Abstract A numerical solution procedure of three-dimensional constrained microstretch (microdilatation) elastic problem is presented. The approach called local point interpolation–boundary element method (LPI–BEM) uses a partition of the kinematical variables into complementary and particular parts. The complementary fields are obtained by isotropic boundary element method. The particular integrals are determined by solving the corresponding strong form differential equations using local radial point interpolation. The effectiveness and accuracy of the approach are proven on some simple examples. The latter are also used in order to highlight some peculiarities and potentialities of such extended continuum mechanical approach.