Prediction of Forming Limit Diagrams for Materials with HCP Structure

doi:10.1007/s40195-015-0344-3

Abstract

Abstract:

The forming limit diagram (FLD) is an important tool to be used when characterizing the formability of metallic sheets used in metal forming processes. Experimental measurement and determination of the FLD is time-consuming and therefore the analytical prediction based on theory of plasticity and instability criteria allows a direct and efficient methodology to obtain critical values at different loading paths, thus carrying significant practical importance. However, the accuracy of the plastic instability prediction is strongly dependent on the choice of the material constitutive model^[1,2,3]. Particularly for materials with hexagonal close packed (H-P) crystallographic structure, they have a very limited number of active slip systems at room temperature and demonstrate a strong asymmetry between yielding in tension and compression^{[4, 5]}. Not only the magnitude of the yield locus changes, but also the shape of the yield surface is evolving during the plastic deformation^[4]. Conventional phenomenological constitutive models of plasticity fail to capture this unconventional mechanical behavior^{[4, 6]}. Cazacu and Plunkett^[6] have proposed generic yield criteria, by using the transformed principal stress, to account for the initial plastic anisotropy and strength differential (SD) effect simultaneously. In this contribution, a generic FLD MATLAB script was developed based on Marciniak-Kuczynski analytical theory and applied to predict the localized necking. The influence of asymmetrical effect on the FLD was evaluated. Several yield functions such as von Mises, Hill, Barlat89, and Cazacu06 were incorporated into analysis. The paper also presents and discusses the influence of different hardening laws on the formability of materials with HCP crystal structures. The findings indicate that the plastic instability theory coupled with Cazacu model can adequately predict the onset of localized necking for HCP materials under different strain paths.

Key words: Forming limit diagram (FLD), Marciniak-Kuczynski analysis, Plastic instability, Yield function, Hardening law

Sheng-Hua Wu, Nan-Nan Song, Francisco M. Andrade Pires, Abel D. Santos. Prediction of Forming Limit Diagrams for Materials with HCP Structure[J]. Acta Metallurgica Sinica (English Letters), 2015, 28(12): 1442-1451.

Figures/Tables 14

Fig.1 Schematic diagram of M-K analysis [19]

Fig.2 Tensile stress-strain response

Table 1 Material parameters of AZ31B

Modulus E (GPa)	Poissons ratio ͼ/em>	$Y = K\left( {\bar{\varepsilon } + \varepsilon_{0} } \right)^{n}$			$Y = A + B\left( {1 - e^{{ - C\bar{\varepsilon }}} } \right)$
Modulus E (GPa)	Poissons ratio ͼ/em>	K	$\varepsilon_{0}$	n	A	B	C
45	0.35	406.8	0.0078	0.187	195.78	117.57	18.46

Table 2 AZ31B coefficients corresponding to the yield surface evolution

$\bar{\varepsilon }$	$L_{11}$	$L_{22}$	$L_{33}$	$L_{12}$	$L_{13}$	$L_{23}$	$L_{44}$	k
0.01	1.3386	1.3028	0.7254	0.2097	-0.3170	-0.2595	1.1262	0.8988
0.02	1.3382	1.3024	0.6851	0.2114	-0.3012	-0.2522	1.1260	0.9899
0.05	1.3669	1.2640	0.7453	0.1279	-0.2824	-0.2327	1.2265	0.6481
0.1	0.8363	1.2747	0.3461	0.2766	-0.2072	0.4730	1.1108	?0.0069
0.15	0.9259	1.2302	0.11	0.2902	-0.13	0.5734	1.0953	?0.1305

Fig.3 The function $\beta$ ($\alpha$) with different $k$ values

Fig.4 The function f($\alpha$) corresponding to the calculated FLD with different k values

Fig.5 The function g($\alpha$) corresponding to the calculated FLD with different k values

Fig.6 Influence of the asymmetrical effect on FLD

Fig.7 Influence of the asymmetrical effect on FLSD

Fig.8 Influence of the initial imperfection $f_0$ on FLD

Fig.9 Influence of the initial imperfection $f_0$ on FLSD

Fig.10 Influence of hardening law on FLD

Fig.11 Influence of yield criterion on FLD

Fig.12 Influence of yield criterion on FLSD

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