Second-order estimates of the self-consistent type for viscoplastic polycrystals

The ‘second–order’ homogenization procedure of Ponte Castañeda is used to propose new estimates of the self–consistent type for the effective behaviour of viscoplastic polycrystals. This is accomplished by means of appropriately generated estimates of the self–consistent type for the relevant ‘linear thermoelastic comparison composite’, in the homogenization procedure. The resulting nonlinear self–consistent estimates are the only estimates of their type to be exact to second order in the heterogeneity contrast, which, for polycrystals, is determined by the grain anisotropy. In addition, they satisfy the recent bounds of Kohn and Little for two–dimensional power–law polycrystals, which are known to be significantly sharper than the Taylor bound at large grain anisotropy. These two features combined, suggest that the new self–consistent estimates, obtained from the second–order procedure, may be the most accurate to date. Direct comparison with other self–consistent estimates, including the classical incremental and secant estimates, for the special case of power–law creep, appear to corroborate this observation

Second-order estimates of the self-consistent type for viscoplastic polycrystals

The ‘second–order’ homogenization procedure of Ponte Castañeda is used to propose new estimates of the self–consistent type for the effective behaviour of viscoplastic polycrystals. This is accomplished by means of appropriately generated estimates of the self–consistent type for the relevant ‘linear thermoelastic comparison composite’, in the homogenization procedure. The resulting nonlinear self–consistent estimates are the only estimates of their type to be exact to second order in the heterogeneity contrast, which, for polycrystals, is determined by the grain anisotropy. In addition, they satisfy the recent bounds of Kohn and Little for two–dimensional power–law polycrystals, which are known to be significantly sharper than the Taylor bound at large grain anisotropy. These two features combined, suggest that the new self–consistent estimates, obtained from the second–order procedure, may be the most accurate to date. Direct comparison with other self–consistent estimates, including the classical incremental and secant estimates, for the special case of power–law creep, appear to corroborate this observation

Infinite-contrast periodic composites with strongly nonlinear behavior: Effective-medium theory versus full-field simulations

This paper presents a combined numerical-theoretical study of the macroscopic behavior and local field distributions in a special class of two-dimensional periodic composites with viscoplastic phases. The emphasis is on strongly nonlinear materials containing pores or rigid inclusions. Full-field numerical simulations are carried out using a Fast-Fourier Transform algorithm [H. Moulinec, P. Suquet, C. R. Acad. Sci. Paris II 318, 1417 (1994)] Moulinec, P. Suquet, C. R. Acad. Sci. Paris II 318, 1417 (1994), while the theoretical results are obtained by means of the `second-order' nonlinear homogenization method [P. Ponte Castaneda, J. Mech. Phys. Solids 50, 737 (2002)]. The effect of nonlinearity and inclusion concentration is investigated in the context of power-law (with strain-rate sensitivity m) behavior for the matrix phase under in-plane shear loadings. Overall, the `second-order' estimates are found to be in good agreement with the numerical simulations, with the best agreement for the rigidly reinforced materials. For the porous systems, as the nonlinearity increases (m decreases), the strain field is found to localize along shear bands passing through the voids (the strain fluctuations becoming unbounded) and the effective stress exhibits a singular behavior in the dilute limit. More specifically, for small porosities and fixed nonlinearity m>0, the effective stress decreases linearly with increasing porosity. However, for ideally plastic behavior (m = 0), the dependence on porosity becomes non-analytic. On the other hand, for rigidly-reinforced composites, the strain field adopts a tile pattern with bounded strain fluctuations, and no singular behavior is observed (to leading order) in the dilute limit.Comment: 28 pages, 28 B&W figures, 2 tables of color maps, to be published in International Journal of Solids and Structures (in press

Homogenization-Based Predictions for Texture Evolution in Halite

International audienceThe “variational” homogenization method developed by deBotton and Ponte Castaneda [2] is used here to predict texture development in halite polycrystals at room and high temperatures accounting for hardening and grain shape changes. The new predictions are compared with those of the Taylor and “tangent” model of Molinari et al. [5] for uniaxial tension and compression. The predictions of the “variational” model are found to be intermediate between the Taylor and “tangent” predictions, although not too different from either, as a consequence of the relatively high isotropy of the halite single crystal grains

