## Appendix A Frequency-Dependent Constitutive Equation for the Dissipative Structure
### Compatibility Equation between \(a_{ijkh}\) and \(b_{ijkh}\)  
We recall that \(a_{ijkh}(0)\) is the equilibrium modulus tensor that is the elastic tensor and which is denoted by \(a_{ijkh}^{\rm elas}\),  
\[a_{ijkh}^{\rm elas}=a_{ijkh}(0)\,. \tag{106}\]  
Due to the causality property in the time domain and using the Hilbert transform for causal function [116; 117; 118; 119; 120; 121; 122; 123; 124; 125; 126; 127; 128; 129; 130; 131; 132; 133; 134; 135; 136; 137; 138; 139; 140; 141  
* _(ii) General case._ In the general case for which \(a_{ijkh}(\omega)\) and \(b_{ijkh}(\omega)\) are not derived from such an algebraic representation but correspond to a general integral operator in the time domain (e.g., constructed using experimental curves), a rigorous method of construction is proposed below to satisfy the causality principle.  
For the general case, it is assumed that a part \(\Omega_{\text{visco}}\) of the structure \(\Omega_{S}\) is made of material modeled in the framework of the linear viscoelasticity theory (see after), while the complementary part \(\Omega_{\text{damp}}\) will be modeled with a linear dissipative constitutive equation for modeling damping effects (detailed in Section A.4). We then have \(\Omega_{S}=\Omega_{\text{visco}}\cup\Omega_{\text{damp}}\).  
For the practical construction of the constitutive equation related to \(\Omega_{\text{visco}}\), it is assumed that functions \(\omega\mapsto b_{ijkh}(\mathbf{x},\omega)\) for \(\omega\geq 0\) and equilibrium modulus tensor \(a_{ijkh}^{\text{elas}}(\mathbf{x})\) (which is the symmetric and positive-definite elastic tensor) are given. For real \(\omega\) and for \(\mathbf{x}\) belonging to \(\Omega_{\text{visco}}\), functions \(\omega\mapsto a_{ijkh}(\mathbf{x},\omega)\) can then be constructed.  
* The given functions \(\omega\mapsto b_{ijkh}(\mathbf{x},\omega)\) cannot be arbitrarily chosen, but must satisfy some hypotheses to ensure the coherence of the viscoelastic model: (1) For all fixed \(\mathbf{x}\) and \(\omega\), the tensor \(\{b_{ijkh}(\mathbf{x},\omega)\}_{ijkh}\) must be symmetric and positive definite. (2) For \(\omega\rightarrow+\infty\), Equation (A3) must hold, which means that functions \(b_{ijkh}(\mathbf{x},\omega)\) decrease at infinity at least in \(\omega^{-\alpha}\) with \(\alpha>1\). (3) Functions \(\omega\mapsto b_{ijkh}(\mathbf{x},\omega)\) that satisfy (1) and (2) are then extended to \(\omega<0\) using the even property defined by Equation (A6).
* For all fixed \(\mathbf{x}\) and \(\omega\), the tensor \(\{a_{ijkh}(\mathbf{x},\omega)\}_{ijkh}\) must be symmetric and positive definite. For all \(\omega\geq 0\), functions \(\omega\mapsto a_{ijkh}(\mathbf{x},\omega)\) are then constructed using the following equation (A12), \[a_{ijkh}(\mathbf{x},\omega)=a_{ijkh}^{\text{elas}}(\mathbf{x})+\frac{\omega}{\pi }\text{ p.v}\int_{-\infty}^{+\infty}\frac{b_{ijkh}(\mathbf{x},\omega^{\prime})}{ \omega-\omega^{\prime}}\,d\omega^{\prime}\,,\] (A14) and are extended to \(\omega<0\) using the even property.
* As seen above, for all fixed \(\mathbf{x}\) and \(\omega\), symmetric tensor \(\{a_{ijkh}(\mathbf{x},\omega)\}_{ijkh}\) must be positive definite. This property must then be checked at the end of the construction, and if it is not satisfied, functions \(\omega\mapsto b_{ijkh}(\mathbf{x},\omega)\) must be modified. In [101], it has been shown that the following sufficient condition allows this property to be satisfied: if functions \(\omega\mapsto b_{ijkh}(\mathbf{x},\omega)\) are decreasing functions for \(\omega\geq 0\), then the property is verified.