Spring-back evaluation of automotive sheets based on isotropic-kinematic hardening laws and non-quadratic anisotropic yield functions: Part I: theory and formulation

In order to improve the prediction capability of spring-back in automotive sheet forming processes, the modified Chaboche type combined isotropic-kinematic hardening law was formulated based on the modified equivalent plastic work principle to account for the Bauschinger effect and transient behavior. As for the yield stress function, the non-quadratic anisotropic yield potential, Yld2000-2d, was utilized under the plane stress condition. Besides the theoretical aspect of the constitutive law including the general plastic work principle for monotonously proportional loading, the method to determine hardening parameters as well as numerical formulations to update stresses were developed based on the incremental deformation theory and the consistency requirement as summarized in Part I, while the characterization of material properties and verifications with experiments are discussed in Part II and III, respectively.     

Spring-back evaluation of automotive sheets based on isotropic–kinematic hardening laws and non-quadratic anisotropic yield functions,part III: applications

In order to improve the prediction capability of spring-back in the computational analysis of automotive sheet forming processes, the modified Chaboche type combined isotropic–kinematic hardening law was formulated to account for the Bauschinger and transient behavior in Part I. As for the yield stress function, the non-quadratic anisotropic yield potential, Yld2000-2d, was utilized under the plane stress condition. Experimental procedures to obtain the material parameters of the combined hardening law and the yield potential were presented in Part II for three automotive sheets. For verification purposes, comparisons of simulations and experiments were performed here for the unconstrained cylindrical bending, the 2-D draw bending and the modified industrial part (the double-S rail). For all three applications, simulations showed good agreements with experiments. Simplified one-dimensional plane strain analytical and numerical methods were also developed here to better understand the spring-back in forming processes.     

A novel approach for anisotropic hardening modeling. Part II: Anisotropic hardening in proportional and non-proportional loadings,application to initially isotropic material

The modeling of anisotropic hardening, in particular for non-proportional loading paths, is a challenging task for advanced macroscopic models. The complex distortion of the yield locus is related to the activation and cross-hardening of different slip systems, depending on crystallographic orientations. These physical mechanisms can be taken into account in polycrystalline models but the computation times are enormous. The novel approach detailed in Part I (Rousselier et al., 2009) consists in: (i) drastically reducing the number of crystallographic orientations to save the computation cost, (ii) applying a parameter calibration procedure to obtain a good agreement with the experimental database. This methodology is first applied here to the anisotropic hardening in the proportional loadings of the strongly anisotropic aluminum alloy of Part I. Very good modeling is achieved with only eight crystallographic orientations. Different levels of additional hardening in biaxial proportional loading as compared to uniaxial loading can be modeled with the same polycrystalline model. For this, only the parameter calibration has to be performed with different databases. The same methodology has also been applied for the modeling of isotropic behavior. The best compromise between model accuracy and numerical cost is obtained with fourteen orientations. The deviations from isotropy are acceptable in all loading directions. Different levels of hardening in orthogonal loading: simple shear followed by simple tension, are achieved without any modification of the model equations. Only the parameter calibration has to be performed with different hardening levels in the database. FE calculations of a deep drawing test have been performed. The CPU time of the polycrystalline model is only five times larger than that with the simple von Mises model. The CPU time with texture evolution is further increased by a factor of two. The effects of texture evolution in rolling of the initially isotropic fcc material have been investigated. The resulting texture and hardening are qualitatively good.     

Model identification and FE simulations: Effect of different yield loci and hardening laws in sheet forming

The bi-axial experimental equipment [Flores, P., Rondia, E., Habraken, A.M., 2005a. Development of an experimental equipment for the identification of constitutive laws (Special Issue). International Journal of Forming Processes] developed by Flores enables to perform Bauschinger shear tests and successive or simultaneous simple shear tests and plane strain tests. Flores investigates the material behavior with the help of classical tensile tests and the ones performed in his bi-axial machine in order to identify the yield locus and the hardening model. With tests performed on one steel grade, the methods applied to identify classical yield surfaces such as [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic materials. Proceedings of the Royal Society of London A 193, 281–297; Hosford, W.F., 1979. On yield loci of anisotropic cubic metals. In: Proceedings of the 7th North American Metalworking Conf. (NMRC), SME, Dearborn, MI, pp. 191–197] ones as well as isotropic Swift type hardening, kinematic Armstrong–Frederick or Teodosiu and Hu hardening models are explained. Comparison with the Taylor–Bishop–Hill yield locus is also provided. The effect of both yield locus and hardening model choices is presented for two applications: plane strain tensile test and Single Point Incremental Forming (SPIF).     

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.     

