide some new results. The formulation and the results given by the classical and the Reissner plate theories for through and part-through cracks are compared. For surface cracks the three-dimensional finite element solution is used as the basis of comparison. The solution is obtained and results are given for the crack/contact problem in a plate with a through crack under pure bending and for the crack interaction problem. Also, a procedure is developed to treat the problem of subcritical crack growth and to trace the evolution of the propagating crack.

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Résumé On considère la propagation dynamique en conditions stable de deux fissures parallèles semi-infinies de Mode III, dans un corps quelconque à visco-élasticité linéaire et à caractéristiques infinie, homogène et isotrope.On suppose exister à l'extrémité de la fissure une zone de rupture de type Barenblatt et on tire une formulation pour la vitesse de relaxation de l'énergie, qui fournit des comparaisons immédiates avec le modèle correspondant pour une fissure simple.On illustre par des calculs numériques l'influence de la vitesse de la fissure, de la distance séparant les fissures et des propriétés du matériau sur la vitesse de relaxation de l'énergie, dans le cas d'un matériau à comportement parabolique, et dans le cas d'un matériau linéaire standard.Comme cas limites de l'analyse, on établit les résultats pour des problèmes élastiques et des problèmes visco-élastiques quasi-statiques.

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This work was supported by NSF under the Grant MSM-8613611 and by NASA-Langley under the Grant NAG-1-713.     

On the problems of crack interactions and crack coalescence

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.     

Application of BEM to generalized plane problems for anisotropic elastic materials in presence of contact

It is in many cases very instructive and useful to have the possibility of treating three-dimensional problems by means of two-dimensional models. It always implies a reduction in computing cost which is particularly significant in presence of non-linearities, derived for instance from the presence of contact between the solids involved in the problem. The term generalized plane problem is adopted for a three-dimensional problem in a homogeneous linear elastic cylindrical body where strains and stresses are the same in all transversal sections. This concept covers many practical cases (for instance in the field of composites), a particular situation called generalized plane strain (strains, stresses and displacements are the same in all transversal sections) being the most frequently analyzed. In this paper, a new formulation is developed in a systematic way to solve generalized plane problems for anisotropic materials, with possible friction contact zones, as two-dimensional problems. The numerical solution of these problems is formulated by means of the boundary element method. An explicit expression of a new particular solution of the problem associated to constant body forces is introduced and applied to avoid domain integrations. Some numerical results are presented to show the performance and advantages of the formulation developed.     

Anti-plane Yoffe-type crack in ferroelectric materials

Extending the polarization saturation (PS) model and Yoffe crack model for ferroelectric materials, a moving PS model is proposed to study the problems of crack propagation considering the electrical nonlinearity. The model is solved using continuous distribution dislocation method. And the explicit expressions of the size of the electric saturation zone, intensity factors and the local energy release rate for the moving PS model are derived. It can be deducted from this model that the intensity factors and the size of the electric saturation zone are independent of the velocity of the crack. The local energy release rate for the moving PS model has the form of that for a stationary crack multiplied by the local energy release rate universal function f(v). And it increases monotonically with increasing v. When the velocity of the crack v ?? 0, the moving PS model will reduce to the static PS model. When the size of the electric saturation zone r ?? 0, the moving PS model is in agreement with the moving linear piezoelectric model.     

Creep crack growth in brittle materials

During steady state crack growth by diffusive cavitation at grain boundaries, crack tip fields are relaxed due to the presence of a cavitation zone. In the present analysis, analytic solutions for the actual crack tip stress fields and the crack velocity in the presence of cavitation zone consisting of continuously distributed cavities ahead of the crack tip are derived using the smeared volume concept. Results indicate that the r ^–1/2 singularity is now attenuated to r ^{–1/2 +} (0<<1/2) singularity. The singularity attenuation parameter is a function of the crack velocity and material parameters. The crack growth rate is related to the mode I stress intensity factor K by K ² at relatively high load, K ⁿ at intermediate load, and approaches zero at small load near K _th. Meanwhile, the cavitation zone extends further into the material due to the stress relaxation at the crack tip and the subsequent stress redistribution. Such relaxation effects become very distinct at low crack velocity and low applied load. Key words: Creep crack growth, brittle material, diffusive cavity growth, sintering stress, crack tip stress field.     

Fatigue crack propagation in polymeric materials

In order to gain a better understanding of matrix-controlled fatigue failure processes in non-metallic materials a series of fatigue tests were performed on several different polymer materials representing different classes of mechanical response. Fatigue crack propagation rates between 5×10^–6 in. cycle^–1 (127 nm cycle^–1) and 4×10^–4 in. cycle^–1 (10 300 nm cycle^–1) were measured in nylon, polycarbonate, ABS resin, low-density polyethylene and polymethyl methacrylate. A strong correlation was found between the fatigue crack propagation rate and the stress intensity factor range prevailing at the advancing crack tip. Whereas metals exhibit comparable fatigue growth rates for a given stress intensity range when normalised with respect to their static elastic modulus, the polymer materials exhibited a 1300-fold difference in crack growth rate for a given normalised stress intensity range. This observation dramatically illustrates the importance of understanding molecular motion and energy dissipation processes in polymer materials as related to their chemistry and architecture. The relative behaviour of the different polymer materials could be generally correlated with their reported damping characteristics.