Boundary element analysis for viscoelastic solids containing interfaces/holes/cracks/inclusions

With the aid of the elastic–viscoelastic correspondence principle, the boundary element developed for the linear anisotropic elastic solids can be applied directly to the linear anisotropic viscoelastic solids in the Laplace domain. Green's functions for the problems of two-dimensional linear anisotropic elastic solids containing holes, cracks, inclusions, or interfaces have been obtained analytically using Stroh's complex variable formalism. Through the use of these Green's functions and the correspondence principle, special boundary elements in the Laplace domain for viscoelastic solids containing holes, cracks, inclusions, or interfaces are developed in this paper. Subregion technique is employed when multiple holes, cracks, inclusions, and interfaces exist simultaneously. After obtaining the physical responses in Laplace domain, their associated values in time domain are calculated by the numerical inversion of Laplace transform. The main feature of this proposed boundary element is that no meshes are needed along the boundary of holes, cracks, inclusions and interfaces whose boundary conditions are satisfied exactly. To show this special feature by comparison with the other numerical methods, several examples are solved for the linear isotropic viscoelastic materials under plane strain condition. The results show that the present BEM is really more efficient and accurate for the problems of viscoelastic solids containing interfaces, holes, cracks, and/or inclusions.     

XFEM simulation of cracks, holes and inclusions in functionally graded materials

The present work aims at the numerical simulation of inhomogeneities/discontinuities (cracks, holes and inclusions) in functionally graded materials (FGMs) using extended finite element method (XFEM). A FGM with unidirectional gradation in material properties is modeled under plane strain condition. The domain contains a major crack either at the center or at the edge of the domain along with multiple minor discontinuities/flaws such as minor cracks and/or voids/inclusions distributed all over the domain. The effect of the variation in stress intensity factor (SIF) of the major crack due to the presence of the minor cracks and voids/inclusions is studied in detail. The simulations show that the presence of minor discontinuities significantly affects the values of SIFs.     

Deformation stability of solids with cracks or inclusions

The method of calculating the limiting state of solids built on known elasticity relations is proposed to describe stable and unstable deformation of the defect-containing material. The basis for the method is evaluating the rigidity distribution over the body volume. The limiting state of the body is associated with the total negative rigidity of the body and loading system. An increase in damaged area sizes that are much smaller than component sizes is shown to result in stability and carrying capacity losses of the whole component at a certain rigidity of this area. Rigidity calculations for the bodies in the form of a sphere, cylinder or plate with inclusions as well as a crack-containing plate are cited.Translated from Problemy Prochnosti, No. 1, pp. 118–135, January–February, 2005.     

Elastic moduli of heterogeneous solids with ellipsoidal inclusions and elliptic cracks

Summary The influence of ellipsoidal inclusions and elliptic cracks on the overall effective moduli of a two-phase composite and of a cracked body, respectively, is investigated by means of Mori-Tanaka's theory for three types of inclusion and four types of crack arrangements: monotonically aligned, 2-D randomly oriented (two kinds for cracks), and 3-D randomly oriented. The effective moduli of the composite in the aligned case are known to coincide with Willis' orthotropic lower (or upper) bounds with a two-point ellipsoidal correlation function if the matrix is the softer (or harder) phase. With 2-D randomly oriented inclusions, the effective moduli are examined under Willis' transversely isotropic bounds with a two-point spheroidal correlation function, and it is found that, as the cross-sectional aspect ratio of the ellipsoidal inclusions flattens from circular shape to disc-shape, the two effective shear moduli and the plane-strain bulk modulus all lie on or within the bounds. The effective bulk and shear moduli of an isotropic composite containing randomly oriented ellipsoidal inclusions also fall on or within Hashin-Shtrikman's bounds as the shape of the ellipsoids changes. The obtained moduli are then extended to a cracked body containing elliptic cracks, which are generated by compressing the thickness of ellipsoidal voids to zero. It is found that only selected components of the effective moduli are dependent upon the crack density parameter . Their dependence on and the crack shape are explicitly established.     

