Theoretical study of acoustoelastic effects caused by plastic anisotropy growth

The general expressions for strain-velocity relations, which are called acoustoelastic effects, of waves propagating in elasto-plastically deformed solids have been derived with aid of Jaumann co-rotational rate. In the previous paper (Kobayashi [1986]), for the slightly orthotropic case, the theoretical results of those effects were presented and it was shown that plastic anisotropy influences the acoustoelastic effects through anisotropic stress increment terms constituting plastic strain increment. In the present paper, those anisotropic stress increment terms are defined by using the unsymmetric yield function proposed by Shriastava, Mróz & Dubey [1973]; the acoustoelastic effects caused by the plastic anistropy growth are analyzed.     

Estimating the plastic strain with the use of acoustic anisotropy

Experimental verification is used to show that reference specimens and structure unloading do not permit obtaining an adequate estimate of plastic strain by measuring the acoustic anisotropy. Analytic estimates of the speed of propagation of a plane acoustic wave of various polarizations in an elastoplastic material in the direction orthogonal to the action of preliminary uniaxial stress are obtained. An analysis of the obtained relations reveala an advantage of using absolute values of the velocity of longitudinal and transverse waves for the plastic strain identification. In contrast to acoustic anisotropy, the velocities vary monotonically in a wider range of plastic strains. At the same time, the elastic strain does not affect the longitude wave velocity, which allows one to use the measurement results to estimate the character of strains.     

Parameter identification of advanced plastic strain rate potentials and impact on plastic anisotropy prediction

In the work presented in this paper, several strain rate potentials are examined in order to analyze their ability to model the initial stress and strain anisotropy of several orthotropic sheet materials. Classical quadratic and more advanced non-quadratic strain rate potentials are investigated in the case of FCC and BCC polycrystals. Different identifications procedures are proposed, which are taking into account the crystallographic texture and/or a set of mechanical test data in the determination of the material parameters.     

Multiscale modelling of the plastic anisotropy and deformation texture of polycrystalline materials

A hierarchical multilevel method is presented for the plastic deformation of polycrystalline materials with texture-induced anisotropy. It is intended as a constitutive material model for finite element codes for the simulation of metal forming processes or for the prediction of forming limits. It consists of macroscopic models of which the parameters are to be identified using the results of two-level (meso/macro) or three-level (micro/meso/macro) models. A few such two-level models are presented, ranging from the full-constraints Taylor model to the crystal-plasticity finite element models, including the grain interaction models GIA, LAMEL and ALAMEL. Validation efforts based on experimental cold rolling textures obtained for steel and aluminium alloys are shortly discussed. An assessment is also given of the assumptions of the LAMEL and ALAMEL models concerning stress and strain rate heterogeneity at grain boundaries, based on the results of a crystal plasticity finite element study. Finally a recent three-level model which also looks at the microscopic level (dislocation substructure) is discussed.     

On strong ellipticity for isotropic hyperelastic materials based upon logarithmic strain

This paper discusses various constitutive restrictions on the strain energy function for an isotropic hyperelastic material derived from the condition of strong ellipticity. The strain energy function is assumed to be a function of a novel set of invariants of the Hencky (logarithmic or natural) strain tensor introduced by Criscione et al. (J. Mech. Phys. Solids 48 (2000) 2445). A key step in the analysis is the derivation of an expression for the Fréchet derivative of the Hencky strain with respect to the deformation gradient that is convenient for analyzing the quadratic form over the space of second order tensors central to establishing strong ellipticity. The theory is illustrated by applying the restrictions to a model for rubber proposed by Criscione et al. (J. Mech. Phys. Solids 48 (2000) 2445) It is shown that while that model can be made to violate strong ellipticity, it does so only for very large strains.     

Subcritical crack growth in strain-hardening materials with allowance for strain anisotropy

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 2, pp. 90–94, February, 1988.     

Localized plastic strain in polycrystalline materials with hole and notches

Analyzed numerically are the localized strain of polycrystalline materials subjected to quasi-static loading. The objective is to study the peculiarities associated with the deformation process close to the stress concentrators such as holes, notches and interfaces of internal structure. Analytical results show that geometry and/or heterogeneous internal structure of material together with the action of maximum shear result in the development of a system of plastically deformed shear bands. Shears and rotations in the regions of strain localization are found to be higher than in other parts of the specimen while rotations are more sensitive to localization.     

Instrumentation to control strain rate for materials undergoing plastic flow

An instrument is described which controls the head velocity and, therefore, the strain rate of specimens tested in electromechanical universal testing machines. Within the available speed limits of these machines, a predetermined strain rate is achieved by automatic adjustment of a variable resistance in the speed-control circuit. This adjustment is achieved by coupling a potentiometer to the crosshead movement with a cam. An exact mathematical solution is derived for general cam profiles which give either nondecreasing or nonincreasing strain rates. Numerical results are presented for cams which achieve constant true strain rate in tensile and compression testing. For these cases, instrument calibration curves and tensile stress-strain curves for iron at room temperature are presented.     

Description of plastic anisotropy effects at large deformations. Part II: the case of transverse isotropy

In a previous paper (see Tsakmakis, 1999) a general thermodynamically consistent (visco-) plasticity theory has been developed, which accounts for anisotropy effects. For simplicity, isotropic hardening has not be regarded, while anisotropy arises from kinematic hardening and orientational evolution of the underlying substructure. In the present paper the capabilities of this theory are discussed for the study case of transverse isotropy. Anisotropy effects are elaborated in the free energy and the yield function by means of structural tensors. Characteristic features of the transversely isotropic model are illustrated for the case of homogeneous simple shear.     

