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Displacement potentials use none none implementation tocreate none on noneusing bar codes in c# functions I0 and K0 if none for none is imaginary. This form of Z(r, z) is particularly useful if boundary conditions are speci ed on cylindrical surfaces, for example, axisymmetric tractions on the side surfaces r = r0 of a rod of radius r0 , or loading of plates containing circular holes. A generalisation of the above formulation leads to the use of Hankel transform methods.

The Hankel transform of order zero of the Love strain function is de ned as Z0 ( , z) =. Web application framework rZ(r, z)J 0 ( r)dr. (6.53). Applying the Hankel tran sform to the biharmonic equation Z(r, z) = 0 and using the differentiation properties of Bessel function J 0 leads to the ordinary differential equation in z for the unknown function d2 2 Z0 ( , z) = 0. dz2 The solution of this equation is given by Z0 ( , z) = (A( ) + B( ) z) exp( z). (6.

55) (6.54). The Hankel transform mus none none t also now be applied to displacement and/or traction boundary conditions of the form (6.47), (6.48) or (6.

49) in order to nd the unknown functions A( ) and B( ). Soutas-Little (1973) gives an example of using this approach to obtain the Boussinesq solution for the concentrated load P acting normal to the surface of a half space. The Hankel transform of Love strain function can be sought in the form Z0 ( , z) = CP [2 + z] exp( z), 2 2 (6.

56). where the constant C is found by satisfying equlibrium between the externally applied force P and internal stresses. Axisymmetric contact problems involving elastic half-spaces are a class of so-called mixed boundary value problems, since on part of the surface the boundary condition is prescribed in terms of displacements, and elsewhere traction boundary conditions apply. The integral transform formulation of Love strain solution provides an effective way of addressing such problems, because it allows the condition on some part of the boundary (e.

g., the traction-free requirement) to be satis ed by construction, thus leading to a single integral equation formulation..

SUMMARY The displacement potenti al approach to the solution of elastic problems is presented. Harmonic Papkovich Neuber potentials are rst used as the basis for the analysis, and various fundamental solutions are considered (Kelvin, Boussinesq, and Cerruti). Biharmonic potentials (Galerkin vector and Love strain function) are introduced next, and their application to the solution of special problems is discussed, for example, problems involving cylindrical or spherical symmetry.

. Exercises EXERCISES 1. Simple stress and str ain states in terms of the Love strain function Using the procedures for identifying biharmonic homogeneous polynomials described in the chapter, generate the following third-order polynomial Love strain function solutions: Az(r2 + z2 ) and B(2z3 3r2 z)..

Compute the displacement none none s arising due to the superposition of these solutions. Derive the strain and stress states, and show that they are uniform. Determine the relationship between constants A and B required to obtain a uniaxial strain state and uniaxial and hydrostatic stress states.

. Hint: See notebook C06 Love 3-poly.nb 2. Kelvin solution using none none Love strain function Demonstrate that the Love strain function of the form Z(r, z) = B r2 + z2 provides the solution to the Kelvin problem about the concentrated force P applied at the origin and acting in the positive z direction. Determine the unknown constant B.

. Hint: See notebook C06 L ove Kelvin.nb and follow the procedures used for deriving this solution in terms of the Papkovich Neuber potentials..

3. Momentless force dipo none for none le Using the Kelvin solution for the concentrated force at the origin as the starting point, derive the Love strain function solution for the momentless force doublet at the origin. Verify that the stress eld is divergence-free and self-equilibrated.

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