Texture predictions using a polycrystal plasticity model incorporating neighbor interactions

A viscoplastic model is presented for distributing the deformation applied to a polycrystal in a non-uniform fashion among the constituent crystals. Interactions with surrounding crystals are incorporated in the calculation of the deformation rate of each crystal through an appropriately defined local neighborhood. A compliance tensor is computed for each crystal based on a viscoplastic constitutive relation for deformation by crystallographic slip. The compliance of the crystal relative to that of its neighborhood provides a means for partitioning the macroscopic deformation rate among the crystals. The deviation of the crystal deformation rate from the macroscopic value is suitably scaled to obtain the crystal spin. Polycrystal simulations of crystallographic texture development using this model are compared to the results of similar calculations using the Taylor model, to finite element simulations of a model polycrystal, and to experimental data. The model incorporating neighbor interactions is shown to result in improved texture predictions, in terms of both the intensity levels and the locations of certain texture components.     

Texture and strain localization prediction using a N-site polycrystal model

Most polycrystal models of plastic deformation rely on the assumption that strain and stress are uniform within the domain of each grain. Comparison between measured and predicted textures suggests that this assumption is realistic for most single-phase aggregates and crystal symmetries. In this paper, we implement a self-consistent N-site model that allows one to account for strain localization and local misorientation near grain boundaries. We apply this model to face centered cubic (fcc) and hexagonal close packed (hcp) aggregates, and analyze the similarities and differences with a one-site model that assumes uniform stress and strain-rate within a grain. We find that the assumption of uniformity is justified in first order. We discuss the implications of the N-site model for the simulation of systems with hard inclusions, orientation correlations, and recrystallization mechanisms.     

Analysis for polycrystal nonproportional cyclic plasticity with a dissipated energy based rule for cross-hardening

Based on a nonclassical hardening law and the Hill’s self-consistent scheme, a new approach is proposed for the analysis of polycrystal nonproportional cyclic plasticity. A novel parameter related to the plastic dissipation on each slip system is proposed and embedded in the Bassani’s definition of cross-hardening. The tangential elastoplastic tensor relating the increments of stress and strain in a single crystal is derived and the corresponding numerical algorithm for polycrystal plasticity is developed. The elastoplastic response of 316 stainless steel subjected to typical biaxial nonproportional strain cycling is analyzed, and the main features are well replicated. The validity of the proposed approach is demonstrated by the satisfactory agreement between the computed results and experimental observation.     

A polycrystal plasticity model based on the mechanical threshold

A temperature and rate-dependent viscoplastic polycrystalmodel is presented.It uses a single crystal constitutive response that is based on the isotropic Mechanical Threshold Stress continuum model. This combination gives us theability to relate the constitutive model parameters between the polycrystaland continuum models. The individual crystal response is used to obtain themacroscopic response through the extended Taylor hypothesis. A Newton-Raphsonalgorithm is used to solve the set of fully implicit nonlinear equations for each crystal. The analysis also uses a novel state variable integration method which renders the analysis time step size independent for constant strain rate simulations. Material parameter estimates are obtained through an identification study, where the error between experimental and computed stress response is minimized. The BFGS method, which is used to solve theidentification problem, requires first-order gradients. These gradients arecomputed efficiently via the direct method of design sensitivity analysis.Texture augmentation is performed in a second identification study by changing crystal weights (volume fractions).     

An examination of the Lin model for polycrystal plasticity by means of two-dimensional finite element analysis

A two-dimensional inhomogeneous problem is calculated by the finite element method, where a nonhardening infinite medium containing inclusions is subjected to a simple tension. The results of plastic strains are substituted into the Lin and KBW models to get the stress fields, which are compared with the results of FEM. The Lin model agrees well with the FEM, but the KBW model does not. Even though the variation of plastic strain exceeds the order of elastic strain, the stresses still remain within the same order as the yielding stress. This fact may suggest that the coefficient matrix relating to the variation of plastic strain to that of stress in the Lin model is near-singular.     

On the stability of the solution to a gonorrhea discrete mathematical model

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ECAP对纯铜循环硬化/软化特性的改变

以纯铜棒材试样为研究对象,通过试验研究1、4、8道次等通道转角挤压(ECAP)后材料的单轴拉压循环行为,探讨ECAP后材料循环特性的变化,得到以下结论:(1)具有循环硬化特性的纯铜进行ECAP挤压后,其循环特性可转变为循环软化;(2)第一道次ECAP挤压对材料循环应力应变响应的强化作用最大,后续道次挤压对强化的效用迅速降低,4道次以后挤压的强化作用似可忽略不计;(3)因循环软化,纯铜经ECAP挤压后的循环应力应变曲线大大低于其单调拉伸应力应变曲线,与未经ECAP挤压的结果相反.论文研究表明,评估ECAP对材料的强化效果需同时考察材料单调加载和循环加载的力学性能.     

