A variational approach to strain gradient damage mechanics with an application to compressed frames

Irreversible phenomena such as corrosion or micro-cracks formation influence theload bearing capacity of a material. We will present a mechanical modelling, basedon a variational approach, for dissipative phenomena in damaged strain gradientmaterials. In particular, we will consider the case of an evolving microstructure dueto damage progression. Strain gradient regularization is adopted to model damageand we aim at studying quasi-static damage propagation in a 2D geometrically linearisotropic continuum by means of a variational inequality formulation. The noveltyof this analysis is in the use of the total deformation energy as a functional, whosesurface density depends upon the strain gradient. The total deformation energyfunctional includes a contribution due to the dissipation energy, a contribution dueto the elastically stored energy and a contribution due to the work made by generalizedforces onto the system. The dissipation energy does not depend upon the gradientor the Laplacian of the damage field. The non-locality is given by the dependenceof the elastic strain energy density upon the second gradient of the displacement.Both first and second gradient elastic moduli are assumed to depend upon the scalardamage field. Results from numerical simulations will be presented and the crucialrole of the inclusion of higher gradient terms in the energy will be discussed.