Continuum Mechanics Description of Plastic Flow Produced by Micro-Shear Bands

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Macroscopic Properties of Open-Cell Foams Based on Micromechanical Modelling

This paper presents a micromechanical analysis for the assessment of macroscopic behaviour of threedimensional open-cell solid foams. The analysis is based on material properties of a solid phase and topological arrangement of cell structure. A foam structure consists of idealized tetrahedral unit cells, which are built of four identical half-struts forming a diamond-like structure and identified as Plateau borders. Such a unit cell represents the essential microstructural features of foam. An analytical formulation of force-displacement relations for struts can be found by considering the affinity of node displacements in tensile, bending, and shear deformation. The elements of the stiffness matrix for a single cell are expressed as functions of the compliance coefficients for stretching and bending of struts. The effective elastic constants for metallic foam considered as isotropic material are determined as functions of foam relative density and compared with available results. In this paper we define an energy-based limit condition of linear elasticity for open-cell foams and calculate the critical energy density pertinent to a particular orthogonal energy state accounting for elementary interactions in a microstructure. The study based on the assumption of linear elasticity leads to simple analytical formulas. Nevertheless, it should be stressed that the proposed theoretical basis of micromechanical modelling could be also applied for the analysis of nonlinear elastic behaviour, plasticity, and failure of foams. Such problems require, however, a more complex numerical approach

Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.

Kinetics of oriented crystallization of polymers in the linear stress-orientation range in the series expansion approach

An analytical formula is derived for the oriented crystallization coefficient governing kinetics of oriented crystallization under uniaxial amorphous orientation in the entire temperature range. A series expansion approach is applied to the free energy of crystallization in the Hoffman-Lauritzen kinetic model of crystallization at accounting for the entropy of orientation of the amorphous chains. The series expansion coefficients are calculated for systems of Gaussian chains in linear stress-orientation range. Oriented crystallization rate functions are determined basing on the ‘proportional expansion’ approach proposed by Ziabicki in the steady-state limit. Crystallization kinetics controlled by separate predetermined and sporadic primary nucleation is considered, as well as the kinetics involving both nucleation mechanisms potentially present in oriented systems. The involvement of sporadic nucleation in the transformation kinetics is predicted to increase with increasing amorphous orientation. Example computations illustrate the dependence of the calculated functions on temperature and amorphous orientation, as well as qualitative agreement of the calculations with experimental results

Development of the multi-scale analysis model to simulate strain localization occurring during material processing

Abstract A detailed description of possibilities given by the developed Cellular Automata&mdash;Finite Element (CAFE) multi scale model for prediction of the initiation and propagation of micro shear bands and shear bands in metallic materials subjected to plastic deformation is presented in the work. Particular emphasis in defining the criterion for initiation of micro shear and shear bands, as well as in defining the transition rules for the cellular automata, is put on accounting for the physical aspects of these phenomena occurring in two different scales in the material. The proposed approach led to the creation of the real multi scale model of strain localization phenomena. This model predicts material behavior in various thermo-mechanical processes. Selected examples of applications of the developed model to simulations of metal forming processes, which involve strain localization, are presented in the work. An approach based on the Smoothed Particle Hydrodynamic, which allows to overcome difficulties with remeshing in the traditional CAFE method, is a subject of this work as well. In the developed model remeshing becomes possible and difficulties limiting application of the CAFE method to simple deformation processes are solved. Obtained results of numerical simulaA detailed description of possibilities given by the developed Cellular Automata&mdash;Finite Element (CAFE) multi scale model for prediction of the initiation and propagation of micro shear bands and shear bands in metallic materials subjected to plastic deformation is presented in the work. Particular emphasis in defining the criterion for initiation of micro shear and shear bands, as well as in defining the transition rules for the cellular automata, is put on accounting for the physical aspects of these phenomena occurring in two different scales in the material. The proposed approach led to the creation of the real multi scale model of strain localization phenomena. This model predicts material behavior in various thermo-mechanical processes. Selected examples of applications of the developed model to simulations of metal forming processes, which involve strain localization, are presented in the work. An approach based on the Smoothed Particle Hydrodynamic, which allows to overcome difficulties with remeshing in the traditional CAFE method, is a subject of this work as well. In the developed model remeshing becomes possible and difficulties limiting application of the CAFE method to simple deformation processes are solved. Obtained results of numerical simulations are compared with the experimental results of cold rolling process to show good predicative capabilities of the developed model.tions are compared with the experimental results of cold rolling process to show good predicative capabilities of the developed model.<br /