Abstract
Trefftz approximation scheme on the structure of subdomains-blocks for the problems of the gradient elasticity is proposed. This scheme based on the analytical representation for the gradient elasticity solutions of Papkovich–Neuber type. Independent in blocks, complete systems of functions are used for approximation, that analytically exact satisfy the initial fourth-order equations. It is shown that the generalized Trefftz scheme allows simultaneously with minimizing the energy functional to stitch together all the necessary quantities on the block boundaries: functions, their derivatives, cohesive moments and surface forces. It is achieved exclusively due to the analytical construction of the used functions. The paper gives a derivation of the Papkovich–Neuber representation for the gradient elasticity and formulates the uniqueness conditions. The analytical representation of the solution has a great advantage over the finite element one, since it opens up the possibility of constructing finite elements on unstructured meshes with independent local shape functions.
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Funding
The first author acknowledge the financial support of the Russian Science Foundation under the grant 20-41-04404 issued to the Institute of Applied Mechanics of Russian Academy of Sciences (theoretical statement and analytical realization of Treftz method for gradient elasticity) and second author acknowledge the financial support of the Russian Foundation of Basic Research by grants no. 18-29-10085 mk (theoretical part in justification of generalized Papkovich–Neuber representation in gradient elasticity).
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Volkov-Bogorodskiy, D.B., Moiseev, E.I. Generalized Trefftz Method in the Gradient Elasticity Theory. Lobachevskii J Math 42, 1944–1953 (2021). https://doi.org/10.1134/S1995080221080321
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DOI: https://doi.org/10.1134/S1995080221080321