by Zhang, Bin; Guo, Wanlin

The out-of-plane constraints Tz around the semi-elliptical surface cracks in an elastic plate subjected to uniform tension loading have been investigated through detailed three-dimensional (3D) finite element (FE) analyses. The distributions of Tz are obtained in the vicinity of the crack border with aspect ratios of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. Tz drops from Poisson’s ratio at the crack tip to approximate zero beyond certain radial distance in the normal plane of the crack front line, and increases gradually from the free surface to the mid-plane at the same radial distance. By fitting the numerical results, empirical formulae are obtained to describe the 3D distributions of Tz for semi-elliptical surface cracks with a sufficient accuracy in the wide aspect ratio range of 0.2≤a/c ≤1.0 except very near the free surface, where Tz is extremely low. Tz, combining with the corresponding K and T or J and Q, can be applied to establish the three-parameter dominated stress field, which can characterize the 3D crack front field completely as an attempt.

DOI: 10.1007/s10704-004-5105-7
Print publication date: 1/1/2005
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by Khalmanova, Dinara; Walton, Jay. R.

The problem considered herein is the dynamic, subsonic, steady-state propagation of a semi-infinite, generalized plane strain crack in an infinite, transversely isotropic, linear viscoelastic body. The corresponding boundary value problem is considered initially for a general anisotropic, linear viscoelastic body and reduced via transform methods to a matrix Riemann–Hilbert problem. The general problem does not readily yield explicit closed form solutions, so attention is addressed to the special case of a transversely isotropic viscoelastic body whose principal axis of material symmetry is parallel to the crack edge. For this special case, the out-of-plane shear (Mode III), in-plane shear (Mode II) and in-plane opening (Mode I) modes uncouple. Explicit expressions are then constructed for all three Stress Intensity Factors (SIF). The analysis is valid for quite general forms for the relevant viscoelastic relaxation functions subject only to the thermodynamic restriction that work done in closed cycles be non-negative. As a special case, an analytical solution of the Mode I problem for a general isotropic linear viscoelastic material is obtained without the usual assumption of a constant Poisson’s ratio or exponential decay of the bulk and shear relaxation functions. The Mode I SIF is then calculated for a generalized standard linear solid with unequal mean relaxation times in bulk and shear leading to a non-constant Poisson’s ratio. Numerical simulations are performed for both point loading on the crack faces and for a uniform traction applied to a compact portion of the crack faces. In both cases, it is observed that the SIF can vanish for crack speeds well below the glassy Rayleigh wave speed. This phenomenon is not seen for Mode I cracks in elastic material or for Mode III cracks in viscoelastic material.

DOI: 10.1007/s10704-004-4270-z
Print publication date: 1/1/2005
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by Rahman, Sharif; Chen, Guofeng

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 the J-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.

DOI: 10.1007/s10704-004-3948-6
Print publication date: 1/1/2005
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by Kim, K.; Sudak, L. J.

The solution for the elastic three-phase circular inclusion problem plays a fundamental role in many practical and theoretical applications. In particular, it offers the fundamental solution for the generalized self-consistent method in the mechanics of composites materials. In this paper, a general method is presented for evaluating the interaction between a pre-existing radial matrix crack and a three-phase circular inclusion. The bonding at the inclusion-interphase interface is considered to be imperfect with the assumption that the interface imperfections are constant. On the remaining boundary, that being the interphase-matrix interface, the bonding is considered to be perfect. Using complex variable techniques, we derive series representations for the corresponding stress functions inside the inclusion, in the interphase layer and the surrounding matrix. The governing boundary value problem is then formulated in such a way that these stress distributions simultaneously satisfy the traction free condition along the crack face, the imperfect interface conditions and the prescribed asymptotic loading conditions. Stress intensity factor (SIF) calculations are performed at the crack tips for different material property combinations, imperfect interface conditions and crack positions. The results illustrate convincingly the role of an interphase layer as well as the effects of an imperfect interface on crack behavior. For instance, when the interphase layer is softer than the inclusion and matrix, the results show that the radial matrix crack will propagate from the nearby crack tip regardless of the level of the imperfect (spring-layer) interface parameter. In comparison, when the interphase layer is stiffer than the inclusion and matrix, the interphase layer will shield the crack from effects of the imperfect (spring-layer) interface. Hence, these results provide a quantitative description of the interaction problem between a three-phase inclusion with interface imperfections and a radial matrix crack.

DOI: 10.1007/s10704-004-3636-6
Print publication date: 1/1/2005
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by Carpinteri, Alberto; Invernizzi, Stefano

In this paper, an attempt is made to find some general relations for the microcutting process in brittle or quasi-brittle materials, under different hypotheses of microscopic failure behavior. Fracture beneath the indenters and sudden chip formation are the main dissipation mechanisms taken into consideration. Fracture patterns in more homogeneous brittle solids are obtained by the Finite Element Method in the framework of Linear Elastic Fracture Mechanics (LEFM). On the other hand, the quasi-brittle response due to microstructural heterogeneities is taken into account by Lattice Model simulations. The analysis is not limited to the more common study of a single indenter. When two indenters are acting in parallel, their mutual distance plays an important role. If the indenters are very close, they behave like a unique larger indenter, whereas if the distance is relatively large, their mechanical interaction vanishes. In addition, when the distance is approximately three to four times their dimension, the mechanism of chipping (with formation of secondary chip between the two parallel grooves) can take place, improving the ratio of removed volume to spent energy and then the demolition ability of the two indenters. Some comparisons are proposed between the presented approach and more sophisticated and computationally demanding models from the literature, as well as with experimental data. The analysis should provide useful hints for the optimal design of super-abrasive tools.

