Strain gradient elasticity is an enriched version of continuum mechanics, where in addition to the first gradient also the second gradient of displacements is considered in the expression for the elastic energy density. This leads to the advantage that size dependent material properties can be explained. However, the determination of the length scale parameters of the theory is a difficult task. Depending on the method used, one derives different values for the material dependent parameters. If one determines the parameters from fits to bending experiments performed with micro cantilever beams, then length scale parameters in the range of a few µm are derived. But if one instead uses the phonon dispersion relation for determination of the parameters, then length scales in the nm range are obtained. The present investigation attempts to resolve this contradiction by introducing additional terms, which are added to the energy density. By analogy to Mindlin’s theory, there is an energy contribution proportional to the square of strain gradients, whereby the correlated proportionality constants have the dimension of a force. However, in the new formulation also square roots of fourth order terms in strain gradients are allowed instead of using just quadratic forms of strain gradients. Consequently, one arrives at a theory with a larger number of material parameters, and it is therefore easier to distinguish different deformation modes from each other. In conclusion, one can describe the stiffness of all relevant deformation modes with one consistent set of material parameters. In order to evaluate this material behaviour, a Finite Element implementation based on the penalty method is elaborated. Finally, the simulation method is demonstrated for typical examples of small scaled structures.