Phenomenological Yield Function Research Articles

A new texture descriptor for data-driven constitutive modeling of anisotropic plasticity

Constitutive modeling of anisotropic plastic material behavior traditionally follows a deductive scheme, relying on empirical observations that are cast into analytic equations, the so-called phenomenological yield functions. Recently, data-driven constitutive modeling has emerged as an alternative to phenomenological models as it offers a more general way to describe the material behavior with no or fewer assumptions. In data-driven constitutive modeling, methods of statistical learning are applied to infer the yield function directly from a data set generated by experiments or numerical simulations. Currently these data sets solely consist of stresses and strains, considering the microstructure only implicitly. Similar to the phenomenological approach, this limits the generality of the inferred material model, as it is only valid for the specific material employed in the virtual or physical experiments. In this work, we present a new generic descriptor for crystallographic texture that allows an explicit consideration of the microstructure in data-driven constitutive modeling. This descriptor compromises between generality and complexity and is based on an approximately equidistant discretization of the orientation space. We prove its ability to capture the structure–property relationships between a variety of cubic–orthorhombic textures and their anisotropic plastic behavior expressed by the yield function Yld2004-18p. Three different machine learning models trained with the descriptor can predict yield loci as well as r-values of unseen microstructures with sufficient accuracy. The descriptor allows an explicit consideration of crystallographic texture, providing a pathway to microstructure-sensitive data-driven constitutive modeling.

Elastoplastic discrete bond cell model for rock-like material with irreversible sliding interfaces

An elastoplastic discrete bond cell model with irreversible sliding interfaces is proposed for rock-like material. It represents the rock-like material with an assembly of linear elastic discrete bond cells. Irreversible sliding interfaces are inserted between the bond cells, which are governed by a simple friction law. Using this method, the complex elastoplastic behaviours of rock-like materials can be well simulated without using the phenomenological yield function and plastic potential of bulk material. In addition, the fracture process is also well simulated. The simulation results suggest that this model can capture the microplasticity and fracture mechanism of rock-like material. It provides a simple and effective approach to the fracture and plasticity simulation of rock-like material.

Enhancing phenomenological yield functions with data: Challenges and opportunities

The formulation of history-dependent material laws has been a significant research and industrial activity in solid mechanics for over a century. A large variety of models has been developed, tailored for the description of different families of materials. However, model selection for a specific problem is a delicate issue and there still remain open problems. In fact, the catalog of yield models is continuously being enriched by experts to meet new needs and tailor models to new experimental evidence. We propose here an alternative approach, that is a flexible model-data-driven plasticity formulation, and analyze its challenges and opportunities. A phenomenological yield model is locally improved by a data-driven correction term. The data-based correction component is described by a surrogate model built from machine-learning techniques. In this regard, the framework is versatile, as shown by the fact that similar performances are obtained with Support Vector Regression, Gaussian Process Regression, and Neural Networks. The convexity of model-data yield functions is guaranteed by employing convex extensions of the adopted machine-learning techniques. The proposed approach is substantiated by reproducing a highly anisotropic yield response with tension/compression asymmetries from a simple phenomenological model enhanced with a limited number of synthetic data points of yield onset. It is shown that a crucial role is played by both model and data, the former allowing to use a limited number of data and the latter significantly enhancing model performance.

On the reliability of yield functions in deep drawing simulations

In sheet metal forming simulations, the yield functions are usually calibrated based on experimental data, and validated by comparing the modelled and measured r-values and (in more advanced models) yield stresses along various in-plane directions. The fitted yield function should ideally reproduce (or interpolate) all the experimental values used for the fitting. However, this requirement does not guarantee accurate results in a forming process simulation, and it can even lead to unexpected results. The performance of a yield function, in addition to the fitting procedure, depends on the active loading modes which the material experiences during the simulation. The active loading modes are in turn determined by the die geometry as well as the process parameters like blank holder force and lubrication. As a consequence, the performance of a yield function, which is fitted to simple experimental data, is not identical for different forming conditions. Therefore, the application of each model is usually limited to a range of materials and processes, and this applicability is often evaluated based on experience. In the present study, we examine three phenomenological yield functions and a new crystal plasticity based material model for cup drawing process simulation with an AA6016-T4 aluminium alloy. These functions are different in their input data types used for the calibration. The results surprisingly shows that the models with more experimental data (in particular, yield stresses in different directions) in their formulation not only predict unexpectedly wrong results, but also show strong sensitivity to some of those additional input data and even to modelling parameters like the friction coefficient.

