Elsevier

International Journal of Plasticity

Volume 101, February 2018, Pages 125-155
International Journal of Plasticity

Anisotropic yield function based on stress invariants for BCC and FCC metals and its extension to ductile fracture criterion

https://doi.org/10.1016/j.ijplas.2017.10.012Get rights and content

Highlights

  • Drucker function is extended into anisotropy using linear transformation tensor.

  • Anisotropic flexibility is enhanced by two approaches: non-AFR and sum of n-components of the anisotropic Drucker function.

  • Comparison demonstrates that anisotropy is accurately modeled for BCC and FCC metals by the anisotropic Drucker function.

  • The enhanced Drucker function can model anisotropy of BCC and FCC metals easily numerical analysis under spatial loading.

  • Drucker function has been extended to ductile fracture criterion with high accuracy from compressive to tension.

Abstract

It is essential to accurately model the anisotropic plastic deformation and ductile fracture of metals in order to guarantee the reliable numerical analysis and optimization of metal forming. For this purpose, the Drucker function is revisited. Effect of the third stress invariant in the Drucker function is analyzed and calibrated for metals with body-centered cubic (BCC) and face-centered cubic (FCC) crystal systems based on the yielding and plastic flow of both crystal plasticity and biaxial tensile experiments. The calibrated Drucker function is extended into an anisotropic form using a fourth order linear transformation tensor. The anisotropic flexibility is enhanced by two approaches: non-associate flow rule (non-AFR) and the sum of n-components of the anisotropic Drucker function. The proposed anisotropic Drucker function is applied to model the anisotropic behavior of both BCC and FCC metals. The predicted anisotropic behavior is compared with experimental results. The comparison demonstrates that the anisotropy is accurately modeled for both BCC and FCC metals by the anisotropic Drucker function. The anisotropic Drucker function is also implemented into numerical analysis of tension of specimens with a central hole to investigate its computation efficiency under spatial loading compared with the Yld2000-18p function. It is found that the proposed anisotropic Drucker function can reduce about 60% of computation time in case that the Yld2000-18p function is substituted by the anisotropic Drucker function in numerical computation due to its simplicity compared to the Yld2000-18p function. A ductile fracture criterion is also developed by coupling the Drucker function with the first stress invariant. The modified Drucker function is reformulated to investigate the effect of the stress triaxiality and the normalized third invariant on ductile fracture. Comparison of the modified Drucker fracture locus with the experimental results of AA2024-T351 demonstrates that the modified Drucker criterion accurately illustrates the fracture stress of the alloy in wide stress states with the stress triaxiality ranging from −0.5 in plane strain compression to 0.6 in tension of notched specimens. The modified Drucker fracture criterion is expected to be less sensitive to the change of strain path considering that the criterion describes fracture in the stress space. Accordingly, the anisotropic Drucker yield function and the pressure-coupled Drucker fracture criterion are suggested to model anisotropic plastic deformation and to predict the onset of failure for both BCC and FCC metals due to simple implementation in numerical analysis under spatial loading and computation efficiency with brick elements.

Introduction

Accurate modeling of anisotropic plastic deformation and ductile fracture has been one of the key issues in sheet metal forming researches. Dozens of anisotropic yield functions were developed with different accuracy and distinct applications. Hill (1948) proposed the first function to describe anisotropic behavior of metals based on the isotropic von Mises yield function. Thereafter, various anisotropic yield functions (i.e. Aretz and Barlat, 2013, Hill, 1948, Hill, 1979, Hill, 1990, Hill, 1993, Hosford, 1979, Barlat and Lian, 1989, Barlat et al., 1991, Barlat et al., 1997, Barlat et al., 2003, Barlat et al., 2005, Banabic et al., 2005, Cazacu et al., 2006, Karafillis and Boyce, 1993, Lou et al., 2013a) were developed to introduce more anisotropic coefficients for the improvement of accuracy. Most of these yield functions can only describe yielding behavior under plane stress, while the other yield functions for spatial loading are generally expressed as a function of principal stresses of a linear transformed stress tensor. However, the computation of the principal stresses is complicated for spatial loading. It is even lengthy to calculate the derivatives of yield functions for their implementation in numerical analysis.

