Lam,'s strain potential method for plane gradient elasticity problems

摘要

The method of Lam,'s strain potential for solving analytically a certain class of problems of classical elasticity is extended here to plane gradient elasticity of the simple Mindlin's type with just one constant (internal length) in addition to the two classical elastic moduli. According to this method, the strains are expressed as second-order derivatives of a scalar function (Lam,'s strain potential). Thus, combining compatibility, equilibrium and stress-strain equations under plane stress or plane strain conditions and zero body forces, one can prove that this strain potential satisfies an equation of the fourth order instead of the second one for the classical case. General solutions of this equation in Cartesian and polar coordinates are provided. Four examples, two in Cartesian and two in axisymmetric polar coordinates, are presented to illustrate the method and demonstrate its advantages and limitations.