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In continuum mechanics, the Michell solution is a general solution to the elasticity equations in polar coordinates (,) developed by John Henry Michell in 1899. [1] The solution is such that the stress components are in the form of a Fourier series in θ {\displaystyle \theta } .
For most boundary conditions several solutions exist simultaneously. When small deflections of a structure are to be analyzed, elastica theory is not required and an approximate solution may be found using the simpler linear elasticity theory or (for 1-dimensional components) beam theory .
In continuum mechanics, the Flamant solution provides expressions for the stresses and displacements in a linear elastic wedge loaded by point forces at its sharp end. This solution was developed by Alfred-Aimé Flamant [ fr ] in 1892 [ 1 ] by modifying the three dimensional solutions for linear elasticity of Joseph Valentin Boussinesq .
In solid mechanics, the linear stability analysis of an elastic solution is studied using the method of incremental deformations superposed on finite deformations. [1] The method of incremental deformation can be used to solve static, [2] quasi-static [3] and time-dependent problems. [4]
In continuum mechanics, Eshelby's inclusion problem refers to a set of problems involving ellipsoidal elastic inclusions in an infinite elastic body. Analytical solutions to these problems were first devised by John D. Eshelby in 1957.
Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are: [1]. Equation of motion: , + = where the (), subscript is a shorthand for () / and indicates /, = is the Cauchy stress tensor, is the body force density, is the mass density, and is the displacement.
The Papkovich–Neuber solution is a technique for generating analytic solutions to the Newtonian incompressible Stokes equations, though it was originally developed to solve the equations of linear elasticity. It can be shown that any Stokes flow with body force = can be written in the form:
In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of thin structures that react elastically to external forces.