An elasto-plastic constitutive model with plastic strain rate potentials for anisotropic cubic metals

An elasto-plastic constitutive model with the plastic strain rate potential was developed for finite element analysis. In the model, isotropic-kinematic hardening was incorporated under the plane stress condition for anisotropic sheet cubic metal forming analysis. The formulation is general enough for any homogeneous plastic strain rate potential (with the first-order homogeneous effective strain rate) but the plastic strain rate potential Srp2004-18p was considered here. Attention was focused on the development of the elasto-plastic transition criterion and the effective stress update algorithm. Also, to assure the quadratic convergence rate in Newton’s method, the elasto-plastic tangent modulus was analytically derived. Accuracy and convergence of the stress update algorithm were assessed by the iso-error maps, whereas stability of the algorithm was confirmed by analytical procedure. Validations were performed for the examples of the circular cup drawing, 2D draw-bending and unconstrained cylindrical bending tests, utilizing aluminum sheet alloys.     

帘线/橡胶复合材料各向异性黏-超弹性本构模型

帘线/橡胶复合材料广泛应用于轮胎等重要工程领域,为了描述其在服役条件下的大变形、非线性、各向异性和高应变率等材料力学行为,基于纤维增强复合材料连续介质力学理论,提出了一种考虑应变率效应的帘线/橡胶复合材料各向异性黏-超弹性本构模型. 该模型中单位体积的应变能被解耦为便于参数识别的基体等容变形能、帘线拉伸变形能、剪切应变能和黏性应变能四部分. 给出了模型参数的确定方法,并通过拟合文献中单轴拉伸、偏轴拉伸实验数据,得到了模型参数. 利用该模型预测了不同加载和变形条件下的力学行为,并将预测结果与实验结果对比分析. 结果表明, 考虑黏性模型和不考虑黏性模型对不同应变率变形条件下的预测结果相差很大,且考虑黏性模型的预测结果与实验结果吻合很好. 因此,与不考虑黏性模型相比,所提出的各向异性黏-超弹性本构模型能更好地表征帘线/橡胶复合材料在大变形、高应变率条件下的力学特性.      

Failure prediction in anisotropic sheet metals under forming operations with consideration of rotating principal stretch directions

An approximate macroscopic yield criterion for anisotropic porous sheet metals is adopted to develop a failure prediction methodology that can be used to investigate the failure of sheet metals under forming operations. Hill's quadratic anisotropic yield criterion is used to describe the matrix normal anisotropy and planar isotropy. The approximate macroscopic anisotropic yield criterion is a function of the anisotropy parameter R, defined as the ratio of the transverse plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial loading conditions. The Marciniak–Kuczynski approach is employed here to predict failure/plastic localization by assuming a slightly higher void volume fraction inside randomly oriented imperfection bands in a material element of interest. The effects of the anisotropy parameter R, the material/geometric inhomogeneities, and the potential surface curvature on failure/plastic localization are first investigated. Then, a non-proportional deformation history including relative rotation of principal stretch directions is identified in a critical element of a mild steel sheet under a fender forming operation given as a benchmark problem in the 1993 NUMISHEET conference. Based on the failure prediction methodology, the failure of the critical sheet element is investigated under the non-proportional deformation history. The results show that the gradual rotation of principal stretch directions lowers the failure strains of the critical element under the given non-proportional deformation history.     

Evolution of plastic anisotropy in amorphous polymers during finite straining

The large strain deformation response of amorphous polymers results primarily from orientation of the molecular chains within the polymeric material during plastic straining. Molecular network orientation is a highly anisotropic process, thus the observed mechanical response is strongly a function of the anisotropic state of these materials. Through mechanical testing and material characterization, the nature of the evolution of molecular orientation under different conditions of state of strain is developed. The role of developing anisotropy on the mechanical response of these materials is discussed in the context of assessing the capabilities of several models to predict the state of deformation-dependent response. A three-dimensional rubber elasticity spring system that is capable of capturing the state of deformation dependence of strain hardening is used to develop a tensorial internal state variable model of the evolving anisotropic polymer response. This fully three-dimensional constitutive model is shown to be successfully predictive of the true stress vs. true strain data obtained in our isothermal uniaxial compression and plane strain compression experiments on amorphous polycarbonate (PC) and polymethylmethacrylate (PMMA) at moderate strain rates. A basis is established for providing the polymer designer with the ability to predict the flow strengths and deformation patterns of highly anisotropic materials. A companion paper by Arruda, Boyce, and Quintus-Bosz [in press] shows how the model developed herein is used to predict various anisotropic aspects of the large strain mechanical response of preoriented materials. Additional work has been done to extend the model to include the effects of strain rate and temperature in Arruda, Jayachandran, and Boyce [in press].     

