Variational formulation of a modified couple stress theory and its application to a simple shear problem |
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Authors: | S K Park X-L Gao |
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Institution: | (1) Department of Mechanical Engineering, Texas A&M University, 3123 TAMU, College Station, TX 77843-3123, USA |
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Abstract: | A variational formulation is provided for the modified couple stress theory of Yang et al. by using the principle of minimum
total potential energy. This leads to the simultaneous determination of the equilibrium equations and the boundary conditions,
thereby complementing the original work of Yang et al. where the boundary conditions were not derived. Also, the displacement
form of the modified couple stress theory, which is desired for solving many problems, is obtained to supplement the existing
stress-based formulation. All equations/expressions are presented in tensorial forms that are coordinate-invariant. As a direct
application of the newly obtained displacement form of the theory, a simple shear problem is analytically solved. The solution
contains a material length scale parameter and can capture the boundary layer effect, which differs from that based on classical
elasticity. The numerical results reveal that the length scale parameter (related to material microstructures) can have a
significant effect on material responses.
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 74A10 74A20 74A60 74B05 |
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