Improved bounds on the energy-minimizing strains in martensitic polycrystals

Abstract

International audienceThis paper is concerned with the theoretical prediction of the energy-minimizing (or recoverable) strains in martensitic polycrystals, considering a nonlinear elasticity model of phase transformation at finite strains. The main results are some rigorous upper bounds on the set of energy-minimizing strains. Those bounds depend on the polycrystalline texture through the volume fractions of the different orientations. The simplest form of the bounds presented is obtained by combining recent results for single crystals with a ho-mogenization approach proposed previously for martensitic polycrystals. However, the polycrystalline bound delivered by that procedure may fail to recover the monocrystalline bound in the homogeneous limit, as is demonstrated in this paper by considering an example related to tetragonal martensite. This motivates the development of a more detailed analysis, leading to improved polycrystalline bounds that are notably consistent with results for single crystals in the homogeneous limit. A two-orientation polycrystal of tetragonal martensite is studied as an illustration. In that case, analytical expressions of the upper bounds are derived and the results are compared with lower bounds obtained by considering laminate textures

Extremal inclusions in nonlinear conductivity

“…A tighter Hashin-Shtrikman type bound than the linear comparison bound κ LC has been proposed by Peigney and Peigney (2017) using the translation method (Milton, 2002;Kohn, 1991;Peigney, 2016). That bound -henceforth denoted by It is possible, however, to derive the first order expansion of κ T in closed form.…”

Bounds on the recoverable deformations of polycrystalline SMAs at finite strain

“…The bound (4) is obtained using known restrictions on Young measures [1,14]. The crucial point is that the function F → Φ(F.a + F * .b) is quasiconvex for all a and b [24]. In (4), C is a given arbitrary subset of R 3×3 × R 3×3 : Each choice of C generates a corresponding bound onK.…”

“…where the boundK + is calculated using (23) with a well chosen class C of tensors (a, b) for which closed-form expressions can be obtained (see [24] for details). The volume fraction χ 1 is set equal to 0.7.…”

Bounds for nonlinear composite conductors via the translation method

“…Originally introduced by Lurie and Cherkaev (1984) and independently by Tartar (1985); Murat and Tartar (1985), the translation method has proved to be a powerful tool in various nonlinear homogenization problems, such as bounding the stress-strain curve of composites (Milton and Serkov, 2000;Talbot and Willis, 2004;Peigney, 2005) or bounding the recoverable strains in shape memory alloys (Bhattacharya and Kohn, 1997;Peigney, 2008Peigney, , 2016. For linear composites, the Hashin-Shtrikman bounds can be elegantly recovered from the translation method by embedding the original problem in a problem of higher dimension (namely, d 2 where d is the dimension of the physical space).…”