1. Variational formulation

1.1. Potential energy and minimization

Potential energy, for which we will look for the local minimum, is written as the sum of the elastic deformation energy (in the cable), of the cohesive energy (at the sheath/cable interface) minus the work of external forces:

\({E}_{\mathit{pot}}(u)\mathrm{=}{E}_{\mathit{el}}(u)+{E}_{\mathit{gc}}(u)\mathrm{-}{W}_{\mathit{ext}}(u)\)

The expression for elastic energy is as follows:

\({E}_{\mathit{el}}(u)\mathrm{=}{\mathrm{\int }}_{\Gamma }\Phi (\varepsilon ({u}_{\mathit{cable}}(s)))\mathit{ds}\)

where \(\Gamma\) is the path (1D) of the cable, \(\Phi\) is the elastic energy density, and \({u}_{\mathit{cable}}\) is the displacement of the cable.

The movement of the cable is defined as follows:

\({u}_{\mathit{cable}}(s)\mathrm{=}{u}_{\mathit{gaine}}(s)+g(s)T(s)\)

where \(T\) represents the tangent to the cable, \({u}_{\mathit{gaine}}\) represents the displacement of the sheath, and \(g\) the relative displacement of the cable with respect to the sheath.

We then have (Cf. Salençon (« mechanics of continuous media — Volume II: Elasticity — Curvilinear environments »):

\(\varepsilon ({u}_{\mathit{cable}}(s))\mathrm{=}\frac{d({u}_{\mathit{gaine}}+gT)}{\mathit{ds}}\mathrm{\cdot }T\mathrm{=}\frac{d({u}_{\mathit{gaine}})}{\mathit{ds}}\mathrm{\cdot }T+\frac{\mathit{dg}}{\mathit{ds}}\)

It should be noted that multiplication by the tangent vector to cable \(T\) makes it possible to pass from an « elongation » vector in the three directions of space to a scalar « elongation » along the cable. In fact, to get this expression we overlook \(\frac{dT}{\mathit{ds}}\).

The following two quantities are defined:

\({\epsilon }_{{u}_{\mathit{gaine}}}({u}_{\mathit{gaine}})=\frac{d({u}_{\mathit{gaine}})}{\mathit{ds}}\cdot T\) and \({\epsilon }_{g}(g)=\frac{\mathit{dg}}{\mathit{ds}}\)

Thus the deformation of the cable is finally written:

\(\epsilon ({u}_{\mathit{cable}}(s))={\epsilon }_{{u}_{\mathit{gaine}}}({u}_{\mathit{gaine}})+{\epsilon }_{g}(g)\)

Cohesive energy (sheath/cable) for its part is written as:

\({E}_{\mathit{gc}}(u)={\int }_{\Gamma }\Pi (g(s))\mathit{ds}\)

where \(\Pi\) is the cohesive energy density.

Given the non-differentiability of \(\Pi\), we will use a decomposition-coordination method to deal with this minimization. This consists in introducing slippage \(\delta\), in asking \(\delta (s)=g(s)\) and in reducing ourselves to the following constrained minimization problem:

\(\underset{\begin{array}{c}u,\delta \\ g=\delta \end{array}}{\mathit{min}}E(u,\delta )\)

with

\(E(u,\delta )\mathrm{=}{\mathrm{\int }}_{\Gamma }\Phi (\varepsilon ({u}_{\mathit{cable}}(s)))\mathit{ds}+{\mathrm{\int }}_{\Gamma }\Pi (\delta (s))\mathit{ds}\mathrm{-}{W}_{{\mathit{ext}}_{\mathit{gaine}}}({u}_{\mathit{gaine}})\mathrm{-}{W}_{{\mathit{ext}}_{\mathit{cable}}}(g)\)

This will make it possible to treat separately (decomposition) the minimization of cohesive energy at the local level (static condensation) while the minimization of elastic energy and work will be treated at the global level.

1.2. Augmented Lagrangian

The problem of minimization under constraints is treated by dualization: we introduce the augmented Lagrangian \(L\) and the multiplier field \(\lambda\) (coordination):

\(L(u,\delta ,\lambda )=E(u,\delta )+{\int }_{\Gamma }\lambda (s)\cdot (g(s)-\delta (s))\mathit{ds}+\frac{r}{2}{\int }_{\Gamma }(g(s)-\delta (s))\mathrm{²}\mathit{ds}\)

with \(r\) penalty coefficient.

1.3. Characterization of the saddle point

The first-order optimality conditions make it possible to write:

\(\forall \delta {u}_{\mathit{gaine}}\mathrm{:}{\int }_{\Gamma }\sigma \mathrm{:}{\epsilon }_{{u}_{\mathit{gaine}}}(\delta {u}_{\mathit{gaine}}(s))\mathit{ds}={W}_{{\mathit{ext}}_{\mathit{gaine}}}(\delta {u}_{\mathit{gaine}})\) with \(\sigma =\frac{\partial \Phi }{\partial \epsilon }(\epsilon )\) [éq. 1.1]

\(\forall \delta g\mathrm{:}{\int }_{\Gamma }\sigma \mathrm{:}{\epsilon }_{g}(\delta g(s))\mathit{ds}+{\int }_{\Gamma }[\lambda (s)+r(g(s)-\delta (s))]\cdot \delta g(s)\mathit{ds}={W}_{{\mathit{ext}}_{\mathit{cable}}}(\delta g)\) [éq. 1.2]

\(\forall \delta \lambda \mathrm{:}{\int }_{\Gamma }[g(s)-\delta (s)]\cdot \delta \lambda (s)\mathit{ds}=0\) [éq. 1.3]

\(\forall \delta \delta \mathrm{:}{\int }_{\Gamma }[t(s)-\lambda (s)-r(g(s)-\delta (s))]\cdot \delta \delta (s)\mathit{ds}=0\) with \(t\mathrm{\in }\mathrm{\partial }\Pi (\delta )\) [éq. 1.4]