Viscoelasticity and rate-dependent continuum damage

Abstract

In classical viscoelasticity, the mechanical behaviour is charac- terized by the relaxation function or the compliance function and the constitutive relationships are formulated in the form of Volterra integral equations [Bazant 1988]. This approach is clearly unsuitable for numerical computations because of its memory and CPU time requirements.

However, it is possible to expand any relaxation function into a Dirichlet series, and retain only a finite number of terms. This achieves a double goal: first, the constitutive laws for the viscoelastic material can be written in terms of a finite num- ber of internal variables, and only these need to be stored from one time step to the next, thus providing huge computational advantages compared to the hereditary integral equations; and secondly, the resulting rheological model can be interpreted as a generalized Maxwell chain, where a number of springs and dashpots are arranged in parallel. Alternatively, the compliance function of concrete can be considered and expanded in a Dirichlet series. This leads to a generalized Kelvin chain with a series arrangement. Although both approaches are completely equivalent (if a large enough number of terms is considered in the Dirichlet series), the first one leads to first order dif- ferential equations to be solved for the evolution of the inter- nal variables, while the second approach leads to second order differential equations [Carol and Bazant 1993]. Therefore, the Maxwell chain model is preferred here.

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