Multiple development of new phase inclusions in elastic solids

We consider an initial stage of a stress-induced phase transformation as a process of heterogeneous deformation due to multiple appearance of new phase inclusions. We develop a model describing how multiple development of interacting new phase inclusions depends on external strains and how the new phase development affects the macro-behavior of a deformable material. We assume that the random fields of the inclusions and strains are symmetric in a statistical sense. We use the notion of the symmetry of the effective inclusion and assume that this symmetry is the symmetry of an ellipsoid. Then we accept the idea that the geometry of the two-phase structure, in contrast to usual composite materials, is adapted to strains so that certain correspondences between the symmetries of the inclusions and strain fields are settled. A symmetry consistent approach developed as an alternative to a self-consistent (effective) field method for materials undergoing phase transformations allows us, similar to the effective field method, to construct average free energy as a function of a new phase volume concentration and two tensors which characterize the orientation and the shape of the effective ellipsoid and the set of inclusions. Then the problem of equilibrium two-phase structure reduces to the free energy minimization on the class of self-organizing two-phase structures controlled, given average strains, by the symmetry restrictions. We show that the average energy minimization requirements are equivalent to the thermodynamic condition written for the effective ellipsoid. We obtain the dependencies of the new phase concentration and the effective ellipsoid tensor on average strains or stresses. We discuss the stability and estimate energy changes due to phase transformations. We construct domains of existence of the described two-phase structures in strain space and emphasize that the new phase development by the mechanism considered is not possible at any deformation path. We also derive average stress-average strain relationships which demonstrate strain softening effect on the path of the phase transformation.     

A generalized comninou contact model for interface cracks in anisotropic elastic solids

An open question of interest to the mechanics of interface fracture is how to generalize the Comninou contact model for interface cracks in isotropic solids to the general anisotropic case. Part of the difficulty lies in that the peculiar oscillatory behavior can not be fully eliminated by Comninou's original assumption of pure pressure contact between the crack surfaces. In this paper, we propose a model that strictly enforces the non-oscillatory condition by allowing the crack face contact force to have a shear component normal to the direction of slip, which is somewhat reminiscent of frictionless slip between a pair of grooved surfaces. Based on that model, complex variable representations are adopted to determine the complete series expansion for the crack-tip field. The solutions are incorporated into a hybrid finite element procedure to develop a special element for closed interfacial crack tips obeying the generalized contact model. Numerical examples involving a partially closed crack between a pair of misoriented cubic crystals are given to illustrate how the special crack-tip element helps in determining the stress intensity factors as well as the contact zone geometry.     

A material point time integration procedure for anisotropic, thermo rheologically simple, viscoelastic solids

This study presents an effective and robust time integration procedure for general anisotropic, thermal rheologically simple viscoelasticity, that is suitable for implementation in a broad spectrum of general purpose nonlinear finite element programs. It features a judicious choice of state variables which record the extent of inelastic flow (creep), a stable backward Euler integration step, and a consistent tangent operator. Numerical examples involving homogeneous stress states such as uniaxial tension and simple shear, and non-uniform stress states such as a beam under tip load, were carried out by incorporating the present scheme into a general purpose FEM package. Excellent agreement with analytical results is observed.     

