5. Discretized writing#
5.1. Ratings#
Remember that elements CABLE_GAINE have 3 knots and that only the end nodes have Lagrange degrees of freedom (cf.).
Displacement shape functions are noted \({n}_{m}\) with \(m\mathrm{\in }\mathrm{\{}\mathrm{1,}\mathrm{2,}3\mathrm{\}}\).
The shape functions of the Lagrange multipliers are noted \({l}_{p}\) with \(p\mathrm{\in }\mathrm{\{}\mathrm{1,}2\mathrm{\}}\).
We note (the quantities are evaluated at each Gauss point):
\({s}_{g}\) the curvilinear abscissa
\({\omega }_{g}\) the weight of Gauss points
\({\sigma }_{g}\) the stress tensor
\({N}_{g}^{3}\) matrix of the values of the shape functions at the Gauss point \(g\) discretizing the movements of the sheath (3 components)
\({N}_{g}^{3}\mathrm{=}(\begin{array}{ccccccccc}{n}_{1}({s}_{g})& 0& 0& {n}_{2}({s}_{g})& 0& 0& {n}_{3}({s}_{g})& 0& 0\\ 0& {n}_{1}({s}_{g})& 0& 0& {n}_{2}({s}_{g})& 0& 0& {n}_{3}({s}_{g})& 0\\ 0& 0& {n}_{1}({s}_{g})& 0& 0& {n}_{2}({s}_{g})& 0& 0& {n}_{3}({s}_{g})\end{array})\)
\({N}_{g}^{1}\) matrix of the values of the shape functions at the Gauss point \(g\) discretizing the relative displacements of the cable (1 component):
\({N}_{g}^{1}\mathrm{=}(\begin{array}{ccc}{n}_{1}({s}_{g})& {n}_{2}({s}_{g})& {n}_{3}({s}_{g})\end{array})\)
\({T}_{g}\) the tangent vector to the Gauss point \(g\)
\(\mathrm{\nabla }{N}_{g}^{1}\) the derivatives of form functions discretizing the relative movements of the cable
\(\mathrm{\nabla }{N}_{g}^{3}\) the derivatives of the shape functions discretizing the movements of the sheath
We also ask to simplify the notations: \({B}_{g}\mathrm{=}{T}_{g}^{T}\mathrm{\nabla }{N}_{g}^{3}\)
\({L}_{g}\) matrix of values of form functions discretizing the Lagrange multiplier at the Gauss point \({L}_{g}\mathrm{=}(\begin{array}{cc}{l}_{1}({s}_{g})& {l}_{2}({s}_{g})\end{array})\)
\(\mathrm{\{}U\mathrm{\}}\) nodal movements (duct + cable)
\(\mathrm{\{}{U}_{\mathit{ga}}\mathrm{\}}\) the nodal movements of the sheath
\(\{{G}_{c}\}\) the relative nodal displacement of the cable
\(\{\Lambda \}\) the nodal Lagrange multiplier
\(\mathrm{\{}{F}_{\mathit{ext}}^{\mathit{ga}}\mathrm{\}}\) the nodal external force vector, dual of sheath movements
\(\{{F}_{\mathit{ext}}^{c}\}\) the nodal external force vector, dual of the nodal relative cable displacement
\(A\) the cable section
5.2. Discretization of optimality conditions#
The equations 1.1, 1.2 and 1.3 (Eq. 1.4 being treated at the level of the law of behavior) that characterize the saddle point are discretized in the form:
\(\mathrm{\sum }{\omega }_{g}({B}_{g}^{T}\mathrm{:}{\sigma }_{g})\mathrm{=}\mathrm{\{}{F}_{\mathit{ext}}^{\mathit{ga}}\mathrm{\}}\)
