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Verification of strain gradient elasticity computation by analytical solutions

Yang, Hua; Timofeev, Dmitry; Abali, B. Emek; Li, Baotong; MĂĽller, Wolfgang H.

As there are different computational methods for simulating problems in generalized mechanics, we present simple applications and their closed-form solutions for verifying a numerical implementation. For such a benchmark, we utilize these analytical solutions and examine three-dimensional numerical simulations by the finite element method (FEM) using IsoGeometric Analysis (IGA) with the aid of open source codes, called tIGAr, developed within the FEniCS platform. A study for the so-called wedge forces and double tractions help to comprehend their roles in the displacement solution as well as examine the significance by comparing to the closed form solutions for given boundary conditions. It is found that numerical results are in a good agreement with the analytical solutions if wedge forces and double tractions are considered. It is also presented how the wedge forces become necessary in order to maintain equilibrium in strain gradient materials.
Published in: ZAMM - Zeitschrift fĂĽr Angewandte Mathematik und Mechanik, 10.1002/zamm.202100023, Wiley