Advances in experiments on metal sheets and tubes in support of constitutive modeling and forming simulations

This work is a review of experimental methods for observing and modeling the anisotropic plastic behavior of metal sheets and tubes under a variety of loading paths, such as biaxial compression tests; biaxial tension tests on metal sheets and tubes using closed-loop electrohydraulic testing machines; the abrupt strain path change method for detecting a yield vertex and subsequent yield loci without unloading; in-plane stress reversal tests on metal sheets; and multistage tension tests. Observed material responses are compared with the predictions of phenomenological plasticity models. Special attention is paid to the plastic deformation behavior of materials commonly used in industry, and to verifying the validity of conventional anisotropic yield criteria for those materials and associated flow rules at large plastic strains. The effects of using appropriate anisotropic yield criteria on the accuracy of simulations of forming defects, such as large springback and fracture, are also presented to highlight the importance of accurate material testing and modeling.     

An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations

The initial yield surface of 2D lattice materials is investigated under biaxial loading using finite element analyses as well as by analytical means. The sensitivity of initial yield surface to the dominant deformation mode is explored by using both low- and high-connectivity topologies whose dominant deformation mode is either local bending or strut stretching, respectively. The effect of microstructural irregularity on the initial yield surface is also examined for both topologies. A pressure-dependent anisotropic yield criterion, which is based on total elastic strain energy density, is proposed for 2D lattice structures, which can be easily extended for application to 3D cellular solids. Proposed criterion uses elastic constants and yield strengths under uniaxial loading, and does not rely on any arbitrary parameter. The analytical framework developed allows the introduction of new scalar measures of characteristic stresses and strains that are capable of representing the elastic response of anisotropic materials with a single elastic master line under multiaxial loading.     

Modelling and experimental characterisation of the Lüders strain in complex loaded ferritic steel compact tension specimens

The use of 3D digital image correlation (DIC) has been used to capture the Lüders strains in a low carbon ferritic steel. Results were used to calibrate and compare with finite element (FE) results based on a constitutive plasticity model, capable of yield drop behaviour and therefore Lüders strains, by Zhang et al. (2001). Tensile tests were carried out at several strain rates to characterise the material behaviour. The results of these tests were used to fit parameters in the constitutive plasticity model. The FE model was then tested on a complex loading situation of in-plane compression of a compact tension (CT) specimen. The FE model predicts the shape and formation of the Lüders bands well. This FE model, using Zhang’s constitutive plasticity model, was used to predict the residual stress profile to compare with standard elastic–plastic isotropic hardening models with no yield point. The yield point reduced both the predicted peak tensile stress, at the notch root, and the amount of plastic strain. In regions where the plastic strain was of a similar size to the Lüders strain the stress profiles were perturbed from flat profiles predicted by the standard elastic–plastic hardening models.     

Plastic spin associated with a non-normality theory of plasticity

A plasticity model using a vertex-type plastic flow rule on a smooth yield surface for an anisotropic solid has been proposed recently. This model is here completed by incorporating the effect of plastic spin. Simple shear with a large shear strain is one of the hardest tests on finite strain anisotropic plasticity models, and it is here shown which plastic spin expression is needed to avoid unrealistic oscillatory behavior of the shear stress under large shear strains. The idea of using non-normality with a smooth yield surface originates from a recent proposal of using an abrupt strain path change to determine the subsequent yield surface shape. For this method both polycrystal plasticity calculations and experiments have shown a vertex-type response on the apparently smooth yield surface.     

Multiaxial yield surface of transversely isotropic foams: Part I—Modeling

A new yield criterion is proposed for transversely isotropic solid foams. Its derivation is based on the hypothesis that the yielding in foams is driven by the total strain energy density, rather than a completely phenomenological approach. This allows defining the yield surface with minimal number of parameters and does not require complex experiments. The general framework used leads to the introduction of new scalar measures of stress and strain (characteristic stress and strain) for transversely isotropic foams. Furthermore, the central hypothesis that the total strain energy density drives yielding in foams ascribes to the characteristic stress an analogous role of von Mises stress in metal plasticity. Unlike the overwhelming majority of yield models in literature the proposed model recognizes the tension–compression difference in yield behavior of foams through a linear mean stress term. Predictions of the proposed yield model are in excellent agreement with the results of uniaxial, biaxial and triaxial FE analyses implemented on both isotropic and transversely isotropic Kelvin foam models.     

A generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials

In this paper, a generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials is derived. The evolution equation for the active yield surface with reference to the memory yield surface is obtained by considering the continuous expansion of the active yield surface during the unloading/reloading process. The incremental constitutive relation based on the associated flow rule is then derived for a general yield function for pressure insensitive and sensitive materials. Detailed incremental constitutive relations for materials based on the Mises yield function, the Hill quadratic anisotropic yield function and the Drucker–Prager yield function are derived as the special cases. The closed-form solutions for one-dimensional stress–plastic strain curves are also derived and plotted for materials under cyclic loading conditions based on the three yield functions. In addition, the closed-form solutions for one-dimensional stress–plastic strain curves for materials based on the isotropic Cazacu–Barlat yield function under cyclic loading conditions are summarized and presented. For materials based on the Mises and the Hill anisotropic yield functions, the stress–plastic strain curves show closed hysteresis loops under uniaxial cyclic loading conditions and the Masing hypothesis is applicable. For materials based on the Drucker–Prager and Cazacu–Barlat yield functions, the stress–plastic strain curves do not close and show the ratcheting effect under uniaxial cyclic loading conditions. The ratcheting effect is due to different strain ranges for a given stress range for the unloading and reloading processes. With these closed-form solutions, the important effects of the yield surface geometry on the cyclic plastic behavior due to the pressure-sensitive yielding or the unsymmetric behavior in tension and compression can be shown unambiguously. The closed form solutions for the Drucker–Prager and Cazacu–Barlat yield functions with the associated flow rule also suggest that a more general anisotropic hardening theory needs to be developed to address the ratcheting effects for a given stress range.     

Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal pressure

The anisotropic plastic deformation behavior of extruded 5000 series aluminum alloy tubes, A5154-H112, of 76 mm outer diameter and 3.9 mm wall thickness is investigated, using a servo-controlled tension-internal pressure testing machine. This machine is capable of applying arbitrary stress or strain paths to a tubular specimen using an electrical, closed-loop control system. Detailed measurements were made of the initial yield locus, contours of plastic work for different levels of work-hardening, and the directions of the incremental plastic strain vectors for both linear and combined stress paths. It is found that the measured work contours constructed in the principal stress space are similar in shape, and that the directions of the incremental plastic strain vectors remain almost constant at constant stress ratios. The work-hardening behavior predicted using Hosford's or the Yld2000-2d yield functions under the assumption of isotropic hardening agrees closely with the observations for both linear and combined stress paths. The material is thus found to work-harden almost isotropically. Both yield functions are effective phenomenological plasticity models for predicting the anisotropic plastic deformation behavior of the material.     

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.     

Path-dependence of the forming limit stresses in a sheet metal

The effect of changing strain paths on the forming limit stresses of sheet metals is investigated using the Marciniak–Kuczyński model and a phenomenological plasticity model with non-normality effects [Kuroda, M., Tvergaard, V., 2001. A phenomenological plasticity model with non-normality effects representing observations in crystal plasticity. J. Mech. Phys. Solids 49, 1239–1263]. Forming limits are simulated for linear stress paths and two types of combined loading: a combined loading consisting of two linear stress paths in which unloading is included between the first and second loadings (combined loading A), and combined loading in which the strain path is abruptly changed without unloading (combined loading B). The forming limit stresses calculated for combined loading A agree well with those calculated for the linear stress paths, while the forming limit curves in strain space depend strongly on the strain paths. The forming limit stresses calculated for the combined loading B do not, however, coincide with those calculated for the linear stress paths. The strain-path dependence of the forming limit stress is discussed in detail by observing the strain localization process.     

考虑路径相关性的非比例循环塑性本构模型

根据非比例加载下金属材料响应的延迟特性及加载路径相关性,选取沿应力迹法向的塑性应变的累积量作为非比例加载影响的度量,相应给出反映非比例附加强化的变量,并假设其模量和强化率与加载路径的几何参数相关．为反映由于非比例加载而引起的材料强化的异向效应,在Valanis的塑性内时响应方程中引入与加载路径几何性质有关的应力项,构成非比例循环塑性本构关系．对316和304不锈钢材料在一些典型非比例循环加载路径下的应力响应进行了理论预测,与Benallal等及McDowell的实验结果取得了良好的一致．     

A mathematical basis for strain-gradient plasticity theory. Part II: Tensorial plastic multiplier

