# New Optimal Design of Multimode Shunt-Damping Circuits for Enhanced Vibration Control
## 2 Methodology
### Finite-Element Model of the Laminated Composite Beam
#### 2.1.2 Constitutive Relations  
The linear constitutive equations of a piezoelectric medium with respect to the plane coordinates \((x,y,z)\) are given as:  
\[\mathbf{\sigma}^{(p)}=\overline{\mathbf{Q}}^{(p)}\mathbf{\epsilon}-\overline{\mathbf{\epsilon}}^ {(p)T}\mathbf{E}^{(p)} \tag{5}\]  
\[\mathbf{D}^{(p)}=\overline{\mathbf{\epsilon}}^{(p)}\mathbf{\epsilon}+\overline{\mathbf{\xi}}^{ (p)T}\mathbf{E}^{(p)} \tag{6}\]  
where \(\mathbf{\sigma}\), \(\mathbf{\epsilon}\), \(\mathbf{D}\) and \(\mathbf{E}\) are the stress and strain vectors, electric displacement and electric field, respectively. Additionally, \(\overline{\mathbf{Q}}\), \(\overline{\mathbf{\epsilon}}\) and \(\overline{\xi}\) denote the elasticity, piezoelectric and permittivity constant matrices, respectively.  
As mentioned above, the piezoelectric patch is polarized in the transverse direction \(z\), and the electric field is applied in the same direction. Such a configuration is characterized by a \"31\" coupling between the transverse electric field and the membrane stresses/strains, i.e., the material is poled in the '3' direction, and the mechanical stress acts in the '1' direction. Furthermore, for the 1D beam, the width in the \(y\)-direction is stress free. Taking into account all these considerations along with the plane stress assumption, the constitutive relations for the piezoelectric layer can be expressed as [25]:  
\[\mathbf{\sigma}^{(p)}=\left\{\begin{array}{l}\sigma_{x}\\ \tau_{xz}\end{array}\right\}^{(p)}=\left[\begin{array}{cc}\widetilde{Q}_{11 }^{(p)}&0\\ 0&k_{sc}\widetilde{Q}_{55}^{(p)}\end{array}\right]\left\{\begin{array}{l} \varepsilon_{x}\\ \gamma_{xz}\end{array}\right\}-\left\{\begin{array}{l}\tilde{\sigma}_{31}^{( p)}\\ 0\end{array}\right\}E_{z}^{(p)} \tag{7}\]  
\[D_{z}^{(p)}=\tilde{\sigma}_{31}^{(p)}\varepsilon_{x}+\tilde{\xi}_{33}^{(p)}E_ {z}^{(p)} \tag{8}\]  
where \(\sigma_{x}\) and \(\tau_{xz}\) denote the normal and shear stress, respectively, \(\varepsilon_{x}\) and \(\gamma_{xz}\) indicate the normal and shear strain, respectively, \(D_{z}^{(p)}\) is the transverse electric displacement, \(\widetilde{Q}_{11}^{(p)}\) and \(\widetilde{Q}_{55}^{(p)}\) are the reduced stiffness coefficients, \(\tilde{\varepsilon}_{31}^{(p)}\) is the piezoelectric constant and \(\tilde{\varepsilon}_{33}^{(p)}\) is the electric permittivity constant. Finally, \(k_{sc}\) is the shear correction coefficient, which is introduced so that results obtained by the first-order shear deformation theory are equal to the exact solution in certain representative benchmark problems [26] and, in this work, is taken to be equal to \(\frac{5}{6}\). The constitutive equations for the elastic beam can be obtained by making their piezoelectric constants vanish as:  
\[\mathbf{\sigma}^{(b)}=\left\{\begin{array}{l}\sigma_{x}\\ \tau_{xz}\end{array}\right\}^{(b)}=\left[\begin{array}{cc}\widetilde{Q}_{11 }^{(b)}&0\\ 0&k_{sc}\widetilde{Q}_{55}^{(b)}\end{array}\right]\left\{\begin{array}{l} \varepsilon_{x}\\ \gamma_{xz}\end{array}\right\} \tag{9}\]  
As is obvious, the superscript \(p\) and \(b\) represent piezoelectric and host beam layers, respectively.