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On the convexity bound of the generalized Drucker’s yield function CB2001 for orthotropic sheets. (English) Zbl 1486.74014

Summary: D. C. Drucker’s sixth-order yield function for isotropic materials [J. Appl. Mech. 16, 349–357 (1949; Zbl 0036.39903)] was extended by O. Cazacu and F. Barlat [Math. Mech. Solids 6, No. 6, 613–630 (2001; Zbl 1128.74303)] for modeling orthotropic sheet metals via a generalization of the two stress invariants according to the theory of representation. The constant \(c\) in the original Drucker’s isotropic yield function was found by B. Dodd and K. Naruse [Int. J. Mech. Sci. 31, No. 7, 511–519 (1989; Zbl 0695.73008)] to be bound between \(-27/8\) and 9/4 per the convexity requirement. In many subsequent modeling applications of orthotropic sheets, the same bound is also tacitly assumed for the constant \(c\) used in this class of the generalized Drucker’s yield function CB2001. No actual proof has, however, been presented in the literature that such a bound is indeed absolutely necessary if not sufficient to guarantee the convexity of the orthotropic CB2001 yield function. In this study, the validity of assuming such a convexity bound on the adjustable constant \(c\) is examined using a recently proposed numerical convexity certification algorithm. Representative orthotropic CB2001 yield functions whose material parameters have been calibrated and reported in the literature for some 12 FCC, BCC, and HCP sheet metals are evaluated. It is found that a single convexity bound between \(-27/8\) and 9/4 on the constant \(c\) does not hold at all for any of those yield functions. This is in contrast to another class of generalized Drucker’s yield function based on linearly transformed stresses where the original convexity bound on the constant \(c\) does still hold.

MSC:

74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
74E10 Anisotropy in solid mechanics
74S99 Numerical and other methods in solid mechanics
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