A phenomenological, flow theory version of gradient plasticity for isotropic and anisotropic solids is constructed along the lines of Gudmundson [Gudmundson, P., 2004. A unified treatment of strain-gradient plasticity. J. Mech. Phys. Solids 52, 1379-1406]. Both energetic and dissipative stresses are considered in order to develop a kinematic hardening theory, which in the absence of gradient terms reduces to conventional J₂ flow theory with kinematic hardening. The dissipative stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition with associated plastic flow. The theory includes interfacial terms: elastic energy is stored and plastic work is dissipated at internal interfaces, and a yield surface is postulated for the work-conjugate stress quantities at the interface. Uniqueness and extremum principles are constructed for the solution of boundary value problems, for both the rate-dependent and the rate-independent cases. In the absence of strain gradient and interface effects, the minimum principles reduce to the classical extremum principles for a kinematically hardening elasto-plastic solid. A rigid-hardening version of the theory is also stated and the resulting theory gives rise to an extension to the classical limit load theorems. This has particular appeal as previous trial fields for limit load analysis can be used to generate immediately size-dependent bounds on limit loads.     

Mechanism-based multi-surface plasticity model for ideal truss lattice materials

The mechanical behavior of ideal truss lattice materials is controlled by the so-called direct action mechanism at the microscale which involves the uniform stretching and compressing of individual truss members. Standard homogenization techniques have been employed to develop a general micromechanics-based finite-strain constitutive model for truss lattice materials. Furthermore, a specialized small-strain plasticity model has been derived. Both models have been implemented in a finite-element program and used to simulate the anisotropic plastic behavior of the octet-truss lattice material in various applications including cyclic uniaxial loading, pure shear, and three-point bending. The constitutive model predictions agree well with the results obtained from discrete finite element models. Regarding the plasticity of the octet-truss lattice material, it has been found that the elastic domain is constrained by twelve pairwise parallel hyperplanes in the six-dimensional stress space. Moreover, the mechanism-based small-strain formulation reveals that the direction of plastic flow is normal to the pressure-dependent macroscopic yield surfaces.     

A finite element formulation based on non-associated plasticity for sheet metal forming

In the present paper, a finite element formulation based on non-associated plasticity is developed. In the constitutive formulation, isotropic hardening is assumed and an evolution equation for the hardening parameter consistent with the principle of plastic work equivalence is introduced. The yield function and plastic potential function are considered as two different functions with functional form as the yield function of Hill [Hill, R., 1948. Theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. A 193, 281–297] or Karafillis–Boyce associated model [Karafillis, A.P. Boyce, M., 1993. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J. Mech. Phys. Solids 41, 1859–1886]. Algorithmic formulations of constitutive models that utilize associated or non-associated flow rule coupled with Hill or Karafillis–Boyce stress functions are derived by application of implicit return mapping procedure. Capabilities in predicting planar anisotropy of the Hill and Karafillis–Boyce stress functions are investigated considering material data of Al2008-T4 and Al2090-T3 sheet samples. The accuracy of the derived stress integration procedures is investigated by calculating iso-error maps.     

汽车薄钢板应力应变曲线及屈服轨迹的研究

采用十字形双向拉伸的实验方法对两种汽车用薄钢板BH220和SPEN进行了不同 加载路径下的双向拉伸试验，得到了不同应力状态下的应力应变关系曲线，同时，根据单位 体积塑性功相等的原则，确定了两种钢板等效塑性应变从0.2\%$\sim$2\%的实验屈服轨迹. 结果分析表明：不同加载路径下板料的应力应变关系不同，随着加载比例由单拉到等双拉状 态，板料的硬化指数逐步增大；实验屈服轨迹呈外凸性，且以等双拉为界的上下部分屈服轨 迹不对称，随着变形程度的增加，屈服轨迹向外扩大，但单拉时强化程度最小，而等双拉 时最大. 对BH220和SPEN钢板的实验屈服轨迹与几种常用理论屈服轨迹的比较发现，Hosford各向 异性屈服准则的理论轨迹与实验结果最为接近，Hill48准则与实验结果相差最大，此外一 向被视为只适用于各向同性材料的Mises准则与实验结果也较为接近，其他几个屈服准则的 理论屈服轨迹与实验点相差较大.     

