SUMMARY Extensions of a new technique for the nite element modelling of cracks with multiple branches, multiple holes and cracks emanating from holes are presented. This extended nite element method (X-FEM) allows the representation of crack discontinuities and voids independently of the mesh. A standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method. A methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed. Computation of the stress intensity factors (SIF) in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique. Copyright ? 2000 John Wiley & Sons, Ltd.

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Arbitrary branched and intersecting cracks with the eXtended Finite Element Method

The foundations of classical homogenization techniques, which aim at predicting the overall behavior of heterogeneous materials from that of their constituents, are reviewed. After introductory definitions and a methodological preamble, attention is focused on linear elasticity, for which the basic principles of estimating and bounding the overall properties are introduced and illustrated. In this context, special recourse is made for that to the solution of the inclusion and inhomogeneity problems as reported by Eshelby in 1957. Approaches proposed recently to account in a better way for the structural morphology of the considered materials are briefly mentioned. The case of linear elasticity with eigenstrains is then discussed: several applications, including heterogeneous thermoelasticity, can be investigated within this framework. Finally, outlines of nonlinear micromechanics are briefly reported from a historical point of view: from rate-independent elastoplasticity to nonlinear elasticity and viscoplasticity, examples of a fruitful interaction between the search for new estimates and the derivation of rigorous bounds are given and the crucial question of the description of intraphase heterogeneity is emphasized. Viscoelastic coupling and rate-dependent effects are briefly discussed in conclusion.

https://hal.archives-ouvertes.fr/hal-00111366/document

Continuum Micromechanics: Survey

Crack coalescence in a rock-like material containing two cracks

A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials

A large strain elastic–viscoplastic self-consistent (EVPSC) model for polycrystalline materials is developed. At single crystal level, both the rate sensitive slip and twinning are included as the plastic deformation mechanisms, while elastic anisotropy is accounted for in the elastic moduli. The transition from single crystal plasticity to polycrystal plasticity is based on a completely self-consistent approach. It is shown that the differences in the predicted stress–strain curves and texture evolutions based on the EVPSC and the viscoplastic self-consistent (VPSC) model proposed by Lebensohn and Tome (1993) are negligible at large strains for monotonic loadings. For the deformations involving unloading and strain path changes, the EVPSC predicts a smooth elasto-plastic transition, while the VPSC model gives a discontinuous response due to lack of elastic deformation. It is also demonstrated that the EVPSC model can capture some important experimental features which cannot be simulated by using the VPSC model.

A finite strain elastic–viscoplastic self-consistent model for polycrystalline materials

A Microcrack Model of Dilatancy in Brittle Materials

Based on Hill’s method, a self-consistent averaging scheme is proposed for estimating the overall, finite deformation response of polycrystalline aggregates consisting of single crystals which undergo plastic flow by rate-dependent crystallographic slip, accompanied by elastic lattice distortion. First, constitutive relations for such single crystals are developed assuming that the slip-rate and the associated resolved shear stress are governed by: (1) a power-law relation, and (2) a viscoplastic relation. Then, Hill’s idea that the constraint imposed on a single crystal by the remaining aggregates may be represented by embedding the single crystal in a homogeneous, infinitely extended matrix having the instantaneous overall moduli, is used to formulate a completely self-consistent averaging procedure, valid for rate-dependent materials at finite strains and rotations. This method includes both the Hill and the Krӧner‒Budiansky‒Wu (K. B. W.) methods as limiting cases; when rate-effects are negligible, it reduces to Hill’s self-consistent method as formulated by Iwakuma and Nemat-Nasser for finite deformations, while it reduces to a generalized finite deformation version of the K. B. W. method for strongly rate-dependent materials. Illustrative numerical examples are presented for a plane uniaxial deformation, using a two-dimensional poly crystalline model. These examples clearly show that the rate-dependent crystallographic slip on the level of single crystals produces a more stable overall behaviour of poly crystals. This supports similar results arrived at by other investigators for single crystals and for polycrystals, by using the Taylor averaging scheme. It is shown that, while Taylor’s averaging scheme gives accurate estimates of the incremental quantities at large strains, the total overall quantities differ considerably from the ones obtained by the self-consistent method.

Rate-dependent, finite elasto-plastic deformation of polycrystals

Branched crack problems are analyzed in two-dimensional, anisotropically elastic homogeneous solids. The method of analysis is based on the complex variable approach of Savin and Lekhnitskii. The Hilbert problem in an anisotropic body is defined, and a pair of singular integral equations are derived for dislocation density functions associated with a branched crack. For both symmetric and nonsymmetric geometries, and under symmetric and antisymmetric loads, the stress intensity factors and the energy release rate are computed numerically by extrapolation for infinitesimally small lengths of branched cracks. The results are compared with those of the isotropic case given in the literature and the effects of anisotropy are discussed.

Branched cracks in anisotropic elastic solids

This paper investigates the elastic-plastic seismic behavior of long span cable-stayed steel bridges through the plane finite-element model Both geometric and material nonlinearities are involved in the analysis The geometric nonlinearities come from the stay cable sag effect, axial force-bending moment interaction, and large displacements Material nonlinearity arises when the stiffening steel girder yields The example bridge is a cable-stayed bridge with a central span length of 605 m The seismic response analyses have been conducted from the deformed equilibrium configuration due to dead loads Three strong earthquake records of the Great Hanshin Earthquake of 1995 in Japan are used in the analysis These earthquake records are input in the bridge longitudinal direction, vertical direction, and combined longitudinal and vertical directions To evaluate the residual elastic-plastic seismic response, a new kind of seismic damage index called the maximum equivalent plastic strain ratio is proposed The results show that the elastic-plastic effect tends to reduce the seismic response of long span cable-stayed steel bridges The elastic and elastic-plastic seismic response behavior depends highly on the characteristics of input earthquake records The earthquake record with the largest peak ground acceleration value does not necessarily induce the greatest elastic-plastic seismic damage

Elastic-Plastic Seismic Behavior of Long Span Cable-Stayed Bridges

Thin-walled steel columns composed of stiffened plate elements are often used in Japan as piers for elevated highway bridges in urban areas, because of their small cross-sectional areas and high earthquake resistance. An accurate finite-element-method-based (FEM-based) analysis to predict the ultimate hysteretic behavior of thin-walled steel columns subjected to seismic loading is presented in this paper. For this purpose, a three-surface cyclic metal plasticity model is developed. The three-surface model includes less material parameters and internal variables for the ease of its implementation in the FEM analysis. This model is characterized by a discontinuous surface inserted between the yield surface and the bounding surface. The discontinuous surface is used to express the yield plateau and the change of hardening coefficient. Most material parameters in the three-surface model can be determined by the tensile coupon test. However, three parameters independent of material types are calibrated by the existing cyclic loading test results of thin-walled steel columns. This is to improve the accuracy of this analysis specifically in the simulation of the localized buckling behavior, which accompanies extremely large plastic strains.

Makoto Obata

Papers

A Microcrack Model of Dilatancy in Brittle Materials

Rate-dependent, finite elasto-plastic deformation of polycrystals

Branched cracks in anisotropic elastic solids

Elastic-Plastic Seismic Behavior of Long Span Cable-Stayed Bridges

Fem analysis for hysteretic behavior of thin-walled columns