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Pointwise convexity conditions of the Legendre-Hadamard type for energy minimizers are derived in the context of a special Cosserat elasticity model for fiber-reinforced solids [1]. In this approach, the Cosserat rotation tensor accounts for the kinematics of embedded fibers, regarded as continuously distributed Kirchhoff rods. Firstly, we obtain new Legendre-Hadamard inequalities for three-dimensional fiber-reinforced elastic solids using variational calculus as in [2]. Secondly, we consider nonlinear elastic shells reinforced by a family of fibers [3] and we derive the counterparts of these necessary conditions for energy minimizers in the theory of shells.
References:
[1] M. Shirani and D.J. Steigmann. A Cosserat model of elastic solids reinforced by a family of curved and twisted fibers. Symmetry, 12: 1133, 2020.
[2] M. Shirani, D.J. Steigmann and M. Birsan. Legendre-Hadamard conditions for fiber-reinforced materials with one, two or three families of fibers. Mechanics of Materials, 184: 104745, 2023.
[3] M. Birsan, M. Shirani and D.J. Steigmann. Convexity conditions for fiber-reinforced elastic shells, Mathematics and Mechanics of Solids, https://doi.org/10.1177/10812865241261485, 2024.
