A general solution of equations of equilibrium in linear elasticity

Palaniappan, D.

doi:10.1016/j.apm.2011.01.041

Cited by 18 publications

(19 citation statements)

References 6 publications

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“…By expanding thev 0,L (r, t) in Zernike polynomials R 1 2k+1 (r/a) in line with (44), the end boundary condition for different truncation orders are presented in Figure 7(a). The linear term k = 0 corresponds to the result for elementary theory.…”

Section: Boundary Value Problemsmentioning

confidence: 99%

“…Note in this static case that the exact results may not be obtained directly from the static counterpart of the dynamic solution presented in Section 7.2. Instead, the methods presented in [43,44] are to be used.…”

Section: Boundary Value Problemsmentioning

confidence: 99%

“…The end boundary conditions are obtained from truncation of (44). Hence, the prescribed fields at the ends (both for tractiont θ and displacementv) are to be expanded in Zernike polynomials R 1 2k+1 (r/a).…”

Section: End Boundary Conditionsmentioning

confidence: 99%

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A hierarchy of dynamic equations for solid isotropic circular cylinders

Abadikhah

Folkow

2014

Wave Motion

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This work considers homogeneous isotropic circular cylinders adopting a power series expansion method in the radial coordinate. Equations of motion together with consistent sets of end boundary conditions are derived in a systematic fashion up to arbitrary order using a generalized Hamilton's principle. Time domain partial differential equations are obtained for longitudinal, torsional, and flexural modes, where these equations are asymptotically correct to all studied orders. Numerical examples are presented for different sorts of problems, using exact theory, the present series expansion theories of different order, and various classical theories. These results cover dispersion curves, eigenfrequencies and the corresponding displacement and stress distributions, as well as fix frequency motion due to prescribed end displacement or lateral distributed forces. The results illustrate that the present approach may render benchmark solutions provided higher order truncations are used, and act as engineering cylinder equations using low order truncation.

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Section: Boundary Value Problemsmentioning

confidence: 99%

Section: Boundary Value Problemsmentioning

confidence: 99%

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A hierarchy of dynamic equations for solid isotropic circular cylinders

Abadikhah

Folkow

2014

Wave Motion

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“…To solve (140), the displacement is expressed in terms of the Galerkin-Papkovich vectors [43,[55][56][57]…”

Section: Interpolating Temperature and Body Force Separatelymentioning

confidence: 99%

Hybrid Fundamental Solution Based Finite Element Method: Theory and Applications

Cao

Qin

2015

Advances in Mathematical Physics

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An overview on the development of hybrid fundamental solution based finite element method (HFS-FEM) and its application in engineering problems is presented in this paper. The framework and formulations of HFS-FEM for potential problem, plane elasticity, three-dimensional elasticity, thermoelasticity, anisotropic elasticity, and plane piezoelectricity are presented. In this method, two independent assumed fields (intraelement filed and auxiliary frame field) are employed. The formulations for all cases are derived from the modified variational functionals and the fundamental solutions to a given problem. Generation of elemental stiffness equations from the modified variational principle is also described. Typical numerical examples are given to demonstrate the validity and performance of the HFS-FEM. Finally, a brief summary of the approach is provided and future trends in this field are identified.

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“…Three methods used in the formulation of the mathematical theory of elasticity problems are the displacement method, stress method and mixed (hybrid) methods (Ojedokun and Olutoge, 2012;Westergaard, 1964;Barbar, 2010;Sadd, 2014;Bowles, 1997). The displacement methods of the theory of elasticity presented by Navier and Lamé involve a reformulation of the system of fifteen governing equations of equilibrium, stress -strain and kinematics such that only the three components of displacement (in a three dimensional problem) become the primary unknown variables (Kachanov et al, 2003;Sitharam and Govinda Reju, 2017;Palaniappan, 2011;Hazel, 2015). The advantage of the displacement formulation is the obvious reduction in the number of equations to be solved from fifteen to three coupled equations in terms of the three displacement components.…”

Section: Introductionmentioning

confidence: 99%

Hankel transformation method for solving the Westergaard problem for point, line and distributed loads on elastic half-space

Ike

2019

Lat. Am. j. solids struct.

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The Hankel transformation method was used in this work to determine the normal and shear stress distributions due to point, line and distributed loads applied to the surface of an elastic media. The elastic media considered in this study was assumed to be inextensible in the horizontal directions, and the only non-vanishing displacement component is the vertical component. Such materials were first considered by Westergaard as models of elastic half-space with alternating layers of soft and stiff materials with the stiff materials of negligible thickness and so closely spaced that the composite material characteristics is idealised as isotropic and homogeneous. The study adopted a displacement formulation. The Hankel transformation was applied to the governing Cauchy-Navier differential equation of equilibrium to reduce the problem to a second order ordinary differential equation (ODE) in terms of the deflection function in the Hankel transform space. Solution of the ODE subject to the boundedness condition yielded bounded deflection function in the Hankel transform space. Equilibrium of the internal vertical forces and the external applied load, was used to obtain the constant of integration, for the deflection function in the Hankel transform space. Inversion yielded the deflection in the physical domain variable. The stress displacement equations were then used to determine the Cauchy stresses. The vertical stress distributions due to line and distributed load over a rectangular area were also determined by using the point load solution for vertical stresses as Green functions, and then performing integration along the line and over the rectangular area of the load. The results obtained for vertical stresses due to point, line and distributed loads were determined in terms of dimensionless influence coefficients which were presented. The results obtained for the deflection, normal and shear stresses due to point, line and distributed load agreed with the solutions originally presented by Westergaard who used a stress function method.

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A general solution of equations of equilibrium in linear elasticity

Cited by 18 publications

References 6 publications

A hierarchy of dynamic equations for solid isotropic circular cylinders

A hierarchy of dynamic equations for solid isotropic circular cylinders

Hybrid Fundamental Solution Based Finite Element Method: Theory and Applications

Hankel transformation method for solving the Westergaard problem for point, line and distributed loads on elastic half-space

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