On the dynamic interaction between a penny-shaped crack and an expanding spherical inclusion in 3-D solid

Three-dimensional stress investigation on the interaction between a penny-shaped crack and an expanding spherical inclusion in an infinite 3-D medium is studied in this paper. The spherical transformation area (the inclusion) expands in a self-similar way. By using the superposition principle, the original physical problem is decomposed into two sub-problems. The transient elastic filed of the medium with an expanding spherical inclusion is derived with the dynamic Green's function. A time domain boundary integral equation method (BIEM) is then adopted to solve the current problem. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a square root crack-tip shape function for elements near the crack tip to describe the proper behavior of the unknown quantities near the crack front. A collocation method as well as a time stepping scheme is applied to solve the BIEs. Numerical examples for the Mode I stress intensity factor are presented to assess the dynamic effect of the expanding inclusion.     

Detachment of an elastic matrix from a rigid spherical inclusion

An approximate theoretical treatment is given for detachment of an elastomer from a rigid spherical inclusion by a tensile stress applied to the elastomeric matrix. The inclusion is assumed to have an initially-debonded patch on its surface and the conditions for growth of the patch are derived from fracture energy considerations. Catastrophic debonding is predicted to occur at a critical applied stress when the initial debond is small. The strain energy dissipated as a result of this detachment, and hence the mechanical hysteresis, are also evaluated. When a reasonable value is adopted for Young's modulus E of the elastomeric matrix, it is found that detachment from small inclusions, of less than about 0.1 mm in diameter, will not occur, even when the level of adhesion is relatively low. Instead, rupture of the matrix near the inclusion becomes the preferred mode of failure at an applied stress given approximately by E/2. For still smaller inclusions, of less than about 1 m in diameter, rupture of the matrix becomes increasingly difficult, due to the increasing importance of a surface energy term. These considerations account for the general features of reinforcement of elastomers. Small-particle fillers become effectively bonded to the matrix, whereas larger inclusions induce fracture near them, or become detached from the matrix, at applied stress that can be calculated from the particle diameter, the strength of adhesion, and the elasticity of the matrix material.     

Eshelby tensors for a spherical inclusion in microelongated elastic fields

Eshelby tensors are found for a spherical inclusion in a microelongated elastic field. Here, a special micromorphic model is introduced to describe the damaged material which defines the damage as the formation and the growth of microcracks and microvoids occurred in the material at the microstructural level. To determine the new material coefficients of the model, an analogy is established between the damaged body and the composite materials and then Mori–Tanaka homogenization technique is considered to obtain overall material moduli. Following this idea, the determination of the Eshelby tensors which establish the relation between the strains of the matrix material and of the inclusion becomes the first task. Introducing the concept of eigenstrain and microeigenstrain, the general constitutive theory is given for a homogeneous isotropic centrosymmetric microelongated media with defects. Then by the use of Green’s functions, micro and macro elastic fields are presented for the case of spherical inclusions embedded in an infinite microelongated material. Thus, the Eshelby tensors are obtained for a microelongated elastic field with a spherical inclusion and it is also shown that the classical Eshelby tensors can be obtained as a limit case of the microelongation.     

Response of an elastic solid to nonuniformly expanding surface loads

Exact expressions are obtained for the displacement field in a homogeneous isotropic elastic half space whose surface is subjected to a unit normal pressure. The load emanates from a point and expands nonuniformly and radially outwards. The displacement field is obtained in the form of triple integrals over finite ranges. Both accelerating and decelerating loads have been considered. Wave front surfaces with their regions of existence have been identified. First motion approximations near different wave arrivals have been obtained by a limiting process and do not involve any integration.     

The displacement field due to an interface crack along an elastic inclusion in a differing elastic matrix

Summary A two dimensional mathematical model of an interface crack which lies along an elastic inclusion embedded in an elastic matrix with different elastic constants is considered. In contrast to a previous study by Toya, which determined only the displacement of the crack faces for a far-field biaxial load, a formula for theEntire displacement throughout the matrix and inclusion is obtained for a far-field biaxial load. Herrmann [16], which considered a fixed rigid inclusion, and this paper are the first solutions for the entire displacement of an interface crack problem with in plane far-field loading. From this expression for the displacement, a natural decomposition of the problem is identified and the extent of the predicted interpenetration of the crack faces is discussed for each case. It is seen, analogous to a Griffith crack, that interpenetration regions always occur and are large for most mixed far-field loads. This confirms the statement in England [10] that it might be expected... that a similar wrinkling and crossover phenomena will be observed near the ends of the crack. This elegant closed-form expression for the displacements throughout both the matrix and the inclusion is of interest either for use as a benchmark for numerical studies of interface problems or to determine a domain of influence for interface cracks in fiber-reinforced and particular composites.     

