Strain gradient crystal plasticity (SGCP) represents a very promising way to account for size effects in miniaturized components, thanks to the intrinsic length scale(s) embedded. Most of the existing SGCP models are based on a quadratic form of defect energy. However, it has recently been shown that this form leads to physically unrealistic results concerning the size-dependence of the mechanical response of miniaturized components. A generalized non-quadratic form is proposed in this work which aims to study the influence of the defect energy order on the global response of size-dependent materials.