by Paynter, R. J. H.; Hills, D. A.; Sackfield, A.

The crack tip stress intensities are found for a penny shaped crack lying beneath the free surface of a half-space, and parallel with it. This is done by employing a novel distributed dislocation approach using axi-symmetric Somigliana dislocations as the kernel of an integral equation, and provides a precise solution at the expense of little computing cost. A comparison is made with the crack tip stress intensity factors for a simple plane crack, and predictions are made for the preferred crack extension direction.

DOI: 10.1007/s10704-006-9042-5
Online Date: 12/21/2006
Print publication date: 11/1/2006
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by Yongdong, Li; Wei, Tan; Hongcai, Zhang

The mechanical model was established for the Dirac-type anti-plane transient fracture problem of the weak-discontinuous interface between two FGMs half-planes. Integral transform was adopted to derive Cauchy singular integral equation and Erdogan’s allocation method was used to calculate transient stress intensity factors numerically. The numerical solutions of the weak-discontinuous case were contrasted with those of the infinitesimal-discontinuous one. Two possible effective methods to diminish the peak values of transient stress intensity factors are discussed. One is to reduce the weak-discontinuity of the interface, i.e., to make the ratio of the two non-homogeneity parameters be close to 1.0 and to avoid the case that the signs of the two non-homogeneity parameters are different. Another is to make a compromise between the weak-discontinuity and the all-continuity, i.e., to make FGMs interface infinitesimal-discontinuous. Simple method was suggested for the realization of the infinitesimal-discontinuity of FGMs interface. From the strong-discontinuous interface to the weak- discontinuous one, and then to the infinitesimal-discontinuous one, this is a law and trend of the development of composite interfaces. To design and manufacture infinitesimal-discontinuous interfaces may be a brand-new effective approach to enhance the reliability of composite structures, and the first rank infinitesimal-discontinuity is enough to improve the mechanical performances of composites notably.

DOI: 10.1007/s10704-006-9041-6
Online Date: 12/21/2006
Print publication date: 11/1/2006
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by Guo, Zaoyang; Bažant, Zdeněk P.

The strength and size effect of a slender eccentrically compressed column with a transverse pre-existing traction-free edge crack or notch is analyzed. Rice and Levy’s spring model is applied to simulate the effect of a crack or notch. An approximate, though accurate, formula is proposed for the buckling strength of the column of variable size. Depending on the eccentricity, the crack at maximum load can be fully opened, partially opened or closed. The size effects in these three situations are shown to be different. The exponent of the power-law for the large-size asymptotic behavior can be −1/2 or −1/4, depending on the relative eccentricity of the compression load. Whether the maximum load occurs at initiation of fracture growth, or only after a certain stable crack extension, is found to depend not only on the column geometry but also on its size. This means that the definition of positive or negative structural geometry (as a geometry for which the energy release rate at constant load increases or decreases with the crack length) cannot be extended to stability problems or geometrically nonlinear behavior. Comparison is made with a previous simplified solution by Okamura and coworkers. The analytical results show good agreement with the available experimental data.

DOI: 10.1007/s10704-006-9037-2
Online Date: 12/21/2006
Print publication date: 11/1/2006
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by El-Borgi, S.; Aloulou, W.; Zghal, A.

In an attempt to simulate buckling of nonuniform coatings, we consider the problem of an embedded crack in a functionally graded coating bonded to a homogeneous substrate subjected to a compressive loading. The coating is graded in the thickness direction and the material gradient is orthogonal to the crack direction which is parallel with the free surface. The loading consists of a uniform compressive strain applied away from the crack region. The graded coating is modeled as a nonhomogeneous medium with an isotropic stress-strain law. Using a nonlinear continuum theory and a suitable perturbation technique, the plane strain problem is reduced to an eigenvalue problem describing the onset of buckling. Using integral transforms, the resulting plane elasticity equations are converted analytically into singular integral equations which are solved numerically to give the critical buckling strain and the corresponding crack opening displacement shapes. The main objective of the paper is to study the influence of material nonhomogeneity on the buckling resistance of the graded layer for various crack positions and coating thicknesses.

DOI: 10.1007/s10704-006-9031-8
Online Date: 12/21/2006
Print publication date: 11/1/2006
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by Zhang, Xi; Liu, Qianchu; Mai, Yiu-Wing

This paper is concerned about void growth and associated deformation models in porous visco-plastic solids under conditions similar to those found in highly stressed regions ahead of a crack. A plane-strain unit cell containing an initially circular void is examined to simulate the stress states during dynamic fracture of a metal. Two proportional loading rates are prescribed in the two directions of the cell and their ratio is called the “strain biaxiality” expressed in a monotonic relation with stress triaxility. Finite element analysis is performed for the effective stress–strain curves of the porous solids during void growth for a range of initial porosities, strain biaxialities, strain rates and thermal softening coefficients. Numerical results show that the void evolution and the associated non-uniform deformation depend in a complex fashion on these factors. The local zone of high stress concentration which emanates from the void spreads out in the cell to trigger non-uniform deformation and plastic yielding. Subsequently, a small zone with intense plastic strain and heating either expands smoothly near the growing voids or propagates in a specific direction determined by its interaction with the boundary conditions of the cell such as strain biaxility. At low strain biaxiality and for small voids, formation and propagation of zones with intense plastic strain and heating is localized. However, high strain biaxiality leads to rapid uniform expansion of small voids as observed experimentally. It is found that the intense heating zone follows the zone of high plastic strain concentration and diffuses with imposed strain. Thermal softening which reduces the overall stress can be neglected at the early stage of void growth, but it is magnified past the peak stress by accelerating the void growth. But in the long term, the void growth rate is insensitive to thermal softening coefficient. Increasing strain rates can promote void growth and the rate of which tends to be proportional to the eventual strain rate.

