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Constrained variational problems
Published in Louis Komzsik, Applied Calculus of Variations for Engineers, 2018
which, due to the fundamental lemma of calculus of variations, results in the relevant Euler-Lagrange differential equation ∂h∂y−ddx∂h∂y′=0.
Energy Principles and Variational Methods
Published in J. N. Reddy, Theories and Analyses of Beams and Axisymmetric Circular Plates, 2022
The fundamental lemma of calculus of variations is useful in obtaining differential equations from variational principles involving integral statements. The lemma can be stated as follows:
The Kantorovich method applied to bending, buckling, vibration, and 3D stress analyses of plates: A literature review
Published in Mechanics of Advanced Materials and Structures, 2019
Pairod Singhatanadgid, Thanyarat Singhanart
A bending problem of a microplate based on the modified strain gradient elasticity theory was investigated by Movassagh and Mahmoodi [62]. The solutions of the proposed model were compared with those of the classical and modified couple stress models. The governing equation of the problem was derived based on the minimum potential energy principle utilizing the fundamental lemma of calculus of variations. The extended Kantorovich method was then applied to the governing equation, with the out-of-plane displacement function assumed in the form of a single-term and separable function. The obtained ODEs were solved analytically as closed-form solutions. For specimens with small thickness, the deflections of the deformed plate from the proposed model were substantially different from those of the modified couple stress plate and the classical plate models. The difference decreased as the plate thickness increased.