The strain rate intensity factor in the plane strain compression of thin anisotropic metal strip

Using an available analytic solution for instantaneous plane strain compression of a plastically anisotropic strip between two parallel plates the strain rate intensity factor is found assuming Hill’s quadratic yield criterion. The distribution of material properties is uniform. The effect of parameters characterizing plastic anisotropy of the strip on the magnitude of the strain rate intensity factor is demonstrated. A possibility to replace the strain rate intensity factor with the plastic work rate intensity factor is discussed. Singular behavior of the plastic spin in the vicinity of the friction surface is revealed and discussed.     

Large strain creep analysis of thick-walled cylinders

The finite-strain theory has been used to study the creep behaviour of a thick-walled cylinder under large strains. The analysis is divided into two parts. In part 1 the creep deformation of a thick-walled cylinder of an anisotropic material subjected to internal pressure has been discussed. The effect of the anisotropy has been depicted graphically. It is found that the anisotropy of the material has a significant effect on the axial stress, strain and strain rate. Part 2 of the paper deals with the creep analysis of cylinders of either isotropic or anisotropic materials subjected to combined internal and external pressures. The effect of the anisotropy is found to be similar to that found in part 1. It is seen, however, that the introduction of external pressure results in decreasing the strain rate and thus increasing the life of the cylinder.     

A quasi-three-dimensional approach for homogeneous anisotropic plates undergoing extensional deformations

Summary In present paper, the Kane-Mindlin assumptions are applied to homogeneous anisotropic plates undergoing extensional deformations. A complex potential approach is formulated;to that end, a third complex potential is added to the two Lekhniskii complex potentials, [2]. The components of displacement and stress as well as boundary conditions are represented in terms of these three complex potentials. The advantage of the proposed approach is that problems of anisotropic plates under extensional deformations are treated as a quasi-three-dimensional problem by considering the effect of the transverse normal strain.     

Analysis of stress-strain relations by use of an anisotropic hardening plastic potential

A method of analyzing plastic behavior by use of an anisotropic hardening plastic potential is proposed. The plastic potential surface in deviatoric stress space is assumed to be the same as the equi-plastic-strain surface. Stress-strain relations in combined loading and in multi-axial cyclic loading are calculated by use of the anisotropic hardening plastic potential and the normality rule of the plastic strain increment vector to the plastic potential surface, which are experimentally determined or confirmed by subjecting thinwalled tubular test specimens of $\text{60}\text{40}$ brass to combined axial load, internal pressure and torsion. The calculated results agree fairly well with the experimental observations.     

Effects of initial anisotropy on the finite strain deformation behavior of glassy polymers

Solid phase deformation processing of glassy polymers produces highly anisotropic polymer components as a result of the massive reorientation of molecular chains during the large strain forming operation. Indeed, the polymer preform used as the starting materials is usually anisotropic owing to its prior deformation history. The process end product has often been fashioned for a particular application, i.e. to possess an increased flow strength along a particular axis, thereby exploiting the orientation induced anisotropy effects. The fully three-dimensional issues involved in the use of glassy polymer components include anisotropic flow strenghts, limiting extensibilities, and deformation patterns. These characteristics have been altered by the initial forming operation but are obviously not expected to be enhanced in all directions. The presence of anisotropy in structural components may also lead to premature failure or unexpected shear localization. In this report the effects of initial deformation and the associated anisotropies are investigated through uniaxial compression tests on preoriented polycarbonate (PC) and polymethylmethacrylate (PMMA) specimens. The evolving anisotropy is monitored by testing materials preoriented by various amounts of strain and under different states of deformation. The tensorial nature of the anisotropic material is characterized by examining the preoriented material response in three orthogonal directions. A model for the large strain deformation response of glassy polymers has been shown by Arruda and Boyce [in press] to be well predictive of the evolution of anisotropy during deformation in initially isotropic materials. Here the authors evaluate the ability of the model developed in Arruda and Boyce [in press] to predict several aspects of the anisotropic response of preoriented materials. Using material properties determined from the characterization of the isotropic material response and a knowledge of the anisotropic state of the preoriented material, model simulations are shown to accurately capture all aspects of the large strain anisotropic response including flow strengths, strain hardening characteristics, cross-sectional deformation patterns, and limiting extensibilities. Although anisotropy has been shown to evolve with temperature and strain rate in Boyce, Arruda and Jayachandran [in press] and also state of deformation in Arruda and Boyce [in press], we submit an experimental observation that the subsequent deformation response of preoriented polymers may be predicted using only a measure of optical anisotropy, and not the prior strain or thermal history. Optical anisotropy, as measured for example by birefringence, therefore represents a true internal variable indicative of the evolution of anisotropy with inelastic strain, state of strain, and temperature.     

