Yield Criteria for Shape Memory Materials: Convexity Conditions and Surface Transport

Abstract

We study the yield conditions of phase transformation initiation for shape memory alloys exhibiting asymmetry between tension and compression. An extension of the choice of the classical invariant parameters such as those of Lode is proposed. A necessary and sufficient condition of convexity of these surfaces representing the elastic domain of austenite in the stress space, is established. Moreover the transport of these surfaces in the space of effective transformations strains of martensite is done. Hence, the duality between these two spaces is built. Some applications involving Cu-Al-Be and NiTi shape memory alloys end the purpose.

Yield criteria for quasibrittle and frictional materials: A generalization to surfaces with corners

“…h A.2. The convexity condition given by Laydi and Lexcellent (2009) Laydi and have shown, with the example reported in Fig. 7, that the convexity conditions given by Bigoni and Piccolroaz (2004) does not cover yield surfaces with corners.…”

“…7. Laydi and Lexcellent (2009) incorrectly argued that the Bigoni and Piccolroaz (2004) proposition on convexity had flaws, while the problem lies only in the fact that the deviatoric section of the yield surface described by Eq. (A.7) has corners, a case which is not covered by the Bigoni and Piccolroaz (2004) proposition and has been addressed in the present paper.…”

“…Note that the difference between the two versions of the theorem lies on the two conditions g 0 (0) 6 0 and g 0 (p/3) P 0. A consequence of the following theorem is that the convexity conditions by Laydi and Lexcellent (2009) are only sufficient (but not necessary) for convexity of smooth functions (see Appendix A.2).…”

“…This proof is based on both the hypotheses of smoothness of the function g(h), 4 and validity of the smoothness border conditions g 0 (0) = 0 and g 0 (p/3) = 0, 5 while as noticed by Laydi and Lexcellent (2009), 6 the class of functions (1) -even under the particularizations (3), but allowing parameter c to become equal to 1 -may describe deviatoric yield surfaces with corners ( Fig. 1), in which case the conditions for convexity provided by Bigoni and Piccolroaz (2004) remain only necessary but not sufficient.…”

“…1), in which case the conditions for convexity provided by Bigoni and Piccolroaz (2004) remain only necessary but not sufficient. 7 Since the convexity proposition is fundamental in developing new yield or phase-transformation criteria -and, generally speaking, convexity is the basis of extremum principles in mechanics (see for instance Duvaut and Lions, 1976;Noble and Sewell, 1972), it has immediately attracted a strong attention (Taillard et al, 2008;Laydi and Lexcellent, 2009;Lavernhe-Taillard et al, 2009;Saint-Sulpice et al, 2009;Valoroso and Rosati, 2009). Therefore, the convexity proposition is definitely important in analyzing yield criteria with corners, so that it becomes imperative a generalization to nonsmooth deviatoric yield surfaces, which is obtained in the present article (Theorem 4.3, Section 4).…”