Mixed finite element formulations of strain-gradient elasticity problems |
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| Affiliation: | 1. Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong, PR China;2. City University of Hong Kong Shenzhen Research Institute, Shenzhen 518057, PR China;3. Mechanical Engineering Department, Texas A&M University, College Station, TX 77843-3123, USA;1. Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam;2. Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam;3. Departamento de Engenharia Mecanica, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal;4. Institute of Research and Development, Duy Tan University, Da Nang, Viet Nam;5. Department of Architectural Engineering, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul 05006, South Korea;1. Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam;2. Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam;3. Institute of Structural Mechanics, Bauhaus-Universität Weimar, Marienstraße 15, 99423 Weimar, Germany;4. Dep. of Mechanical Eng, Boston University, Boston, MA 02215, USA;1. Young Researchers and Elite Club, Najafabad Branch, Islamic Azad University, Najafabad, Iran;2. Faculty of Engineering, Shahrekord University, Shahrekord, Iran |
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| Abstract: | Theories on intrinsic or material length scales find applications in the modeling of size-dependent phenomena. In elasticity, length scales enter the constitutive equations through the elastic strain energy function, which, in this case, depends not only on the strain tensor but also on gradients of the rotation and strain tensors. In the present paper, the strain-gradient elasticity theories developed by Mindlin and co-workers in the 1960s are treated in detail. In such theories, when the problem is formulated in terms of displacements, the governing partial differential equation is of fourth order. If traditional finite elements are used for the numerical solution of such problems, then C1 displacement continuity is required. An alternative “mixed” finite element formulation is developed, in which both the displacement and the displacement gradients are used as independent unknowns and their relationship is enforced in an“integral-sense”. A variational formulation is developed which can be used for both linear and non-linear strain-gradient elasticity theories. The resulting finite elements require only C0 continuity and are simple to formulate. The proposed technique is applied to a number of problems and comparisons with available exact solutions are made. |
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