First time, an analytical Two-dimensional (2D) elasticity solution for arbitrarily supported axially functionally graded (FG) beam is developed. Linear gradation of the material property along the axis of the beam is considered. Using the strain displacement and constitutive relations, governing partial differential equations (PDEs) is obtained by employing Ressiner mixed variational principle. Then PDEs are reduced to Two set of ordinary differential equations (ODEs) by using recently developed extended Kantorovich method. The set of 4n ODEs along the z-direction has constant coefficients. But, the set of 4n nonhomogeneous ODEs along x-direction has variable coefficients which is solved using modified power series method. Efficacy and accuracy of the present methodology are verified thoroughly with existing literature and 2D finite element solution. Effect of axial gradation, boundary conditions and configuration lay-ups are investigated. It is found that axial gradation influence vary with boundary conditions. These benchmark results can be used for assessing 1D beam theories and further present formulation can be extended to develop solutions for 2D micro or Nanobeams.

In this work, the Mellin transform method was used to obtain solutions for the stress field components in Two dimensional (2D) elasticity problems in terms of plane polar coordinates. the Mellin transformation was applied to the biharmonic stress compatibility equation expressed in terms of the Airy stress potential function, and the boundary value problem transformed to an algebraic problem which was solved to obtain the Mellin transformed Airy stress potential function. The Mellin transform was similarly used to obtain the Mellin transformed stress field components. The use of Mellin transform inversion formula yielded the solutions to the 2D elasticity problem in the physical space domain variables. Specific illustration was considered of the solution by using the Mellin transform method for the Flamant problem and the Mellin transform solutions found to agree with solutions from the literature.

A novel Two-dimensional elasticity solution is presented in this paper, specifically designed for studying the vibration of functionally graded porous (FGP) beams. The kinetics of the beam are defined by Two-dimensional elasticity theory, and Lagrange’s equations are used to derive the governing equations of motion. The Ritz method devises the expansion of displacement variables in polynomial and trigonometric series in the thickness and axial directions. Furthermore, microvoids can emerge as a result of technical issues during the manufacture of functionally graded materials (FGMs), leading to the development of porosities. The porosity distribution functions, one for three porosity distributions: uniform porosity (UP), non-uniform porosity-I (NUP-I), and non-uniform porosity-II (NUP-II), are considered in the problem. This study investigates the impact of the gradation exponents (p) in the z-direction, the slenderness ratio (L/h), the distribution of porosity, the porosity coefficient (e), and various boundary conditions on the natural frequencies. A comparison with the findings from higher-order shear deformation theory (HSDT) validated the accuracy and effectiveness of the proposed methodology.

This research presents a semi analytical solution of finitely long, simply supported, orthotropic, piezoelectric, radically polarized, shell panel under pressure and electrostatic excitation. The general solution of the governing partial differential equations is obtained by the method of separation of variables. The displacements and electric potential are expanded in appropriate trigonometric Fourier series in the circumferential and axial coordinate to satisfy the boundary conditions at the simply-supported circumferential and axial edges. The governing ordinary differential equations are solved by the Galerkin finite element method. In this procedure, the quadratic shape function is used in each element. Numerical examples are provided for typical external pressure on outer .I'll/face of a single layer piezoelectric panel.

The governing equations of three dimensional elasticity problems include the six Beltrami– Michell stress compatibility equations, the three differential equations of equilibrium, and the six material constitutive relations; and these are usually solved subject to boundary conditions. The system of fifteen differential equations is usually difficult to solve, and simplified methods are usually used to achieve a solution. Stress-based formulation methods, and displacement-based formulation methods are Two common simplified methods for solving elasticity problems. This work adopts a stress-based formulation for a three dimensional elasticity problem. In this work, the Maxwell’ s stress functions for solving three dimensional problems of elasticity theory are derived from fundamental principles. It is shown that the three Maxwell stress functions identically satisfy all the three differential equations of static equilibrium when body force components are ignored. It is further shown that the three Maxwell stress functions are solutions to the six Beltrami-Michell stress compatibility equations if the Maxwell stress functions are potential functions. It is also shown that the Airy’ s stress functions for Two dimensional elasticity problems are special cases of the Maxwell stress functions.