Predictions of forming limit diagrams using a rate-dependent polycrystal self-consistent plasticity model

In the sheet-metal forming industry, forming-limit strains have been a useful tool for quantifying metals formability. However, the experimental measurement of these strains is a difficult, time consuming and expensive process. It would be useful if strains calculated with a theoretical model could replace many of the experimental measurements. In this research, we analyze forming-limit strains of metals using a rate-dependent plasticity, polycrystal, self-consistent (VPSC) model in conjunction with the Marciniak–Kuczynski (M–K) approach. Previous researchers have studied forming limit diagrams (FLDs) based on the full-constraints Taylor model. This is the first time, to the authors’ knowledge, that the self-consistent approach has been introduced to simulate the polycrystal FLD behavior. Numerous microstructural factors characterizing the material have a strong influence on the FLD, so our model includes the effects of slip hardening, strain-rate sensitivity, anisotropy and initial texture. Finally, the calculation of the FLD with a more realistic scale transition successfully predicts some of the experimental tendencies that the Taylor model cannot reproduce for aluminum alloys AA6116-T4 and AA5182-O.     

Lagrangian description and numerical analysis of a discrete model of flexible systems

An approach is proposed to derive the equations of motion for one-dimensional discrete-continuous flexible systems with one-sided deformation characteristics. To implement this approach, the stationarity principle is generalized to dynamic problems. Solution algorithms are based on cubic spline functions. The capabilities of the approach are demonstrated by the example of a beacon buoy connected by a flexible tether to a submersible that moves along a prescribed trajectory.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 107–116, December 2004.     

On the theory of discrete structural model analysis and design

Exclusive theory for analysis of Structural Models (comprising of springs, masses, dash pots, etc.) is presented by adapting the electrical network theory. It commences a brief statement of a new Principle of Quasi Work (PQW) , relevant to this theory. Derivations presented here include theorems addressing maximum displacements, relative flexibilities, sensitivity analysis of global flexibilities, inverse problem of load prediction and interpolation of stiffnesses and flexibilities of the Structural Models. Finally a Design Equation capable of providing a starting point which more or less satisfies all the displacement constraints for iterative design employing a pair of estimated starting points for design iterations (within or outside feasible region) is evolved. Simple substantive illustrations are included to demonstrate the potential of these theoretical developments.     

A comparative study of hardening theories in torsion using the Taylor polycrystal model

A study of five rate-independent hardening rules (from Taylor and Elam (“The Distortion of an Aluminium Crystal during a Tensile Test”, Proc. R. Soc. Lond. (1923), A102, 643) to Bassani and Wu (“Latent Hardening in Single Crystals II. Analytical Characterization and Predictions,” Proc. R. Soc. Lond. (1991), A435, 21)) is presented based upon the classic Taylor polycrystal model in finite strain torsion. Comparisons of aggregate shear stress-strain curves, evolving crystallographic textures, yield loci, and axial stresses among the theories are made; and results are assessed against experiments on polycrystalline copper from the literature.     

A generalized self-consistent polycrystal model for the yield strength of nanocrystalline materials

Inspired by recent molecular dynamic simulations of nanocrystalline solids, a generalized self-consistent polycrystal model is proposed to study the transition of yield strength of polycrystalline metals as the grain size decreases from the traditional coarse grain to the nanometer scale. These atomic simulations revealed that a significant portion of atoms resides in the grain boundaries and the plastic flow of the grain-boundary region is responsible for the unique characteristics displayed by such materials. The proposed model takes each oriented grain and its immediate grain boundary to form a pair, which in turn is embedded in the infinite effective medium with a property representing the orientational average of all these pairs. We make use of the linear comparison composite to determine the nonlinear behavior of the nanocrystalline polycrystal through the concept of secant moduli. To this end an auxiliary problem of Christensen and Lo (J. Mech. Phys. Solids 27 (1979) 315) superimposed on the eigenstrain field of Luo and Weng (Mech. Mater. 6 (1987) 347) is first considered, and then the nonlinear elastoplastic polycrystal problem is addressed. The plastic flow of each grain is calculated from its crystallographic slips, but the plastic behavior of the grain-boundary phase is modeled as that of an amorphous material. The calculated yield stress for Cu is found to follow the classic Hall-Petch relation initially, but as the gain size decreases it begins to depart from it. The yield strength eventually attains a maximum at a critical grain size and then the Hall-Petch slope turns negative in the nano-range. It is also found that, when the Hall-Petch relation is observed, the plastic behavior of the polycrystal is governed by crystallographic slips in the grains, but when the slope is negative it is governed by the grain boundaries. During the transition both grains and grain boundaries contribute competitively.     