A virtual materials testing approach to calibrate anisotropic yield functions for the simulation of earing during deep drawing of aluminium alloy sheet

We present a pragmatic and computationally efficient scheme to combine polycrystal-plasticity models based on crystallographic texture with phenomenological yield functions into a multilevel modelling framework for consideration of the anisotropic materials response during finite-element (FE) simulations of forming operations. Materials tests are conducted virtually using the visco-plastic self-consistent (VPSC) polycrystal-plasticity model to provide the stresses and strain ratios in uniaxial tension, plane-strain tension and under balanced biaxial conditions. These anisotropic materials data are then utilized to calibrate an advanced phenomenological yield function, viz. Yld2011–27p due to Aretz and Barlat, to simulate the anisotropic materials behaviour during subsequent sheet metal forming operations, here, deep-drawing. Since the simulated materials data can be used just as experimental values, no special fitting procedure for the yield function is required and validation of the simulation results is straightforward. Moreover, the followed modelling scheme enables to combine materials data obtained from experimental and virtual tests in a so-called hybrid approach. The modelling scheme was validated by FE-simulations of the earing behaviour of various Al sheet alloys, including AA 1200-O, AA 3104-H19, AA 5182-O and AA 5050A-H44, with distinctively different textures. Since earing is almost entirely controlled by crystallographic texture, analysis of the earing behaviour provides for a meaningful but demanding proof-of-principle for the current hierarchical modelling scheme.

A multi-scale modelling framework for anisotropy prediction in aluminium alloy sheet and its application in the optimisation of the deep-drawing process

This study proposed a crystallographic texture-based multi-scale modelling framework to predict mechanical anisotropy of textured aluminium alloy sheet. The multi-scale scheme was constructed by the combination of a full-field crystal plasticity model in mesoscopic scale and a newly developed phenomenological yield function in continuum scale. In this approach, the mechanical anisotropy of materials is directly obtained from crystal plasticity (CP) modelling without extensive directional tensile tests. The results show that the multi-scale scheme predicted earing profiles coincide well with experimental measurements. This multi-scale modelling could be utilised to design an optimised blank shape with minimum earing for the deep-drawn component, and a convoluted cut-edge was devised accordingly.

Multi-level deep drawing simulations of AA3104 aluminium alloy using crystal plasticity finite element modelling and phenomenological yield function

This paper proposed a hierarchical multi-level model to study the crystallographic texture induced mechanical anisotropy of AA3104-H19 aluminium sheet from mesoscale to continuum scale. In the mesoscale, full-field crystal plasticity finite element method (CPFEM) was used to provide both in-plane and out-of-plane yield stresses and plastic potential points in various deformation modes. In the continuum scale, these materials sampling points were used to determine the parameters of two phenomenological yield functions (Yld2000-2d in plane stress space and Yld2004-18p in 3D stress space) using associated flow rule (AFR) and non-associated flow rule (non-AFR). The results indicate that higher accuracy obtained by Yld2000-2d and Yld2004-18p yield functions associated with non-AFR in comparison with AFR. These phenomenological models were successfully implemented into finite element (FE) code using an explicit integration scheme to simulate sheet metal forming. It is found that the 3D Yld2004-18p model involved with both in-plane and out-of-plane anisotropies is superior to 2D Yld2000-2d model which only accounts for in-plane anisotropy.

A micro‐mechanically motivated phenomenological yield function for cubic crystal aggregates

AbstractA micro‐mechanically motivated phenomenological yield function, for polycrystalline cubic metals is presented. In the suggested yield function microstructure is taken into account by the crystallographic orientation distribution function in terms of tensorial Fourier coefficients. The yield function is presented in a polynomial form in powers of the stress state. Known group‐theoretic results are used to identify isotropic and anisotropic parts in the yield function, whereby anisotropic parts are characterized by tensorial Fourier coefficients. The form of the presented yield function is inspired by the classic, phenomenological von Mises ‐ Hill yield function first published in 1913. For a specific choice of material parameters, both functions coincide, thus a micro‐mechanically motivated generalization of the von Mises ‐ Hill yield function is presented. For the given yield function, two dimensional experimental results are sufficient, to identify a three dimensional anisotropic yield behavior. The work concludes with a treatment of the isotropic special case, i.e. a tension‐compression split in yield behavior as well as parameter ranges for convexity and shapes of the yield surface.