Soare (2008) and Soare and Barlat (2010) tried to model anisotropic plastic deformation using homogeneous polynomials. The yield functions are expressed directly by stress components. These yield functions are user-friendly for numerical application under spatial loading since principal stress components are not required for computation. The problem lies in that the convexity of the polynomials must be considered first for some specific cross sections of a 3D yield locus. Besides, the polynomial yield functions cannot differentiate between face-centered cubic (FCC) and body-centered cubic (BCC) metals.

Another branch of yield functions is developed based on three stress invariants of the stress tensor. Examples of these yield functions are Drucker, 1949, Cazacu and Barlat, 2001, Cazacu and Barlat, 2004, Gao et al., 2011, Yoshida et al., 2013, Yoon et al., 2014, Smith et al., 2015, and Cazacu and Revil-Baudard (2017). The advantage of these yield functions is that the yield functions can be utilized under both plane stress condition and the spatial loading cases conveniently, and the derivatives can be easily computed analytically. Moreover, the convexity is automatically guaranteed by setting a coefficient in these functions in a specified range, which is the merit of these functions over polynomial functions. Their drawback is that they do not differentiate between BCC and FCC metals.

Recently, Tuninetti et al. (2015) modeled the anisotropy and tension-compression asymmetry of Ti-6Al-4V at room temperature. Gawad et al. (2015) developed an evolving plane stress yield criterion based on crystal plasticity virtual experiments. Yoshida et al. (2015) introduced a framework for constitutive modeling of anisotropic hardening and Bauschinger effect for sheet metals. Kuwabara et al. (2017) conducted biaxial tensile tests of AA6016-O and AA6016-T4 and found that the strength under plane strain tension modeled by the quadratic von Mises and Hill48 yield functions is much higher than experimental results. Non-quadratic yield functions with high exponents can provide good description of yield stress around plane strain tensile condition. Raemy et al. (2017) developed a yield function based on Fourier series to model asymmetry and anisotropy in HCP metals under plane stress loading. Li et al. (2017) modeled anisotropic and asymmetrical yielding and its distorted evolution of titanium tubular metals with the shear stress-based CPB′2006 yield function (Cazacu et al., 2006) and a stress invariants-based model (Yoon et al., 2014).

In the last decade, a number of uncoupled ductile fracture criteria were proposed based on the experimental results of Bao and Wierzbicki (2004). Xue (2007) first introduced the effect of the Lode angle on ductile fracture. Bai and Wierzbicki (2008) proposed a phenomenological ductile fracture criterion with dependence on the stress triaxiality and the normalized Lode angle. Bai and Wierzbicki (2010) modified the Mohr-Coulomb criterion for the ductile fracture prediction of metals. Li et al. (2011) comprehensively compared the predictability of various ductile fracture models. Stoughton and Yoon (2011) provided an efficient method for the analysis of necking and fracture limits for sheet metals. Stoughton and Yoon (2012) proposed a new type of forming limit curves based on a polar representation of the effective plastic strain. Khan and Liu, 2012a, Khan and Liu, 2012b established a phenomenological fracture criterion using the magnitude of stress vector and the first invariant of stress tensor and considered effect of strain rate and temperature on ductile fracture in the proposed model. Lou et al., 2012, Lou et al., 2014, Lou et al., 2017 proposed a series of models to describe ductile fracture taking place along the maximum shear stress for metals with high ratio of strength to density. The Lode dependence of ductile fracture is also correlated with the effect of the maximum shear stress on the coalescence of voids along the maximum shear stress (Lou and Huh, 2013b). These criteria were successfully applied to predict onset of ductile fracture in various metal forming processes, such as limited dome height at the onset of ductile fracture for DP780 (Lou et al., 2013b), edge fracture prediction in hole expansion (Mu et al., 2017), fracture in high velocity perforation (Vershinin, 2015), and a series of independent validation experiments including a hole tension test, a conical and flat punch hole expansion test, and a hemispherical punch test (Anderson et al., 2017). Cao et al. (2014) modified the Lemaitre model to describe ductile fracture at low stress triaxiality. Mohr and Marcadet (2015) proposed a phenomenological Hosford-Coulomb model to depict ductile fracture at low stress triaxiality. Lee et al. (2017) predicted fracture based on a two-surface plasticity law for anisotropic magnesium alloys. Lou and Yoon, 2017a, Lou and Yoon, 2017b modified the DF2012 criterion to take into account the anisotropic behavior of ductile fracture based on linear transformation. Sebek et al. (2017) coupled Lode dependent plasticity with nonlinear damage accumulation for prediction of ductile fracture of aluminum alloy. Cao et al. (2018) developed a modified elliptical fracture criterion based on two kinds of strain energy density.

