Gradient damage models applied to dynamic fragmentation of brittle materials

This paper is devoted to the use of gradient damage models in a dynamical context. After the setting of the general dynamical problem using a variational approach, one focuses on its application to the fragmentation of a brittle ring under expansion. Although the 1D problem admits a solution where the damage field remains uniform in space, numerical simulations show that the damage field localizes in space at a certain time and then a fragmentation of the ring rapidly occurs. To understand this phenomenon from a theoretical point of view, one develops a stability analysis of the homogeneous response by studying the growth of small perturbations. A dimensional analysis shows that the problem essentially depends on two dimensionless parameters $$\tilde{\ell }$$ and $$\tilde{\dot{\varepsilon }}_0$$, $$\tilde{\ell }$$ being related to the characteristic length present in the damage model and $$\tilde{\dot{\varepsilon }}_0$$ to the applied expansion rate. Then, since the product $$\tilde{\ell }\tilde{\dot{\varepsilon }}_0$$ is small in practice, the problem of stability is solved in a closed form by using asymptotic expansions. The comparison with the numerical results allows us to conclude that the time at which the damage localizes and the number of fragments are really governed by the growth of the imperfections. To conclude, a numerical simulation of the fragmentation of a 2D ring is presented.