Incompatible Deformations in Hyperelastic Plates

Sergey Lychev (), Alexander Digilov, Vladimir Bespalov and Nikolay Djuzhev
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Sergey Lychev: Ishlinsky Institute for Problems in Mechanics RAS, 119526 Moscow, Russia
Alexander Digilov: Ishlinsky Institute for Problems in Mechanics RAS, 119526 Moscow, Russia
Vladimir Bespalov: MEMSEC R&D Center, National Research University of Electronic Technology (MIET), 124498 Moscow, Russia
Nikolay Djuzhev: MEMSEC R&D Center, National Research University of Electronic Technology (MIET), 124498 Moscow, Russia

Mathematics, 2024, vol. 12, issue 4, 1-20

Abstract: The design of thin-walled structures is commonly based on the solutions of linear boundary-value problems, formulated within well-developed theories for elastic plates and shells. However, in modern appliances, especially in MEMS design, it is necessary to take into account non-linear mechanical effects that become decisive for flexible elements. Among the substantial non-linear effects that significantly change the deformation properties of thin plates are the effects of residual stresses caused by the incompatibility of deformations, which inevitably arise during the manufacture of ultrathin elements. The development of new methods of mathematical modeling of residual stresses and incompatible finite deformations in plates is the subject of this paper. To this end, the local unloading hypothesis is used. This makes it possible to define smooth fields of local deformations (inverse implant field) for the mathematical formalization of incompatibility. The main outcomes are field equations, natural boundary conditions and conservation laws, derived from the least action principle and variational symmetries taking account of the implant field. The derivations are carried out in the framework of elasticity theory for simple materials and, in addition, within Cosserat’s theory of a two-dimensional continuum. As illustrative examples, the distributions of incompatible deformations in a circular plate are considered.

Keywords: plates; incompatible deformations; hyperelasticity; Lagrangian approach; least action principle; variational symmetries; non-linear boundary-value problem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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