The purpose of this paper is twofold. On the one hand, the stress invariant-based Drucker function is revisited. The effect of the third stress invariant is calibrated for BCC and FCC metals for the Drucker yield function based on the comparison of the Drucker yield surface with the non-quadratic Hershey function. The calibrated Drucker yield function for BCC and FCC metals is applied to model the yielding and plastic flow computed for randomly oriented polycrystals for the further verification of the calibrated effect of the third stress invariant. The Drucker function for BCC and FCC metals is extended into an anisotropic form using a fourth order linear transformation tensor. Two approaches are proposed to improve the predictability of the anisotropic Drucker function. Both approaches are applied to model anisotropic behavior of BCC and FCC metals for the verification of the calibrated Drucker-based function for the modeling of anisotropic yielding and plastic flow directions. The proposed anisotropic Drucker yield function is further implemented into numerical analysis of tension of specimens with a central hole to investigate its computational efficiency. On the other hand, the Drucker function is coupled with the first stress invariant to model ductile fracture characteristics of metals. The modified Drucker function is reformulated with dependence on the stress triaxiality and the normalized third stress invariant to analyze the effect of the stress triaxiality and the normalized third stress invariant on ductile fracture. The pressure-coupled Drucker fracture criterion is then applied to model the fracture stress of AA2024-T351 in various loading conditions from compressive upsetting to tension of notched specimens to investigate the predictability of ductile fracture.

Section snippets

Stress invariant-based yield functions

For isotropic materials, any stress states denoted by the stress tensor σij can be solely determined by its stress invariants which are defined as follows:I1=Tr(σij)=σ11+σ22+σ33I2=σ11σ22+σ22σ33+σ11σ33σ122σ232σ132I3=σ11σ22σ33+2σ12σ23σ13σ11σ232σ22σ132σ33σ122

The stress tensor can be decomposed into two parts: a hydrostatic stress tensor, 13I1δij which is responsible for the volume change of a stressed body; and a deviatoric stress tensor sij which tends to distort the body. Then the stress

Calibration of the Drucker function for BCC and FCC metals

The Drucker function is proposed to couple the effect of the third stress invariant on plastic yielding for pressure-insensitive metals. It is further transformed into anisotropy by Cazacu and Barlat (2001) and Yoshida et al. (2013). But the effect of the third stress invariant is not calibrated for BCC and FCC metals with the Drucker function. The difference of the yield surfaces between BCC and FCC metals modeled by non-quadratic functions lies in the different strength under plane strain

Extension to anisotropy

After the calibration of the value of c for BCC and FCC metals, the Drucker function is extended into an anisotropic form. Cazacu and Barlat (2001) extended the Drucker yield function into an anisotropic form by the theory of representation for the second and third stress invariants. But it is challenging to control the effect of the third invariant on yielding. Accordingly, the Drucker yield function is extended into anisotropy based on the linear transformation tensor for both the second and

Application of the proposed yield function under AFR (associated flow rule) to a BCC metal

The anisotropic Drucker yield function is first utilized to model anisotropic behavior of a BCC metal of 719B (Stoughton and Yoon, 2009). A good accuracy is achieved by setting n=4 in Eq. (27) which means that four linear transformations are used to improve its flexibility. AFR is employed since the flexibility is improved by the sum of four components of the anisotropic Drucker yield function. c is set to 1.226 considering that 719B is a BCC metal. Then there are 24 anisotropic coefficients to