Boundary element analysis of three‐dimensional cracks in anisotropic solids

This paper presents a boundary element analysis of linear elastic fracture mechanics in three‐dimensional cracks of anisotropic solids. The method is a single‐domain based, thus it can model the solids with multiple interacting cracks or damage. In addition, the method can apply the fracture analysis in both bounded and unbounded anisotropic media and the stress intensity factors (SIFs) can be deduced directly from the boundary element solutions. The present boundary element formulation is based on a pair of boundary integral equations, namely, the displacement and traction boundary integral equations. While the former is collocated exclusively on the uncracked boundary, the latter is discretized only on one side of the crack surface. The displacement and/or traction are used as unknown variables on the uncracked boundary and the relative crack opening displacement (COD) (i.e. displacement discontinuity, or dislocation) is treated as a unknown quantity on the crack surface. This formulation possesses the advantages of both the traditional displacement boundary element method (BEM) and the displacement discontinuity (or dislocation) method, and thus eliminates the deficiency associated with the BEMs in modelling fracture behaviour of the solids. Special crack‐front elements are introduced to capture the crack‐tip behaviour. Numerical examples of stress intensity factors (SIFs) calculation are given for transversely isotropic orthotropic and anisotropic solids. For a penny‐shaped or a square‐shaped crack located in the plane of isotropy, the SIFs obtained with the present formulation are in very good agreement with existing closed‐form solutions and numerical results. For the crack not aligned with the plane of isotropy or in an anisotropic solid under remote pure tension, mixed mode fracture behavior occurs due to the material anisotropy and SIFs strongly depend on material anisotropy. Copyright © 2000 John Wiley & Sons, Ltd.     

Computation of mixed-mode stress intensity factors for curved cracks in anisotropic elastic solids

A numerical procedure, based on the concept of the J_k-integrals, is presented for computation of the mixed-mode stress intensity factors for curved cracks. Since no auxiliary solutions are required in practice, this approach can be applicable for problems containing cracks of arbitrary curved shape in generally anisotropic elastic solids. Nevertheless, while these integrals are demonstrated to be path-independent in a modified sense, it is noted that the near-tip region is always inevitably included. Attention is therefore addressed to modeling of the singular behavior in the vicinity of the crack tip in such manner that the calculation is based on a combination of full-field finite element analysis and near-tip asymptotic results. With this, no particular singular elements are required in the calculation.     

Transient dynamic analysis of interface cracks in layered anisotropic solids under impact loading

Transient elastodynamic crack analysis in two-dimensional (2D), layered, anisotropic and linear elastic solids is presented in this paper. A time-domain boundary element method (BEM) in conjunction with a multi-domain technique is developed for this purpose. Time-domain elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids are applied in the present time-domain BEM. The spatial discretization of the boundary integral equations is performed by a Galerkin-method, while a collocation method is adopted for the temporal discretization of the arising convolution integrals. An explicit time-stepping scheme is developed to compute the unknown boundary data and the crack-opening-displacements (CODs). To show the effects of the crack configuration, the material anisotropy, the layer combination and the dynamic loading on the dynamic stress intensity factors and the scattered elastic wave fields, several numerical examples are presented and discussed.     

A self-consistent mechanics method for solids containing inclusions and a general distribution of cracks

Summary The damage in a composite material due to a distribution of cracks manifests itself as a reduction of moduli and/or change in elastic constants. This paper presents the effective elastic moduli of a solid containing inclusions and a general distribution of tunnel cracks. Both in-plane and out-of-plane elastic constants are determined. In addition to crack density and inclusion volume fraction, the effective elastic constants are found to depend on a function (), which characterizes the crack orientation distribution, while the anisotropy of a cracked composite is solely induced by the crack orientation distribution. It is established that the effect of inclusions and microcracks on effective moduli is decoupled, i.e., one can obtain the moduli of a solid containing microcracks and inclusions by the corresponding moduli of the solids with microcracks only and with inclusions only. For a solid containing a crack distribution with mirror symmetry, the effective elastic constants can be greatly simplified and can be expressed in terms of two scalar quantities rather than a general function (). This conclusion is particularly useful in the analysis of the micromechanical model. The effect of the asymmetry of () on the effective elastic constants is also investigated.     

Direct boundary integral procedure for a Boltzmann viscoelastic plane with circular holes and elastic inclusions

A direct boundary integral method in the time domain is presented to solve the problem of an infinite, isotropic Boltzmann viscoelastic plane containing a large number of randomly distributed, non-overlapping circular holes and perfectly bonded elastic inclusions. The holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can, in general, be different. The method is based on a direct boundary integral approach for the problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described by Crouch and Mogilevskaya [1], and a time marching strategy for viscoelastic analysis described by Mesquita and Coda [2–8]. Benchmark problems and numerical examples are included to demonstrate the accuracy and efficiency of the method.     