\(\mathrm{\sum }{\omega }_{g}\mathrm{[}{\mathrm{[}\mathrm{\nabla }{N}_{g}^{1}\mathrm{]}}^{T}\mathrm{:}{\sigma }_{g}+{\mathrm{[}{N}_{g}^{1}\mathrm{]}}^{T}\mathrm{\cdot }\mathrm{[}\mathrm{[}{L}_{g}\mathrm{]}\mathrm{\{}\Lambda \mathrm{\}}+r\mathrm{[}{N}_{g}^{1}\mathrm{]}\mathrm{\{}{G}_{c}\mathrm{\}}\mathrm{-}r\delta (\mathrm{\{}{G}_{c}\mathrm{\}},\mathrm{\{}\Lambda \mathrm{\}})\mathrm{]}\mathrm{]}\mathrm{=}\mathrm{\{}{F}_{\mathit{ext}}^{c}\mathrm{\}}\)
\(\mathrm{\sum }{\omega }_{g}\mathrm{[}{\mathrm{[}{L}_{g}\mathrm{]}}^{T}\mathrm{\cdot }\mathrm{[}\mathrm{[}{N}_{g}^{1}\mathrm{]}\mathrm{\{}{G}_{c}\mathrm{\}}\mathrm{-}\delta (\mathrm{\{}{G}_{c}\mathrm{\}},\mathrm{\{}\Lambda \mathrm{\}})\mathrm{]}\mathrm{]}\mathrm{=}\mathrm{\{}0\mathrm{\}}\)
5.3. Internal forces#
The notations are introduced:
\(({f}_{u}^{g})\mathrm{=}({B}_{g}^{T}\mathrm{:}{\sigma }_{g})\)
\(({f}_{{g}_{c}}^{g})\mathrm{=}{\mathrm{[}\mathrm{\nabla }{N}_{g}^{1}\mathrm{]}}^{T}\mathrm{:}{\sigma }_{g}+{\mathrm{[}{N}_{g}^{1}\mathrm{]}}^{T}\mathrm{\cdot }\mathrm{[}\mathrm{[}{L}_{g}\mathrm{]}\mathrm{\{}\Lambda \mathrm{\}}+r\mathrm{[}{N}_{g}^{1}\mathrm{]}\mathrm{\{}{G}_{c}\mathrm{\}}\mathrm{-}r\delta (\mathrm{\{}{G}_{c}\mathrm{\}},\mathrm{\{}\Lambda \mathrm{\}})\mathrm{]}\)
\(({f}_{\lambda }^{g})\mathrm{=}{\mathrm{[}{L}_{g}\mathrm{]}}^{T}\mathrm{\cdot }\mathrm{[}\mathrm{[}{N}_{g}^{1}\mathrm{]}\mathrm{\{}{G}_{c}\mathrm{\}}\mathrm{-}\delta (\mathrm{\{}{G}_{c}\mathrm{\}},\mathrm{\{}\Lambda \mathrm{\}})\mathrm{]}\)
The contributions to the forces, for \(i\mathrm{\in }\mathrm{\{}\mathrm{1,}\mathrm{2,}3\mathrm{\}}\), \(m\mathrm{\in }\mathrm{\{}\mathrm{1,}\mathrm{2,}3\mathrm{\}}\), and \(p\mathrm{\in }\mathrm{\{}\mathrm{1,}2\mathrm{\}}\), are then written:
\({({f}_{u})}_{i,m}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}{({f}_{{u}_{\mathit{gaine}}}^{g})}_{i,m}\)
\({({f}_{{g}_{c}})}_{m}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}{({f}_{{u}_{\mathit{cable}}}^{g})}_{m}\)
\({({f}_{\lambda })}_{p}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}{({f}_{\lambda }^{g})}_{p}\)
where
\({({f}_{u}^{g})}_{i,m}\) is the \((3(m\mathrm{-}1)+i)\) component of \(({f}_{u}^{g})\),
\({({f}_{{g}_{c}}^{g})}_{m}\) is the \(m\) component of \(({f}_{{g}_{c}}^{g})\) and
\({({f}_{\lambda }^{g})}_{p}\) is the \(p\) component of \(({f}_{\lambda }^{g})\)
5.4. Stiffness matrix#
Recall the Gauss one-point notations:
the relative displacement of the cable: \({g}_{g}\mathrm{=}{N}_{g}^{1}\mathrm{\{}{G}_{c}\mathrm{\}}\)
cable deformation: \({\varepsilon }_{g}\mathrm{=}{B}_{g}\mathrm{\{}{U}_{\mathit{ga}}\mathrm{\}}+(\mathrm{\nabla }{N}_{g}^{1})\mathrm{\{}{G}_{c}\mathrm{\}}\)
the Lagrange multiplier: \({\lambda }_{g}\mathrm{=}({L}_{g})\mathrm{\{}\Lambda \mathrm{\}}\)
Assuming \({\tau }_{g}\mathrm{=}{\lambda }_{g}+{\mathit{rg}}_{g}\), the following two paragraphs give the contributions to the tangent matrix for the adherent and slippery cases and for the rubbing case.
