Stress integration method for a nonlinear kinematic/isotropic hardening model and its characterization based on polycrystal plasticity

Sheet metal forming processes generally involve non-proportional strain paths including springback, leading to the Bauschinger effect, transient hardening, and permanent softening behavior, that can be possibly modeled by kinematic hardening laws. In this work, a stress integration procedure based on the backward-Euler method was newly derived for a nonlinear combined isotropic/kinematic hardening model based on the two-yield’s surfaces approach. The backward-Euler method can be combined with general non-quadratic anisotropic yield functions and thus it can predict accurately the behavior of aluminum alloy sheets for sheet metal forming processes. In order to characterize the material coefficients, including the Bauschinger ratio for the kinematic hardening model, one element tension–compression simulations were newly tried based on a polycrystal plasticity approach, which compensates extensive tension and compression experiments. The developed model was applied for a springback prediction of the NUMISHEET’93 2D draw bend benchmark example.     

Effects of texture on shear band formation in plane strain tension/compression and bending

In this study, effects of typical texture components observed in rolled aluminum alloy sheets on shear band formation in plane strain tension/compression and bending are systematically studied. The material response is described by a generalized Taylor-type polycrystal model, in which each grain is characterized in terms of an elastic–viscoplastic continuum slip constitutive relation. First, a simple model analysis in which the shear band is assumed to occur in a weaker thin slice of material is performed. From this simple model analysis, two important quantities regarding shear band formation are obtained: i.e. the critical strain at the onset of shear banding and the corresponding orientation of shear band. Second, the shear band development in plane strain tension/compression is analyzed by the finite element method. Predictability of the finite element analysis is compared to that of the simple model analysis. Third, shear band developments in plane strain pure bending of a sheet specimen with the typical textures are studied. Regions near the surfaces in a bent sheet specimen are approximately subjected to plane strain tension or compression. From this viewpoint, the bendability of a sheet specimen may be evaluated, using the knowledge regarding shear band formation in plane strain tension/compression. To confirm this and to encompass overall deformation of a bent sheet specimen, including shear bands, finite element analyses of plane strain pure bending are carried out, and the predicted shear band formation in bent specimens is compared to that in the tension/compression problem. Finally, the present results are compared to previous related studies, and the efficiency of the present method for materials design in future is discussed.     

Analytical springback model for lightweight hexagonal close-packed sheet metal

Experiments have shown that magnesium alloy sheet a common hexagonal close-packed metal, exhibits mechanical behavior unlike that of sheets made of cubic metals (X.Y. Lou et al., 2007, Int. J. Plasticity, 24, 44). The unique stress–strain response includes a strong asymmetry in the initial yield and subsequent plastic hardening. In other words, the stress–strain curves in tension and compression are significantly different. A proper representation of the constitutive relationships is crucial for the accurate evaluation of springback, which occurs due to the residual moment distribution through the sheet thickness after bending. In this paper, we propose an analytical model for asymmetric elasto-plastic bending under tension followed by elastic unloading in order to evaluate the bending moment, which is equivalent to the springback amount. To simplify the calculations, the experimentally measured stress–strain curve of the magnesium alloy sheet was approximated with discrete linear hardening in each deformation region, and the material properties were characterized according to several simplifying assumptions. The bending moment was calculated analytically using the approximate asymmetric stress–strain relationship up to the prescribed curvature corresponding to the radius of the tool in sheet metal forming operations. A numerical example showed an unusual springback increase, even with an increase in the applied force; this is an unexpected result for conventional symmetric materials. We also compared the calculated springback amounts with the results of physical measurements. This showed that the proposed model predicts the main trends of the springback in magnesium alloy sheets reasonably well considering the simplicity of the analytical approach.     

