An Experimental Methodology to Characterize the Plasticity of Sheet Metals from Uniaxial to Plane Strain Tension

Abstract

Background

Although accurate knowledge of material behavior in plane strain tension is important for the modelling of sheet metal forming processes, it is often overlooked in yield function calibration because of experimental characterization challenges. Plane strain notch tensile tests, though experimentally convenient, are subject to stress and strain gradients across the gauge width that complicate the analysis.

Objective

A novel experimental integration methodology was developed to exploit these stress and strain gradients to locally calibrate the arc of an anisotropic yield surface from uniaxial-to-plane strain tension.

Methods

Constraining the anisotropic yield surface at the plane strain point, to be consistent with pressure-independent plasticity, enables the local arc to be governed by a single parameter. The arc shape is largely independent of the choice of yield function and can be optimized using a cutting line approach and full-field optical strain measurements. The accuracy of the method was evaluated using finite-element simulations of isotropic and anisotropic materials with different hardening behaviors.

Results

The methodology was applied to a dual phase DP1180 steel and AA5182-O aluminum alloy in the rolling, transverse, and diagonal directions. Data along each of the three locally calibrated arcs was included in calibrations of Yld2000 and Yld2004 yield surfaces.

Conclusions

The plane strain yield strength and arc shape had significant implications on the calibration of advanced anisotropic yield criteria. The yield exponent of the DP1180 agreed with the common value of six for BCC metals while the AA5182 yield surface approximated a Tresca-shape with local yield exponents in excess of 20.

Appendix: Evaluation of Alternative Yield Functions for Plane Strain Characterization Hill48 Yield Function

The anisotropic Hill48 criterion [54] in plane stress is attractive for its simplicity but cannot be adopted in the present study because it exhibits a non-physical drifting of the plane strain stress state. To demonstrate, the Hill48 model is expressed using the reference and transverse R-values, R₀ and R₉₀, as:

$$\sigma _{{eq}}^{{Hill48}} = \frac{1}{{\sqrt {1 + R_{0} } }}\sqrt {\sigma _{1}^{2} + \frac{{R_{0} }}{{R_{{90}} }}\sigma _{2}^{2} + R_{0} (\sigma _{1} - \sigma _{2} )^{2} }$$

(20)

The uniaxial tensile yield stress and R-value in the reference direction are automatically satisfied. Plane strain tension occurs at a principal stress ratio of:

$$\left( {\frac{{\sigma _{2} }}{{\sigma _{1} }}} \right)_{{PST}}^{{Hill48}} = \frac{{R_{{90}} }}{{1 + R_{{90}} }}$$

(21)

The plane strain constraint is satisfied when ${R}_{90}=1$ but there is no flexibility remaining in the model to calibrate the plane strain yield strength. If the plane strain constraint were ignored, the transverse R-value could be taken as a free parameter for calibration, to adjust the plane strain yield strength in the reference direction. However, adjusting R₉₀ can significantly shift the plane-strain stress ratio as demonstrated in Fig. 29. The Hill48 yield criterion will produce non-unique solutions for the constitutive response by violating the plane strain constraint.

Fig. 29

Variation of the plane stress Hill48 yield surface with the tensile R-value for a fixed plane strain yield strength. The stress state for plane strain varies with the R-value and is inconsistent with pressure-independent plasticity

Hosford-1979 (HF79) Yield Function

The non-quadratic extension of Hill48 for plane stress conditions with planar isotropy by Hosford [55], denoted as HF79, appears promising since the yield exponent can be varied while the R-value and uniaxial stress in the reference direction are automatically satisfied. The HF79 model is

$$\sigma _{{eq}}^{{HF79}} = \left( {\frac{{\left| {\sigma _{1} } \right|^{m} + \left| {\sigma _{2} } \right|^{m} + R\left| {\sigma _{1} - \sigma _{2} } \right|^{m} }}{{1 + R}}} \right)^{{1/m}}$$

(22)

where $m$ is the yield function exponent. Unfortunately, the plane strain constraint is only satisfied for isotropy when R = 1 with plane strain occurring at a stress ratio of:

$$\left( {\frac{{\sigma _{2} }}{{\sigma _{1} }}} \right)_{{PST}}^{{HF79}} = \frac{{R^{{\frac{1}{{m - 1}}}} }}{{1 + R^{{\frac{1}{{m - 1}}}} }}$$

(23)

The drifting of the plane strain location for HF79 with the R-value and exponent is illustrated in Fig. 30. The errors are significant for a quadratic exponent when HF79 reverts to the Hill48 model with normal anisotropy. Increasing the exponent flattens the yield surface in the tensile quadrant as it converges to a Tresca model which helps to mitigate the error in the HF79 model.

