Exact solutions for axisymmetric flexural free vibrations of inhomogeneous circular Mindlin plates with variable thickness

Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and fiexural rigidity with the consideration of the effect of Poisson＇s ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.

Analytical solutions for inflation of pre-stretched elastomeric circular membranes under uniform pressure

Elastomeric membranes are frequently used in several emerging fields such as soft robotics and flexible electronics.For convenience of the structural design,it is very attractive to find simple analytical solutions to well describe their elastic deformations in response to external loadings.However,both the material/geometrical nonlinearity and the deformation inhomogeneity due to boundary constraints make it much challenging to get an exact analytical solution.In this paper,we focus on the inflation of a prestretched elastomeric circular membrane under uniform pressure,and derive an approximate analytical solution of the pressure-volume curve based upon a reasonable assumption on the shape of the inflated membrane.Such an explicit expression enables us to quantitatively design the material and geometrical parameters of the pre-stretched membrane to generate a target pressure-volume curve with prescribed peak point and initial slope.This work would be of help in the simplified mechanical design of structures involving elastomeric membranes.

Interaction energy of interface dislocation loops in piezoelectric bi-crystals

Interface dislocations may dramatically change the electric properties, such as polarization, of the piezoelectric crystals. In this paper, we study the linear interactions of two interface dislocation loops with arbitrary shape in generally anisotropic piezoelectric bi-crystals. A simple formula for calculating the interaction energy of the interface dislocation loops is derived and given by a double line integral along two closed dislocation curves. Particularly, interactions between two straight segments of the interface dislocations are solved analytically, which can be applied to approximate any curved loop so that an analytical solution can be also achieved. Numerical results show the influence of the bi-crystal interface as well as the material orientation on the interaction of interface dislocation loops.