DOI: 10.1007/s10704-006-9030-9
Online Date: 12/21/2006
Print publication date: 11/1/2006
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by Wen, P. H.; Aliabadi, M. H.

A displacement discontinuity formulation is presented for modeling cracks in orthotropic Reisnner plates. Fundamental solutions for displacement discontinuity are derived for the first time using a Fourier transform method. Boundary integral equations are presented in terms of discontinuity rotations on the crack surfaces for opening mode problems. As the fundamental solutions have singularity of O (1/r
²), Chebyshev polynomials of the second kind are used to evaluate the integral equations. By solving for coefficients of the Chebyshev polynomials, the stress intensity factors at the crack tips are obtained directly. Comparisons are made with solutions using the finite element method to demonstrate that the displacement discontinuity method is an efficient and accurate method for solving crack problems in orthotropic Reissner plates.

DOI: 10.1007/s10704-006-9023-8
Online Date: 12/20/2006
Print publication date: 11/1/2006
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by Mastorakos, I. N.; Zbib, H. M.

The interaction between a penny-shape crack and a dislocation in crystalline materials is investigated within the framework of dislocation dynamics. The long-range and singular stress field resulting from the crack is determined by modeling the crack as continuous distribution of dislocation loops. This distribution is determined by satisfying the traction boundary condition at the crack face, resulting into a singular integral equation of the first kind that is solved numerically. This crack model is integrated with the dislocation dynamics simulation technique to yield the stress field of the combine system of crack and different types of dislocations situated at different positions in a three dimensional space. The integrated system is then used to investigate the dislocation behavior and its influence on the crack opening displacement and the characteristic of the stress field near the crack tip. It is shown that, depending on the relative position of the dislocation and its character, the dislocation may result in reduction in the stress amplitude at the crack tip and in some cases in closure of the crack tip. These analyses yield shielding and amplification zones near the crack providing an insight of the dislocation influence on the crack. The full dislocation dynamic analysis reveals the nature of the crack dislocation interaction and the manner in which the dislocation morphology changes as it is attracted to the crack surfaces, as well as the changes it causes to the crack profile.

DOI: 10.1007/s10704-006-9028-3
Online Date: 12/14/2006
Print publication date: 11/1/2006
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by Clark, Randal J.; Romilly, Douglas P.

The authors develop an eigth-order model for bending of transversally isotropic plates and use integral transforms and a collocation method to form a line-spring model for a cracked plate. The eigth-order model allows satisfaction of the three standard plate bending boundary conditions; the normal moment, twisting moment, and transverse shear force, and an additional shear stress resultant that allows analysis of transverse normal stresses near the crack tip. The line-spring model is used to develop geometry correction factors for bending of finite-thickness plates, accounting for transverse shear deformation and pressurization of the plate near the crack tip. The line-spring model is then applied to the problem of a plate with a reinforced crack, and the results are used to validate an interpolation solution based on an energy method. While not explicitly analysed, the models are applicable to many problems, including bending of bonded repairs, fracture and fatigue of composite and layered materials, surface cracks, crack tip plasticity and crack closure or crack face interaction.

DOI: 10.1007/s10704-006-9027-4
Online Date: 12/14/2006
Print publication date: 11/1/2006
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by Charles, Yann; Hild, François; Roux, Stéphane; Vandembroucq, Damien

An extensive experimental study of indentation and crack arrest statistics is presented for four different brittle materials (alumina, silicon carbide, silicon nitride, glass). Evidence is given that the crack length statistics is described by a universal (i.e., material independent) distribution. The latter directly derives from results obtained when modeling crack propagation as a depinning phenomenon. Crack arrest (or effective toughness) statistics appears to be fully characterized by two parameters, namely, an asymptotic crack length (or macroscopic toughnes) value and a power law size-dependent width. The experimental knowledge of the crack arrest statistics at one given scale thus gives access to its knowledge at all scales.

DOI: 10.1007/s10704-006-9022-9
Online Date: 12/14/2006
Print publication date: 11/1/2006
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by Tvergaard, Viggo

Crack growth along an interface between two adjacent elastic–plastic materials in a layered solid is analysed, using special interface elements to represent the fracture process ahead of the crack-tip. These interface elements account for ductile failure by the nucleation and growth of voids to coalescence. In these elements the stress components normal to the interface and the shear stresses are given by equilibrium with the surrounding material, and the stress component tangential to the interface is determined by the requirement of compatibility with the surrounding material in the tangential direction. It is assumed that the layers are sufficiently thick, so that the plastic regions around the crack-tip are much smaller than the thickness of the nearest layers. The analyses focus on the effect of initial residual stresses in the layered material, or on T-stress components induced during loading. The results show that the value of the T-stress component in the softer material adjacent to the interface crack plays the dominant role, such that a negative value of this stress component gives a significant increase of the interface fracture toughness.

DOI: 10.1007/s10704-006-9019-4
Online Date: 12/14/2006
Print publication date: 11/1/2006
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