On the propagation of waves in an electromagnetic elastic solid

The coupling of electromagnetic and elastic waves is considered from the standpoint of linear elasticity and a linearized electromagnetic theory. The problem of plane waves traveling through a uniform magnetostatic field is considered and couplings of the waves are studied. An investigation of the same problem for a uniform electrostatic field shows that the usual plane waves propagate without any change in their phase velocities but that the mechanical waves are accompanied by small fluctuating electromagnetic fields. The problem of the vibration of a free infinite elastic plate in a large magnetostatic field is examined under the assumption that the resulting electromagnetic fields are quasistationary. Frequency equations are obtained for both symmetric and antisymmetric vibrations and the damping caused by the field for both the first two symmetric and antisymmetric modes is obtained as a linear correction to the usual free plate frequencies.     

A crack along the interface of a rigid circular inclusion embedded in an elastic solid

The two-dimensional problem of a crack opened under uniform internal pressure and lying along the interface of a rigid circular inclusion embedded in an infinite elastic solid is examined. Based on the complex variable method of Musk helishvili closed form solutions of the stresses and displacements around the crack are obtained and these are then combined with the Griffith's virtual work argument to give a criterion of a crack extension, or decohesion of the interface. The critical pressure is expressed explicitly by a function of the radius of the inclusion and the central angle sublended by the half length of the crack; especially it is inversely proportional to the square root of the radius of the inclusion.

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Zusammenfassung Das zweidimensionale Problem eines unter gleichmässigem internen Druck geöffneten Risses der sich längs der Übergangsfläche von einer steifen kreisförmigen Inklusion befindet, die in einem unendlichen elastischen Körper liegt, wird untersucht.Man erhält Lösungen von geschlossener Form für die Spannungen und die Verschiebungen, die sich auf das Verfahren der komplexen Veränderlichen von Muskhelishvili gründet. Diese Resultate werden dann mit der Hypothese der virtuellen Arbeit von Griffith verbunden um ein Kriterium für eine Rißausbreitung, oder eine Auftrennung der Übergangsfläche zu erhalten.Der kritische Druck wird durch eine Funktion des Radius der Inklusion, und des Mittelpunktwinkels der die Hälfte der Rißlänge einschließt ausgedrückt. Besonders ist er umgekehrt proportional zur Quadratwurzel des Radius der Inklusion.

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Résumé On s'est penché sur le problème à deux dimensions d'une fissure s'ouvrant sous une pression interne uniforme, et située sur l'interface d'une inclusion circulaire rigide noyée dans un solide élastique infini.En se basant sur la méthode des variables complexes de Muskhelishvili, on a obtenu des solutions de forme fermée pour les contraintes et les déplacements au voisinage de la fissure. Ces solutions ont été combinées à l'hypothèse de travail virtuel de Griffith, pour conduire à un critère d'extension de la fissure ou de décohésion de l'interface.La pression critique s'exprime de façon explicite en fonction du rayon de l'inclusion et de l'angle central soustendu par la demi-longueur de la fissure. En particulier, elle est inversément proportionnelle à la racine carrée du rayon de l'inclusion.

     

Thermostatics of an infinite mixture of two elastic solids and a spherical thermal inclusion problem

This work is concerned with the linear theory of a binary mixture of two elastic solids. With the help of displacement potentials, two differential equations that govern the displacement of each constituent are obtained. Then, more generalised forms of Betti's reciprocal theorem and Maysel's formula for the mixture are found. Finally, a solution of spherical thermal inclusion problem in an infinite mixture of two elastic solids is obtained by using generalised Maysel's formula and by direct integration of the governing differential equations.     

An n-layered spherical inclusion model for predicting the elastic moduli of concrete with inhomogeneous ITZ

An n-layered spherical inclusion model is presented in this paper for predicting the elastic moduli of concrete with inhomogeneous interfacial transition zone (ITZ). In this model, concrete is represented as a three-phase composite material, composed of the aggregate, bulk paste, and an inhomogeneous ITZ. An analytical solution for the ITZ volume fraction is derived for the general aggregate gradation. By constituting a semi-empirical initial cement gradient model, the local water/cement ratio, degree of hydration, and porosity at the ITZ are estimated. The inhomogeneous ITZ is then divided into a series of homogenous concentric shell elements of equal thickness. The elastic moduli of concrete are determined by solving the n-layered spherical inclusion problem. Finally, the validity of the model is verified with three independent sets of experimental data and the effects of the maximum aggregate diameter, aggregate gradation, and ITZ thickness on the Young’s modulus of concrete are evaluated in a quantitative manner. The paper concludes that the proposed n-layered spherical inclusion model can be used to predict the elastic moduli of concrete.     