Plane strain slip line theory for anisotropic rigid/plastic materials

A slip line theory governing states of incipient plane flow at the yield point load is developed for anisotropic rigid/plastic materials which exhibit a reduced yield criterion, governing states of plane flow, that depends only on the deviatoric parts of the in-plane stress tensor. It is shown that every homogeneous, incompressible material which complies with the principle of maximum plastic work, but which is of otherwise arbitrary anisotropy, is of this class. The stress equilibrium requirements are seen to take a remarkably simple form expressing the constancy of the quantities mean in-plane normal stress plus or minus arc length around the governing yield contour in a Mohr stress plane along members of the two slip line families. Further, this generalization of the Hencky equations is valid for every material of the considered class. Some special features of yield contours containing corners and flat segments are discussed, and velocity equations are given for materials complying with the maximum work inequality. The theory is applied to obtain the solution for indentation of an arbitrarily anisotropic half-space with a flat-ended, rigid, frictionless punch. A simple, universal formula, involving only geometrical dimensions of the governing yield contour, is derived for the yield point indentation pressure.     

Evolving internal length scales in plastic strain localization for granular materials

A numerical model in the Cosserat continuum for strain localization phenomena in granular materials is developed and proposed in this paper. The model assumes a constant internal length scale that is used to describe the shear band thickness. However, it is observed that the internal length scales need to change to accommodate the possible change in the contact surface between the particles, damage of the particles or/and any change in the local void ratio within the domain, which will change the shear band thickness. The mathematical formulations used in the present numerical model were equipped with evolution equations for the length scales through the Micropolar theory, those formulations are proposed and discussed in this paper. The evolution equations of the internal length scales describe any possible change in the contact surface between the particles, damage of the particles if exists and/or any change in the local void ratio within the domain. Hence, the strain localization described by the enhanced model with evolving internal length scales is more accurate and closer to the real solution. The solution for the shear bands thickness shows more accurate correlation with the experimental results and less dependency on the mesh size when such evolution equations are used. Moreover, the shear band thickness and inclination evolve during the deformation process.     

《Elaticity of Transversely Isotropic Materials》一书评介

浙江大学土木系丁皓江教授和陈伟球教授及澳大利亚悉尼大学航空、机械与机电工程学院章亮炽教授的专著“Elasticity of Transversely Isotropic Ma- terials”(ISBN：1-4020-4033-4),2006年由Springer公司出版,该书是加拿大著名力学家G．M．L．Gladwell     

Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals

A Taylor-like polycrystal model is adopted here to investigate the plastic behavior of body centered cubic (b.c.c.) sheet metals under plane-strain compression and the subsequent in-plane biaxial stretching conditions. The <111> pencil glide system is chosen for the slip mechanism for b.c.c. sheet metals. The {110} <111> and {112} <111> slip systems are also considered. Plane-strain compression is used to simulate the cold rolling processes of a low-carbon steel sheet. Based on the polycrystal model, pole figures for the sheet metal after plane-strain compression are obtained and compared with the corresponding experimental results. Also, the simulated plane-strain stress—strain relations are compared with the corresponding experimental results. For the sheet metal subjected to the subsequent in-plane biaxial stretching and shear, plastic potential surfaces are determined at a given small amount of plastic work. With the assumption of the equivalence of the plastic potential and the yield function with normality flow, the yield surfaces based on the simulations for the sheet metal are compared with those based on several phenomenological planar anisotropic yield criteria. The effects of the slip system and the magnitude of plastic work on the shape and size of the yield surfaces are shown. The plastic anisotropy of the sheet metal is investigated in terms of the uniaxial yield stresses in different planar orientations and the corresponding values of the anisotropy parameter R, defined as the ratio of the width plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial tensile loading. The uniaxial yield stresses and the values of R at different planar orientations from the polycrystal model can be fitted well by a yield function recently proposed by Barlat et al. (1997b).     

Shear modulus of transversely isotropic materials

A new and simple method is presented to determine the inde-pendent shear modulus of transversely isotropic material.Mathematical formulation,derivation and solution are given,and test apparatus and results are presented.The method wastested on one of such materials-Green River Formation oilshale.Comparisons wich other approximate results and acou-stical methods are made.Confirmation of the test methodwith materials having known shear moduli is also presented.     

A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F=F^eF^p [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion F^e contributes to a standard elastic free-energy ψ^(e), while , the energetic part of F^p, contributes to a defect energy ψ^(p) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since F^e=FF^p-1 and , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for F^p and – the two flow rules of the theory.We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.     

Analytical solutions of mixed mode plane strain crack problems in elastic perfectly plastic materials

In the present paper analytical solutions concerning the stress state at the tip of a crack in an elastic-perfectly plastic body, subjected to mixed mode loadings under plane strain conditions, are presented. Analytical solutions of the nonlinear ordinary differential equations are obtained and the dominant singularity is completely determined with the aid of suitable boundary conditions. The obtained results are in perfect agreement with those given by other investigators, both analytical and numerical. The novel aspect here is the methodology used for the solution, as well as the direct determination of the plastic zones. As a consequence, the resulting analytical solutions cover many more problems in the mathematical theory of plasticity compared to similar existing methods and they may be proved of importance in various applications.