各向异性材料疲劳损伤模拟

根据连续损伤理论，考虑到了损伤能释放率等引起损伤的重要因素，提出了一个各向异性疲劳损伤模型。此模型结构简单，适用于各种各向异性材料。利用该模型对玻璃聚酯复合材料层板在单向受力情况下进行了寿命预测，预测值与实验值符合较好     

A plane stress anisotropic plastic flow theory for orthotropic sheet metals

A phenomenological theory is presented for describing the anisotropic plastic flow of orthotropic polycrystalline aluminum sheet metals under plane stress. The theory uses a stress exponent, a rate-dependent effective flow strength function, and five anisotropic material functions to specify a flow potential, an associated flow rule of plastic strain rates, a flow rule of plastic spin, and an evolution law of isotropic hardening of a sheet metal. Each of the five anisotropic material functions may be represented by a truncated Fourier series based on the orthotropic symmetry of the sheet metal and their Fourier coefficients can be determined using experimental data obtained from uniaxial tension and equal biaxial tension tests. Depending on the number of uniaxial tension tests conducted, three models with various degrees of planar anisotropy are constructed based on the proposed plasticity theory for power-law strain hardening sheet metals. These models are applied successfully to describe the anisotropic plastic flow behavior of 10 commercial aluminum alloy sheet metals reported in the literature.     

A general perturbation method for inhomogeneities in anisotropic and piezoelectric solids with applications to quantum-dot nanostructures

By introducing a homogeneous piezoelectric material and its Green’s function, we present a new semi-analytical three-dimensional perturbation method for general inhomogeneity problems in anisotropic and piezoelectric solids. This method removes the limitations associated with previous analytical methods, which often ignore the anisotropic properties or the difference between the material properties of the inhomogeneity and its surrounding matrix. As an important application, the proposed theory is employed to calculate the elastic and electric fields in a truncated pyramidal InAs/GaAs quantum-dot (QD) nanostructure. Numerical results demonstrate that the anisotropy of the materials and the difference between the material constants of the QD and the matrix have a significant influence on the strain and electric fields. The relative differences of the strain and electric field inside the QD between the simplified isotropic and homogeneous model and the real anisotropic and heterogeneous one may reach 22% and 53%, respectively. The accuracy of the calculated elastic strain and electric fields is improved greatly by a second order approximate solution (OAS). Since the third OAS nearly coincides with the second one, good convergence of the iteration procedure is demonstrated. Moreover, contours of the hydrostatic strain and electric potential within and around the QD are also presented and analyzed.     

Formability of TWIP (twinning induced plasticity) automotive sheets

The main objective of this work is to experimentally and numerically evaluate the macro-performance of the automotive TWIP (twinning induced plasticity) sheet in conjunction with formability. In order to characterize the mechanical properties, the simple tension and compression tests were performed for anisotropic properties, while the strain rate test was carried out to evaluate strain rate sensitivity. The forming limit diagram was measured and incorporated into the simulation program, while the theoretical prediction of the diffuse and localized necking was also carried out utilizing Hill’s and the M-K theories as well as Dorn’s and Swift’s diffuse theories. Note that the generalized criteria of Hill’s, Dorn’s and Swift’s theories were derived for general anisotropic sheets as well in this work. For numerical simulations, the anisotropic yield functions Yld2000-2d and Hill48 as well as the isotropic Mises yield function were selectively applied along with the isotropic hardening law. Formability verification was performed, utilizing Yld2000-2d, for the hemispherical dome stretching, notch and simple tension tests with specimens selectively prepared by milling and punching, while anisotropic properties were verified through the three point bending and cylindrical cup drawing tests, comparing the performance of the three yield functions.     

