Effective potentials in nonlinear polycrystals and quadrature formulae

Jean-Claude Michel (1) , Pierre Suquet (1)

1 LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]

This study presents a family of estimates for effective potentials in nonlinear polycrystals. Noting that these potentials are given as averages, several quadrature formulae are investigated to express these integrals of nonlinear functions of local fields in terms of the moments of these fields. Two of these quadrature formulae reduce to known schemes, including a recent proposition (Ponte Castañeda Proc. R. Soc. Lond. A 471) obtained by completely different means. Other formulae are also reviewed that make use of statistical information on the fields beyond their first and second moments. These quadrature formulae are applied to the estimation of effective potentials in polycrystals governed by two potentials, by means of a reduced-order model proposed by the authors (Nonuniform Transformation Field Analysis). It is shown how the quadrature formulae improve on the tangent second-order approximation in porous crystals at high stress triaxiality. It is found that, in order to retrieve a satisfactory accuracy for highly nonlinear porous crystals under high stress triaxiality, a quadrature formula of higher order is required.

Mots clés: Homogenization; model reduction; numerical integration; field statistics; crystal plasticity

Domaines: Mécanique des solides [physics.class-ph]

Jean-Claude Michel, Pierre Suquet. Effective potentials in nonlinear polycrystals and quadrature formulae. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2017, 473, pp.20170213. ⟨10.1098/rspa.2017.0213⟩. ⟨hal-01578482⟩