Explicit and implicit gradient series in damage mechanics

Gradient enhancement series are studied in the context of damage mechanics. Distinction is made between so-called explicit series and implicit series, both of which can be derived from a nonlocal damage model. The paper focuses on the difference between second-order and fourth-order truncations for either series. Dispersion analysis and numerical simulations are used to compare the various models. It is shown that for the explicit series the fourth-order term has a detrimental influence on the response, while for the implicit series the fourth-order term leads to a slightly closer approximation of the nonlocal model. The role of the critical wave length as it emerges from the dispersion analysis is shown to be decisive. When the critical wave length acts as an upper bound, a stable response is obtained and the critical wave length equals the width of the damaging zone. On the other hand, when the critical wave length acts as a lower bound, oscillations may appear of which the periodicity is set by this critical wave length.