Multi-axial behavior of shape-memory alloys undergoing martensitic reorientation and detwinning

Recently, a rate-independent, finite-deformation-based crystal mechanics constitutive model for martensitic reorientation and detwinning in shape-memory alloys has been developed by Thamburaja [Thamburaja, P., 2005. Constitutive equations for martensitic reorientation and detwinning in shape-memory alloys. Journal of the Mechanics and Physics of Solids 53, 825–856] and implemented in the ABAQUS/Explicit [Abaqus reference manuals. 2005. Providence, RI] finite-element program. In this work, we show that the aforementioned model is able to quantitatively predict the experimental response of an initially textured and martensitic polycrystalline Ti–Ni rod under a variety of uniaxial and multi-axial stress states. By fitting the material parameters in the model to the stress–strain response in simple tension, the constitutive model predicts the stress–strain curves for experiments conducted under simple compression, torsion, proportional-loading tension–torsion, and path-change tension–torsion loading conditions to good accord. Furthermore the constitutive model also reproduces the force–displacement response for an indentation experiment to reasonable accuracy.     

A combined necking and shear localization analysis for aluminum sheets under biaxial stretching conditions

A combined necking and shear localization analysis is adopted to model the failures of two aluminum sheets, AA5754 and AA6111, under biaxial stretching conditions. The approach is based on the assumption that the reduction of thickness or the necking mode is modeled by a plane stress formulation and the final failure mode of shear localization is modeled by a generalized plane strain formulation. The sheet material is modeled by an elastic-viscoplastic constitutive relation that accounts for the potential surface curvature, material plastic anisotropy, material rate sensitivity, and the softening due to the nucleation, growth, and coalescence of microvoids. Specifically, the necking/shear failure of the aluminum sheets is modeled under uniaxial tension, plane strain tension and equal biaxial tension. The results based on the mechanics model presented in this paper are in agreement with those based on the forming limit diagrams (FLDs) and tensile tests. When the necking mode is suppressed, the failure strains are also determined under plane strain conditions. These failure strains can be used as guidances for estimation of the surface failure strains on the stretching sides of the aluminum sheets under plane strain bending conditions. The estimated surface failure strains are higher than the failure strains of the forming limit diagrams under plane strain stretching conditions. The results are consistent with experimental observations where the surface failure strains of the aluminum sheets increase significantly on the stretching sides of the sheets under bending conditions. The results also indicate that when a considerable amount of necking is observed for a sheet metal under stretching conditions, the surface failure strains on the stretching sides of the sheet metal under bending conditions can be significantly higher.     

Flexural behavior of strain-softening solids

The flexural behavior of a beam is investigated in an attempt to establish a correlation between the tensile and bending properties of strain-softening solids. Given the complete uniaxial stress—strain relations, including the post-peak tension-softening portion, it is possible to predict the flexural behavior in moment—curvature and load—deflection relations. The results indicate that strain-softening gives rise to enhanced bending strength in agreement with experimental data. Conversely, given the bending responses together with the softening characteristics the complete tensile behavior can be determined. Since bending experiments are easier to perform than uniaxial tensile tests, this well-defined correlation provides a feasible means to obtain the entire tensile behavior of strain-softening solids such as concrete, rocks and ceramics.     

Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets

Magnesium alloy sheets have been extending their field of applications to automotive and electronic industries taking advantage of their excellent light weight property. In addition to well-known lower formability, magnesium alloys have unique mechanical properties which have not been thoroughly studied: high in-plane anisotropy/asymmetry of yield stress and hardening response. The reason of the unusual mechanical behavior of magnesium alloys has been understood by the limited symmetry crystal structure of HCP metals and thus by deformation twinning. In this paper, the phenomenological continuum plasticity models considering the unusual plastic behavior of magnesium alloy sheet were developed for a finite element analysis. A hardening law based on two-surface model was further extended to consider the general stress–strain response of metal sheets including Bauschinger effect, transient behavior and the unusual asymmetry. Three deformation modes observed during the continuous in-plane tension/compression tests were mathematically formulated with simplified relations between the state of deformation and their histories. In terms of the anisotropy and asymmetry of the initial yield stress, the Drucker–Prager’s pressure dependent yield surface was modified to include the anisotropy of magnesium alloy. The numerical formulations and characterization procedures were also presented and finally the correlation of simulation with measurements was performed to validate the proposed theory.     