A new boundary element method for mixed boundary value problems involving cracks and holes: Interactions between rigid inclusions and cracks

This paper derives a new boundary integral equation (BIE) formulation for plane elastic bodies containing cracks and holes and subjected to mixed displacement/ traction boundary conditions, and proposes a new boundary element method (BEM) based upon this formulation. The basic unknown in the formulation is a complex boundary function H(t), which is a linear combination of the boundary traction and boundary displacement density. The present BIE formulation can be related directly to Muskhelishvili's formalism. Singular interpolation functions of order r ^–1/2 (where r is the distance measured from the crack tip) are introduced such that singular integrand involved at the element level can be integrated analytically. By applying the BEM, the interaction between a rigid circular inclusion and a crack is investigated in details. Our results for the stress intensity factor are comparable with those given by Erdogan and Gupta (1975) and Gharpuray et al. (1990) for a crack emanating from a stiff inclusion, and with those by Erdogan et al. (1974) for a crack in the neighborhood of a stiff inclusion.     

A novel boundary-domain element method of initial stress, finite deformation and discrete cracks in multilayered anisotropic elastic solids

We introduce a novel boundary-domain element method of initial stress, finite deformation (due to large rotation) and discrete cracks in multilayered anisotropic elastic solids. Because the special Green’s function that satisfies the interfacial continuity and surface boundary conditions is employed, the numerical discretization is reduced to be along one side of the cracks and over the subdomains of finite deformation. Two examples are presented. First, the process of interfacial delamination is simulated around a growing through-thickness crack in a pre-stretched film bonded to a flexible substrate. It is shown that the progression of delamination damage is stable but the initiation of delamination crack can be a snap-back instability. This simultaneous damage and fracture process is approached by the cohesive zone model. Second, the postbuckling of a circular delaminated and pre-compressed film is simulated on a flexible substrate. It is shown that the compliance of substrate can play a significant role on the critical behavior of buckling. If the substrate is more compliant or stiffer than the film, the instability initiates as a subcritical hard or a supercritical soft bifurcation. The critical magnitude of pre-strain for the initiation of buckling increases with substrate stiffness. Also, the transition of buckling from the first to the second mode is captured in the simulation.     

The relationship between cracks,holes and surface dislocations

International Journal of Fracture - It has been shown that surface cracks as well as holes can be represented in terms of surface dislocations. These surface dislocations exist in order to insure...     

A time domain direct boundary integral method for a viscoelastic plane with circular holes and elastic inclusions

This paper considers the problem of an infinite, isotropic viscoelastic plane containing an arbitrary number of randomly distributed, non-overlapping circular holes and isotropic elastic inclusions. The holes and inclusions are of arbitrary size. All inclusions are assumed to be perfectly bonded to the material matrix but the elastic properties of the inclusions can be different from one another. The Kelvin model is employed to simulate the viscoelastic plane. The numerical approach combines a direct boundary integral method for a similar problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described in [Crouch SL, Mogilevskaya SG. On the use of Somigliana's formula and Fourier series for elasticity problems with circular boundaries. Int J Numer Methods Eng 2003;58:537–578], and a time-marching strategy for viscoelastic material analysis described in [Mesquita AD, Coda HB, Boundary integral equation method for general viscoelastic analysis. Int J Solids Struct 2002;39:2643–2664]. Several numerical examples are given to verify the approach. For benchmark problems with one inclusion, results are compared with the analytical solution obtained using the correspondence principle and analytical Laplace transform inversion. For an example with two holes and two inclusions, results are compared with numerical solutions obtained by commercial finite element software—ANSYS. Benchmark results for a more complicated example with 25 inclusions are also given.     