Effect of microstructure on the hardening and softening behaviors of polycrystalline shape memory alloys Part II: Numerical simulation under axisymmetrical loading

Based on the microstructure-based constitutive model established in Part I, a detailed numerical investigation on the role of each microstructure parameter in the kinematical and kinetic evolution of polycrystalline SMA under axisymmetrical tension loading is performed. Some macroscopic constitutive features of stress-induced martensite transformation are discussed. The subject supported by the Research Grant Committee (RGC) of Hong Kong SAR, the National Natural Science Foundation of China and the Provincial Natural Science Foundation of Jiangxi Province of China     

Inflation and burst of aluminum tubes. Part II: An advanced yield function including deformation-induced anisotropy

In a recent study [Korkolis, Y.P., Kyriakides, S., 2008. Inflation and burst of anisotropic aluminum tubes for hydroforming applications. Int’l. J. Plasticity 24, 509–543], the formability of aluminum tubes was investigated using a combination of experimental and numerical approaches. The tubes were loaded to failure under combined internal pressure and axial load along radial paths in the engineering stress space. The experiments were then simulated using appropriate FE models and two established anisotropic yield functions. It was found that for some loading paths the computed deformations did not agree with the experimental ones, whereas rupture was generally overpredicted. In the current study the problem is tackled using a more advanced yield function [Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E., 2003. Plane stress function for aluminum alloy sheets – part I: theory. Int’l. J. Plasticity 19, 1297–1319]. Three different calibration schemes of this function are employed, in two of which the experimentally observed deformation-induced anisotropy is taken into account. It is demonstrated that both deformation and failure can ultimately be predicted successfully, albeit arduously, using a hybrid procedure detailed herein.     

Relaxational interactions and viscoelasticity of polymer melts: Part II. Rheological properties in shear and elongational flows

The rheological equation of state derived in Part I on the basis of relaxation equations of chain dynamics is analyzed for the steady and transient shear and uniaxial elongational flows of monodisperse polymers. The effect of superslow relaxation processes associated with basic macro-molecular motions that occur on a characteristic scale essentially greater than the so-called distance between entanglements was investigated in these flows. It is shown that the relaxation times in the region of linear and non-linear viscoelasticity are self-consistent. Theoretical predictions are in good agreement with the experimental data for melts of nearly monodisperse flexible polymers.     

The effect of contact conditions and material properties on plastic yield inception in a spherical shell compressed by a rigid flat

The onset of plastic yielding in a spherical shell loaded by a rigid flat is studied for stick and slip contact conditions using finite element analysis. The effect of various material properties on the critical normal load, critical interference and critical contact area at the onset of plastic yielding is investigated and the location where plastic yielding first occurs is determined. A comparison is made with results obtained previously for slip contact condition. Substantial differences are found at low to medium Poisson’s ratio values, while some similarities are found to occur for high Poisson’s ratio values. In particular, a spherical shell is more prone to yield under stick than under slip contact condition.     

Effect of microstructure on the hardening and softening behavIors of polycrystalline shape memory alloys Part I: Micromechanics constitutive modeling

The effects of microstructure and its evolution on the macroscopic superelastic stress-strain response of polycrystalline Shape Memory Alloy (SMA) are studied by a microstructure-based constitutive model developed in this paper. The model is established on the following basis: (1) the transformation conditions of the unconstrained single crystal SMA microdomain (to be distinguished from the bulk single crystal), which serve as the local criterion for the derivation of overall transformation yield conditions of the polycrystal; (2) the micro- to macro-transition scheme by which the connection between the polycrystal aggregates and the single crystal microdomain is established and the macroscopic transformation conditions of the polycrystal SMA are derived; (3) the quantitative incorporation of three microstructure factors (i.e., nucleation, growth and orientation distribution of martensite) into the modeling. These microstructural factors are intrinsic of specific polycrystal SMA systems and the role of each factor in the macroscopic constitutive response is quantitatively modeled. It is demonstrated that the interplay of these factors will result in different macroscopic transformation kinematics and kinetics which are responsible for the observed macroscopic stress-strain hardening or softening response, the latter will lead to the localization and propagation of transformation bands in TiNi SMA. The project supported by the Research Grant Committee (RGC) of Hong Kong SAR, the National Natural Science Foundation of China and the Provincial Natural Foundation of Jiangxi Province of China     

Extremal properties of endochronic plasticity,part II: Extremal path of the constitutive equation with a yield surface and application

Extremal paths for endochronic constitutive equations without using a yield surface and the corresponding principle of minimum potential work were obtained in Part I of this article. In this paper, the extremal properties of endochronic constitutive equation with a yield surface and the corresponding method for deformation bound analysis are proposed. An example is presented that demonstrates that the application of endochronic constitutive models to simplified analysis is not significantly different from classical models due to the derived extremal properties. The adopted constitutive model involves both nonlinear isotropic and kinematic hardening, which may provide more accurate results in simplified and bounding analysis.     