DOI: 10.1007/s10704-004-3635-7
Print publication date: 1/1/2005
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by Carpinteri, Alberto; Dimastrogiovanni, Luciano; Pugno, Nicola

Drilling perforations and tool wear are intimately and mutually connected by fracture propagations at different size-scales. To study this interaction phenomenon, we propose an ad hoc developed fractal coupled theory. Describing the two processes in terms of drilling and wear velocities, the theory is able to predict the relation between these two quantities. The result is represented by a power law between wear and drilling velocities with exponent comprised between 2/3 and 3/2. Some experimental tests on different materials like mortar, concrete and reinforced concrete have also been performed. Theoretical predictions and experimental results agree satisfactorily.

DOI: 10.1007/s10704-004-3634-8
Print publication date: 1/1/2005
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by Aono, Yuuta; Noguchi, Hiroshi

In this paper, a method to predict fatigue limit reliability of specimens with 2D complex rough surface is proposed. First, a effective surface profile on fatigue limit is proposed. This is obtained from the ineffective crack length against the fatigue limit. Next, an equivalent notch depth is proposed to replace a rough profile to a smooth profile with a notch. To calculate the stress concentration of the notch and to determine the equivalent notch depth, an exact solution is given for a problem of an infinite plate with a complex profile under tension. The solution is obtained with the complex variable method. Finally, a method to predict the fatigue limit reliability is discussed. The Linear Notch Mechanics and
$\sqrt{area}$ parameter model is used to predict the fatigue limit of a smooth profile with a notch, and then the fatigue limit reliability is estimated with the fatigue limit of many simulated surfaces. Moreover, rotating bending fatigue tests of 0.1% carbon steel with a complex surface are carried out. The experimental fatigue limit data is compared with the present estimated value. As results, the validity of the present method is examined.

DOI: 10.1007/s10704-004-3638-4
Print publication date: 1/1/2005
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by Broberg, K. B.

In general, numerical determination of T-stresses requires careful handling, because of their location in the vicinity of singular points. Discretization methods, such as common finite element methods, may not lead to accurate results. However, the well known technique of using dislocation arrays for determination of stress intensity factors may also be used for determination of T-stresses. Except for a few simple cases, this technique leads to a Fredholm equation, that can be solved very accurately. The general method is described and the technique is demonstrated by examples.

DOI: 10.1007/s10704-004-3637-5
Print publication date: 1/1/2005
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by Pommier, Sylvie; Risbet, Marion

Predicting fatigue crack growth in metals remains a difficult task because available models are based on cycle-derivative equations, such as the Paris law, while service loads are often far from being cyclic. The main objective of this paper is therefore to propose a set of time-derivative equations for fatigue crack growth. The model is based on the thermodynamics of dissipative processes. For this purpose, three global state variables are introduced in order to characterize the state of the crackthe crack length a, the plastic blunting at crack tip ρ and the intensity of crack opening C. Thermodynamics counterparts are introduced for each variable. Special attention is paid to the elastic energy stored inside the crack tip plastic zone, because, in practice, residual stresses at crack tip are known to considerably influence fatigue crack growth. The stored energy is included in the energy balance equation, and this leads to the appearance of a kinematics hardening term in the yield criterion for the cracked structure. No dissipation is associated with crack opening, but to crack growth and to crack tip blunting. Finally, the model consists in two laws: a crack propagation law, which is a relationship between dρ dt and da/dt and which observes the inequality stemmed from the second principle, and an elastic-plastic constitutive behaviour for the cracked structure, which provides dρ dt versus applied-load. The model was implemented and tested. It reproduces successfully the main features of fatigue crack growth as reported in the literature, such as the Paris law, the stress-ratio effect and the overload retardation effect.

DOI: 10.1007/s10704-004-3633-9
Print publication date: 1/1/2005
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by Su, R. K. L.; Sun, H. Y.

The stress intensity factors (SIFs) and the T-stress for a planar crack with anisotropic materials are evaluated by the fractal finite element method (FFEM). The FFEM combines an exterior finite element model and a localized inner model near the crack tip. The mesh geometry of the latter is self-similar in radial layers around the tip. A higher order displacement series derived from Laurent series and Goursat functions is used to condense the large numbers of nodal displacements at the inner model near the crack tip into a small set of unknown coefficients. In this study, the variations of the SIFs and the T-stress with material properties and orientations of a crack are presented. The separation of the analytical displacement series into four fundamental cases has shown to be necessary in order to cover all the material variations and the orientations of a crack in the plate with general rectilinear anisotropic materials.

DOI: 10.1007/s10704-004-3366-9
Print publication date: 1/1/2005
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