Numerical Nonlinear Elastic Analysis of Orthotropic Spherical Shells with an Elliptic Cutout

Study is made of how the nonlinear properties of orthotropic composites and the ellipticity of an opening affect the stress–strain state of thin spherical shells subjected to internal pressure. An analysis of numerical results reveals compression on the inside surface of the shell near the broad of the elliptical opening. It is pointed out that as the ellipticity increases, the deformation process near the opening transforms from uniform tension and bending, typical of a shell with a circular opening, to dominated tension near the narrow of the opening and to dominated bending near the broad.     

一种基于电磁霍普金森杆的材料动态包辛格效应测试装置及方法

金属材料在复杂载荷条件下的动态力学行为研究一直备受关注,但受限于实验设备,金属材料的动态包辛格效应响应一直都难以获得。为了探究金属材料的包辛格效应与应变率效应之间的关系,本文中提出一种基于电磁霍普金森杆(electromagnetic split Hopkinson bar,ESHB) 的非同步加载实验技术,为测试金属材料在高应变率加载下的包辛格效应提供了一种有效的实验方法。本文中,首先介绍了非同步加载装置的主要特点,即可以用两列由脉冲发生器产生的应力波对受载试样进行连续的一次动态拉-压循环加载,且加载过程保证了应力波的一致性。分析了应力波对试样加载过程中的波传播历程,确保了加载过程的连续性。随后介绍了动态加载过程,数据处理方法和波形分离手段,并对动态加载过程进行应力平衡性分析,论证了实验装置的可靠性。最后采用该方法测试了5%预应变下6061铝合金动态压缩-动态拉伸的包辛格效应,并与准静态下的实验结果进行对比。实验结果表明,该材料单轴压缩没有明显的应变率效应,但其包辛格效应具有应变率依赖性,高应变率下材料的包辛格应力影响因子由0.07增大至0.17,具有显著的提升,这对传统意义上铝合金材料应变率不敏感的结论提出了挑战。     

Dual-phase steel sheets under cyclic tension–compression to large strains: Experiments and crystal plasticity modeling

In this work, we develop a physically-based crystal plasticity model for the prediction of cyclic tension–compression deformation of multi-phase materials, specifically dual-phase (DP) steels. The model is elasto–plastic in nature and integrates a hardening law based on statistically stored dislocation density, localized hardening due to geometrically necessary dislocations (GNDs), slip-system-level kinematic backstresses, and annihilation of dislocations. The model further features a two level homogenization scheme where the first level is the overall response of a two-phase polycrystalline aggregate and the second level is the homogenized response of the martensite polycrystalline regions. The model is applied to simulate a cyclic tension–compression–tension deformation behavior of DP590 steel sheets. From experiments, we observe that the material exhibits a typical decreasing hardening rate during forward loading, followed by a linear and then a non-linear unloading upon the load reversal, the Bauschinger effect, and changes in hardening rate during strain reversals. To predict these effects, we identify the model parameters using a portion of the measured data and validate and verify them using the remaining data. The developed model is capable of predicting all the particular features of the cyclic deformation of DP590 steel, with great accuracy. From the predictions, we infer and discuss the effects of GNDs, the backstresses, dislocation annihilation, and the two-level homogenization scheme on capturing the cyclic deformation behavior of the material.     