一种碳纤维织物增强复合材料应变率相关的各向异性强度准则

为了获得一种碳纤维二维正交平纹机织布增强树脂基复合材料在一维应变状态下的强度准则,在已完成的准静态和动态压缩实验的基础上,拟合出了单轴压缩下三个主方向上的计及应变率的应力-应变关系式,进而得到初始屈服应力和压缩破坏强度与应变率相关性表达式。依据该表达式,得到了该复合材料在一维应变下考虑应变率效应的Tsai-Hill屈服强度和破坏强度准则方程。通过计算,考察了Tsai-Hill屈服强度和破坏强度准则随应变率的变化规律。结果表明,本文中研究的复合材料的强度性能,不但存在应变率效应,而且这种效应是各向异性的。     

One parameter model for creep in a whisker reinforced anisotropic rotating disc of Al–SiCw composite

Steady state creep in a rotating disc of anisotropic aluminum silicon carbide whisker composite has been studied in the present study. The creep behavior is described by Norton's power law. Stress and strain rate distributions for anisotropic discs have been calculated and compared with those obtained for isotropic disc. It is concluded that the radial strain rate which always remained compressive for the isotropic composite ($\alpha =1.0$) and anisotropic disc ($\alpha =1.3$), becomes tensile in the middle region of the disc when the anisotropy parameter $\alpha =0.7$. Also if α is reduced from 1.3 to 0.7 the variation of tensile strain rate in the tangential direction remains similar but the magnitude reduces by five orders of magnitude. The study revealed that anisotropy introduces significant change in the strain rates although its effect on the resulting stress distribution may be relatively small.     

A solution of torsional problem by energy method in case of anisotropic cross-section

We proposed an original method to investigate the problem of torsion of anisotropic cross-section. We implemented an energy method to calculate the stress function represented by infinite series of trigonometric functions adapted to rectangular cross-section. After validation, we implemented a parametric sensitivity study to investigate the influence of the cross-section aspect ratio and the anisotropy level on the stress function, the strain energy density and the torsion stiffness. The process showed a fast convergence with a very good accuracy. The model showed a potential interest for the experimental identification of anisotropic material properties.     

On non-associative anisotropic finite plasticity fully coupled with isotropic ductile damage for metal forming

In this work, non-associative finite strain anisotropic elastoplasticity fully coupled with ductile damage is considered using a thermodynamically consistent framework. First, the kinematics of large strain based on multiplicative decomposition of the total transformation gradient using the rotating frame formulation, is recalled and different objective derivatives defined. By using different anisotropic equivalent stresses (quadratic and non-quadratic) in yield function and in plastic potential, the evolution equations for all the dissipative phenomena are deduced from the generalized normality rule applied to the plastic potential while the consistency condition is still applied to the yield function. The effect of the objective derivatives and the equivalent stresses (quadratic or non-quadratic) on the plastic flow anisotropy and the hardening evolution with damage is considered. Numerical aspects mainly related to the time integration of the fully coupled constitutive equations are discussed. Applications are made to the AISI 304 sheet metal by considering different loading paths as tensile, shear, plane tensile and bulge tests. For each loading path the effect of the rotating frame, the equivalent stress (quadratic or non-quadratic) and the normality rule (with respect to yield function or to the plastic potential) are discussed on the light of some available experimental results.     

Channel flow of a liquid crystal polymer around an obstacle: Weld line structure and strain field

 We have studied by in situ microscopy the flow of a lyotropic liquid crystal polymer, hydroxypropylcellulose (HPC) in water, around an obstacle placed in a rectangular flow channel. The obstacle separates the flow into two parts which rejoin downstream of the obstacle, resulting in the formation of a `weld-line'. Measuring the velocity field in the vicinity of the weld-line beyond the obstacle, we find as expected a positive elongational strain (acceleration) along the weld (parallel to the flow direction). For an anisotropic (concentrated) HPC solution we observe in addition a significant shear strain in the weld-line region, there being an important velocity gradient perpendicular to the plane of the weld line. Isotropic (lower concentration) solutions of the same polymer demonstrate no visible weld line, a larger elongational strain rate near the obstacle, and no shear component of strain downstream of the obstacle. These results are similar to observations reported for fluids reinforced by macroscopic fibres. Polarised light observations of the anisotropic solution show that the strain field generates a generally increased degree of orientation of the liquid crytalline polymer near the weld (generally reduced crossed-polariser transmitted intensity when the polariser is parallel to the flow direction), however there is also a striking fine birefringent colour variation in the weld-line region, reminiscent of the structure observed at the channel side walls in rectangular channel flow (Haw and Navard 2000). The results show that the simple concept of weld-line structure as confined to an enhanced alignment along the weld due to elongational strain is incomplete; the two-dimensional shear strain field must also be taken into account for the anisotropic fluid. Received: 22 December 1999/Accepted: 4 January 2000