# Static Response of Double-Layered Pipes via a Perturbation Approach
## 2 Model Description
\begin{split} u(s)=&\tilde{u}_{s}\\ v(\tilde{s})=&\tilde{v}_{\tilde{s}}\\ \vartheta(s)=&\tilde{\vartheta}_{g}\\ a_{j}(s)=&\tilde{u}_{j_{k}}\end{split}  \tag{2}\]  
where \(\tilde{u}_{\tilde{s}}\), \(\tilde{v}_{\tilde{s}}\), \(\tilde{\vartheta}_{\tilde{s}}\), \(\tilde{a}_{j_{k}}\) are imposed displacements. Time derivative (indicated with the dot) of Equation (1) provides the strain-rates:  
\[ \begin{split}\dot{\epsilon}_{0}&=\dot{u}^{\prime} \cos\vartheta+\dot{v}^{\prime}\sin\vartheta+\gamma_{0}\dot{\vartheta}\\ \dot{\gamma}_{0}&=-\dot{u}^{\prime}\cos\vartheta+ \dot{v}^{\prime}\sin\vartheta-(1+\epsilon_{0})\dot{\vartheta}\\ \dot{\kappa}_{0}&=\dot{\vartheta}^{\prime}\\ \dot{\alpha}_{j}&=\dot{a}_{j}\\ \dot{\beta}_{j}&=\dot{a}_{j}^{\prime}\end{split}  \tag{3}\]  
which are functional to define the internal virtual power, after introducing mechanical quantities dual to each strain-rate component: normal force \(N\), shear force \(T\) and bending moment \(M\), and distortion forces \(D_{j}\) and bi-forces \(B_{j}\). Imposing the validity of the virtual power theorem, i.e., equating the internal power to the external one, where imposed external forces spend power in the velocity field, one gets:  
\[ \begin{split}&\int\limits_{0}^{l}\Big{(}N\epsilon_{0}+T\gamma_{ 0}+M\kappa_{0}+\sum\limits_{j=p,g,w}(D_{j}\dot{\kappa}_{j}+B_{j}\dot{\beta}_{ j})\Big{)}ds\\ &=\int\limits_{0}^{l}\Big{(}p_{u}\dot{u}+p_{v}\dot{v}+c\dot{ \vartheta}+\sum\limits_{j=p,g,w}q_{j}\dot{a}_{j}\Big{)}ds+\sum\limits_{s=l,g,w }\Big{(}P_{\kappa_{s}}u(s)+P_{\kappa_{s}}v(\tilde{s})+C_{\delta}\dot{\vartheta }(s)+\sum\limits_{j=p,g,w}Q_{j}\dot{a}_{j}(s)\Big{)}\end{split}  \tag{4}\]  
where \(p_{u},p_{v}\) are external forces per unit length in direction \(\bar{\mathbf{a}}_{x}\), \(\bar{\mathbf{a}}_{y}\), respectively, \(c\) is external bending moment per unit length, \(q_{j}\) are external distortion forces per unit length whereas, at cross-section \(\tilde{s}=0\) and/or \(\tilde{s}=l\), \(P_{\mu_{\ell}},P_{\
u_{\tilde{s}}}\) are external tip forces in direction \(\bar{\mathbf{a}}_{x},\bar{\mathbf{a}}_{y}\), respectively, \(C_{\tilde{s}}\) is external tip bending moment, \(Q_{j_{k}}\) are external tip distortion forces. After localization of Equation (4), the following equilibrium equations are obtained:  
\[ \begin{split}&(N^{\prime}-\kappa_{0}T)\cos\vartheta-(T^{\prime}+ \kappa_{0}N)\sin\vartheta+p_{u}=0\\ &(N^{\prime}-\kappa_{0}T)\sin\vartheta+(T^{\prime}+\kappa_{0}N) \cos\vartheta+p_{v}=0\\ & M^{\prime}+(1+\epsilon_{0})T-\gamma_{0}N+c=0\\ & B_{j}^{\prime}-D_{j}+q_{j}=0\end{split}  \tag{5}\]together with the boundary conditions:  
\[ \begin{split}&(P_{u_{s}}\pm[N\cos\vartheta-T\sin\vartheta]_{s})\dot{u} (\hat{s})=0\\ &(P_{p_{s}}\pm[N\sin\vartheta+T\cos\vartheta]_{s})\dot{v}(\hat{s}) =0\\ &(C_{\xi}\pm M(\hat{s}))\dot{\vartheta}(\hat{s})=0\\ &(Q_{js}\pm B_{h}(\hat{s}))\dot{u}_{j}(\hat{s})=0\qquad\qquad \text{ for }j=p,w,g\end{split}  \tag{6}\]  
It is worth noting that in both the strain-displacement equations, Equations (1) and (2), and the equilibrium equations, Equations (5) and (6), the global variables (i.e., those relevant to the Timoshenko beam model) are uncoupled to the local variables (i.e., those relevant to the cross-section distortion). The constitutive relationship is evaluated in case of hyperelastic materials and comes from the definition of a strain potential, whose expression is identified after a homogenization procedure from a suitable three-dimensional continuum (see [20]). It turns out to describe a nonlinear elastic material (up to the third order), where coupling between global and local variables actually occurs:  
\[