A Hierarchy in Partial Di_x000B_erential Equations on Material Modeling

A Hierarchy in Partial Di_x000B_erential Equations on Material Modeling

Chan, Youn-Sha; Paulino, Glaucio H.

Abstract:

Various elasticity theories including linear and nonlinear theories on material modeling are reviewed. In the case of linear theories, a structure of hierarchy in the governing partial di erential equations (PDEs) are observed. The structure of PDE hierarchy includes two sets of comparisons: (1) homogeneous materials versus nonhomogeneous materials, and (2) classical linear elasticity theory versus strain-gradient elasticity theory. We then found that crack problems can be used to simplify the formidable look of the governing PDEs. We also show that the fourth order PDE in the higher order strain-gradient elasticity theory converges to the second order PDE in classical linear elastic fracture mechanics (CLEFM). In the case of nonlinear theories, we observe that some nonlinear elasticity theory may not be applicable to formulate crack problems.

Various elasticity theories including linear and nonlinear theories on material modeling are reviewed. In the case of linear theories, a structure of hierarchy in the governing partial di erential equations (PDEs) are observed. The structure of PDE hierarchy includes two sets of comparisons: (1) homogeneous materials versus nonhomogeneous materials, and (2) classical linear elasticity theory versus strain-gradient elasticity theory. We then found that crack problems can be used to simplify the formidable look of the governing PDEs. We also show that the fourth order PDE in the higher order strain-gradient elasticity theory converges to the second order PDE in classical linear elastic fracture mechanics (CLEFM). In the case of nonlinear theories, we observe that some nonlinear elasticity theory may not be applicable to formulate crack problems.

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Referências bibliográficas

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Como citar:

Chan, Youn-Sha; Paulino, Glaucio H.; "A Hierarchy in Partial Di_x000B_erential Equations on Material Modeling", p-50-53. In: Proceedings of the 13th International Symposium on Multiscale, Multifunctional and Functionally Graded Materials [=Blucher Material Science Proceedings, v.1, n.1]. São Paulo: Blucher, 2014.
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