Anisotropic strain hardening behavior in simple shear for cube textured aluminum alloy sheets

Finite element (FE) simulations of the simple shear test were conducted for 1050-O and 6022-T4 aluminum alloy sheet samples. Simulations were conducted with two different constitutive equations to account for plastic anisotropy: Either a recently proposed anisotropic yield function combined with an isotropic strain hardening law or a crystal plasticity model. The FE computed shear stress–shear strain curves were compared to the experimental curves measured for the two materials in previous works. Both phenomenological and polycrystal approaches led to results consistent with the experiments. These comparisons lead to a discussion concerning the assessment of anisotropic hardening in the simple shear test.     

Multiaxial ratcheting with advanced kinematic and directional distortional hardening rules

Ratcheting is defined as the accumulation of plastic strains during cyclic plastic loading. Modeling this behavior is extremely difficult because any small error in plastic strain during a single cycle will add to become a large error after many cycles. As is typical with metals, most constitutive models use the associative flow rule which states that the plastic strain increment is in the direction normal to the yield surface. When the associative flow rule is used, it is important to have the shape of the yield surface modeled accurately because small deviations in shape may result in large deviations in the normal to the yield surface and thus the plastic strain increment in multi-axial loading. During cyclic plastic loading these deviations will accumulate and may result in large errors to predicted strains.This paper compares the bi-axial ratcheting simulations of two classes of plasticity models. The first class of models consists of the classical von Mises model with various kinematic hardening (KH) rules. The second class of models introduce directional distortional hardening (DDH) in addition to these various kinematic hardening rules. Directional distortion describes the formation of a region of high curvature on the yield surface approximately in the direction of loading and a region of flattened curvature approximately in the opposite direction. Results indicate that the addition of directional distortional hardening improves ratcheting predictions, particularly under biaxial stress controlled loading, over kinematic hardening alone.     

The effect of load biaxiality on crack growth in non-linear materials

The influence of load biaxiality on the stress field and fracture behavior of a cracked plate is investigated. Considered is a square plate containing a central through the thickness crack and subjected to a biaxial loading perpendicular and parallel to the crack plane. The stress field of the plate is analyzed by a finite element code based on incremental plasticity and the von Mises yield condition. A method based on the strain energy density theory is used to determine the critical stress for crack initiation. It was found that the equi-biaxial loading mode induces the smallest plastic zones, while the critical applied stress for crack initiation becomes maximum. Quite the contrary happens for the shear loading system which causes the largest plastic zones and the minimum applied stress values fro crack growth. Results showing the dependence of the above quantities on the biaxiality of the applied stress are presented in graphical form.     

Plastic deformation modes in perforated sheets and their relation to yield and limit surfaces

Construction of mechanism-based plasticity theories for the homogenized response of heterogeneous materials requires identification of plastic deformation modes as a function of loading direction relative to the microstructural details. Herein, we employ an efficient homogenization theory to identify for the first time such deformation modes in plates under plane stress with hexagonal arrays of circular holes at small and intermmediate pore volume fractions, and establish their relation to the branches of initial and subsequent yield and limit surfaces. Newly introduced maps of the intrinsic geometric features of point-wise yield surfaces provide full-field picture of the investigated microstructures’ propensity for plastic strain initiation and localization. The identified characteristic plastic modes provide a rational explanation for the evolving geometric features of subsequent yield and limit surfaces whose branches represent different plastic flow mechanisms, as well as a basis for the construction of a mechanism-based homogenized plasticity theory for use in structural analysis algorithms. The results suggest the need for composite yield surfaces comprised of multiple branches in the construction of mechanism-based homogenized plasticity theory for the investigated class of porous materials.     

An exact semi-analytic solution for residual stresses and strains within a thin hollow disc of pressure-sensitive material subject to thermal loading

There is considerable current interest in the development of constitutive equations for pressure-dependent plastic materials. In particular, in contrast to classical plasticity there is no commonly accepted relation to connect stress and strain or strain rate for such materials. Analytic and semi-analytic solutions are convenient to compare qualitative features of boundary value problems solved for different models. Such comparative studies can be useful to choose this or that model for specific applications. Analytic and semi-analytic solutions are also necessary to verify numerical codes. In the present paper, a new semi-analytic solution for a thin hollow disc subject to thermal loading is developed. A numerical method is only necessary to solve transcendental equations. The constitutive equations for connecting the plastic portion of the strain rate tensor and the stress tensor consist of the Drucker-Prager yield criterion and its associated flow rule. Therefore, the main distinguished feature of the solution is that the material is plastically compressible.