5.4.1. Adherent and slippery case#
In adherent and slippery cases, the tangent matrix is symmetric (minimization of a saddle point):
\({K}_{uu}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}{B}_{g}^{T}\frac{d\sigma }{d\varepsilon }{B}_{g}\)
\({K}_{u{g}_{c}}\mathrm{=}{({K}_{{g}_{c}u})}^{T}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}{B}_{g}^{T}\frac{d\sigma }{d\varepsilon }(\mathrm{\nabla }{N}_{g}^{1})\)
\({K}_{u\lambda }\mathrm{=}{({K}_{\lambda u})}^{T}\mathrm{=}0\)
\({K}_{{g}_{c}{g}_{c}}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}{(\mathrm{\nabla }{N}_{g}^{1})}^{T}\frac{d\sigma }{d\varepsilon }(\mathrm{\nabla }{N}_{g}^{1})+{({N}_{g}^{1})}^{T}r({N}_{g}^{1})\mathrm{-}{({N}_{g}^{1})}^{T}\mathit{r²}\frac{d{\delta }_{g}}{d{\tau }_{g}}({N}_{g}^{1})\mathrm{]}\)
\({K}_{{g}_{c}\lambda }\mathrm{=}{({K}_{\lambda {g}_{c}})}^{T}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}{({N}_{g}^{1})}^{T}({L}_{g})+{({N}_{g}^{1})}^{T}r\frac{d{\delta }_{g}}{d{\tau }_{g}}({L}_{g})\mathrm{]}\)
\({K}_{\lambda \lambda }\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}\mathrm{-}{({L}_{g})}^{T}\frac{d{\delta }_{g}}{d{\tau }_{g}}({L}_{g})\mathrm{]}\)
It is specified that \(\frac{d{\delta }_{g}}{d{\tau }_{g}}\) is obtained from the law of friction behavior.
5.4.2. Rubbing case#
In the case of friction, there is a dependence of the slip on the tension in the cable. The following expressions are changed as a result:
\({K}_{{g}_{c}u}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}{(\mathrm{\nabla }{N}_{g}^{1})}^{T}\frac{d\sigma }{d\varepsilon }{B}_{g}\mathrm{-}r{({N}_{g}^{1})}^{T}\frac{\mathrm{\partial }{\delta }_{g}}{\mathrm{\partial }N}A\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }\varepsilon }{B}_{g}\mathrm{]}\)
\({K}_{\lambda u}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}\mathrm{-}{({L}_{g})}^{T}\frac{\mathrm{\partial }{\delta }_{g}}{\mathrm{\partial }N}A\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }\varepsilon }{B}_{g}\mathrm{]}\)
\({K}_{{g}_{c}{g}_{c}}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}{(\mathrm{\nabla }{N}_{g}^{1})}^{T}\frac{d\sigma }{d\varepsilon }(\mathrm{\nabla }{N}_{g}^{1})+{({N}_{g}^{1})}^{T}r({N}_{g}^{1})\mathrm{-}{({N}_{g}^{1})}^{T}\mathit{r²}\frac{d{\delta }_{g}}{d{\tau }_{g}}({N}_{g}^{1})\mathrm{-}{({N}_{g}^{1})}^{T}r\frac{\mathrm{\partial }{\delta }_{g}}{\mathrm{\partial }N}A\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }\varepsilon }(\mathrm{\nabla }{N}_{g}^{1})\mathrm{]}\)
\({K}_{\lambda {g}_{c}}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}{({L}_{g})}^{T}({N}_{g}^{1})+{({L}_{g})}^{T}r\frac{d{\delta }_{g}}{d{\tau }_{g}}({N}_{g}^{1})\mathrm{-}{({L}_{g})}^{T}\frac{\mathrm{\partial }{\delta }_{g}}{\mathrm{\partial }N}A\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }\varepsilon }(\mathrm{\nabla }{N}_{g}^{1})\mathrm{]}\)
(The other expressions are unchanged from 5.4.1)
It can be seen that the matrix is no longer symmetric.