The present work deals with a continuum damage theory, developed within a thennodynamical framework, able to perform a macroscopic description of material degradation induced by deformation in brittle elastic structures (i.e. failure occurs without permanent deformations). In the modeling, the Helmholtz free energy is supposed to depend not only on the strain and on the absolute temperature but on a damage variable and its gradient as well. Besides, to account for microscopic effects, the power of internal forces depends not only on the velocity and its gradient, but also on the damage velocity and its gradient. Despite the mechanical sophistication of the theory, a simple numerical technique, based on the únite element method, is proposed to approximate the solution of the resulting non linear mathematical problems. The coupling between damage and strain variables in these problems is circumvented by means of a splitting technique. In order to analyse the physical coherence of the model and to access the main features of the numerical method, a number of examples is presented showing that the numerical computations are not mesh dependent.