A generalized, computationally versatile plasticity model framework - Part II: Theory and verification focusing on shear anisotropy

In this work the onset of failure caused by particle-matrix debonding in aluminum alloy 2090-T3 and magnesium alloy AZ31 under either overall uniaxial plane strain tension or compression load is studied numerically using a unit cell model. Especially for magnesium AZ31, experiments show a significant difference in the tensile-compressive yield stresses, the so-called Strength-Differential-effect (SD-effect). This SD-effect cannot be captured by any isotropic yield function and only by recent anisotropic yield functions. To accurately model AA 2090-T3 and magnesium AZ31 the general yield function by Yoon et al. (2014) is adopted in addition to the isotropic von Mises plasticity yield function and the anisotropic yield function of Hill (1948). The numerically predicted strain at which void nucleation initiates is generally found to be larger for magnesium AZ31 than for AA 2090-T3. However, depending on the orientation of the principal axes of plastic anisotropy this void nucleation strain varies significantly. This holds true for both materials, but the void nucleation strain of AA 2090-T3 in compression is numerically predicted to be practically identical whether or not isotropic or anisotropic plasticity is considered. In tension the void nucleation strain of AA 2090-T3 based on anisotropic plasticity is found to be more than double the strain obtained by von Mises plasticity. The numerically predicted void nucleation strain of magnesium AZ31 is significantly affected by the yield function adopted. For the von Mises yield function the value of the void nucleation strain is in between the predictions based on the anisotropic yield functions. For both tension and compression, the yield function of Yoon et al. (2014) results in the largest void nucleation strain, whereas the yield function of Hill results in the smallest strain value. This holds true for all five orientations of the principal axes of plastic anisotropy investigated.

On anisotropic plasticity models using linear transformations on the deviatoric stress: Physical constraints on plastic flow in generalized plane strain

Many metallic sheets exhibit a certain initially plastic anisotropy that influences considerably the mechanical behaviour of the sheet material in forming operations. Nowadays, computer modelling based on the finite element method (FEM) in conjunction with the concept of anisotropic yield functions is successfully utilized to design sheet forming processes. The yield function applied in the FEM analysis affects very strongly the computed results such as material flow, springback, wrinkling and limiting strains. Consequently, in order to increase the accuracy of the FEM simulations the yield function used should be as accurate as possible in order to capture all the important anisotropy effects.In this paper, the new plane stress yield function for orthotropic sheet metals recently introduced by the author (Aretz H 2003 J. Mater. Process. Technol. submitted) is applied to orthotropic steel and aluminium sheet metals. The yield function contains eight anisotropy parameters that may be fitted by means of a Newton solver or a least square solver to selected experimental input data. The main attractiveness of the new yield function lies in its very simple mathematical form, which makes it very effective and thus quite interesting for implementation in finite element codes. The capabilities of the new yield function are demonstrated by applications to selected orthotropic sheet materials.

Effect of anisotropic yield functions on the accuracy of hole expansion simulations

Currently available models describing superplastic deformation are mostly based on uniaxial tensile test data and assume isotropic behavior, thus leading to limited predictive capabilities of material deformation and failure. In this work we present a multi-axial microstructure-based constitutive model that describes the anisotropic superplastic deformation within the continuum theory of viscoplasticity with internal variables. The model accounts for microstructural evolution and employs a generalized anisotropic dynamic yield function. The anisotropic yield function can describe the evolution of the initial state of anisotropy through the evolution of unit vectors defining the direction of anisotropy during deformation. The generalized model is then reduced to the plane stress condition to simulate sheet metal stretching in superplastic blow forming using pressurized gas. Different ratios of biaxial stretching were investigated, including the case simulating the uniaxial loading condition, where the model successfully captured the uniaxial experimental data. The model is also used to develop a new forming pressure profile that accounts for anisotropy and microstructural evolution.

Forming of AA5182-O and AA5754-O at elevated temperatures using coupled thermo-mechanical finite element models

Recent results (Khraisheh et al., 1995 and 1997) have indicated that superplastic materials exhibit a strong degree of anisotropy and that the plastic flow cannot be described by the isotropic von Mises flow rules. In this study, the yield potential for the model Pb-Sn superplastic alloy is constructed experimentally for different effective strain rates using combined tension/torsion tests. A generalized anisotropic “dynamic” yield function is also proposed to represent the experimentally constructed yield potentials. The anisotropic function is not only capable of describing the initial anisotropic state of the yield potential, it can also describe its evolution through the evolution of unit vectors defining the direction of anisotropy. The anisotropic yield function includes a set of material constants which determine the degree of deviation of the yield potential from the isotropic von Mises yield surface. It is shown that the anisotropic yield function successfully represents the experimental yield potentials, especially in the superplastic region. [S0094-4289(00)01401-8]

Nonquadratic anisotropic yield functions have been developed in recent years for many lightweight automotive sheet metals. Realization of the improved performance of these advanced anisotropic yield functions depends in part on a careful calibration of their material constants via simple mechanical tests. This paper describes a novel approach on the parameter identification of plane–stress polynomial anisotropic yield functions by recasting them in a special form in terms of two principal stresses and one loading orientation angle. Independent mechanical tests for the parameter identification can thus be classified according to the plane stress state and the in-plane loading orientation, respectively. Parameter identification options have been examined in detail for a fourth-order homogeneous polynomial anisotropic yield function using the proposed approach. Some new insights have been gained on the permissible types and the number of independent mechanical test measurements per type that are needed for fully calibrating the material constants of the fourth-order yield function. The use of equal biaxial plastic strain ratio instead of equal biaxial yield stress makes the experimental calibration of the yield function more feasible as only simple in-plane uniaxial tension and out-of-plane compression tests of sheet metal samples are required.

