Anisotropic yield function based on stress invariants for BCC and FCC metals and its extension to ductile fracture criterion
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 can be solely determined by its stress invariants which are defined as follows:
The stress tensor can be decomposed into two parts: a hydrostatic stress tensor, which is responsible for the volume change of a stressed body; and a deviatoric stress tensor 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 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. 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 , while Eq. (33) with 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 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 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 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: and . 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 of
The equivalent stress to fracture denoted by 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.
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