Some recent developments on integrity assessment of pipes and elbows. Part II: Experimental investigations

Integrity assessment of piping components is very essential for safe and reliable operation of power plants. Over the last several decades, considerable work has been done throughout the world to develop a methodology for integrity assessment of pipes and elbows, appropriate for the material involved. However, there are scope of further development/improvement of issues, particularly for pipe bends, that are important for accurate integrity assessment of piping. Considering this aspect, a comprehensive Component Integrity Test Program was initiated in 1998 at Reactor Safety Division (RSD) of Bhabha Atomic Research Centre (BARC), India in collaboration with MPA, Stuttgart, Germany through Indo-German bilateral project. In this program, both theoretical and experimental investigations were undertaken to address various issues related to the integrity assessment of pipes and elbows. The important results of the program are presented in this two-part paper. In the part II of the paper, the experimental investigations are discussed. Part I covered the theoretical investigations. Under the experimental investigations, fracture mechanics tests have been conducted on pipes and elbows of 200–400 mm diameter with various crack configurations and sizes under different loading conditions. Tests on small tensile and three point bend specimens, machined from the tested pipes, have also been done to evaluate the actual stress–strain and fracture resistance properties of pipe/elbow material. The load–deflection curve and crack initiation loads predicted by non-linear finite element analysis matched well with the experimental results. The theoretical collapse moments of throughwall circumferentially cracked elbows, predicted by the recently developed equations, are found to be closer to the test data compared to the other existing equations. The role of stress triaxialities ahead of crack tip is also shown in the transferability of J–resistance curve from specimen to component.     

Mechanics of submerged jet cavitating action: material properties,exposure time and temperature effects on erosion

Experimental setup with a submerged cavitating jet has been used for the study of influences of material, exposure time and working fluid temperature on the erosion process. Each of the parameters has been varied separately, and the results of erosion are analyzed in detail. Additionally, comparison of experiments with nitrated and non-nitrated material has been made in order to study the enhancement (mostly reflected as the prolonged incubation time) of erosion resistance achieved by nitrating the specimen surface.     

Prediction of forming limit strains under strain-path changes: Application of an anisotropic model based on texture and dislocation structure

Strain-path changes strongly influence the forming limit strains of sheet metals. The value of the limit strains is greatly affected by material-related effects such as initial anisotropy, transient. hardening, Bauschinger effect and cross hardening. A model which can describe these mechanical behaviours has been developed on the physical basis of texture and dislocation structure, and applied in conjunction with the Marciniak-Kuczynski analysis of the forming limit strains. The results are represented in forming limit diagrams (FLDs) in which the forming limit strains are indicated. The calculation successfully predicts some of the experimental tendencies which cannot be reproduced by conventional phenomenological models. Furthermore, the model has been used to discuss the effects of texture and dislocation structure on the FLDs. Especially, it is suggested that transient hardening caused by the latent part of the persistent dislocation structure significantly reduces the forming limit strain for a strain-path change from equi-biaxial stretching to uniaxial tension.     

Scaling laws of nanoporous gold under uniaxial compression: Effects of structural disorder on the solid fraction,elastic Poisson's ratio,Young's modulus and yield strength

In this work the relationship between the structural disorder and the macroscopic mechanical behavior of nanoporous gold under uniaxial compression was investigated, using the finite element method. A recently proposed model based on a microstructure consisting of four-coordinated spherical nodes interconnected by cylindrical struts, whose node positions are randomly displaced from the lattice points of a diamond cubic lattice, was extended. This was done by including the increased density as result of the introduced structural disorder. Scaling equations for the elastic Poisson's ratio, the Young's modulus and the yield strength were determined as functions of the structural disorder and the solid fraction. The extended model was applied to identify the elastic–plastic behavior of the solid phase of nanoporous gold. It was found, that the elastic Poisson's ratio provides a robust basis for the calibration of the structural disorder. Based on this approach, a systematic study of the size effect on the yield strength was performed and the results were compared to experimental data provided in literature. An excellent agreement with recently published results for polymer infiltrated samples of nanoporous gold with varying ligament size was found.     

Effect of dentin tubules on the mechanical properties of dentin. Part I: Stress-strain relations and strength criterion

As known, there is a large number of dentin tubules in dentin. These tubules have varying radii and are shaped into radially parallel pattern. The anisotropy of microstructure of dentin shows that dentin should be treated as a material of varying transverse isotropy. In this Part, the elastic stress-strain relations and the quadratic strength criterion are established in the form of having varying transverse isotropy, in the framework of micromechanics to take into account of the effect of the microstructures-dentin tubules. Simplified forms for isotropic and homogeneous cases, as well as the corresponding plane stress form of the stress-strain relations are also given. These theoretical models are very well supported by the experiments shown later in the continued paper (Part II). The project supported by the National Natural Science Foundation of China (19525207).