Fig. 30

Variation of the stress ratio where plane strain tension occurs in the reference direction for the HF79 yield criterion

Drucker-Yoshida

The anisotropic extension of the Drucker yield surface [69] by Yoshida et al. [58] is:

$$\sigma _{{eq}}^{{{\text{Drucker - Yoshida}}}} = \left[ {(\tilde{J}_{2}^{{}} )^{3} - c(\tilde{J}_{3}^{{}} )^{2} } \right]^{{1/6}}$$

(24)

where ${\stackrel{\sim }{J}}_{2}$ and ${\stackrel{\sim }{J}}_{3}$ are the second and third invariants of the linearly transformed deviatoric stress tensor. The Drucker-Yoshida model is a simplification of the model of Cazacu and Barlat [70] who used independent linear transformations for each invariant. The shape of the yield function is controlled by the parameter, c which weights the contribution of ${\stackrel{\sim }{J}}_{3}$ and is bounded by convexity to fall between $-27/8\le c\le 9/4$ with the isotropic yield surface shown in Fig. 31. The Drucker model becomes equivalent to the von Mises yield function by setting c = 0 and using ${c}_{1-6}^{VM}=\sqrt{3}$ as the anisotropy parameters. Lou and Yoon [71] found that the Drucker model can closely approximate the isotropic FCC (m = 8) and BCC (m = 6) Hosford models using the coefficients c = 2.0 and ${c}_{1-6}=1.8365$, and c = 1.226 and ${c}_{1-6}=1.7909$, respectively.

The main advantage of the isotropic Drucker model is that it can capture larger plane strain yield strengths than the Hosford-based functions as shown in Fig. 32. However, the Drucker model does not reduce to the Tresca model and thus cannot describe plane strain yield strength ratios close to unity as a Hosford-based model can. The normalized plane strain yield strength in the Drucker-Yoshida model, with the plane strain constraint enforced in the reference direction is:

$$\left( {\frac{{\sigma _{1} }}{{\sigma _{0} }}} \right)_{{PST}}^{{{\text{Yoshida - Drucker}}}} = \frac{6}{{c_{1} + 2c_{2} }}$$

(25)

Fig. 31

Variation of the isotropic Drucker [69] yield function between its convexity bounds

Fig. 32

Variation of the normalized plane strain yield strength as a function of R-value and c-parameter for the Drucker-Yoshida model. The Drucker-Yoshida form can obtain higher plane strain yield strengths than HF85-PSC model but does not converge to Tresca for materials with plane strain yield strengths close to uniaxial tension

Coincidence of Drucker-Yoshida and HF85-PSC with the Plane-strain Constraint

Both the HF85-PSC and Drucker-Yoshida models provide the flexibility required for local calibration and simultaneous enforcement of the plane strain constraints. Ideally all calibrated yield criteria should reproduce the same physically observed yield behavior for a given material, independent of their functional forms and free of any calibration bias. Two extreme cases have been considered to compare the HF85-PSC and the Drucker-Yoshida models. The first fictional material is loosely based upon 6000-series aluminum alloys with an R-value of 0.50 and plane strain yield strength ratio of 1.05 [34]. The second fictional material is based upon low carbon steel with an R-value of 2.00 and a relatively large plane strain yield strength ratio of approximately 1.20 [40]. A comparison of the Drucker-Yoshida and HF85-PSC yield surfaces, in the tensile quadrant for both fictional materials, is presented in Fig. 33 with the parameters summarized in Table 10.

Fig. 33

Comparison of the predicted shape of the yield surface in the tensile quadrant from uniaxial to plane strain tension for two fictional materials using the HF85-PSC and Drucker-Yoshida models

Table 10 Parameters of HF85-PSC and Drucker – Yoshida yield functions used to demonstrate uniqueness of the calibrated yield surface arc due to the plane strain constraints

Both yield functions, despite their different underlying forms, reproduce nearly identical local yield surfaces when subjected to the plane strain constraints. The subsequent shape of the associated yield surface calibrated from uniaxial-to-plane strain tension can be approximated as a unique solution for the purpose of constitutive characterization. It is emphasized that no uniqueness in the material parameters of the yield functions is implied. Other combinations of material parameters in models such as Yld2000 could theoretically be used to obtain the same local shape of the yield surface. Either yield function could therefore be used for local calibration as long as the plane strain yield strength falls within the allowable range for the function.