A 3D phenomenological yield function with both in and out-of-plane mechanical anisotropy using full-field crystal plasticity spectral method for modelling sheet metal forming of strong textured aluminum alloy

In this paper, a multi-scale model for AA3104-H19 and AA2024-T3 aluminium alloy with strong cold rolled crystallographic texture is proposed. This model consists of two components: (i) a full-field crystal plasticity spectral method using fast Fourier transformation (CPFFT) that can be used to predict anisotropic mechanical properties in both the in and out-of-plane uniaxial and multi-axial stress states and (ii) a 3D new phenomenological anisotropic yield function with enough flexibility to describe the in and out-of-plane mechanical anisotropies of strongly textured aluminium alloy sheet. This multi-scale model was implemented into a commercial finite element (FE) software Abaqus via a user-defined material subroutine UMAT to predict the anisotropic plastic deformation in a multi-step forming process (deep drawing). Results showed that this new modelling approach is able to accurately predict earing profiles with multiple ears in deep drawn cups based on the crystallographic texture information data obtained from orientation density functions (ODFs).

Numerical investigation of evolution of earing, anisotropic yield and plastic potentials in cold rolled FCC aluminium alloy based on the crystallographic texture measurements

The mechanical anisotropy of FCC aluminium alloy is influenced by the polycrystalline orientations transformed by cold rolling. The relationship between mechanical anisotropy and evolution of crystallographic textures was established using the crystal plasticity finite element method (CPFEM) based on measured polycrystalline orientation density functions (ODFs) and used to determine the parameters of a phenomenological convex anisotropic yield function which are solved with an implicit finite element (FE) stress integration code. A multi-scale modelling system is generated by combining CPFEM and the phenomenological yield function enabling accurate simulations of forming operations without the need for conventional physical mechanical testing of the material.

Evaluation and calibration of anisotropic yield criteria in shear Loading: Constraints to eliminate numerical artefacts

Many anisotropic phenomenological yield functions have been proposed in the literature in which their predictive capabilities strongly depend on the experimental calibration data as well as the calibration procedure to identify the anisotropy parameters. In this paper, emphasis is placed upon the experimental and numerical calibration procedure of anisotropic yield functions in the region of shear loading (zero hydrostatic stress with equal and opposite in-plane principal strains and stresses). Conventional anisotropic calibration procedures are shown to introduce non-physical artefacts into constitutive models which manifest as a non-zero hydrostatic stress or through-thickness strains generated under in-plane shear stress that violate the definition of the shear loading condition. To overcome this issue, a new physically necessary constraint is applied on the plastic potential to enforce equal and opposite principal strains in the shear state and correct the shear region of anisotropic yield functions. Using this necessary constraint, the widely used Yld2000-2d anisotropic yield function was calibrated using an associated flow rule for aluminum alloy sheet using published data for AA2090-T3 to demonstrate how enforcing this constraint can be readily implemented to correct the shear region of the anisotropic yield surface. Furthermore, to investigate the influence of the shear constraint, an AA7075-T6 alloy was experimentally characterized in uniaxial tension, equal-biaxial tension and shear. It was revealed that with the additional shear constraints, non-physical artefacts of plane-stress anisotropic yield functions such as Yld2000-2d can be removed during the calibration procedure. However, due to the additional shear constraints, available anisotropic models may become over-constrained and alternate yield functions with more flexibility or non-associated flow rules may be required.