Modelling the transient interaction of a thin elastic shell with an exterior acoustic field

Coupled finite and boundary element methods for solving transient fluid–structure interaction problems are developed. The finite element method is used to model the radiating structure, and the boundary element method (BEM) is used to determine the resulting acoustic field. The well‐known stability problems of time domain BEMs are avoided by using a Burton–Miller‐type integral equation. The stability, accuracy and efficiency of two alternative solution methods are compared using an exact solution for the case of a thin spherical elastic shell. The convergence properties of the preferred solution method are then investigated more thoroughly. Copyright © 2007 John Wiley & Sons, Ltd.     

Determination of stress field in an elastic solid weakened by parallel penny-shaped cracks

Summary This paper investigates the stress intensity factors of two penny-shaped cracks with different sizes in a three-dimensional elastic solid under uniaxial tension. The two cracks are symmetrically parallel located in the isotropic solid. Based on Eshelby's equivalent inclusion method and the superposition pronciple of the elasticity theory, a closed-form analytical elastic solution for the stress intensity factors (SIFs) on the boundaries of the cracks is obtained when the center distance between the two cracks is much larger than the crack sizes. A numerical method is employed to extract the solution for small center distance case. It is found that, due to the interaction between the two cracks, the first and second kinds of SIFs exist at the same time even if the applied stress is pure tension. Numerical examples are given for different configurations and it is clearly shown that the SIFs are strongly determined by the distance between the centers of the two cracks.     

Resonance scattering of elastic waves by an elastic inclusion in an elastic matrix. Numerical calculations

The scattering of an incident plane ultrasonic (longitudinal) wave by an elastic spherical inhomogeneity contained within an elastic matrix is studied. The emphasis is on the computation and analysis of basic multiple cases that result when different material behaviours are present in the matrix and in the inclusion. These calculations are useful in underwater acoustic applications. The behaviors are dominated by the soft or rigid backgrounds of the resonance scattering theory (RST). The first three multiple coefficients appearing in the expansions for the total elastodynamic fields developed around the inhomogeneity during the scattering process have been calculated in suitable frequency bands in all the cases considered. The examination of modulus, the real parts, and the imaginary parts of these (complex) coefficients under the RST approach allows the quantitative assessment of the conditions under which monopole or dipole resonances will occur and their relative magnitudes. The decomposition of the multiple coefficients into their resonance and background portions shows that it is the upward frequency shift of the background curves that controls the dominance of either radial (monopole) or translation (dipole) oscillations of the inclusion. This has an effect on the dispersion curves of the composite, which develop optical as well as acoustical branches. The real and imaginary parts of the multiple coefficients are respectively proportional to the attenuation and the effective wave speed in this simple inhomogeneous composite.     

A slit-like inclusion in an orthorhombic piezoelectric solid

Summary A slit-like inclusion in an unbounded orthorhombic piezoelectric solid is considered. We demonstrate that the associated polarization tensor correct to the first order of the aspect ratio of the inclusion can be obtained in an analytical form. As an application, the results are exploited to estimate the effective moduli of a cracked piezoelectric solid. Micromechanical models of the self-consistent and Mori-Tanaka methods are implemented in the present case. The methods offer simple approaches to estimate the stiffness changes due to the presence of aligned longitudinal slit-like cracks.     

On the crack-tip fields of a dynamically propagating crack in an anisotropic elastic solid

General expressions of the crack-tip fields for a dynamically extending crack through an anisotropic elastic material are given. Based on the crack-tip fields, the associated dynamic energy release rate is also derived. Explicit results are given for transversely isotropic materials.

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Résumé On fournit des expressions générales des champs régnant à l'extrémité d'une fissure en expansion dynamique dans un matériau élastique anisotrope. En se basant sur ces champs, on déduit également la vitesse de relaxation de l'énergie dynamique qui y est associée. Des résultats explicites sont fournis pour des matériaux à isotropie transversale.

     

Eshelby's formula for an ellipsoidal elastic inclusion in anisotropic poroelasticity and thermoelasticity

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.     

Elastodynamic analysis of a half plane crack in a transversely isotropic solid under 3-D transient loading

Summary A three-dimensional analysis of a half plane crack in a transversely isotropic solid is performed. The crack is subjected to a pair of suddenly-applied normal line loadings on its faces. Transform methods are used to reduce the boundary value problem to a single integral equation that can be solved by the Wiener-Hopf technique. The Cagniard-de Hoop method is employed to invert the transforms. An exact expression is derived for the mode I stress intensity factor as a function of time and position along the crack edge. Numerical results are given.