A Modified Couple Stress Theory For Composite Laminated Plates/Beams And Vibrational Size Effects

A new modified couple stress theory for anisotropic elasticity is suggested. Differingfrom the modified couple stress theory (C1theory), the couple stress constitute relationshipsare introduced for anisotropic elasticity, in which the curvature (rotation-gradient) tensor isasymmetric, but the couple stress tensor is symmetric and under isotropic case both of themodified couple stress theories can be identical. A detailed variational formulation is providedfor this theory by using the principle of minimum total potential energy. The differences andrelationships of standard, modified and new modified couple stress theories are given. It canbe shown both the equilibrium equations and boundary conditions expressed by using stressesare the same for the three couple stress theories, while the displacement forms are not.The modified couple stress theory for composite laminated plate can be derived from thenew modified couple stress theory for anisotropic elasticity. Based on the virtual workprinciple, a new model for composite laminated thin plate containing two material lengthscale parameters is established for the first time. In this theory, the form of the new curvaturetensor is asymmetric, however it is identical to the modified couple stress theory for theisotropic elasticity. More simplified bending and vibration engineering model for cross-plycomposite laminated thin plate of couple stress theory with one material length constant isestablished by ignoring the material length constant related to the matrix. Also, the newmodified couple stress theory for anisotropic elasticity is adopted to establish the newmodified couple stress theory for composite laminated beam. Based on this theory, amicro-scale free vibration analysis of composite laminated Timoshenko beam (CLTB) modelis developed. A new anisotropic constitutive relation is defined for modeling the CLTB model.This theory uses rotation-displacement as dependent variable and contains only one materiallength scale parameter. Hamilton’s principle is employed to derive the governing equations ofmotion and boundary conditions. This new model can be reduced to composite laminatedBernoulli-Euler beam model of the couple stress theory. Example analyses of free vibration of the cross-ply simply supported thin square plateand CLTB models are adopted and explicit expressions of analysis solution are given.Numerical results show that the present models (both the beam model and the plate model)can capture the scale effects of the natural frequencies of microstructures. Moreover, thenatural frequencies of the nonclassical models are always higher than those of the classicalmodels.