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Action principles
Published in Bijan Kumar Bagchi, Advanced Classical Mechanics, 2017
An action principle is basically a variational principle with the central idea derived by defining a functional, for two states of a physical system, corresponding to their initial and final configurations as the system evolves in time, which yields a stationary value when taken over the actual path which links these configurations, as compared to the neighboring varied paths having the same configurations, provided the total energy remains the same in the varied motion as in the actual motion. In simple terms, the action is defined in terms of the time integral of the Lagrangian between two fixed values of time that are identified with the initial and final position of the particle during the course of its motion. Action principles are of paramount importance in classical mechanics in that almost all formulations of physics admit an action principle interpretation while conversely any formalism resulting from an action principle is termed as well defined. We first introduce the principle of stationary action.
Theory of Plates
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
The variational principle is based on the introduction of certain types of approximations that satisfy boundary conditions and at the same time minimize the strain energy. The most popular choice of variational methods is called the Rayleigh-Ritz method. Before we consider this method, we would further consider some special forms of the strain energy. Let us consider the following integration: ∬[∂2w∂x2∂2w∂y2-(∂2w∂y∂x)2]dxdy=∬[∂∂x(∂w∂x∂2w∂y2)-∂∂y(∂w∂x∂2w∂y∂x)]dxdy $$ \iint {[\frac{{\partial ^{2} w}}{{\partial x^{2} }}\frac{{\partial ^{2} w}}{{\partial y^{2} }} - (\frac{{\partial ^{2} w}}{{\partial y\partial x}})^{2} ]}dxdy = \iint {[\frac{\partial }{{\partial x}}(\frac{{\partial w}}{{\partial x}}\frac{{\partial ^{2} w}}{{\partial y^{2} }}) - \frac{\partial }{{\partial y}}(\frac{{\partial w}}{{\partial x}}\frac{{\partial ^{2} w}}{{\partial y\partial x}})]}dxdy $$
Large amplitude vibration analysis of functionally graded laminated skew plates in thermal environment
Published in Mechanics of Advanced Materials and Structures, 2019
Sanjay Singh Tomar, Mohammad Talha
The previous studies reveal that the contributions of various authors in the field of nonlinear analysis of the FGM and skew FGM structures. Based on best of the author’s knowledge, large amplitude nonlinear vibration response of the functionally graded skew laminated plates has not been reported in the literature. In the present study, the material properties are assumed to be temperature dependent and very in the thickness direction following the power law distribution. The formulation has been done with Reddy’s higher-order shear deformation theory, including the von-Karman nonlinear strain assumptions. The governing equations have been derived using the variational principle, which is the generalization of principle of virtual work. The plate domain has been discretized with a C0 continuous nine-noded isoparametric element. The direct iterative method has been used to calculate the nonlinear frequency response. Convergence and comparison studies have been performed to prove the reliability of the present formulation. The effects of various system parameters such as skew angle (Ψ), aspect ratios (a/b), thickness ratios (a/h), volume fraction index (n), BCs have been analyzed, and presented in tabular and graphical form.
A bi-Helmholtz type of two-phase nonlocal integral model for buckling of Bernoulli-Euler beams under non-uniform temperature
Published in Journal of Thermal Stresses, 2021
With this motivation, this paper aims to study the thermomechanical buckling response of the Bernoulli-Euler beams under a non-uniform temperature environment by using a well-posed two-phase nonlocal integral model with a bi-Helmholtz kernel. Governing equation is derived by invoking the variational principle of virtual work. The temperature effect is equivalent to a thermal load along the axial direction to address the nonlocal thermal characteristic. The numerical results are validated by comparing with those of local elasticity theory. Several parameter analyses are carried out on the buckling loads of the beam under four typical boundary conditions. Finally, the conclusions are summarized.