Application of the proposed yield function under non-AFR to a BCC metal

The anisotropic plasticity of the BCC metal of 719B steel sheet (Stoughton and Yoon, 2009) is also modeled by the proposed anisotropic Drucker yield function under non-AFR. The anisotropic yielding is modeled by the anisotropic yield function in Eq. (27) with n=2, while Eq. (33) with n=2 is employed to model the anisotropy in plastic flow. The Lode dependent parameter in the anisotropic Drucker yield function and proposed plastic potential is set to 1.226 since the material is a BBC metal.

Application of the proposed yield function under AFR to an FCC metal

The anisotropic Drucker function is applied to an FCC metal of AA 6022 T4E32 (Stoughton and Yoon, 2009). In this application, anisotropic behavior of AA 6022 T4E32 is modeled by the anisotropic yield function in Eq. (27) with n=3 under AFR, which means three linear transformations. Accordingly, there are 12 anisotropic coefficients related with in-plane anisotropic yielding and plastic flow. These anisotropic parameters are calibrated by seven uniaxial tensile yield stress at every 15° from RD,

Application of the proposed yield function under non-AFR to an FCC metal

Plasticity of AA 2090 T3 with strong anisotropy is also modeled by the anisotropic Drucker function for its verification. In this case, the anisotropic Drucker function in Eq. (27) with n=2 is used to model anisotropic yielding while the anisotropy in plastic flow is modeled with the assumption of non-AFR of the proposed plastic potential with n=2 in Eq. (33). Therefore, it means two linear transformations for the proposed yield function to fit stress anisotropy and two linear transformations

Application of the proposed yield function to depict biaxial tension of an FCC metal

The proposed anisotropic Drucker yield function is applied to model the biaxial tensile yielding of a 5154-H112 extruded aluminum alloy tube of 76.3 mm outer diameter with 3.9 mm wall thickness (Kuwabara et al., 2005). Two components are used in the anisotropic Drucker yield function in Eq. (27) to improve its flexibility. There are seven coefficients under plane stress for biaxial loading: c,c1'(1),c2'(1),c3'(1),c1'(2),c2'(2) and c3'(2). c is set to 2 since AA5154-T112 is an FCC metal. The

Implementation into numerical simulation

In order to investigate the computation efficiency of the proposed anisotropic Drucker function, user subroutines are written for ABAQUS/Explicit for both anisotropic Drucker function (n = 2) under non-AFR and the Yld2000-18 function. The Euler's backward integration and return mapping technique are utilized to compute the plastic strain increment. Both yield functions are used to model tension of specimens with a central hole. Taking the advantage of its symmetry, one eighth of the specimen is

Extension to ductile fracture criterion based on stress invariants

Micro-mechanisms of ductile fracture include nucleation, growth and coalescence of voids. Growth of voids is strongly affected by the hydrostatic pressure as concluded by McClintock (1968) and Rice and Tracey (1969). Accordingly, the Drucker yield function is coupled with the hydrostatic pressure to model ductile fracture of a metal in a form ofσ¯f(σij)=a(bI1+(J23cJ32)1/6)

The equivalent stress to fracture denoted by σ¯f is formulated as a function of three stress invariants. In Eq. (36), three

Conclusions

A plasticity model and a fracture criterion are developed based on the Drucker function. On the one hand, the Drucker yield function is calibrated for BCC and FCC metals. The Drucker function is then transformed into an anisotropic form. Two approaches are used to improve the flexibility of the anisotropic Drucker function: non-AFR and the summation of several Drucker equations. The anisotropic Drucker function with improved flexibility is applied to depict the anisotropic behavior of several

Acknowledgement

The authors would like to thank the ARC linkage project: LP150101027, for their supports of this study. The authors are also thankful for A/Prof. Saijun Zhang at South China Univeristy of Technology to help the simulation of tension specimen with a hole and for Prof. Toshihiko Kuwabara at Tokyo University of Agriculture and Technology to provide the data for biaxial yield stress and flow directions.

