Finite elements for elasticity with microstructure and gradient elasticity

We present a general finite element discretization of Mindlin’s elasticity with microstructure. A total of 12 isoparametric elements are developed and presented, six for plane strain conditions and six for the general case of three-dimensional deformation. All elements interpolate both the displacement and microdeformation fields. The minimum order of integration is determined for each element, and they are all shown to pass the single-element test and the patch test. Numerical results for the benchmark problem of one-dimensional deformation show good convergence to the closed-form solution. The behaviour of all elements is also examined at the limiting case of vanishing relative deformation, where elasticity with microstructure degenerates to gradient elasticity. An appropriate parameter selection that enforces this degeneration in an approximate manner is presented, and numerical results are shown to provide good approximation to the respective displacements and strains of a gradient elastic solid. Copyright q 2007 John Wiley & Sons, Ltd