Nonlinear Vibration Analysis of Shear Deformable Functionally Graded Ceramic-Metal Plates Using an Improved Higher Order Theory

2010 ◽  
Author(s):  
Mohammad Talha ◽  
B. N. Singh
Keyword(s):  
Nonlinear Vibration ◽  
Vibration Analysis ◽  
Order Theory ◽  
Higher Order ◽  
Functionally Graded ◽  
Transverse Displacement ◽  
Normal Strain ◽  
Higher Order Terms ◽  
Higher Order Theory ◽  
Shear Deformable

In the present study, an improved higher order theory in conjunction with finite element method (FEM) is presented and is applied to study the nonlinear vibration analysis of shear deformable functionally graded material (FGMs) plates. The present structural model kinematics assumes the cubically varying in-plane displacement over the entire thickness, while the transverse displacement varies quadratically to achieve the accountability of normal strain and its derivative in calculation of transverse shear strains. The theory also satisfies zero transverse strains conditions at the top and bottom faces of the plate, and the geometric nonlinearity is based on Green-Lagrange assumptions. All higher order terms appearing from nonlinear strain displacement relations are incorporated in the formulation. The material properties of the plates are assumed to vary smoothly and continuously throughout the thickness of the plate by a simple power-law distribution in terms of the volume fractions of the constituents. A C0 continuous isoparametric nonlinear FEM with 13 degrees of freedom per node is proposed for the accomplishment of the improved elastic continuum. Numerical results with different system parameters and boundary conditions are accomplished, to show the importance and necessity of the higher order terms in the nonlinear formulations.

Canonical formulation of Pais–Uhlenbeck action and resolving the issue of branched Hamiltonian

2017 ◽  
Vol 14 (03) ◽  
pp. 1750038 ◽  
Author(s):  
Kaushik Sarkar ◽  
Nayem Sk ◽  
Ranajit Mandal ◽  
Abhik Kumar Sanyal
Keyword(s):  
Scalar Curvature ◽  
Canonical Transformation ◽  
Auxiliary Variable ◽  
Order Theory ◽  
Higher Order ◽  
End Points ◽  
Canonical Formulation ◽  
Higher Order Terms ◽  
Theory Of Gravity ◽  
Higher Order Theory

Canonical formulation of higher order theory of gravity requires to fix (in addition to the metric), the scalar curvature, which is acceleration in disguise, at the boundary. On the contrary, for the same purpose, Ostrogradski's or Dirac's technique of constrained analysis, and Horowit'z formalism, tacitly assume velocity (in addition to the co-ordinate) to be fixed at the end points. In the process when applied to gravity, Gibbons–Hawking–York term disappears. To remove such contradiction and to set different higher order theories on the same footing, we propose to fix acceleration at the endpoints/boundary. However, such proposition is not compatible to Ostrogradski's or Dirac's technique. Here, we have modified Horowitz's technique of using an auxiliary variable, to establish a one-to-one correspondence between different higher order theories. Although, the resulting Hamiltonian is related to the others under canonical transformation, we have proved that this is not true in general. We have also demonstrated how higher order terms can regulate the issue of branched Hamiltonian.

Linear Thermoelastic Higher-Order Theory for Periodic Multiphase Materials

2001 ◽  
Vol 68 (5) ◽  
pp. 697-707 ◽  
Author(s):  
J. Aboudi ◽  
M.-J. Pindera ◽  
S. M. Arnold
Keyword(s):  
Finite Element ◽  
Predictive Accuracy ◽  
Local Stress ◽  
Computer Code ◽  
Order Theory ◽  
Higher Order ◽  
Functionally Graded ◽  
Infinite Matrix ◽  
Homogenization Technique ◽  
Higher Order Theory

A new micromechanics model is presented which is capable of accurately estimating both the effective elastic constants of a periodic multiphase composite and the local stress and strain fields in the individual phases. The model is presently limited to materials characterized by constituent phases that are continuous in one direction, but arbitrarily distributed within the repeating unit cell which characterizes the material’s periodic microstructure. The model’s analytical framework is based on the homogenization technique for periodic media, but the method of solution for the local displacement and stress fields borrows concepts previously employed by the authors in constructing the higher-order theory for functionally graded materials, in contrast with the standard finite element solution method typically used in conjunction with the homogenization technique. The present approach produces a closed-form macroscopic constitutive equation for a periodic multiphase material valid for both uniaxial and multiaxial loading which, in turn, can be incorporated into a structural analysis computer code. The model’s predictive accuracy is demonstrated by comparison with reported results of detailed finite element analyses of periodic composites as well as with the classical elasticity solution for an inclusion in an infinite matrix.

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