preprints.org > engineering > mechanical engineering > doi: 10.20944/preprints202401.0840.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 9 January 2024 / Approved: 10 January 2024 / Online: 11 January 2024 (02:30:27 CET)

The design of thin-walled structures is commonly based on the solutions of linear boundary-valued 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 substantial non-linear effects, that significantly change the deformation properties of thin plates, are the effects of residual stresses caused by incompatibility of deformations, which inevitably arise during the manufacture of ultra-thin elements. The development of new methods of mathematical modeling of residual stresses and incompatible finite deformations in plates is the subject of the paper. To this end the local unloading hypothesis is used. It makes 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 least action principle and variational symmetries with account of implant field. The derivations were carried out in the framework of elasticity theory for simple materials and, over and above, within the Cosserat’s theory of a two-dimensional continuum. As illustrative examples the distributions of incompatible deformations in a circular plate are considered.

Plates; incompatible deformations; hyperelasticity; Lagrangian approach; least action principle; variational symmetries; non-linear boundary-value problem

Engineering, Mechanical Engineering

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