Meshless methods: An overview and recent developments

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

https://orbilu.uni.lu/bitstream/10993/13726/1/phu-meshless.pdf

Review: Meshless methods: A review and computer implementation aspects

Publisher Summary This chapter describes the material failure by coalescence of microscopic voids. The voids nucleate mainly at second phase particles, by decohesion of the particle-matrix interface or by particle fracture, and subsequently the voids grow because of plastic straining of the surrounding material. The growth of voids to coalescence by plastic yielding of the surrounding material involves so large geometry changes that finite strain formulations of the field equations are a necessary tool. A convected coordinate formulation of the governing equations is used. Convected coordinates are introduced, which serve as particle labels. The convected coordinate net can be visualized as being inscribed on the body in the reference state and deforming with the material. It is found that after nucleation, cavities elongate along the major tensile axis and that two neighboring cavities coalesce when their length has grown to the order of magnitude of their spacing. This local failure occurs by the development of slip planes between the cavities or simply necking of the ligament. A detailed micromechanical study of shear band bifurcation that accounts for the interaction between neighboring voids and the strongly nonhomogeneous stress distributions around each void has been carried out, and are also elaborated in this chapter.

Material Failure by Void Growth to Coalescence

Uniaxial compression of plates of brittle materials containing pre-existing planar cracks oriented at certain angles with respect to the direction of overall compression has revealed that the relative frictional sliding of the faces of the pre-existing cracks may produce, at their tips, tension cracks which deviate at sharp angles from the sliding plane. These tension cracks then continue to grow in a stable manner with increasing axial compression, curving toward an orientation parallel to the direction of axial compression. Within the framework of linear fracture mechanics, the out-of-plane extension of a pre-existing straight crack, induced by overall far-field compression, is analyzed, and various parameters which characterize the growth process are quantified. It is shown analytically that, for a wide range of pre-existing crack orientations, the out-of-plane crack extension initiates at an angle close to 70° from the direction of the pre-existing crack; the exact value of this angle, of course, depends on the friction factor and the orientation of the pre-existing crack. It is found that the growth process is stable initially, but the rate of increase of the length of the extended portion with respect to the increasing axial compression dramatically increases after a certain extension length is attained, and in fact, this length becomes unbounded if a small lateral tension also exists. Various limiting cases are examined and the corresponding analytical estimates are compared with the numerical results, arriving at good correlations. A series of qualitative experiments is performed on thin plates of Columbia Resin CR 39, arriving at excellent agreement with the analytical results. In light of the analysis, the phenomena of axial splitting, exfoliation (or sheet fracture), and rockburst are examined, and it is suggested that they may all be the results of the out-of-plane (tensile) extension of pre-existing cracks, induced by large overall far-field compressions. This assertion is then supported by a series of experiments which show that the relative frictional sliding of the faces of one or even an array of pre-existing cracks does not result in coplanar (sliding mode) crack growth, but rather leads to the formation of tension cracks which grow in the direction of maximum compression. Moreover, a pre-existing crack close to a free boundary grows in a similar manner under compression parallel to the boundary, and shows no tendency to move toward the free surface. Possible lateral buckling which may result, and which may cause further unstable crack extension, is illustrated experimentally, and discussed in an effort to shed light on the phenomena of rockburst and surface spalling.

Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst

The objective of this study was to estimate the strength and deformability of corroded steel plates under quasi-static uniaxial tension. In order to accurately simulate this problem, we first estimated the true stress–strain relationship of a flat steel plate by introducing a vision sensor system to the deformation measurements in tensile tests. The measured true stress–stain relationship was then applied to a series of nonlinear implicit three-dimensional finite element analyses using commercial code LS-DYNA. The strength and deformability of steel plates with various pit sizes, degrees of pitting intensity, and general corrosion were estimated both experimentally and numerically. The failure strain in relation to the finite element mesh size used in the analyses was clarified. Two different steels having yield ratios of 0.657 and 0.841 were prepared to examine the material effects on corrosion damage. The strength and deformability did not show a clear dependence on the yield ratios of the present two materials, whereas a clear dependence was shown with respect to the surface configuration such as the minimum cross-sectional area of the specimens, the maximum depth of the pit cusp from the mean corrosion diminution level, and pitting patterns. Empirical formulae for the reduction of deformability and the reduction of energy absorption of pitted plates were proposed which may be useful in strength assessment when examining the structural integrity of aged corroded structures.

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Strength and deformability of corroded steel plates under quasi-static tensile load

The problem of a slightly non-collinear, quasi-static crack growth in a finite brittle solid is examined. A solution method which accounts for the finite geometry and the applied boundary conditions is presented. The approach uses the results of the first order perturbation solution for non-collinear crack extension from the tip of a semi-infinite straight crack, in an alternating solution scheme which takes into account the effect of the finite geometry and the boundary conditions in an iterative manner. For illustration, a slightly slanted Griffith crack under biaxial loading is examined.

On crack branching and curving in a finite body

Unstable growth of tension cracks in brittle solids: Stable and unstable bifurcations, snap-through, and imperfection sensitivity

It is well known that Wagner's theory for the impact between a flat body and a water surface cannot be applied to very small impact angles because of the effects of the trapped air. We focus our attention on the impact problem with a small impact angle, which has not been studied in detail owing to theoretical and experimental difficulties. In order to investigate the transitional impact behavior from a trapped-air impact to a Wagner-type impact, we carried out precise pressure and strain measurements by dropping a plate and increasing the impact angle, β, from 0° to 4° by increments of 0.5°. Based on the experimental results, the time histories of the measured pressures were identified as belonging to three patterns: the Wagner type, the trapped-air type, and the intermediate type. During the transitional impact process, the Wagner-type pattern was observed near the keel at the beginning of the impact, with the trapped-air pattern toward the edge of the plate. The Wagner-type pressure pattern dominated with increasing impact angle. Although high peak pressures appeared in the transitional impact process, the maximum strains measured in the plate were not so sensitive to the impact angle. It was found that the structural response can be estimated by using the average pressure at impact, which leads to a new design approach for small impact angles.

Yoichi Sumi

Papers

Strength and deformability of corroded steel plates under quasi-static tensile load

On crack branching and curving in a finite body

Unstable growth of tension cracks in brittle solids: Stable and unstable bifurcations, snap-through, and imperfection sensitivity

On the water impact and elastic response of a flat plate at small impact angles

Computational crack path prediction