Mechanical characterization and constitutive parameter identification of anisotropic tubular materials for hydroforming applications

This paper aims to identify the constitutive parameters of anisotropic tubular materials and to verify the accuracy of models׳ prediction. The identification of the constitutive parameters is based on information obtained from tensile tests, performed on samples cut from the tubes, and from the free tubular bulge test, using a home-developed bulge forming machine. Two tubular materials exhibiting different anisotropic behaviour and work hardening characteristics are investigated: a mild steel S235 seamed tube and an aluminium alloy AA6063 extruded tube. It is shown that advanced phenomenological yield functions, including a large number of anisotropy parameters, can accurately describe the plastic flow of highly anisotropic tubular materials during the tube hydroforming process. However, parameter identification procedure of advanced yield criteria requires a high number of experimental tests. Thus, in order to enable the parameter identification of these yield criteria when using a reduced set of experimental results, the present study develops a method that combines tensile tests with (i) a free bulge test, which is used to characterize the biaxial stress state experienced by the tube during the bulge testing, and (ii) some generated artificial input data. Finally, the proposed method shows an excellent agreement between numerical predictions and experimental results.

MULTIAXIAL STRESS TESTS FOR METAL SHEETS AND TUBES FOR ACCURATE MATERIAL MODELING AND FORMING SIMULATIONS

This paper is a review of the mechanical testing methods developed by the author's research group: multiaxial stress test methods using a cruciform test piece and a tubular test piece. The former is useful for small strain ranges under several percent while the latter is useful for larger strain ranges (from yielding to fracture). These test methods are useful to determine appropriate materials model for performing accurate metal forming simulations. Special attention is given to the measurement and modeling of the anisotropic plastic deformation behavior of sheet metals commonly used in industry and to the validation of the material models based on phenomenological yield functions for large plastic strain ranges. The effects of material models used in metal forming simulations on the improvement of the predictive accuracy for forming defects are also discussed.

Orientation and Path Dependence of Formability in the Stress- and the Extended Stress-Based Forming Limit Curves

This article analyzes the formability data sets for aluminum killed steel (Laukonis, J. V., and Ghosh, A. K., 1978, “Effects of Strain Path Changes on the Formability of Sheet Metals,” Metall. Trans. A., 9, pp. 1849–1856), for Al 2008-T4 (Graf, A., and Hosford, W., 1993, “Effect of Changing Strain Paths on Forming Limit Diagrams of Al 2008-T4,” Metall. Trans. A, 24A, pp. 2503–2512) and for Al 6111-T4 (Graf, A., and Hosford, W., 1994, “The Influence of Strain-Path Changes on Forming Limit Diagrams of Al 6111 T4,” Int. J. Mech. Sci., 36, pp. 897–910). These articles present strain-based forming limit curves (ϵFLCs) for both as-received and prestrained sheets. Using phenomenological yield functions, and assuming isotropic hardening, the ϵFLCs are transformed into principal stress space to obtain stress-based forming limit curves (σFLCs) and the principal stresses are transformed into effective stress and mean stress space to obtain the extended stress-based forming limit curves (XSFLCs). A definition of path dependence for the σFLC and XSFLC is proposed and used to classify the obtained limit curves as path dependent or independent. The path dependence of forming limit stresses is observed for some of the prestrain paths. Based on the results, a novel criterion that, with a knowledge of the forming limit stresses of the as-received material, can be used to predict whether the limit stresses are path dependent or independent for a given prestrain path is proposed. The results also suggest that kinematic hardening and transient hardening effects may explain the path dependence observed in some of the prestrain paths.

Finite Element analysis of aluminum tube hydroforming based on non-quadratic yield function

Although very accurate in representing the material anisotropy, finite element analysis based on direct use of a polycrystal model is CPU-intensive (Guan et al., 2006a). In the present study, this Taylor-type polycrystalline model was firstly used to predict the anisotropy coefficients of Yld96 yield function. Then this phenomenological anisotropic yield function was implemented into ABAQUS/Standard finite element code, using UMAT, for the hydroforming simulation of a 6061-T4 extruded aluminum tube. It was shown that compared with von Mises (1913) and Hill's (1948) yield function, the Yld96 material model's predictions are in better agreement with experimental data.

Linking Meso- and Macroscale Simulations: Crystal Plasticity of hcp Metals and Plastic Potentials

A crystal plasticity model has been used to simulate channel die experiments on both, pure magnesium single crystals and polycrystalline textured rolled plates. Deformation mechanisms and slip system activity can be identified by FE-analyses of single crystals. The role of twinning can be understood and modeled phenomenologically by an additional slip system. Simulations of polycrystalline aggregates are used to obtain a representation of the material's phenomenological yield function in order to describe the plastic deformation behavior using the framework of continuum mechanics. This allows for accounting for the specific texture and thus for its optimization. The tension- compression asymmetry, which is typical for mechanically processed magnesium material, can be reproduced by means of the crystal plasticity and a phenomenological model.