Application of strain rate potentials with multiple linear transformations to the description of polycrystal plasticity

This paper reviews a class of anisotropic plastic strain-rate potentials, based on linear transformations of the plastic strain-rate tensor. A new formulation is proposed, which includes former models as particular cases and allows for an arbitrary number of linear transformations, involving an increasing number of anisotropy parameters. The formulation is convex and fully three-dimensional, thus being suitable for computer implementation in finite element codes. The parameter identification procedure uses a micromechanical model to generate evenly distributed reference points in the full space of possible loading modes. Material parameters are determined for several anisotropic, fcc and bcc sheet metals, and the gain in accuracy of the new models is demonstrated. For the considered materials, increasing the number of linear transformations leads to a systematic improvement of the accuracy, up to a number of five linear transformations. The proposed model fits very closely the predictions of the micromechanical model in the whole space of plastic strain-rate directions. The r-values, which are not directly used in the identification procedure, served for the validation of the models and to demonstrate their improved accuracy.     

Similarity analysis and exact solutions for a general discrete two-velocity model of Boltzmann equation

Summary This paper concerns with the similarity analysis for a general discrete two-velocity model of the Boltzmann equation introduced by Illner [12]. We find the general groups of invariance and we get some exact solutions, recovering general results which contain either solutions extensively described in the literature or undiscovered ones.

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Sommario In questa nota si applica l'analisi dei gruppi infinitesimi di trasformazione ad un modello generale discreto a due velocità dell'equazione di Boltzmann introdotto da Illner [12]. Si trovano i più generali gruppi di invarianza e si ottengono alcune soluzioni esatte, ritrovando risultati generali che contengono sia soluzioni ampiamente descritte in letteratura che nuove soluzioni.

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Work supported by the C.N.R. through the G.N.F.M.     

Static and dynamic studies for coupling discrete and continuum media; Application to a simple railway track model

In this paper, we present a formulation for coupling discrete and continuum models for both dynamic and static analyses. This kind of formulation offers the possibility of carrying out better simulations of material properties than the discrete calculations, and with both larger length scales and longer times. Using only a discrete approach to simulate a large medium composed of many degrees of freedom seems very difficult in terms of calculation and implementation. Moreover, using only a continuum approach does not give an accurate solution in a zone where particular and localized phenomena can occur. A direct application of our coupling approach to the case of railway track models subjected to an external load, is proposed for its validation.     

A finite deformation analysis of polycrystal by considering the influence of grain boundary

By combining grain boundary (GB) and its influence zone, a micromechanic model for polycrystal is established for considering the influence of GB. By using the crystal plasticity theory and the finite element method for finite deformation, numerical simulation is carried out by the model. Calculated results display the microscopic characteristic of deformation fields of grains and are in qualitative agreement with experimental results.The project is supported by National Natural Science Foundation of China.     

Constitutive modeling of cyclic plasticity deformation of a pure polycrystalline copper

Cyclic plasticity experiments were conducted on a pure polycrystalline copper and the material was found to display significant cyclic hardening and nonproportional hardening. An effort was made to describe the cyclic plasticity behavior of the material. The model is based on the framework using a yield surface together with the Armstrong–Frederick type kinematic hardening rule. No isotropic hardening is considered and the yield stress is assumed to be a constant. The backstress is decomposed into additive parts with each part following the Armstrong–Frederick type hardening rule. A memory surface in the plastic strain space is used to account for the strain range effect. The Tanaka fourth order tensor is used to characterize nonproportional loading. A set of material parameters in the hardening rules are related to the strain memory surface size and they are used to capture the strain range effect and the dependence of cyclic hardening and nonproportional hardening on the loading magnitude. The constitutive model can describe well the transient behavior during cyclic hardening and nonproportional hardening of the polycrystalline copper. Modeling of long-term ratcheting deformation is a difficult task and further investigations are required.