With the damage models, the stiffness of the material is decreased with increasing material damage:

with the damage variable . The evolution of the damage depends on the selected damage model, and the damage can only increase. The initial value is , and the maximal allowed value for the damage is . When unloaded, the material deforms elastically (with the decreased stiffness) back to its initial state.

Two damage models are available in GeoDict: The Exponential Damage model and Mazars’ Damage model (see Mazars, 1981, Alfano et al. 2001 and Michel and Suquet, 2016 for more details). Both models are very similar and differ only in the way they depend on the strain.

Mazars' Damage: The damage depends on the strain, and the degradation begins as soon as the strain reaches the threshold :

with : minimal strain for damage, : damage hardening modulus, : damage evolution.

Exponential Damage: The damage depends on the strain energy

which is the sum of the elastic and plastic strain energies and . If no plastic deformations occur, the strain energy depends only on the elastic strains. The degradation begins as soon as the minimal energy for damage is reached:

with : minimal strain for damage, : damage hardening modulus, : damage evolution.

The damage evolution formula is the same for both models, the significant difference is that the damage is driven by the strain for Mazars’ Damage and driven by the elastic strain energy in the Exponential Damage model. In other words, the damage criterion for Mazars depends linearly on the strains, and a quadratic relation holds for the exponential damage model.