一种用于金属薄板轴向拉压疲劳试验的防弯夹具

在金属薄板的轴向疲劳S-N曲线测试中,研究将一种侧向防弯曲夹具应用于存在压向载荷的试验。通过粘贴应变片方法测量试样的表面应力,对试样的受力情况做了定量的分析。测量结果表明试样安装防弯曲夹具后,基本消除了由压缩失稳产生的弯曲应力。且通过对不同拧紧方式的测量,表明一定的夹紧力下不对试验力产生影响。试验夹具设计成对试样中心轴线的支撑而让边缘疲劳敏感部位处在非接触状态,试样断口表明疲劳起源在这些并没有与试样接解的部位。用钛和铝两种材料的薄板在不同试验机上进行了不同寿命和频率的试验,试验结果与正应力比试验同时给出以对比,各种研究表明本试验有效解决了薄板疲劳受压时的失稳问题。     

Experimental study of the plastic yielding of rolled sheet metals with the cruciform plate specimen

In the current paper, an experimental technique for the evaluation of the in-plane yield loci of sheet metals with the cruciform plate specimen is presented. The measurement system is shown to conform to the optimized design concept proposed by other researchers. Finite element analysis demonstrates a reasonably wide area of uniform stress distribution in the center of the cruciform specimen, which allows the measurement of in-plane strain field by using a stacked strain rosette. Based on the designed apparatus, the yield loci of the 1100-F aluminum sheets corresponding to the as-received condition, and 25% and 50% thickness reductions by further rolling, were constructed, respectively, by applying biaxial loadings along the two principal axes of the cruciform specimen. A set of uniaxial tension tests were also performed to determine the plastic properties of the aluminum sheet along different directions with respect to the rolling direction. Finally, Hill's 1990 yield criterion is examined based on the experimental data from both biaxial and uniaxial tension tests.     

Modeling stress state dependent damage evolution in a cast Al–Si–Mg aluminum alloy

Internal state variable rate equations are cast in a continuum framework to model void nucleation, growth, and coalescence in a cast Al–Si–Mg aluminum alloy. The kinematics and constitutive relations for damage resulting from void nucleation, growth, and coalescence are discussed. Because damage evolution is intimately coupled with the stress state, internal state variable hardening rate equations are developed to distinguish between compression, tension, and torsion straining conditions. The scalar isotropic hardening equation and second rank tensorial kinematic hardening equation from the Bammann–Chiesa–Johnson (BCJ) Plasticity model are modified to account for hardening rate differences under tension, compression, and torsion. A method for determining the material constants for the plasticity and damage equations is presented. Parameter determination for the proposed phenomenological nucleation rate equation, motivated from fracture mechanics and microscale physical observations, involves counting nucleation sites as a function of strain from optical micrographs. Although different void growth models can be included, the McClintock void growth model is used in this study. A coalescence model is also introduced. The damage framework is then evaluated with respect to experimental tensile data of notched Al–Si–Mg cast aluminum alloy specimens. Finite element results employing the damage framework are shown to illustrate its usefulness.     

Study of the Effects of Thermomechanical Coupling Near Stress Concentrators in Viscoelastoplastic Bodies

A model is developed to investigate thermal effects in elastoviscoplastic bodies. It allows us to obtain some quantitative and qualitative information on the stress–strain state of a body and to carry out a nondestructive diagnostics of internal defects, which are stress and strain concentrators. The problem on the plane stress state of a rectangle with a hole is considered as an example. The plate is made of steel and aluminum alloy. The variation in the temperature on the external contour of the body under monotonous and cyclic tension in the elastic and plastic domains is studied     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part II: A very high work hardening aluminum alloy (annealed 1100 Al)

