Effect Scale Analysis Of Micro-laminated Beam Based On Global And Local Theory

The size effect of the Reddy beam is analyzed based on the global-local theory in thispaper. The global-local theory is different from other theories such as the classical platetheory and first order deformation theory. Mainly, the theory always satisfies free surfaceconditions and stresses continuity conditions at interfaces, so it can really express the sizeeffect of the shear stress of the laminated beam.On the basis of the modified couple stress theory for isotropic material, the new modifiedcouple stress theory for anisotropic material is suggested and the constitutive relations for thetheory are proposed in the paper. In the modified couple stress theory, the rotation variablesare known as curvatures. Compared with the classical couple stress theory, the curvaturetensors are symmetric and the modified couple stress theory includes only one materialcharacteristic length constant. Different from the modified couple stress theory, the newmodified couple stress theory for anisotropic material includes three material characteristiclength constants i. According to the assumptions of the plate and beam, the constitutiverelations for new modified couple stress theory can be reduced to the constitutive relations ofplate and beam which contain two material characteristic length constants.In terms of the new modified couple stress theory, the model of Reddy beam based onthe global-local theory can be formulated. We can use the principle of virtue work to identifythe equilibrium conditions and the corresponding boundary conditions. According to theHamilton’s principle, we can reduce the governing motion equations of the laminated beamand the boundary conditions. Finally we give the analytical solutions of deflection, normalstress and frequency with simple examples. Additionally, the numerical results show that theReddy beam model can capture the scale of the deflection, normal stress and free frequency.As the material characteristic length constant increases, the displacement and stress of thelaminated beam will decrease and free frequency of the laminated beam will increase.