References (73)

  • M. Brünig et al.

    Micro-mechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile fracture

    Int. J. Plast.

    (2013)
  • O. Cazacu et al.

    A criterion for description of anisotropy and yield differential effects in pressure-insensitive metals

    Int. J. Plast.

    (2004)
  • O. Cazacu et al.

    Orthotropic yield criterion for hexagonal closed packed metals

    Int. J. Plast.

    (2006)
  • O. Cazacu et al.

    New analytic criterion for porous solids with pressure-insensitive matrix

    Int. J. Plast.

    (2017)
  • J. Cao et al.

    A modified elliptical fracture criterion to predict fracture forming limit diagrams for sheet metals

    J. Mater. Process. Technol.

    (2018)
  • T.-S. Cao et al.

    A Lode-dependent enhanced Lemaitre model for ductile fracture prediction at low stress triaxiality

    Eng. Fract. Mech.

    (2014)
  • X.S. Gao et al.

    On stress-state dependent plasticity modeling: significance of the hydrostatic stress, the third invariant of stress deviator and the non-associated flow rule

    Int. J. Plast.

    (2011)
  • J. Gawad et al.

    An evolving plane stress yield criterion based on crystal plasticity virtual experiments

    Int. J. Plast.

    (2015)
  • R. Hill

    Constitutive modeling of orthotropic plasticity in sheet metal

    J. Mech. Phys. Solids

    (1990)
  • R. Hill

    A user-friendly theory of orthotropic plasticity in sheet metals

    Int. J. Mech. Sci.

    (1993)
  • A.S. Khan et al.

    A new approach for ductile fracture prediction on Al 2024-T351 alloy

    Int. J. Plast.

    (2012)
  • A.S. Khan et al.

    Strain rate and temperature dependent fracture criteria for isotropic and anisotropic metals

    Int. J. Plast.

    (2012)
  • A.P. Karafillis et al.

    A general anisotropic yield criterion using bounds bad a transformation weighting tensor

    J. Mech. Phys. Solids

    (1993)
  • T. Kuwabara et al.

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

    Int. J. Plast.

    (2005)
  • T. Kuwabara et al.

    Material modeling of 6016-O and 6016-T4 aluminum alloy sheets and application to hole expansion forming simulation

    Int. J. Plast.

    (2017)
  • R.A. Lebensohn et al.

    A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to Zirconium alloys

    Acta Metall. Mater.

    (1993)
  • J.-Y. Lee et al.

    Piecewise linear approximation of nonlinear unloading-reloading behaviors using a multi-surface approach

    Int. J. Plast.

    (2017)
  • H. Li et al.

    Ductile fracture: experiments and computations

    Int. J. Plast.

    (2011)
  • R.W. Logan et al.

    Upper-bound anisotropic yield locus calculations assuming <111>-pencil glide

    Int. J. Mech. Sci.

    (1980)
  • Y.S. Lou et al.

    New ductile fracture criterion for prediction of fracture forming limit diagrams of sheet metals

    Int. J. Solids Struct.

    (2012)
  • Y.S. Lou et al.

    Prediction of ductile fracture for advanced high strength steel with a new criterion: experiments and simulation

    J. Mater. Process. Technol.

    (2013)
  • Y.S. Lou et al.

    Extension of a shear controlled ductile fracture model considering the stress triaxiality and the Lode parameter

    Int. J. Solids Struct.

    (2013)
  • Y.S. Lou et al.

    Evaluation of ductile fracture criteria in a general three-dimensional stress state considering the stress triaxiality and the lode parameter

    Acta Mech. Solida Sin.

    (2013)
  • Y.S. Lou et al.

    Consideration of strength differential effect in sheet metals with symmetric yield functions

    Int. J. Mech. Sci.

    (2013)
  • Y.S. Lou et al.

    Modeling of shear ductile fracture considering a changeable cut-off value for the stress triaxiality

    Int. J. Plast.

    (2014)
  • Y.S. Lou et al.

    Modeling of ductile fracture from shear to balanced biaxial tension for sheet metals

    Int. J. Solids Struct.

    (2017)
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