Real materials exhibit a substructure (microstructure), which can be captured by characteristic material internal lengths. The microstructure can strongly effect the macroscopic response in many applications. Such phenomena have been observed e.g. in statics, in form of lengthscale effects in the material respon- se, and in dynamics in form of non–classical dispersion relations. The examples cover, among others, in the microscopic range the response of thin films, nanotubes and adhesive joints, and in the macroscopic range the response of earth plates and lining of tunnels in soil. The obseved effects can be addressed adequately in the linear range by constitutive laws of gradient elasticity. The present work is concerned with a Gradientelasticity model, which is based on both, the Laplacian of the strain and the Laplacian of the stress. The main investigations focus on the near tip fields within Mode– I – and Mode– I I –crack– problems. It is proved, that, in contrast to the perspective of some others, the model does not avoid the well known singularities in classical elasticity. Nevertheless there are significant differences in compari- son to classical elasticity when considering crack problems. Such differences are discussed in detail on the basis of closed form analytical solutions as well as finite element simulations.