3. Viscoelastic damping model with internal variables

The internal variable viscoelastic damping model can be used to treat the harmonic and transient responses of structures with viscoelastic materials. This model is based on the existence of a law of behavior that makes it possible to determine the stress state according to the history of deformations:

\(\sigma ={E}_{\infty }\varepsilon (t)-{\int }_{0}^{t}{E}_{v}(t-\tau )\frac{\partial \varepsilon (\tau )}{\partial \xi }d\tau\)

where \({E}_{\infty }\) represents the high frequency Young’s modulus, and \({E}_{v}\) represents the relaxation module. This module can be represented in the time domain by a Prony series.

\({E}_{v}(t)={E}_{r}+{\sum }_{k=1}^{N}{E}_{k}\mathrm{exp}(-t/{\tau }_{k})\),

or by a sum of first-order rational fractions in the frequency domain.

\({E}_{v}(s)={E}_{r}+{\sum }_{k=1}^{N}\frac{{\omega }_{k}{E}_{k}}{s+{\omega }_{k}}\).

To take into account these laws of behavior, internal variables are introduced that make it possible to build an equivalent linear model of order two, compatible with*Code_Aster*. These variables make the link between the degrees of physical freedom impacted by the presence of a viscoelastic material and the quantities defining the behavior. Thus, as many \({q}_{\mathrm{vk}}\) vectors as internal parameters are introduced. These variables are governed by the temporal evolution equations

\({\tau }_{k}\dot{{q}_{\mathrm{vk}}}+{q}_{\mathrm{vk}}-q=0\)

Or, in frequency

\(s{q}_{\mathrm{vk}}+{\omega }_{k}({q}_{\mathrm{vk}}-q)=0\)

We can illustrate the relationships describing this behavior by considering a mass « spring » system for which the spring has a behavior that can be represented by \(N\) internal variables:

The equilibrium equations for this system are written

\(\{\begin{array}{}\text{}M\ddot{q}+{K}_{0}q+{\sum }_{k=1}^{N}{K}_{k}(q-{q}_{\mathrm{vk}})=0\text{}\\ \text{}\\ \text{}{C}_{k}\dot{{q}_{\mathrm{vk}}}+{K}_{k}({q}_{\mathrm{vk}}-q)=0\text{}\forall k\in [\mathrm{1,}N]\end{array}\)

The dynamic system for this oscillator can therefore be put in the classical form of a second-order model.

\(\left[\begin{array}{cccc}M& 0& \text{...}& 0\\ 0& 0& \ddots & ⋮\\ ⋮& \ddots & 0& ⋮\\ 0& \mathrm{...}& \mathrm{...}& 0\end{array}\right]\ddot{\left\{\begin{array}{}q\\ {q}_{\mathrm{v1}}\\ {q}_{\mathrm{vk}}\\ {q}_{\mathrm{vN}}\end{array}\right\}}+\left[\begin{array}{cccc}0& 0& \text{...}& 0\\ 0& {C}_{1}& \ddots & ⋮\\ ⋮& \ddots & {C}_{k}& 0\\ 0& \mathrm{...}& 0& {C}_{N}\end{array}\right]\dot{\left\{\begin{array}{}q\\ {q}_{\mathrm{v1}}\\ {q}_{\mathrm{vk}}\\ {q}_{\mathrm{vN}}\end{array}\right\}}+\left[\begin{array}{cccc}{K}_{0}+{\sum }_{k=1}^{N}{K}_{k}& -{K}_{1}& -{K}_{k}& -{K}_{N}\\ -{K}_{1}& {K}_{1}& 0& 0\\ -{K}_{k}& 0& {K}_{k}& 0\\ -{K}_{N}& 0& 0& {K}_{n}\end{array}\right]\left\{\begin{array}{}q\\ {q}_{\mathrm{v1}}\\ {q}_{\mathrm{vk}}\\ {q}_{\mathrm{vN}}\end{array}\right\}=\left\{\begin{array}{}f\\ 0\\ 0\\ 0\end{array}\right\}\)

As it stands, Code_Aster does not allow such laws of behavior to be taken into account for linear dynamic analyses. The solution adopted to solve this problem is to build a reduction base adapted to the problem.

This reduction base is built around natural modes and the static responses of the structure to the viscoelastic forces generated by the modes. Under these conditions, we have

\([T]=\left[\begin{array}{ccc}{\Phi }_{l}& {\Phi }_{e}& {T}_{p}\\ {\Phi }_{\mathrm{lv1}}& 0& {T}_{\mathrm{v1}}\\ {\Phi }_{\mathrm{lvk}}& 0& {T}_{\mathrm{kv}}\\ {\Phi }_{\mathrm{lvN}}& 0& {T}_{\mathrm{vN}}\end{array}\right]\)

The T submatrices are constructed from the static problem

\(\left[\begin{array}{cccc}{K}_{0}+{\sum }_{k=1}^{N}{K}_{k}& -{K}_{1}& -{K}_{k}& -{K}_{N}\\ -{K}_{1}& {K}_{1}& 0& 0\\ -{K}_{k}& 0& {K}_{k}& 0\\ -{K}_{N}& 0& 0& {K}_{n}\end{array}\right]\left\{\begin{array}{}{T}_{p}\\ {T}_{\mathrm{v1}}\\ {T}_{\mathrm{vk}}\\ {T}_{\mathrm{vN}}\end{array}\right\}=\left\{\begin{array}{}0\\ {F}_{\mathrm{v1}}\\ {F}_{\mathrm{vk}}\\ {F}_{\mathrm{vN}}\end{array}\right\}\)

with the load set by

\(\left\{\begin{array}{}0\\ {F}_{\mathrm{v1}}\\ {F}_{\mathrm{vk}}\\ {F}_{\mathrm{vN}}\end{array}\right\}=\left[\begin{array}{cccc}0& 0& \text{...}& 0\\ 0& {C}_{1}& \ddots & ⋮\\ ⋮& \ddots & {C}_{k}& 0\\ 0& \mathrm{...}& 0& {C}_{N}\end{array}\right]\dot{\left\{\begin{array}{}\Phi \\ {\Phi }_{\mathrm{v1}}\\ {\Phi }_{\mathrm{vk}}\\ {\Phi }_{\mathrm{vN}}\end{array}\right\}}\)

The projected problem can then be solved on the basis \(T\), a problem that takes into account the global viscoelastic behavior defined for the materials.

Details for implementing these techniques in*Code_Aster* are detailed in the documentation on building scale models for U2.06.04 dynamics.

This approach is currently reserved for experienced users, since model reduction in the presence of internal states can lead to singular behaviors. On the other hand, the proposed method of reduction is not necessarily sufficient, and it is sometimes necessary to enrich the base in this way to build a reasonable model.

Searching for eigenvalues can only be done with certain settings of the eigenvalue search operator CALC_MODES [U4.52.02].

The reference documentation [R5.01.02] specifies the types of problems that can be treated and the possible settings.

However, for the calculation of the eigenmodes of the reduced model, it is recommended to use the solid solver (METHODE =”QZ”) by looking for all the eigenvalues (OPTION =” TOUT “), since otherwise, it is not possible to access the real eigenvalues, representing relaxation, which are important components of the behavior of this type of material.