A variational approach for a nonlinear 1-dimensional second gradient continuum damage model

A 1-dimensional second gradient damage continuum theory is presented within the framework of the variational approach. The action is intended to depend not only with respect to the first gradient of the displacement field and to a scalar damage field, but also to the field of the second gradient of the displacement. Constitutive prescriptions of the stiffness (the constitutive function in front of the squared first gradient term in the action functional) and of the microstructural material length (i.e., the square of the constitutive function in front of the squared second gradient term in the action functional) are prescribed in terms of the scalar damage parameter. On the one hand, as in many other works, the stiffness is prescribed to decrease as far as the damage increases. On the other hand, the microstructural material length is prescribed, in contrast to a certain part of the literature, to increase as far as the damage increases. This last assumption is due to the interpretation that a damage state induces a microstructure in the continuum and that such a microstructure is more important as far as the damage increases. At a given value of the damage parameter, the behavior is referred to second gradient linear elastic material. However, the damage evolution makes the model not only nonlinear but also inelastic. The second principle of thermodynamics can be considered by assuming that the scalar damage parameter does not decrease its value in the process of deformation, and this implies a dissipation for the elastic strain energy. It is finally remarked that damage initiation, in this second gradient continuum damage model, can be induced not only from a prescribed initial lack of stiffness, but also from an external concentrated double force or from suitable boundary conditions, and these last options have advantages that are discussed in the paper. For example, in this last case, it is possible to initiate the damage in the neighborhood of the boundaries, that is, the case in most of the empirical situations. Simple numerical simulations are also presented in order to show the exposed concepts.