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In this paper, we consider isotropic Mindlin-Toupin strain gradientelasticity theory in which the equilibrium equations contain two additionallength-scale parameters and have the fourth order. For this theory we developedan extended form of Boussinesq-Galerkin (BG) and Papkovich-Neuber (PN) generalsolutions. Obtained form of BG solution allows to define the displacement fieldthrough the single vector function that obeys the eight-orderbi-harmonic/bi-Helmholtz equation. The developed PN form of the solutionprovides an additive decomposition of the displacement field into the classicaland gradient parts that are defined through the standard Papkovich stressfunctions and modified Helmholtz decomposition, respectively. Relations betweendifferent stress functions and completeness theorem for the derived generalsolutions are established. As an example, it is shown that a previously knownfundamental solution within the strain gradient elasticity can be derived byusing the developed PN general solution.

Complete general solutions for equilibrium equations of isotropic strain gradient elasticity