An effective computational algorithm for rate-independent crystal plasticity based on a single crystal yield surface with an application to tube hydroforming

Up to now, several computational methods have been proposed for crystal plasticity models. The main objective of these computational methods has been to overcome the problem with the non-uniqueness of active slip systems during the plastic deformation of a single crystal. Crystal plasticity models based on a single crystal yield function have been proposed as alternative algorithms to overcome this problem. But the problem with these models is that they use a highly non-linear yield function for the crystal, which makes them computationally expensive. In this paper, a computational method is proposed that would modify a single crystal yield function in order to make it computationally efficient. Also to better capture experimental data, a new parameter is introduced into the single crystal yield function to make it more flexible. For verification, this crystal plasticity model was directly applied for the simulation of hydroforming of an extruded aluminum tube under complex strain paths. It was found that the current model is considerably faster than the previous crystal plasticity model based on a power-law type single crystal yield surface. Due to its computational efficiency, the current crystal plasticity model can also be used to calculate the anisotropy coefficients of phenomenological yield functions.

Numerical analysis of diffuse and localized necking in orthotropic sheet metals

In the present paper the diffuse and localized necking models according to Swift [Swift, H.W., 1952. Plastic instability under plane stress, Journal of the Mechanics and Physics of Solids, 11–18], Hill [Hill, R., 1952. On discontinuous plastic states, with special reference to localized necking in thin sheets. Journal of the Mechanics and Physics of Solids 1, 19–30] and Marciniak and Kuczyński [Marciniak, Z., Kuczyński, K., 1967. Limit strains in the process of stretch-forming sheet metal. International Journal of Mechanical Sciences 9, 609–620], respectively, are considered. A theoretical framework for the mentioned models is proposed that covers rigid–plastic as well as elastic–plastic constitutive modelling using various advanced phenomenological yield functions that are able to account very accurately for plastic anisotropy. The mentioned necking models are applied to different orthotropic sheet metals in order to assess their predictive capabilities and to stress out some potential sources for discrepancies between simulations and experiments. In particular, the impact of the applied hardening curve and the equibiaxial r-value, which was recently introduced by Barlat [Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Choi, S.-H., Pourboghrat, F., Chu, E., Lege, D.J., 2003. Plane stress yield function for aluminium alloy sheets – part 1: theory. International Journal of Plasticity 19, 297–1319], on formability prediction is investigated. Furthermore, the impact of the Portevin–LeChatelier effect on the formability of aluminum sheet metals is discussed.

Correlation of Texture and Plastic Anisotropy in the Al-Mg Alloy AA 5005

The control of the plastic anisotropy during forming of a metallic sheet requires detailed knowledge on its microstructure and, especially, crystallographic texture. In the present paper the correlation of texture and plastic anisotropy in the Al-Mg Alloy AA 5005 is studied. Two different material states with different textures are compared, namely 1.7 mm thick soft-annealed material (O-temper) and 1.0 mm cold rolled strip (state H14). Tensile tests were carried out at 0°, 45° and 90° with respect to the sheet rolling direction and the resulting in-plane anisotropy in yield strength, work hardening and strain rate ratios (r-values) was determined. On a more general approach, texture-derived yield loci are compared with both phenomenological yield functions and experimental data.

A non-quadratic yield function for polymeric foams

The yield behavior of a closed cell polymeric foam is investigated under multiaxial loadings. A phenomenological yield function is developed to characterize the initial yield behavior of the closed cell polymeric foam under a full range of loading conditions. The principal stresses of a relative stress tensor and the second invariant of the deviatoric stress tensor are the main parameters in the yield function. The yield function is a linear combination of non-quadratic functions of the relative principal stresses and the second invariant of the deviatoric stress tensor. The convexity of the yield surface based on the non-quadratic yield function is proved. The non-quadratic yield function is shown to well characterize the yield behavior of a closed cell polymeric foam in [Deshpande, V.S., Fleck, N.A., 2001. Multi-axial yield behavior of polymer foams. Acta Materialia 49, 1859–1866] under a full range of loading conditions. Finally, a comparison of different phenomenological yield functions to characterize the yield behavior of the foam is presented.