Rate dependent damage model for concrete in dynamics

This paper presents a rate dependent damage model and its implementation in transient dynamics. The rate dependent model is derived from a rate independent damage model simply by changing the expression of the damage evolution equation. The principle of the model follows the format of Perzyna viscoplasticity whereby the definition of the plastic multiplier is modified compared to a standard rate independent plasticity model. The rate dependent model can reproduce very well the major effects of the rate of loading on the response of concrete; that is, an increase of the material strength and a decrease of the softening tangent modulus with increasing strain rate. The wave propagation problem is assessed. We show that the resulting equations of motion are unconditionally hyperbolic. Well-posedness and mesh objectivity are verified numerically on a simple one-dimensional (1D) problem. As an example of implementation in a finite element calculation, the computation of the response of a reinforced concrete beam subjected to an impact is presented.