## 2 Effective Material Properties and Constitutive Relations
Based on the previous displacement relations, the strain components are determined as  
\[\varepsilon_{rr}\ =\ \frac{\partial u_{r}}{\partial r}\ =\ 0 \tag{11}\]  
\[\varepsilon_{\theta\theta}\ =\ \frac{u_{r}}{r}+\frac{1}{r}\frac{\partial u_{ \theta}}{\partial\theta}\ =\ \frac{u_{r}}{R+\zeta}+\frac{1}{R+\zeta}\frac{\partial u_{ \theta}}{\partial\theta}+\frac{\zeta}{R+\zeta}\frac{\partial\chi}{\partial\theta} \tag{12}\]  
\[\varepsilon_{r\theta}\ =\ \frac{1}{r}\frac{\partial u_{r}}{\partial\theta}+\frac{ \partial u_{\theta}}{\partial r}-\frac{u_{\theta}}{r}\ =\ \frac{1}{R+\zeta}\frac{\partial u_{r}}{\partial\theta}+\chi-\frac{u_{ \theta}}{R+\zeta}-\frac{\zeta}{R+\zeta}\chi \tag{13}\]  
The whole analysis is based on the differential constitutive formulation [53; 54; 55; 56], which can be considered equivalent to the original Eringen integral convolution only for nonlocal problems defined in unbounded domains, involving screw dislocations and surface waves [53]  
\[\big{(}1-\mu^{2}\
abla^{2}\big{)}\sigma^{(k)}_{\theta\theta}\ =\ {E}^{(k)}_{C} \varepsilon^{(k)}_{\theta\theta} \tag{14}\]  
\[\big{(}1-\mu^{2}\
abla^{2}\big{)}\sigma^{(k)}_{r\theta}\ =\ {G}^{(k)}_{C} \varepsilon^{(k)}_{r\theta} \tag{15}\]  
where \(\mu\) is the nonlocal parameter and \(G^{(k)}_{C}\) is the shear modulus at each layer. Thus, the size-dependent governing equations of the problem are derived by applying [PERSON]'s principle, as follows  
\[\int\delta(U-T-V)dt\ =\ 0 \tag{16}\]  
whose variation of the strain energy is expressed as [50]  
\[\delta U\ =\ \iiint(\sigma_{\theta\theta}\delta e_{\theta\theta}+\sigma_{r \theta}\delta e_{r\theta})dV \tag{17}\]  
By definition of the volume element as \(dV\ =\ brdrd\theta\ =\ b(R+\zeta)d\zeta d\theta\), we can redefine the variation of the strain energy as  
\[\delta U\ =\ \iiint_{V}\big{\{}\sigma_{\theta\theta}\big{[}\frac{\delta u_{r}} {R+\zeta}+\frac{1}{R+\zeta}\frac{\partial u_{\theta}}{\partial\theta}+\frac{ \zeta}{R+\zeta}\frac{\partial\chi}{\partial\theta}\big{]}+k_{s}\sigma_{r \theta}\big{[}\frac{1}{R+\zeta}\frac{\partial\zeta u_{r}}{\partial\theta}+ \frac{R}{R+\zeta}\delta\chi-\frac{\delta u_{\theta}}{R+\zeta}\big{]}\big{]}b(R +\zeta)d\zeta d\theta \tag{18}\]  
where we set the shear stress correction factor to \(k_{s}\ =\ 5/6\). By definition of resultant components, the variation of strain energy is expressed as follows  
\[u_{\theta}(r,\theta,t)\ =\ u_{\theta}(\theta,t)+\zeta_{\chi}(\theta,t) \tag{10}\]0  
where the resultant components are expressed as in the following  
\[u_{\theta}(r,\theta,t)\ =\ u_{\theta}(\theta,t)+\zeta_{\chi}(\theta,t) \tag{10}\]1By combination of Equations (12)-(16),(19),(20), we get the following relations between the resultant components and the displacement field components, namely  
\[ \begin{array}{lcl}\Big{(}1-\mu^{2}\
abla^{2}\Big{)}N_{\theta\theta}&=&A_{4}u_{r }+A_{4}\frac{\partial u_{\theta}}{\partial\theta}+A_{5}\frac{\partial\chi}{ \partial\theta}\\ \Big{(}1-\mu^{2}\
abla^{2}\Big{)}N_{r\theta}&=&-A_{6}u_{\theta}+A_{6}\frac{ \partial u_{r}}{\partial\theta}+A\gamma\chi\\ \Big{(}1-\mu^{2}\
abla^{2}\Big{)}M_{\theta\theta}&=&A_{5}u_{r}+A_{5}\frac{ \partial u_{\theta}}{\partial\theta}+A_{8}\frac{\partial\chi}{\partial\theta} \end{array}  \tag{21}\]  
More details about the analytical expressions for the integration constants of Equation (21) \(A_{j}\) can be found in Appendix A. The external work related to the Pasternak foundation is defined as  
\[\delta V\ =\ \iint\limits_{A}\Big{(}R_{f}\Big{)}\delta u_{r}dA \tag{22}\]  
\[R_{f}\ =\ k_{1}u_{r}-k_{2}\
abla^{2}u_{r} \tag{23}\]  
where \(R_{f}\) is the reaction force of the [PERSON] foundation, \(k_{1}\) and \(k_{2}\) are the associated elastic parameters, and \(\
abla^{2}\) is the Laplacian operator in polar coordinate, namely, \(\
abla^{2}\ =\ \frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}\).