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# A Two-dimensional Finite Element Implementation of a Special Form of Gradient Elasticity

L. Teneketzis Tenek1, E.C. Aifantis1,2,3
Laboratory of Mechanics and Materials, Polytechnic School
Aristotle University of Thessaloniki, GR-54006, Thessaloniki, Greece
Center for Mechanics of Materials and Instabilities
Michigan Technological University, Houghton MI 49931, USA
Corresponding Author

Computer Modeling in Engineering & Sciences 2002, 3(6), 731-741. https://doi.org/10.3970/cmes.2002.003.731

### Abstract

A two-dimensional finite element implementation of a special form of gradient elasticity is developed and a connection between classical and the proposed gradient elasticity theory is established. A higher-order constitutive equation is adopted which involves a gradient term of a special form; the higher-order term is precisely the second gradient of the lower-order term. A weak form of the equilibrium equations, based on the principle of virtual work, is formulated for the classical problem. The problem in hand, is solved by means of the finite element method in two steps. First, the displacement field of classical elasticity is computed. Then, the gradient equation is solved using the displacement field extracted from the solution of the classical problem. In concert with the natural-mode finite element method, sets of rigid-body and straining modes are assigned to plane triangular elements for the classical problem. All elemental matrices are integrated explicitly. The theory is applied to two tensile notch specimens including a relatively large and a smaller notch, respectively. Numerical results reveal that the gradient theory yields more closely spaced deformation bands and a stiffer structural response. Strains for both theories are computed. The higher strain concentrations are localized at the notch tip for both specimens while the difference in strain patterns as obtained from both theories are more pronounced for the specimen with the smaller notch. The effect of the specimen size on the structural response and the corresponding size effect is also assessed.

Tenek, L. T., Aifantis, E. (2002). A Two-dimensional Finite Element Implementation of a Special Form of Gradient Elasticity. CMES-Computer Modeling in Engineering & Sciences, 3(6), 731–741. This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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