Comparative evaluation of non-associated quadratic and associated quartic plasticity models for orthotropic sheet metals

In metallurgical processes, as for example cold rolling or deep drawing of sheet metal, it is frequently observed that the crystallographic texture, and with it the anisotropic mechanical properties of a material, evolve dynamically. Hence, to describe such processes, it is necessary to model the functional dependence of anisotropic material parameters on the texture, which itself can vary locally with the different plastic strain histories. In this work, we present a new data-oriented approach to parametrize the anisotropic yield function Barlat Yld2004-18p from micromechanical simulations for different textures. This is accomplished by applying supervised machine learning (ML) methods to express the relationship between different crystallographic textures and the material parameters of the yield function. The crystallographic textures are chosen to vary continuously between a random texture on the one hand side, and a unimodal Goss or Copper texture the other. These crystallographic textures are rather common in sheet metal forming. In this way, furthermore, the transition from isotropic plasticity to a rather severe case of anisotropy can be modeled, which is thought to mimic the dynamical evolution of the texture in a metallurgical process. It is found that a regularization strategy is necessary to circumvent the known non-uniqueness between Yld2004-18p parameters and the resulting plastic yield behavior. After this regularization, a unique relationship between the material parameters and the yield onset is established, making it possible to train different ML models with excellent accuracy and generalization properties to anisotropic plastic material behavior. The trained ML models are able to reliably predict the coefficients of unknown textures even with a small amount of training data and, thus, to correctly represent the yield behavior resulting from the various textures. The proposed method represents an efficient extension of the description of anisotropic plastic yielding as it establishes a data-oriented way to explicitly consider microstructural parameters in the material description, which opens new pathways to formulate material models that include the process history.

An elastic-plastic analysis of anisotropic laminated composite plate with a hole is carried out using a two dimensional finite element model based on higher order shear deformation theory (HSDT) of anisotropic laminated composite plates. The variation in material properties through thickness is made possible by discrete layer approach. The generalized Huber-Mises yield criterion, developed by Hill, for anisotropic material, is used to determine the onset of plastic flow. The inclusion of anisotropic parameters of plasticity generalizes the plastic yield function. These anisotropic parameters of plasticity are updated during the work hardening history of anisotropic laminated composite plate. The plastic potential, which is an anisotropic yield function, is used to determine an associated flow rule. Finally, an elastic-plastic incremental constitutive relation is obtained using finite element method. By means of an incremental and iterative procedure, the numerical solution of deformation problems in nonlinear anisotropic laminated plates is achieved based on first order shear deformation theory and higher order shear deformation theory. The transverse shear effects on the load deformation characteristics and on the spread of plastic zones around the circular hole are discussed. This method yields accurate values for transverse shear stresses as compared to other shear deformation theories and hence the growth of plastic zones.

Orthotropic yield criteria for description of the anisotropy in tension and compression of sheet metals

An accurate description of the directional dependence of uniaxial tensile yielding and plastic flow in advanced high-strength steel sheets may require either a nonassociated plasticity model with separate quadratic yield function and flow potential or an associated plasticity model with nonquadratic yield function. In this paper, Gotoh's fourth-order homogeneous polynomial yield function is applied to model two advanced high-strength steel sheets in an associated plasticity model. Both the parameter selection for calibrating Gotoh's yield function and its positivity and convexity verification are given in some detail. Similarities and differences between the associated plasticity model presented here and the nonassociated one appeared in the literature are discussed in terms of the directional dependence of yield stresses and plastic strain ratios under uniaxial tension and yield stresses under biaxial tension loading.

For finite strain plasticity with both anisotropic yield functions and anisotropic hyperelasticity, we use the Kröner‐Lee decomposition of the deformation gradient to obtain a differential‐algebraic system (DAE) in the semi‐implicit form and solve it by an implicit Richardson‐extrapolated method based on intermediate substeps. The source is here the right Cauchy‐Green tensor and the consistent Jacobian of the second Piola‐Kirchhoff stress is determined with respect to this source. The system is composed by a smooth nonlinear first‐order differential equation and a non‐smooth algebraic equation. The development of a Richardson‐extrapolated implicit integrator for any hyperelastic case and any yield function is the goal of this work. The integration makes use of a backward‐Euler method for the flow law complemented by the solution of a yield constraint. The resulting system is solved by the Newton‐Raphson method to obtain the plastic multiplier and the elastic right Cauchy‐Green tensor . To ensure power consistency, we make use of the elastic Mandel stress construction. Iso‐error maps for three yield functions and three numerical examples are presented.

Modeling of aluminum alloy sheets based on new anisotropic yield functions