On cracks in rectilinearly anisotropic bodies

The general equations for crack-tip stress fields in anisotropic bodies are derived making use of a complex variable approach. The stress-intensity-factors, which permit concise representation of the conditions for crack extension, are defined and are evaluated directly from stress functions. Some individual boundary value problem solutions are given in closed form and discussed with reference to their companion solutions for isotropic bodies.It is found that an elastic stress singularity of the order r ^–1/2 is always present at the crack tip in a body with rectilinear anisotropy (r being the radial distance from the crack front). This result and some additional consideration of the crack-tip stress fields imply that it is possible to extend current fracture mechanics methods to the representation of fracture conditions for anisotropic bodies with cracklike imperfections.

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Zusammenfassung Die aligemeinen Gleichungen für Spannungsfelder an der Spitze eines Risses in anisotropischen Körpern werden näherungsweise unter Benützung einer komplexen Variabele abgeleitet. Die Spannungsintensitätsfaktoren, die eine kurzgefasste Darstellung der Bedingungen für Rissausbreitung gestatten, werden definiert and unmittelbar aus Spannungsfunktionen bestimmt. Einige individuellen Lösungen von Grenzwert-Problemen werden in geschlossener Form gegeben and bezüglich ihrer analogen Lösungen für isotrope Körper diskutiert.Es ergibt sick, dass eine elastische Spannungssingularität der Ordnung r ^–1/2 an der Spitze eines Risses in einem Körper mit rectilinearischer Anisotropie immer da ist. (r ist die Radiusabstand von der Rissfront). Dieses Ergebnis, und einige weiteren Betrachtungen der Spannungsfelder an der Rissspitze weisen darauf hin, dass es möglich ist, die geläufigen Methoden der Bruchmechanik für die Darstellung von Bruchbedingungen anisotropischer Körper mit rissartigen Fehlern zu erweitern.

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Résumé On dérive l'Equation générale des champs de tension d'une pointe de fissure dans un corps anisotrope par une approximation de variable complexe. On a dévini et évalué les facteurs d' intensité de tension qui permettent la représentation concise des conditions pour l'extension d'une fissure, directement des fonctions de tension. On donne quelques solutions de problèmes à valeur limite en forme implicite et on les discute en correspondance avec les solutions analogues pour corps isotropes.On a trouvé qu'une singularité de tension élastique de l'ordre de r ^–1/2 (où r est la distance radiale depuis le front de la fissure) est toujours présente à la pointe d'une fissure dans un corps rectilinéairement anisotrope. Ce résultat et quelques considérations additionelles des champs de tension de la pointe d' une fissure impliquent la possibilité d'étendre les applications courantes de théories de mécanisme de fracture à la représentation de conditions de fracture de corps anisotropes avec des imperfections en forme de fissures.

     

Kinked cracks in anisotropic elastic materials

The problem of a kinked crack is analysed for the most general case of elastic anisotropy. The kinked crack is modelled by means of continuous distributions of dislocations which are assumed to be singular both at the crack tips and at the kink vertex. The resulting system of singular integral equations is solved numerically using Chebyshev polynomials and the reciprocal theorem. The stress intensity factors for modes I, II and III and the generalised stress intensity factor at the vertex are obtained directly from the dislocation densities. This revised version was published online in July 2006 with corrections to the Cover Date.     

A finite element constitutive update scheme for anisotropic,viscoelastic solids exhibiting non‐linearity of the Schapery type

This study presents a new scheme for performing integration point constitutive updates for anisotropic, small strain, non‐linear viscoelasticity, within the context of implicit, non‐linear finite element structural analysis. While the basic scheme has been presented earlier by the authors for linear viscoelasticity, the present work illustrates the generality of the underlying fundamentals by extending to Schapery's non‐linear model. The method features a judicious choice of state variables, a stable backward Euler integration step, and a consistent tangent operator. Its greatest strength lies in ready incorporation into existing FEM codes. Numerical examples involving homogeneous stress states such as uniaxial extension and simple shear, and non‐uniform stress states such as a beam under tip load, were carried out by incorporating the present scheme into a general purpose FEM package. Excellent agreement with analytical results is observed. Copyright © 1999 John Wiley & Sons, Ltd.