Results are presented on the evolution of subsequent yield surfaces with finite deformation in a very high work hardening annealed 1100 aluminum alloy. In Part I [Khan, A.S., Kazmi, R., Stoughton, T., Pandey, A., 2009a. Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part 1: a very low work hardening aluminum alloy (Al-6061–T6511) 25, 1611–1625.] of this paper, similar results are presented for a very low work hardening aluminum alloy. Those results were very different from the present ones, and all the results were for proportional loading paths. The subsequent yield surfaces are determined in tension, free end torsion and combined tension–torsion proportional and non-proportional loading paths, using 10 με deviation from linearity definition of yield. Yield surfaces are also determined after linear, bi-linear, and non-linear unloading paths after finite deformation under tension, free end torsion, and combined tension–torsion loading. The initial yield surface is closer to the von-Mises surface and the subsequent yield surfaces show distortion, expansion, positive cross-effect, and “nose” in the loading direction. Additionally, the subsequent yield surfaces after non-proportional loading paths show shrinkage and compounded distortion. The yield surfaces after unloading depict strong anisotropy, positive cross-effect and exhibits different proportion of distortion in each loading conditions. The Young’s and shear modulus decrease with plastic deformation and this decrease is much less than those reported in the published literature.     

Four-Point Bending Test of Determining Stress-Strain Curves Asymmetric between Tension and Compression

A method of determining both uniaxial tension and compression stress-strain curves from the result of a single four-point bending test was demonstrated. Stress-strain curves of magnesium showing tension-compression asymmetry due to twinning deformation and those of an S45C steel due to the Bauschinger effect were calculated. The Mayville-Finnie equation was modified slightly for this calculation. The calculation is sensitive to small change in the slope of bending curve, revealing an aspect of inverse problem.     

Micromechanics of Short-Term Thermal Damageability

The structural theory of microdamageability of a homogeneous material is generalized to the case of a thermal action. The theory is based on the stochastic thermoelastic equations of a medium with micropores, hollow or filled with particles of a damaged material. This medium models a material with dispersed microdamages. The Schleicher–Nadai fracture criterion is used as the condition of origin of a micropore in a microvolume of an undestroyed material. It is assumed that the particles of the damaged material in the micropores do not resist shear and triaxial tension and behave as the undamaged material under triaxial compression. The porosity balance equation is corrected for the thermal component and together with the relations between macrostresses, macrostrains, and temperature forms a closed system describing the concurrent action of deformation and microdamage. Nonlinear stress–strain diagrams and dependences of microdamage on macrostrain and temperature are constructed     

Modeling Transverse Compression of Cylindrical Bodies in Bending

A nonclassical theory of the stress–strain state of composite beams in bending is developed. Both transverse shear and transverse compression are taken into account. The results demonstrate the high accuracy of the theory in investigation of short transversally isotropic beams     

Effect of Processes at the Compression Pulse Front on Spalling Strength of a Material and Resistance to High-Velocity Penetration

Tests on backward spalling of 38KhN3MFA structural steel, D-16 aluminum alloy, M-2 copper, 02Kh18K9M5-VI maraging steel, KhN75VMYu alloy, beryllium, and other materials show that spalling strength correlates with the threshold of structural instability of a material to compression at the leading edge of the compression pulse. It is shown that the threshold of structural instability to compression obtained in experiments on uniaxial deformation of flat targets determines the strength of resistance to high-velocity penetration in the Alekseevskii–Tate model.     

Experimental and numerical study on formability of friction stir welded TWB sheets based on hemispherical dome stretch tests

In order to investigate formability performance and also to obtain guidelines for the stamping process design of friction stir welded TWB (tailor welded blank) sheets, the hemispherical dome stretching test was experimentally performed and the results of the base and friction stir welded samples were compared. Also, in order to better understand the experimental results, numerical analysis was performed. In this work, five automotive sheets, 6111-T4, 5083-H18, 5083-O aluminum alloy, dual-phase steel (DP590) and AZ31 magnesium alloy sheets were considered by (friction stir) welding the same materials. To represent mechanical properties for the numerical analysis, the non-quadratic orthotropic yield function, Yld2000-2d, was utilized for the aluminum alloy and DP590 sheets, while the Cazacu anisotropic/asymmetric yield function was applied for the AZ31 sheet considering different hardening behavior in tension and compression.