Plane problems of concentrated forces for half-linear material

V. M. Mal’kov ; Yu. V. Mal'kova

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2013), pp. 83-96

Résumé

The plane problems of nonlinear elasticity (the plane strains and the plane stresses) are considered for a plane and a half-plane under the action of concentrated forces. Mechanical properties are described by a half-linear material model. Using a harmonious material model has allowed to apply the methods of theory complex functions and receive exact analytical global solutions of problems, including: concentrated force on the interface of materials of a two-componential plane and concentrated force on the border of a half-plane (problems of Flamant and Michel). From global solutions asymptotic stresses and displacements in a vicinity of a force application point are constructed. The comparison of the results obtained with the solutions of Flamant and Michel linear problems has shown that stresses and displacements have identical singularities in a vicinity of a force application point $-1/r$, displacements have logarithmic singularity $-\ln r$. At the same time there are also principal differences: in linear problems only radial stresses are distinct from zero, and in nonlinear and shear stresses they are not equal to zero. Besides, factors at singular members in nonlinear and linear problems are different. Bibliogr. 10.

Détail

Keywords: plane problems, concentrated force, half-linear material, complex functions method, asymptotic series.