Problems of the Flamant–Boussinesq and Kelvin Type in Dipolar Gradient Elasticity

Abstract

This work studies the response of bodies governed by dipolar gradient elasticity to concentrated loads. Two-dimensional configurations in the form of either a half-space (Flamant-Boussinesq type problem) or a full-space (Kelvin type problem) are treated and the concentrated loads are taken as line forces. Our main concern is to determine possible deviations from the predictions of plane-strain/plane-stress classical linear elastostatics when a more refined theory is employed to attack the problems. Of special importance is the behavior of the new solutions near to the point of application of the loads where pathological singularities and discontinuities exist in the classical solutions. The use of the theory of gradient elasticity is intended here to model material microstructure and incorporate size effects into stress analysis in a manner that the classical theory cannot afford. A simple but yet rigorous version of the generalized elasticity theories of Toupin (Arch. Ration. Mech. Anal. 11:385-414, 1962) and Mindlin (Arch. Ration. Mech. Anal. 16:51-78, 1964) is employed that involves an isotropic linear response and only one material constant (the so- called gradient coefficient) additional to the standard Lame constants (Georgiadis et al., J. Elast. 74:17-45, 2004). This theory, which can be viewed as a first-step extension of the classical elasticity theory, assumes a strain-energy density function, which besides its dependence upon the standard strain terms, depends also on strain gradients. The solution method is based on integral transforms and is exact. The present results show departure from the ones of the classical elasticity solutions (Flamant-Boussinesq and Kelvin plane-strain solutions). Indeed, continuous and bounded displacements are predicted at the points of application of the loads. Such a behavior of the displacement fields is, of course, more natural than the singular behavior present in the classical solutions.