Path Integral Approach to Strongly Nonlinear Composite

We study strongly nonlinear disordered media using a functional method. We solve exactly the problem of a nonlinear impurity in a linear host and we obtain a Bruggeman-like formula for the effective nonlinear susceptibility. This formula reduces to the usual Bruggeman effective medium approximation in the linear case and has the following features: (i) It reproduces the weak contrast expansion to the second order and (ii) the effective medium exponent near the percolation threshold are $s=1$, $t=1+\kappa$, where $\kappa$ is the nonlinearity exponent. Finally, we give analytical expressions for previously numerically calculated quantities.Comment: 4 pages, 1 figure, to appear in Phys. Rev.

Analytical and numerical analyses of the micromechanics of soft fibrous connective tissues

State of the art research and treatment of biological tissues require accurate and efficient methods for describing their mechanical properties. Indeed, micromechanics motivated approaches provide a systematic method for elevating relevant data from the microscopic level to the macroscopic one. In this work the mechanical responses of hyperelastic tissues with one and two families of collagen fibers are analyzed by application of a new variational estimate accounting for their histology and the behaviors of their constituents. The resulting, close form expressions, are used to determine the overall response of the wall of a healthy human coronary artery. To demonstrate the accuracy of the proposed method these predictions are compared with corresponding 3-D finite element simulations of a periodic unit cell of the tissue with two families of fibers. Throughout, the analytical predictions for the highly nonlinear and anisotropic tissue are in agreement with the numerical simulations

Effect of stress-triaxiality on void growth in dynamic fracture of metals: a molecular dynamics study

The effect of stress-triaxiality on growth of a void in a three dimensional single-crystal face-centered-cubic (FCC) lattice has been studied. Molecular dynamics (MD) simulations using an embedded-atom (EAM) potential for copper have been performed at room temperature and using strain controlling with high strain rates ranging from 10^7/sec to 10^10/sec. Strain-rates of these magnitudes can be studied experimentally, e.g. using shock waves induced by laser ablation. Void growth has been simulated in three different conditions, namely uniaxial, biaxial, and triaxial expansion. The response of the system in the three cases have been compared in terms of the void growth rate, the detailed void shape evolution, and the stress-strain behavior including the development of plastic strain. Also macroscopic observables as plastic work and porosity have been computed from the atomistic level. The stress thresholds for void growth are found to be comparable with spall strength values determined by dynamic fracture experiments. The conventional macroscopic assumption that the mean plastic strain results from the growth of the void is validated. The evolution of the system in the uniaxial case is found to exhibit four different regimes: elastic expansion; plastic yielding, when the mean stress is nearly constant, but the stress-triaxiality increases rapidly together with exponential growth of the void; saturation of the stress-triaxiality; and finally the failure.Comment: 35 figures, which are small (and blurry) due to the space limitations; submitted (with original figures) to Physical Review B. Final versio

Phase field modeling of nonlinear material behavior

Materials that undergo internal transformations are usually described in solid mechanics by multi-well energy functions that account for both elastic and transformational behavior. In order to separate the two effects, physicists use instead phase-field-type theories where conventional linear elastic strain is quadratically coupled to an additional field that describes the evolution of the reference state and solely accounts for nonlinearity. In this paper we propose a systematic method allowing one to split the non-convex energy into harmonic and nonharmonic parts and to convert a nonconvex mechanical problem into a partially linearized phase-field problem. The main ideas are illustrated using the simplest framework of the Peierls-Nabarro dislocation model.Comment: 12 pages, 4 figures. v1: as submitted. v2: as published (conclusion added, unessential part of appendix removed, minor typesetting revisions). To appear in: K. Hackl (ed.), Proceedings of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, September 22-26, 2008, Bochum. (Springer-Verlag, 2010 presumably