Appendix C: tangent matrix =========================== The expressions considered always relate only to terms related to touch-friction. It is therefore a question of discretizing linearized expressions. The linearization process reveals matrix terms that we are going to discretize here. Preliminary matrix quantities ------------------------------------ We start by describing the discretized form of a certain number of quantities. Discretizing management :math:`{\stackrel{ˆ}{\tau }}_{k}` will give us: .. math:: : label: eq-388 {\ stackrel {} {\ tau}}} _ {k}\ mathrm {=}\ frac {{\ stackrel {} {h}} _ {\ tau, k}} {\ mathrm {\ parallel}} {\ parallel} {\ parallel} {\ parallel} {\ parallel} {\ parallel}} {\ stackrel {\ parallel}} Discretization of the projection operator on the unit ball: .. math:: :label: eq-389 {\ underline {\ underline {\ stackrel {} {} {P}}}}}} _ {k} ^ {B (\ mathrm {0.1})}\ stackrel {\ text {Discretization}}} {\ to}}\ left [{\ stackrel {} {P}}} _ {k} ^ {B}\ right] The process of linearizing the quantity relative to the unit ball (§ :ref:`3.2.4 `, see ()) gives us: .. math:: : label: eq-390 {\ underline {\ underline {\ stackrel {} {} {P}}}}}} _ {k} ^ {B (\ mathrm {0.1})}\ mathrm {\ frac {1} {\ mathrm {\ parallel} {\ mathrm {\ parallel}} {\ parallel}} {\ mathrm {\ parallel}} {\ parallel}} {\ mathrm {\ parallel}} {\ mathrm {\ parallel}} {\ parallel}} {\ mathrm {\ parallel}} {\ mathrm {\ parallel}} {\ parallel}} {\ mathrm {\ parallel}} {\ parallel}} {\ mathrm {\ parallel}} {\ parallel}} {\ mathrm {\ parallel}} {\ parallel}} {\ underline {1}}\ mathrm {-}\ frac {\ underline {\ underline {\ underline {{\ stackrel {} {h}}} _ {\ tau, k}\ mathrm {\ otimes} {\ times} {\ times} {\ times} {\ times} {\ times} {\ stackrel {} {h}} {h}} _ {\ tau, k}}} {\ mathrm {\ parallel} {\ stackrel {\ otimes} {\ times} {\ stackrel {} {h}} {h}} _ {\ tau, k}}} {\ mathrm {\ parallel} {\ stackrel {\ otimes} {\ times} {\ stackrel {} {h}} {h}}}} _ {\ tau, k}\ mathrm {\ parallel}} ^ {2}}) In passing, we will note the identity: .. math:: : label: eq-391 {\ underline {\ underline {\ stackrel {} {} {P}}}}}} _ {k} ^ {B (\ mathrm {0.1})}\ mathrm {\ frac {1} {\ mathrm {\ parallel} {\ mathrm {\ parallel}} {\ parallel}} {\ mathrm {\ parallel}} {\ parallel}} {\ mathrm {\ parallel}} {\ mathrm {\ parallel}} {\ parallel}} {\ mathrm {\ parallel}} {\ mathrm {\ parallel}} {\ parallel}} {\ mathrm {\ parallel}} {\ parallel}} {\ mathrm {\ parallel}} {\ parallel}} {\ mathrm {\ parallel}} {\ parallel}} {\ underline {{1} _ {}}}\ mathrm {-}\ mathrm {-}\ underline {\ underline {\ stackrel {}} {\ tau}} _ {k}\ mathrm {\ otimes} {\ otimes} {\ times} {\ times} {\ times}} {\ times} {\ times} {\ times} {\ times} {\ times} {\ times} {\ times} {\ times} {\ times} {\ times} {\ stackrel {\ times} {\ times} {\ times} {\ times} {\ times} {\ times Discretization uses the definition (), with () we can therefore write: .. math:: : label: eq-392 {\ underline {\ underline {\ stackrel {} {} {P}}}}}} _ {k} ^ {B (\ mathrm {0.1})}\ stackrel {\ text {discretization}} {\ to}}\ left [{\ to}\\ left [{\ stackrel {} {P}}}}} _ {k} ^ {B}\ right]\ mathrm {=}\ frac {1}\ right]\ mathrm {=}\ frac {1}\ mathrm {\ parallel} {\ stackrel {} {} {h}}} _ {\ tau, k}\ mathrm {\ parallel}} (\ left [1\ right]\ mathrm {-}\ left\ {{\ stackrel {} {} {\ stackrel {} {\ tau}} {\ tau}} {\ tau}} _ {\ tau}}} _ {\ tau}} _ {\ tau}}} _ {\ tau}}} _ {\ tau}}} _ {\ tau}}}\ rangle) Quantities for balance -------------------------- We start by considering the matrix terms resulting from the linearization of the equilibrium equation. The discretization of Jacobian women () and (): .. math:: : label: eq-393 {J} _ {uu} ^ {\ text {c}\ mathrm {,1}}}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {ee}}} {\ mathit {ee}}}} ^ {\ mathit {ee}}}} ^ {\ mathit {ee}}}} ^ {\ text {e}}} ^ {\ text {c}} ^ {\ text {c}} ^ {\ text {c}} ^ {\ text {c}}\ right],\ left [{K}} _ {\ mathit {em}} _ {\ mathit {ee}}} _ {\ mathit {ee}}}} {\ mathit {ee}}}} {\ mathit {ee}}}}\ mathrm {,1}}\ right],\ left [{K} _ {\ mathit {me}}} ^ {\ text {c}\ mathrm {,1}}\ right],\ left [{K} _ {\ K} _ {\ mathit {mm}} _ {\ mathit {mm}}}} ^ {\ text {c}}\ right] {J} _ {uu} ^ {\ text {c}\ mathrm {,2}}}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {ee}}} {\ mathit {ee}}}}} ^ {\ mathit {ee}}}} ^ {\ mathit {ee}}}} ^ {\ text {e}}} ^ {\ text {c}} ^ {\ text {c}} ^ {\ text {c}} ^ {\ text {c}}\ mathrm {,2}}\ right],\ left [{K} _ {\ mathit {me}}} ^ {\ text {c}\ mathrm {,2}}\ right],\ left [{K} _ {\ K} _ {\ mathit {mm}} _ {\ mathit {mm}}}} ^ {\ text {c}}\ right] We start with (): .. math:: : label: eq-394 {J} _ {uu} ^ {\ text {c}\ mathrm {c}\ mathrm {2}}}\ mathrm {=} {\ mathrm {\ int}} _ {\ gamma} _ {c}} {\ rho}} {\ rho}} _ {\ rho}} _ {rho}} _ {rho}} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} delta} _ {t} {\ stackrel {} {u}}} _ {k} {u}}} _ {k} ^ {e}\ mathrm {-} {\ stackrel {} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} _ {k} {u}}} {\ stackrel {} {n}} _ {k}\ mathrm {\ cdot} ({\ Delta} _ {t} {u} ^ {e}\ mathrm {-} {\ Delta} _ {\ Delta} _ {\ Delta} _ {Delta} _ {Delta} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} So: .. math:: : label: eq-395 \ left [{K} _ {\ mathit {uu}}} ^ {\ text {uu}}} ^ {\ mathit {uu}}}} ^ {\ omega} _ {c} {J} _ {J} _ {c} {J} _ {c} {\ rho}} _ {n} _ {n}} _ {n}} _ {u, k} ^ {{g} {J}} _ {n}} _ {n}} (\ mathrm {\ langle} {\ delta} _ {t} {u} _ {u} _ {h} _ {h}} ^ {u}\ right]} ^ {e}\ right]} ^ {T}\ mathrm {-}\ mathrm {-}\ mathrm {-} _ {\ langle} {\ langle} {\ delta} _ {h} {e}\ right]} ^ {m}\ mathrm {-}\ mathrm {-}\ mathrm {-}\ mathrm {-}\ mathrm {-}\ mathrm {-}\ mathrm {-}\ mathrm {-}\ mathrm {-}\ mathrm {-}\ mathrm {-}\ mathrm {-}\ mathrm\ rangle} {\ left [{\ stackrel {} {N}}} ^ {m}\ right]} ^ {T})\ mathrm {\ {} {n} _ {h}\ mathrm {\}}\ mathrm {\}}}\ mathrm {\ langle} {N}} {\}}\ mathrm {\ langle} {n}}}\ mathrm {\ langle} {n}}}\ mathrm {\ langle} {n}}}\ mathrm {\ langle} {n}}}\ mathrm {\ langle} {n}}\ right\ mathrm {\ {} {\ Delta u} _ {h} _ {h} ^ {e}}\ mathrm {\}}\ mathrm {-}\ left [{\ stackrel {} {N}}}}} ^ {m} {N}}}}} ^ {N}}}} ^ {N}}}} ^ {N}}}} ^ {N}}}} ^ {N}}}} ^ {N}}} ^ {N}}} ^ {N}}} ^ {N}}} ^ {N}}} ^ {N}}} ^ {N}}} ^ {N}}} ^ {N}}} ^ {N}}} ^ {N}}} ^ {N}}} We use :math:`\left[{P}^{n}\right]\mathrm{=}\mathrm{\{}{n}_{h}\mathrm{\}}\mathrm{\langle }{n}_{h}\mathrm{\rangle }`. Finally: .. math:: : label: eq-396 \ left [{K} _ {\ mathit {ee}}}} ^ {\ text {ee}} ^ {\ mathit {ee}}}} ^ {\ omega} _ {c} {J} _ {J} _ {c} {J} _ {c} {\ rho} _ {c} {\ rho}} _ {n}} _ {n}} _ {\ stackrel {2}}} {\ stackrel {2}}} {\ stackrel {2}}}\ right]\ mathrm {=} + {\ omega} _ {c} {J}} _ {c} {J} _ {J} _ {n} _ {n}} _ {n}} {\ rho} _ {n}} _ {n}} {\ rho} _ {n} left [{N} ^ {e}\ right]} ^ {T}}\ left [{P} ^ {n}\ right]\ left [{N} ^ {e} ^ {e}\ right] \ left [{K} _ {\ mathit {mm}}} ^ {\ text {mm}} ^ {\ mathit {mm}}} ^ {\ omega} _ {c} {J} _ {J} _ {c} {J} _ {c} {\ rho} _ {c} {\ rho}} _ {n} _ {n}} _ {n}} _ {\ stackrel {2}}} {\ stackrel {2}}} {\ stackrel {2}}}\ right]\ mathrm {=} + {\ omega} _ {c} {J}} _ {c} {J} _ {J} _ {n} _ {n}} _ {n}} {\ rho} _ {n}} _ {\ rho} _ {n}} {left [{\ stackrel {} {N}}} ^ {m}\ right]} ^ {m}\ right]} ^ {T}\ left [{\ stackrel {} {N}} {N}}} ^ {m}\ right]\ left [{\ stackrel {} {} {N}} {N}}\ right] \ left [{K} _ {\ mathit {em}}} ^ {\ text {em}} ^ {\ text {c}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}}}} _ {c} {J}} _ {c} {J}} _ {c} {J}} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J}} _ {c} {J}} _ {c} {J}} _ {\ g} _ {g} _ {n}} {\ left [{N} ^ {e}\ right]}} ^ {T}\ left [{P} ^ {n}\ right]\ left [{\ stackrel {} {} {N} {N}}}} ^ {m}\ right] \ left [{K} _ {\ mathit {me}}} ^ {\ text {me}} ^ {\ text {c}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}}}} _ {c} {J} _ {J} _ {c} {J}} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J}} _ {c} {J}} _ {c} {J}} _ {n}} {\ left [{\ stackrel {} {} {N}}} ^ {m}\ right]} ^ {T}\ left [{P} ^ {n}\ right]\ right]\ left [{N} {n} {n} {right] Starting at (): .. math:: :label: eq-397 \ begin {array} {cc} {J} _ {uu} _ {uu} ^ {\ text {c} ^ {\ array} {cc} {\ mathrm {\ int}} _ {{\ Gamma} _ {\ Gamma} _ {c}} _ {c}}} {c}}} {\ stackrel {c}}} {c}}} {\ stackrel {\}} {c}}} {\ stackrel {\}} {c}}} {\ stackrel {}} {S}}} {\ stackrel {} {g}} _ {n, k} {\ stackrel {} {u} {n}}} _ {k}\ mathrm {\ cdot}\ left\ {({\ delta} _ {t} {t} {\ stackrel {} {u}} {u}} _ {k} {u}} _ {k} {e}\ mathrm {-} {\ delta} _ {t} {t} {t} {\ stackrel {} {u}}} _ {k} ^ {m})\ mathrm {\ cdot}\ frac {\ mathrm {\ partial}\ overline {x}} {\ mathrm {\ partial} {\ zeta} {\ zeta} ^ {\ beta}} ^ {\ beta}} {\ beta}} {m}} {m} {m}} {m} _ {m} _ {m} _ {m} _ {m} _ {m} _ {m} _ {m} _ {m} _ {m} _ {m} _ {m} _ {\ beta}} {m} _ {m} _ {m} _ {m} _ {\ beta}} {m} _ {m} _ {m} _ {\ beta}} {m} _ {m} _ {m} _ {m})} {\ mathrm {\ partial} {\ zeta} {\ zeta} _ {\ alpha}}\ right\} d {\ Gamma} _ {\ partial}} _ {{\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {c} _ {c}}} {c}}} {\ stackrel {} {S}}} {\ stackrel {} {S}}} _ {{g} _ {n}}} {\ gamma} _ {n}} {n}} {\ gamma} _ {n}} {c}}} {\ stackrel {} {S}}} {\ stackrel {} {S}}} {\ stackrel {} {g}} _ {n, k} {\ stackrel {\ k} {n}}} _ {k}\ mathrm {\ cdot}\ left\ {\ frac {\ mathrm {\ partial}} ({\ mathrm {\ partial}} {\ partial} {\ partial}} {\ mathrm {\ zeta} {\ partial}} _ {\ partial} (\ partial}) ({\ delta}} _ {\ alpha}} m} _ {\ alpha\ beta}\ frac {\ mathrm {\ partial}\ overline {\ partial}} {\ mathrm {\ partial} {\ zeta} ^ {\ beta}}\ mathrm {\ cdot}} ({\ cdot} ({\ dot}} ({\ delta}} ({\ delta}} ({\ dot}) ({\ delta}} ({\ dot}) ({\ delta}} ({\ delta}} ({\ delta}) ({\ delta}} ({\ delta}) ({\ delta}) ({\ delta}) ({\ delta}} ({\ delta}) ({\ delta}) ({\ delta}) ({\ delta}) ({\ delta}) ({\ delta}) ({\ delta}))\ right\} d {\ Gamma} _ {c} _ {c}\\\ & {\ mathrm {\ int}} _ {\ gamma} _ {c}} {\ stackrel {} {S}}} _ {u, k} _ {k}} ^ {u, k}} _ {u, k}} _ {u, k}} _ {u, k} _ {u, k}} _ {u, k} _ {u, k}} _ {u, k}} _ {u, k} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {stackrel {} {u}} _ {k} ^ {m} ^ {e}\ mathrm {-} {\ delta} _ {\ stackrel {} {u}}} _ {k} ^ {m})\ mathrm {\ cdot}\ mathrm {\ cdot}\ frac {\ mathrm {\ partial}}\ overline {x}} { \ mathrm {\ partial} {\ zeta} ^ {\ beta}} {\ beta}} {m} {m} _ {\ alpha\ beta} {m} _ {\ alpha\ beta}\ frac {\ alpha\ beta}\ frac {\ mathrm {\ beta}\ frac {\ mathrm {\ partial}}\ mathrm {\ partial}} {\ zeta} _ {\ alpha\ beta}}\ frac {\ mathrm {\ partial}}\ mathrm {\ partial}} {\ cdot} ({\ Delta} _ {t} {u} {u} ^ {e}\ mathrm {-} {\ Delta} _ {t} {u} ^ {m}) d {\ Gamma} {m}) d {\ Gamma} _ {u} _ {c}\ end {array} Let's consider the first integral, by discretizing, we get: .. math:: : label: eq-398 {J} _ {uu} ^ {\ text {c}}\ mathrm {,1} a}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {em}}} {\ mathit {em}}}}} ^ {\ text {em}}}} ^ {c}} ^ {c}} ^ {c}}\ mathrm {,1} a}\ right],\ left [{K} _ {\ mathit {mm}}} {\ mathit {em}}}} ^ {\ text {m}}} ^ {c}} ^ {c}}\ mathrm {c}}\ mathrm {,1} a}\ right],\ left [{K} _ {\ mathit {mm}}} {\ c}\ mathrm {,1} a}\ right] With the expressions: .. math:: : label: eq-399 \ left [{K} _ {\ mathit {em}}} ^ {\ text {em}} ^ {\ text {c}}\ mathrm {1} a}\ right]\ mathrm {=} + {\ omega} _ {c} {J}} _ {c} _ {c} {c} _ {J}} _ {c} {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {j} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J}}} _ {n, k} {\ left [{N}} ^ {e}\ right]} ^ {T}\ left [G\ right]\ left [{\ stackrel {} {N} {N}}} ^ {m}\ right] \ left [{K} _ {\ mathit {mm}}} ^ {\ text {mm}} ^ {\ text {c}}\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {\ omega} _ {c} {J}} _ {c} _ {c} _ {\ stackrel {} {S}}} _ {u, k} ^ {{g} _ {\ omega}} _ {\ omega} _ {\ omega} _ {\ omega} _ {n}} _ {n}} {n}} {J} _ {J}} _ {c} _ {\ stackrel} {S}}} _ {u, k} ^ {{g} _ {\ omega}} _ {\ omega}} _ {\ omega} _ {\ omega} rel {} {g}} _ {n, k} {\ left [{\ stackrel {} {N}}} ^ {m}\ right]} ^ {T}\ left [G\ right]\ left\ left [{\ stackrel {\ stackrel {} {N}}} ^ {m}\ right] With the :math:`\left[G\right]` matrix such as: .. math:: : label: eq-400 \ begin {array} {c}\ left [G\ right]\ mathrm {=} {m} _ {11}\ left [{\ stackrel {} {B}}} _ {1} ^ {m}\ right]\ mathrm {\ m}\ right]\ mathrm {\}\ right]\ mathrm {\ m}}\ mathrm {\ langle}\ right]\ mathrm {\ langle}\ right]\ mathrm {\ {m}\ right]\ mathrm {\ langle} {n}\ right]\ mathrm {\ {m}\ right]\ mathrm {\ langle} {n}\ right]\ mathrm {\ {m}\ right]\ mathrm {\ langle} {n}\ right] {h}\ mathrm {\ rangle}\ left [{\ stackrel {} {B}}} _ {1} ^ {m}\ right] + {m} _ {21}\ left [{\ stackrel {} {} {B} {B}}}} _ {2} ^ {m}\ right]\ mathrm {\ {} {x} _ {h} ^ {m} {m} {m} {m}\ right]\ mathrm {\}}\ mathrm {\ langle} {n} {n} _ {h}\ mathrm {\ rangle}\ left [{\ stackrel {} {B}}} _ {1} ^ {m}\ right]\\ + {m}\ right]\ + {m} _ {m}\ right] _ {1} ^ {m}\ right] _ {12}\ left [{\ stackrel {} {B}}} _ {1} ^ {m}\ right}\ right]\ mathrm {\ {} {x} _ {h} ^ {m}\ mathrm {\}}\ mathrm {\ langle} {n} _ {h}\ mathrm {\ rangle}\ left [{\ stackrel {} {m} {M} {B}}}\ left [{\ stackrel {} {B}}}\ left [{\ stackrel {} {B}} {B}} _ {22}\ left [{\ stackrel {} {B} {B}}}} _ {2} ^ {m}\ right]\ mathrm {\ {} {x} _ {h}} ^ {m} ^ {m}\ mathrm {\}}\ mathrm {\ langle} {n} _ {h}\ mathrm {\ rangle}\\ left [{\ stackrel {} ^ {array}\ left [{\ stackrel {}} {m}\ left [{\ stackrel {} {m}}\ left [{\ stackrel {} {b}}} _ {2} ^ {m}\ right]\ end {array}\ left [{\ stackrel {} {m}}\ left [{\ stackrel {} {m}}\ Let us consider the second integral, by discretizing, we get: .. math:: : label: eq-401 {J} _ {uu} ^ {\ text {c}\ mathrm {,2} a}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {me}}} {\ mathit {me}}}} ^ {\ text {me}}}} ^ {c}} ^ {c}} ^ {c}}\ mathrm {2} a}\ right],\ left [{K} _ {\ mathit {mm}}} {\ mathit {me}}}} ^ {\ text {c}} ^ {c}}\ mathrm {c}}\ mathrm {,2} a}\ right],\ left [{K} _ {\ mathit {mm}} _ {\ mathit {me}}}} c}\ mathrm {,2} a}\ right] With the expressions: .. math:: : label: eq-402 \ left [{K} _ {\ mathit {me}}} ^ {\ text {me}} ^ {\ text {c}}\ mathrm {2} a}\ right]\ mathrm {=} + {\ omega} _ {c} {J}} _ {c} _ {c} {c} _ {J}} _ {c} {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {j} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J}}} _ {n, k} {\ left [{\ stackrel {} {N}}} ^ {m}\ right]} ^ {T}\ left [H\ right]\ left [{H\ right]\ left [{N}\ right] \ left [{K} _ {\ mathit {mm}}} ^ {\ text {mm}} ^ {\ text {c}}\ mathrm {2} a}\ mathrm {=}\ mathrm {-} {\ omega} _ {\ omega} _ {c} _ {c} _ {\ stackrel {} {S}}} _ {u, k} ^ {{g} _ {\ omega}} _ {\ omega} _ {\ omega} _ {\ omega} _ {c} {J}} _ {\ stackrel} _ {n}} {n}} {n}} {n}} {\ stackrel} _ {\ stackrel} {S}}} _ {u, k} ^ {{g} _ {\ omega}} _ {\ omega}} rel {} {g}} _ {n, k} {\ left [{\ stackrel {} {N}}} ^ {m}\ right]} ^ {T}\ left [H\ right]\ left\ left [{\ stackrel {\ stackrel {} {N}}} ^ {m}\ right] With the :math:`\left[H\right]` matrix such as: .. math:: :label: eq-403 \ left [H\ right]\ mathrm {=} {m} {m} _ {m} _ {11} _ {11} {11} {1} ^ {m}\ right]} ^ {T}\ mathrm {\ {T}\ mathrm {\ {m}}\ mathrm {\} {n} _ {h} {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {m}}\ mathrm {\ langle} {x} _ {h} ^ {m} {m}}\ mathrm {\ rangle} {\ left [{\ stackrel {} {B}}} _ {1} ^ {m}\ right]} ^ {T} + {m} _ {21} {\ left [{\ stackrel {} {n} {n} {B}}} _ {2} ^ {m}\ right]} ^ {T}\ mathrm {\ {} {n} {n} _ {n} _ {h}\ mathrm {\}}\ mathrm {\ langle} {x} {x} _ {h} ^ {m}\ mathrm {\ rangle} {\ left [{\ stackrel {} {B} {B}}}} _ {B}}} _ {B}}} _ {12} {\ left [{\ stackrel {} {B}} {B}}} _ {1} ^ {m}\ right]} ^ {T}} ^ {T}\ mathrm {\ {} {n} _ {h}\ mathrm {\ langle} {x} _ {h} _ {h} _ {h} ^ {m} ^ {m} ^ {m}}\ m} ^ {m}}\ right}\ mathrm {\ rangle} {\ left [{\ stackrel {} {B}}}} _ {2} ^ {m}\ right]} ^ {T} + {m} _ {22} {\ left [{\ stackrel {} {B}}} _ {2} ^ {m}\ right]} ^ {T}\ mathrm {\ {} {t}\ mathrm {\}}\ mathrm {\ langle} {x} _ {h} _ {h} ^ {m} ^ {m}\ m}\ mathrm {\ rangle} {\ m} {m}} ^ {m}\ right} {m} {m} {m} {m} {m} {m} {m} {m} {m}} {m} {m} {m} {m} {m} {m} {m} {m} {m}} {m} {m} {m}} {m} {m} {m}} {m} {m} {m}} {m} {m} {m} {m} {m} {^ {T} Finally, the last contribution: .. math:: : label: eq-404 {J} _ {uu} ^ {\ text {c}\ mathrm {0,3} a}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {me}}} {\ mathit {me}}}} ^ {\ text {me}}}} ^ {c}} ^ {c}} ^ {c}}\ mathrm {3} a}\ right],\ left [{K} _ {\ mathit {mm}}} {\ mathit {me}}}} ^ {\ text {c}} ^ {c}}\ mathrm {c}}\ mathrm {3} a}\ right],\ left [{K} _ {\ mathit {mm}}} {\ mathit {me}}}} c}\ mathrm {0,3} a}\ right],\ left [{K}],\ left [{K} _ {\ mathit {ee}} ^ {\ text {c}\ mathrm {0,3} a}\ right],\ left [{K}} _ {\ K} _ {\ mathit {em}} _ {\ mathit {em}}} ^ {\ text {c}\ mathrm {0,3} a}\ right],\ left [{K} _ _ {\ mathit {em}}}} ^ {\ text {c}\ mathrm {0,3} a}\ right],\ left [{K}} _ {\ K} _ {\ mathit {em}}} With the expressions: .. math:: : label: eq-405 \ left [{K} _ {\ mathit {ee}}} ^ {\ text {ee}} ^ {\ text {c}}\ mathrm {3} a}\ right]\ mathrm {=} + {\ omega} _ {c} {J}} _ {c} _ {c} {c} _ {J}} _ {c} {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J}}} _ {n, k} {\ left [{N} ^ {e}\ right]} ^ {T}\ left [L\ right]\ left [{N} ^ {e}\ right] \ left [{K} _ {\ mathit {mm}}} ^ {\ text {mm}} ^ {\ text {c}}\ mathrm {3} a}\ right]\ mathrm {=} + {\ omega} _ {c} {J}} _ {c} _ {c} {c} _ {J}} _ {c} {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {J} _ {j} _ {J} _ {J} _ {J} _ {J}}} _ {n, k} {\ left [{\ stackrel {} {N}}} ^ {m}\ right]} ^ {T}\ left [L\ right]\ left [{\ stackrel {} {} {N}}} ^ {m}\ right] \ left [{K} _ {\ mathit {em}}}} ^ {\ text {em}} ^ {\ mathit {em}}}} ^ {\ omega} _ {\ omega} _ {c} {J}} _ {J} _ {c} _ {c} _ {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c}} {\ stackrel {rel {} {g}} _ {n, k} {\ left [{N} ^ {e}\ right]} ^ {T}\ left [L\ right]\ left [{\ stackrel {} {} {N}} {N}}} ^ {m}\ right] \ left [{K} _ {\ mathit {me}}} ^ {\ text {me}} ^ {\ text {c}}\ mathrm {2} a}\ mathrm {=}\ mathrm {-} {\ omega} _ {\ omega} _ {c} _ {c} _ {\ stackrel {} {S}}} _ {u, k} ^ {{g} _ {\ omega}} _ {\ omega} _ {\ omega} _ {\ omega} _ {c} {J}} _ {\ stackrel} _ {n}} {n}} {n}} {n}} {\ stackrel} _ {\ stackrel} {S}}} {\ stackrel {rel {} {g}} _ {n, k} {\ left} {\ left [{\ stackrel {} {N}}} ^ {m}\ right]} ^ {T}\ left [L\ right]\ left [L\ right]\ left [{N} ^ {e}\ right] With the :math:`\left[L\right]` matrix such as: .. math:: :label: eq-406 \ left [H\ right]\ mathrm {=} {m} _ {m} _ {11} {\ kappa} _ {11} {m} _ {11} {\ left [{\ stackrel {} {B} {B}}}} _ {1} {B} {B}}}} _ {1} {B}}}} _ {1} ^ {m}\ right] + {m} _ {21} {\ kappa} _ {21} {21} {m} {m} _ {21} {\ left [{\ stackrel {} {B}}} _ {2} ^ {m}\ right]}}} ^ {t}} ^ {T}}\ mathrm {\}}} ^ {right]}} ^ {t}} ^ {T}}\ mathrm {\} _ {h} ^ {m}\ mathrm {\ rangle}\ left [{\ stackrel {} {B}}} _ {1} ^ {m}\ right] + {m} _ {12} {\ kappa} _ {12} {m}} _ {12} _ {12} {12} {\ left [{\ stackrel {}} {B}}} _ {1} ^ {m}\ right]} ^ {m}\ right]} ^ {T}\ mathrm {\ {} {n} _ {h}\ mathrm {\}}\ mathrm {\ langle} {x} _ {h} ^ {m}\ mathrm {\ rangle}\ left [{\ stackrel {}}\ left [{\ stackrel {} {B} {B}}}} _ {2} ^ {m}\ right] + {m} _ {22} {\ kappa}}\ left [{\ stackrel {} {B} {B}}}} _ {22} {\ kappa} _ {22} {\ kappa} _ {22}\ left [{\ stackrel {} {B} {B}}}} {m} _ {22} {\ left [{\ stackrel {} {B}}} _ {2} ^ {m}\ right]} ^ {T}\ mathrm {\ {} {n} {n} _ {h}\ mathrm {\}}\ mathrm {\ langle} {x} _ {h} ^ {m}\ mathrm {\ rangle}}\ left [{\ stackrel {\ rangle}\ left [{\ stackrel {}} {B}}} _ {2} ^ {m} ^ {m} {m}\ m}\ m}\ m}\ right] We now consider the discretization of Jacobian women () and (): .. math:: : label: eq-407 {J} _ {uu} ^ {\ text {f}\ mathrm {,1}}}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {ee}}} {\ mathit {ee}}}} ^ {\ mathit {ee}}}} ^ {\ mathit {ee}}}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}}\ mathrm {,1}}\ right],\ left [{K} _ {\ mathit {me}}} ^ {\ text {f}\ mathrm {,1}}\ right],\ left [{K} _ {\ K} _ {\ mathit {mm}} _ {\ mathit {mm}}}} ^ {\ text {f}}\ right] {J} _ {uu} ^ {\ text {f}\ mathrm {,2}}}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {ee}}} {\ mathit {ee}}}} ^ {\ mathit {ee}}}} ^ {\ mathit {ee}}}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}}\ mathrm {,2}}\ right],\ left [{K} _ {\ mathit {me}}} ^ {\ text {f}\ mathrm {,2}}\ right],\ left [{K} _ {\ K} _ {\ mathit {mm}} _ {\ mathit {mm}}}} ^ {\ text {f}}\ right] The first part concerns the adherent case (:math:`{\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1` and :math:`{\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}1`): .. math:: : label: eq-408 {J} _ {uu} ^ {\ text {f}\ mathrm {f}\ mathrm {,1}}\ mathrm {=}\ mathrm {\ int}} _ {{\ Gamma} _ {c}} _ {c}}}\ mu {\ stackrel {\ c}}}\ mathrm {1}}}\ mathrm {=}} {\ mathrm {\ int}} _ {\ lambda} _ {\ lambda}} _ {k}}\ left [{\ underline {\ underline {P}}}} ^ {\ tau}}} ^ {\ tau} ({\ delta} _ {} {u}} _ {k} ^ {e}\ mathrm {-}\ mathrm {-} {\ delta}} _ {k} ^ {m})\ mathrm {-}}\ mathrm {-} {\ delta} _ {k} ^ {m})\ mathrm {-} {\ delta} _ {k} ^ {m})\ mathrm {-} {\ delta}} _ {k} ^ {m})\ mathrm {-} {\ delta}} _ {k} ^ {m})\ mathrm {-}} {\ mathrm {-} {\ delta}} _\ cdot}\ left [{\ underline {\ underline {P}}}}} ^ {\ tau} ({\ Delta} _ {t} {u} ^ {e}\ mathrm {-} {\ Delta} _ {\ Delta} _ {\ Delta} _ {Delta} _ {Delta} _ {c} _ {c} _ {c} _ {c} _ {c} The discretization of :math:`{\underline{\underline{P}}}^{\tau }` and travel gives us: .. math:: :label: eq-409 \ left [{K} _ {\ mathit {ee}}} ^ {\ text {ee}} ^ {\ text {f}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}} _ {J} _ {J} _ {c} _ {c}\ mu {\ stackrel {}} {\ rho}} _ {\ stackrel {\ omega} _ {\ omega} _ {c} {J} _ {J} _ {J} _ {c} _ {c} _ {c}} _ {\ stackrel {\ lambda}} _ {k} {\ left [{N} ^ {e}\ right]} ^ {e}\ right]} ^ {T}\ left [{P}\ right]\ left [{P} ^ {\ tau} ^ {\ tau}\ right]\ right]\ right]\ left [{N} ^ {e}\ right] \ left [{K} _ {\ mathit {mm}}} ^ {\ text {mm}} ^ {\ text {f}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}} _ {J}} _ {c} _ {c}\ mu {\ stackrel {\ rho}}} _ {\ stackrel {\ omega} _ {\ omega} _ {c} {J}} _ {c} {J} _ {c}} _ {c}} _ {\ stackrel {\ lambda}} _ {k} {\ left [{N} ^ {m}\ right]} ^ {m}\ right]} ^ {T}\ left [{P}\ right]\ left [{P} ^ {\ tau} ^ {\ tau}\ right]\ right]\ right]\ left [{N} ^ {m}\ right] \ left [{K} _ {\ mathit {em}}} ^ {\ text {em}} ^ {\ text {f}}\ right]\ mathrm {=} + {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c} _ {c}\ c}\ c} _ {k}\ mu {\ stackrel {}} {\ rho}} _ {k}\ left [{N} ^ {e}\ right]} ^ {T}}\ left [{P} ^ {\ tau}\ right]\ left [{P} ^ {\ tau}\ right]\ right]\ left [{N} ^ {m}\ right] \ left [{K} _ {\ mathit {me}}} ^ {\ text {me}} ^ {\ text {f}}\ right]\ mathrm {=} + {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c} _ {c}\ c}\ c} _ {k}\ mu {\ stackrel {}} {\ rho}} _ {k}\ left [{N} ^ {m}\ right]} ^ {T}} ^ {T}\ left [{P} ^ {\ tau}\ right]\ left [{P} ^ {\ tau}\ right]\ right]\ left [{N} ^ {e}\ right] The second part concerns the slippery case (:math:`{\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1` and :math:`{\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}0`): .. math:: : label: eq-410 {J} _ {uu} ^ {\ text {f}\ mathrm {f}\ mathrm {2}}\ mathrm {=}\ mathrm {\ int}} _ {{\ Gamma} _ {c}} _ {c}}}\ mu {\ stackrel {\ c}}}\ mathrm {2}}\ mathrm {2}} {\ mathrm {\ int}} _ {\ lambda} _ {\ lambda}} _ {k}}\ left [{\ underline {\ underline {P}}}} ^ {\ tau}}} ^ {\ tau} ({\ delta} _ {} {u}} _ {k} ^ {e}\ mathrm {-}\ mathrm {-} {\ delta}} _ {k} ^ {m})\ mathrm {-}}\ mathrm {-} {\ delta} _ {k} ^ {m})\ mathrm {-} {\ delta} _ {k} ^ {m})\ mathrm {-} {\ delta}} _ {k} ^ {m})\ mathrm {-} {\ delta}} _ {k} ^ {m})\ mathrm {-}} {\ mathrm {-} {\ delta}} _\ cdot}\ left [{\ underline {\ underline {P}}}}} ^ {\ tau} {\ underline {\ underline {\ stackrel {} {P}}}}} _ {k} ^ {B (\ mathrm {0.1}} {B (\ mathrm {0.1}}} {B (\ mathrm {0.1}})}} ^ {B (\ mathrm {0.1})}} (\ mathrm {0.1})}} (\ mathrm {0.1}})} (\ mathrm {0.1})} (\ mathrm {0.1})}} (\ mathrm {0.1})} (\ mathrm {0.1})} (\ mathrm {0.1})} (\ mathrm {0.1})}} ({u} ^ {m})\ right] d {\ Gamma} _ {c} This is written, in particular using the discretization of the projection on the unit ball (): .. math:: : label: eq-411 \ left [{K} _ {\ mathit {ee}}} ^ {\ text {ee}} ^ {\ text {f}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}} _ {J} _ {J} _ {c} _ {c}\ mu {\ stackrel {\ rho}} _ {\ rho}} _ {\ stackrel {\ omega} _ {c} {J} _ {J} _ {J} _ {c} _ {c} _ {c}} _ {\ stackrel {\ lambda}} _ {k} {\ left [{N} ^ {e}\ right]} ^ {e}\ right]} ^ {T}\ left [{\ tau}\ right]\ left [{\ stackrel {} {P}}}} _ {k} ^ {e}}\ right] _ {k} ^ {e}\ right] _ {k} ^ {e}\ right] \ left [{K} _ {\ mathit {mm}}} ^ {\ text {mm}} ^ {\ text {f}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}} _ {J}} _ {c} _ {c}\ mu {\ stackrel {\ rho}}} _ {\ stackrel {\ omega} _ {\ omega} _ {c} {J}} _ {c} {J} _ {c}} _ {c}} _ {\ stackrel {\ lambda}} _ {k} {\ left [{N} ^ {m}\ right]} ^ {m}\ right]} ^ {T}\ left [{\ tau}\ right]\ left [{\ stackrel {} {P}}}} _ {k} {m}}}}} _ {m}\ right}} _ {k} ^ {m}\ right] _ {k} ^ {m}\ right] \ left [{K} _ {\ mathit {em}}} ^ {\ text {em}} ^ {\ text {f}}\ right]\ mathrm {=} + {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c} _ {c}\ c}\ c} _ {k}\ mu {\ stackrel {}} {\ rho}} _ {k}\ left [{N} ^ {e}\ right]} ^ {T}}} ^ {T}\ left [{P} ^ {\ tau}\ right]\ left [{\ stackrel {} {P}}} _ {k} ^ {B} {B}\ right]\ right]\ left [{P} ^ {m} {B}\ right] \ left [{K} _ {\ mathit {me}}} ^ {\ text {me}} ^ {\ text {f}}\ right]\ mathrm {=} + {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c} _ {c}\ c}\ c} _ {k}\ mu {\ stackrel {}} {\ rho}} _ {k}\ left [{N} ^ {m}\ right]}} ^ {T}}\ left [{P} ^ {\ tau}\ right]\ left [{\ stackrel {} {P}}} _ {k} ^ {B} {B}\ right]\ right]\ left [{P} ^ {e}\ right]\ left [{N} ^ {e}\ right] We will now discretize the quantities concerning the second column, relating to the contact pressure. From the contact reaction and the Jacobian reaction (): .. math:: : label: eq-412 {J} _ {uc} ^ {\ text {c}}\ mathrm {c}}\ mathrm {=}\ mathrm {-} {\ mathrm {\ int}} _ {c}} {\ stackrel {}}} {\ stackrel {} {S}}} {\ stackrel {S}} {S}} {\ stackrel {S}}} _ {\ delta} _ {t}} {\ stackrel {} {S}} {\ stackrel {S}}} {\ stackrel {S}} {S}}} {\ stackrel {S}} {\ stackrel {S}}} {u}} _ {k} ^ {e}\ mathrm {-} {-} {\ delta} _ {t} {\ stackrel {} {u}} _ {k} ^ {m})\ mathrm {\ cdot} {\ cdot} {n} {n} {n} _ {k}\ Delta\ lambda d {\ gamma} _ {m})\ mathrm {\ cdot} {n} {n}} _ {c} We must discretize: .. math:: :label: eq-413 {J} _ {uc} ^ {\ text {c}}}\ stackrel {\ text {c}}}\ stackrel {\ text {c}}}\ stackrel {\ text {c}}}\ right],\ left [{K}}}\ right],\ left [{K}} _ {\ mathit {mc}}} ^ {\ text {c}}} ^ {\ text {c}}} ^ {\ text {c}}\ right] This gives us: .. math:: :label: eq-414 \ left [{K} _ {\ mathit {ec}}} ^ {\ text {c}}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c}} {c} {c}} {c} {c} _ {c} _ {e} _ {e} _ {n}} _ {n}} _ {n}} _ {n}} _ {n}} _ {n}} _ {N}} _ {N} {e}\ right]} ^ {T}\ mathrm {\ {} {n} {n} _ {h}\ mathrm {\}}\ langle\ psi\ rangle \ left [{K} _ {\ mathit {mc}}} ^ {\ text {c}}}\ right]\ mathrm {=} + {\ omega} _ {J} _ {c} {c} {\ stackrel {} {mc}}} {\ stackrel {} {mc}}} {\ stackrel {} {mc}} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {\ stackrel {}\ mathrm {\ {} {n} _ {h}\ mathrm {\}}\ langle\ psi\ rangle From the friction reaction, that is to say Jacobians () and (): .. math:: : label: eq-415 {J} _ {uc} ^ {\ text {f}\ mathrm {,1}}}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {ec}}} {\ mathit {ec}}}} ^ {\ mathit {ec}}}} ^ {\ mathit {ec}}}} ^ {\ text {f}}} ^ {\ text {f}} ^ {\ text {f}}\ ^ {\ text {f}}\ ^ {\ text {f}}\ right],\ left [{K} _ {\ mathit {mc}}} {\ mathit {ec}}}} ^ {\ mathit {ec}}}} ^ {\ text {f}}} ^ {\ text {f}}} ^ mathrm {,1}}\ right] {J} _ {uc} ^ {\ text {f}\ mathrm {,2}}}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {ec}}} {\ mathit {ec}}}} ^ {\ mathit {ec}}}} ^ {\ mathit {ec}}}} ^ {\ text {f}}} ^ {\ text {f}} ^ {\ text {f}}\ ^ {\ text {f}}\ ^ {\ text {f}}\ right],\ left [{K} _ {\ mathit {mc}}} {\ mathit {ec}}}} ^ {\ mathit {ec}}}} ^ {\ text {f}}} ^ {\ text {f}}} ^ mathrm {,2}}\ right] The first part concerns the adherent case (:math:`{\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1` and :math:`{\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}1`): .. math:: : label: eq-416 The discretization of :math:`{\underline{\underline{P}}}^{\tau }`, :math:`u` travel, and :math:`\lambda` contact pressure gives us: .. math:: :label: eq-417 \ left [{K} _ {\ mathit {ec}}} ^ {\ text {ec}} ^ {\ text {f}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}}} _ {J}} _ {J} _ {J} _ {J} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} right]\ left\ {\ psi\ right\} \ left [{K} _ {\ mathit {mc}}} ^ {\ text {mc}} ^ {\ text {f}}\ right]\ mathrm {=} + {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _\ psi\ right\} The second part concerns the slippery case (:math:`{\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1` and :math:`{\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}0`): .. math:: : label: eq-418 {J} _ {uc} ^ {\ text {f}\ mathrm {2}}\ mathrm {,2}}\ mathrm {=}\ mathrm {\ int}} _ {{\ Gamma} _ {c} _ {c}}\ mu\ c}}}\ mathrm {,2}}}\ mathrm {2}}}\ mathrm {=}}\ mathrm {-} {\ int}} _ {\ Gamma} _ {c} _ {c}} {c}}\ mu\ left}\ mu\ left [{\ underline {\ underline {P}}}} ^ {\ underline {P}}}} ^ {\ tau} ({\ delta}} _ {t} {t} {\ stackrel}} {\ stackrel} {u}} _ {k} ^ {e}\ mathrm {-} {-} {\ delta} _ {t} {\ stackrel {} {u}} _ {k} ^ {m})\ right]\ right]\ mathrm {\ cdot} {\ cdot} {\ dot} {\ dot} {\ dot} {\ stackrel {} {\ tau}} _ {k}\ delta\ lambda d {\ gamma} {m})\ right]\ mathrm {\ cdot} {\ dot} {\ dot} {\ dot} {\ dot} {\ stackrel {} {\ tau}} _ {k}\ delta\ lambda d {\ Gamma} _ {c} This is written, in particular using the discretization of the projection on the unit ball (): .. math:: : label: eq-419 \ left [{K} _ {\ mathit {ec}}} ^ {\ text {ec}} ^ {\ text {f}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}}} _ {J}} _ {J} _ {J} _ {J} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} right]\ left\ {{\ stackrel {} {\ tau}} {\ tau}} _ {k}\ right\}\ left\ {\ psi\ right\} \ left [{K} _ {\ mathit {mc}}} ^ {\ text {mc}} ^ {\ text {f}}\ right]\ mathrm {=} + {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {\ stackrel {} {\ tau}}} _ {k}\ right\}\ left\ {\ psi\ right\} We will now discretize the quantities concerning the third column, relating to the friction pressure. From the friction reaction and the Jacobians () and (): .. math:: : label: eq-420 {J} _ {uf} ^ {\ text {f}\ mathrm {,1}}}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {ef}}} {\ mathit {ef}}}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}}\ mathrm {,1}}\ right] {J} _ {uf} ^ {\ text {f}\ mathrm {,2}}}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {ef}}} {\ mathit {ef}}}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}} ^ {\ text {f}}\ mathrm {,2}}\ right] The first part concerns the adherent case (:math:`{\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1` and :math:`{\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}1`): .. math:: : label: eq-421 {J} _ {uf} ^ {\ text {f}\ mathrm {f}\ mathrm {,1}}\ mathrm {=}\ mathrm {\ int}} _ {{\ Gamma} _ {c} _ {c}}}\ mu {\ stackrel {1}}}}\ mathrm {,1}}}}\ mathrm {=}}\ mathrm {-}} {\ mathrm {\ int}}} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {c}} _ {c}}}}\ c}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}}}\ mu {\ stackrel {{\ tau} ({\ delta} _ {t} {t} {\ stackrel {} {u}}} _ {k} ^ {e}\ mathrm {-} {\ delta} _ {t} {\ stackrel {} {u}} {u}}} {u}}} _ {u}}} _ {u}} _ {u}} _ {u}} _ {u}} _ {u}} _ {u}} _ {m})\ right]\ mathrm {\ cdot}\ Delta\ Lambda d {\ gamma} _ {u}} The discretization of :math:`{\underline{\underline{P}}}^{\tau }`, the movements :math:`u` and the friction pressure :math:`\Lambda` gives us: .. math:: :label: eq-422 \ left [{K} _ {\ mathit {ef}}} ^ {\ text {f}} ^ {\ text {f}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}} _ {J} _ {J} _ {c} _ {c}\ c} _ {\ stackrel}} {\ lambda}} _ {\ lambda}} _ {\ left [{N} {\ omega} _ {n} {\ omega} _ {c} {J} _ {J} _ {J} _ {c}} _ {c}} _ {\ stackrel}}\ right]} ^ {T}\ left [{P} ^ {\ tau} ^ {\ tau}\ right]\ left [T\ right]\ left [\ psi\ right] \ left [{K} _ {\ mathit {mf}}} ^ {\ text {f}} ^ {\ text {f}}\ right]\ mathrm {=} + {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c}\ c}\ c} _ {\ stackrel {f}} {c} _ {\ stackrel {f}} {c} _ {\ stackrel {f}} {c} _ {c} {J} _ {c} _ {c}} _ {\ stackrel {f}} {c} _ {c}} _ {\ stackrel {f}} {c} _ {c} {J} _ {c} _ {c}} _ {\ stackrel {f}}\ right]\ left [{P} ^ {\ tau}\ right]\ right]\ left [T\ right]\ left [\ psi\ right] The second part concerns the slippery case (:math:`{\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1` and :math:`{\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}0`): .. math:: :label: eq-423 {J} _ {uf} ^ {\ text {f}\ mathrm {f}\ mathrm {2}}\ mathrm {=}\ mathrm {\ int}} _ {{\ Gamma} _ {c} _ {c}}}\ mu {\ stackrel {2}}}\\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {\ tau} ({\ delta} _ {t} {t} {\ stackrel {} {u}}} _ {k} ^ {e}\ mathrm {-} {\ delta} _ {t} {\ stackrel {} {u}} {u}}} {u}}}} _ {k} ^ {m}}}} _ {k} ^ {m})\ right]\ mathrm {\ cdot}\ left [{\ underline {\ underline {\ underline {\ stackrel {} {u}}}} _ {k} ^ {m}})\ right]\ mathrm {\ cdot}\ left [{\ underline {\ underline {\ underline {\ stackrel {} {u}}} {u}}} ackrel {} {P}}}}} _ {k}} ^ {B (\ mathrm {0.1})}\ Delta\ Lambda\ right] d {\ Gamma} _ {c} This is written, in particular using the discretization of the projection on the unit ball (): .. math:: : label: eq-424 \ left [{K} _ {\ mathit {ef}}} ^ {\ text {f}} ^ {\ text {f}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}} _ {J} _ {J} _ {c} _ {c}\ c} _ {\ stackrel {2}}\ right]} ^ {T}\ left [{P} ^ {\ tau} ^ {\ tau}\ right]\ left [{\ stackrel {} {P}}} _ {k} ^ {B}\ right]\ right]\ left [T\ right]\ left [\ psi\ right] \ left [{K} _ {\ mathit {mf}}} ^ {\ text {f}} ^ {\ text {f}}\ right]\ mathrm {=} + {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c}\ c}\ c}\ c} _ {\ stackrel {f}} {c} _ {\ stackrel {f}} {c} _ {\ stackrel {f}} {c} _ {c} {J} _ {c} _ {c}} _ {\ stackrel {f}} {c} _ {c} {J} _ {c} _ {c}} _ {\ stackrel {f}} {c} _ {c} {J} _ {c} _ {c}\ left [{P} ^ {\ tau}\ right]\ right]\ left [{\ stackrel {} {P}}} _ {k} ^ {B}\ right]\ left [T\ right]\ left [\ right]\ left [\ psi\ right] Quantities for the law of contact -------------------------------- We consider the matrix terms resulting from the linearization of the contact law and therefore from the second line of the global matrix. The Jacobian discretization () corresponding to the first column of the global matrix: .. math:: : label: eq-425 {J} _ {cu}\ stackrel {\ text {Discretization}} {\ text {Discretization}}} {\ to}\ left [{K}}\ right],\ left [{K} _ {\ mathit {cm}}\ right] Starting with the Jacobian: .. math:: : label: eq-426 {J} _ {cu}\ mathrm {=}\ mathrm {=}\ mathrm {-} {\ mathrm {\ int}} _ {\ gamma} _ {c}} {\ stackrel {} {S}}}} _ {u, k}} _ {u, k}} _ {k}}} _ {u, k}} _ {u, k}} _ {u, k}} ^ {k}} ^ {e}} ^ {e}\ mathrm {-} {\ Delta} _ {t} {u} {u} ^ {m})\ mathrm {\ cdot} {n} _ {k} d {\ Gamma} _ {c} Two quantities are obtained: .. math:: : label: eq-427 \ left [{K} _ {\ mathit {ce}}\ right]\ mathrm {=}\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {\ stackrel {} {S}} {S}}}\ right]\ mathrm {{S}}}\ {u, k}} _ {\ g} _ {n}}\ left\ {\ psi\ right\}}\ mathrm {\ langle} n} _ {h}\ mathrm {\ rangle}\ left [{N} ^ {e}\ right] \ left [{K} _ {\ mathit {cm}}\ right]\ mathrm {=}} + {\ omega} _ {c} {J} _ {c} {c} {c} {S}}} _ {u, k}}} _ {u, k}}\ {u, k}}\ {u, k}} _ {s}} _ {h} _ {h} _ {h}\ mathrm {\ rangle}\ left [{N} ^ {m}\ right] By discretizing the Jacobian (), we obtain the expression of the quantities corresponding to the second column: .. math:: :label: eq-428 {J} _ {cc}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {cc}}}\ right] Starting with the Jacobian: .. math:: : label: eq-429 {J} _ {cc}\ mathrm {=}\ mathrm {-}\ mathrm {-}\ frac {1} {\ rho} _ {\ int}} {\ mathrm {\ int}} _ {\ Gamma} _ {c}} _ {{\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {c} _ {c}}}}\ Delta\ Lambda}}}\ Delta\ Lambda (1\ mathrm {-}} {\ stackrel {-} {S}}} _ {u, k} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {c} _ {c}}} _ {n}})\ delta\ lambda d {\ Gamma} _ {c} We get: .. math:: : label: eq-430 \ left [{K} _ {\ mathit {cc}}\ right]\ mathrm {=}\ mathrm {=}\ mathrm {-} {\ omega} _ {c}\ frac {1} {1} {{\ rho}} {{1} {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {n}})\ left\ {\ psi\ right\}\ mathrm {\ langle}\ psi\ mathrm {\ rangle} Quantities for the law of friction ----------------------------------- We consider the matrix terms resulting from the linearization of the law of friction and therefore from the third row of the global matrix. The Jacobian discretization () and () corresponding to the first column of the global matrix: .. math:: : label: eq-431 {J} _ {fu} ^ {1}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {fe}} ^ {1}\ right],\ left [{1}\ right],\ left [{K}} _ {\ mathit {fm}}} ^ {1}\ right] {J} _ {fu} ^ {2}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {fe}}} ^ {2}\ right],\ left [{2}\ right],\ left [{K}} _ {\ mathit {fm}}} ^ {2}\ right] The first part concerns the adherent case (:math:`{\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1` and :math:`{\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}1`): .. math:: :label: eq-432 {J} _ {fu} ^ {1}\ mathrm {=}\ mathrm {=}\ mathrm {-} {\ mathrm {\ int}} _ {\ Gamma} _ {c}}\ mu {\ stackrel {}}}\ mathrm {\ stackrel {}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}} {}} {}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}}} {\ stackrel {}} {} ^ {\ tau} ({\ Delta} _ {t} {m} _ {m} _ {m})\ mathrm {-} {\ Delta} _ {t} {u} ^ {m})\ right] d {\ Gamma} {u} {u} {u} {u} {u} {u} {u} {m})\ right] d {\ Gamma} _ {c} The discretization of :math:`{\underline{\underline{P}}}^{\tau }`, the movements :math:`u` and the friction pressure :math:`\Lambda` gives us: .. math:: :label: eq-433 \ left [{K} _ {\ mathit {fe}}} ^ {1}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J} _ {c}\ mu {\ c}\ mu {\ stackrel {}}\ mu {\ stackrel {}}\ mu {\ stackrel {}} {\ stackrel {}} {\ stackrel {}} {\ stackrel {}} {\ stackrel {}} {\ lambda}} {\ lambda}} _ {\ left [T\ right]} ^ {T} {\ left [T\ right]} ^ {T} {\ left [T\ right]} ^ {T} {\ left [T\ right]} ^ {{T}\ left [{P} ^ {\ tau}\ right]\ left [{N} ^ {e}\ right] \ left [{K} _ {\ mathit {fm}}} ^ {1}\ right] ^ {1}\ right]\ mathrm {=} + {\ omega} _ {c}\ mu {\ stackrel {} {\ stackrel {}} {\ lambda}} {\ lambda}} {\ lambda}}} _ {\ left [{P} ^ {\ tau}\ right]\ right]\ left [{N} ^ {m}\ right] The second part concerns the slippery case (:math:`{\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1` and :math:`{\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}0`): .. math:: :label: eq-434 {J} _ {fu} ^ {2} =- {\ int} =- {\ int} _ {{\ mathrm {\ Gamma}} _ {c}}\ mathrm {\ mu} {\ widehat {\ mathrm {\ lambda}}}} _ {\ int}} _ {\ int}} _ {k}\ mathrm {\ delta}\ mathrm {\ Lambda}}\ mathrm {\ Lambda}} {\ widehat {\ mathrm {\ lambda}} {\ mathrm {\ lambda}} {\ widehat {\ mathrm {\ lambda}} {\ mathrm {\ lambda}} {\ underline {\ underline}} P}}} ^ {\ mathrm {\ tau}}} {\ underline {\ underline {\ underline {\ widehat {P}}}}} _ {k} ^ {B (\ mathrm {0.1})}\ left ({\ mathrm {\ tau}}}}}\ left ({\ mathrm {\ tau}})}\ left ({\ mathrm {\ tau}}}}\ left ({\ mathrm {\ tau}}}}\ left ({\ mathrm {\ tau}}}}\ left ({\ mathrm {\ tau}}}}\ left ({\ mathrm {\ tau}}}}\ left ({\ mathrm {\ tau}}}}\ left ({\ mathrm {\ tau}} m}\ right)\ right] d {\ mathrm {\ Gamma}} _ {c} This is written, in particular using the discretization of the projection on the unit ball (): .. math:: :label: eq-435 \ left [{K} _ {\ mathit {fe}}} ^ {2}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J} _ {c}\ mu {\ c}\ mu {\ stackrel {}}\ mu {\ stackrel {}}\ mu {\ stackrel {}} {\ stackrel {}} {\ stackrel {}} {\ stackrel {}} {\ stackrel {}} {\ lambda}} {\ lambda}} _ {\ left [T\ right]} ^ {T} {\ left [T\ right]} ^ {T} {\ left [T\ right]} ^ {T} {\ left [T\ right]} ^ {T}\ left [{P} ^ {\ tau}\ right]\ right]\ left [{\ stackrel {} {P}}} _ {k} ^ {B}\ right]\ right]\ left [{N}\ right]\ left [{N} ^ {e}\ right] \ left [{K} _ {\ mathit {fe}}} ^ {2}\ right] ^ {2}\ right]\ mathrm {=} + {\ omega} _ {c}\ mu {\ stackrel {} {\ stackrel {}} {\ lambda}} {\ lambda}} {\ lambda}}} _ {\ left [{P} ^ {\ tau}\ right]\ left [{\ stackrel {} {P}}} _ {k} ^ {B}\ right]\ left [{N} ^ {m}\ right] The Jacobian discretization () corresponding to the third column of the global matrix: .. math:: :label: eq-436 {J} _ {ff}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {ff}}}\ right] The Jacobian is written: .. math:: :label: eq-437 \ begin {array} {cc} {J} _ {ff} _ {ff}\ mathrm {=}} &\ frac {1} {{\ stackrel {}} {\ rho}} _ {t}} {\ mathrm {\ int}}} {\ mathrm {\ int}}} _ {\ mathrm {\ int}}} _ {\ int}} _ {\ stackrel}} _ {\ stackrel}} _ {\ stackrel}} {} {S}} _ {u, k} ^ {{h} _ {n}} ^ {{g} _ {n}} (1\ mathrm {-} {S}}} _ {f, k} ^ {{h}} _ {{h} _ {\ h} _ {\ tau}}) (\ underline {\ tau}}) (\ underline {\ tau}}) (\ underline {\ tau}}) (\ underline {\ underline {\ tau}}) (\ underline {\ stackrel {-}} {\ underline {\ underline {\ underline {\ stackrel {1}}}\ mathrm {-} {\ underline {\ underline {\ underline {\ underline {\ stackrel}}} {P}}}} _ {k} ^ {B (\ mathrm {0.1})})\ Delta\ Lambda\ mathrm {\ cdot}\ delta\ Lambda d {\ Gamma} _ {c} _ {c}\\\ &\ mathrm {-} {-} {\ mathrm {-}}} _ {\ Gamma} _ {c}}} (1\ mathrm {-}} (1\ mathrm {-}}\ stackrel {} {S}} _ {u, k} _ {u, k} ^ {{g} _ {n}})\ Delta\ Lambda\ mathrm {\ cdot}\ delta\ Lambda d {\ Lambda d {\ Gamma} _ {\ Gamma} _ {c}\ end {array} We prefer to divide it in two. A contribution for the slippery contact case (:math:`{\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1` and :math:`{\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}0`): .. math:: :label: eq-438 {J} _ {ff} ^ {1}\ mathrm {=}\ frac {1} {{\ stackrel {} {\ rho}} _ {t}} {\ mathrm {\ int}}} _ {{\ int}}} _ {\ int}} _ {\ int}}} _ {\ int}}} {\ mathrm {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}}}\ mathrm {-} {\ underline {\ underline {\ underline {\ underline {\ stackrel {} {P}}}}} _ {k} ^ {B (\ mathrm {0.1})})\ Delta\ Lambda {\)}})\ Delta\ Lambda {0.1}}})\ Delta\ Lambda {0.1}}})\ Delta\ Lambda {0.1}}})\ Delta\ Lambda\ Lambda\ And a contribution for the contactless case: .. math:: :label: eq-439 {J} _ {ff} ^ {2}\ mathrm {=}\ mathrm {=}\ mathrm {-} {\ mathrm {\ int}} _ {\ Gamma} _ {c}}\ Delta\ Lambda\ {c}}\ Delta\ Lambda\ {c}}\ Delta\ Lambda d {\ Gamma} _ {c}}\ Delta\ Lambda\ {c}} Writing infinitesimal sliding: GRAND_GLIS --------------------------------- In the case where contact is established and the sliding is active and the latter is constrained by Coulomb's law, it is necessary to linearize the weak form of the force inducing the sliding, the Coulomb law remaining unchanged. :math:`{G}_{f}={\int }_{{\mathrm{\Gamma }}_{c}}\mathrm{\mu }{\widehat{\mathrm{\lambda }}}_{k}{\widehat{S}}_{u,k}^{{g}_{n}}\left(1-{\widehat{S}}_{f,k}^{{h}_{\mathrm{\tau }}}\right){\widehat{\mathrm{\tau }}}_{k}\cdot {\mathrm{\delta }d}_{\mathrm{\tau }}d{\mathrm{\Gamma }}_{c}` So far, relative slippage has been approximated as follows: .. math:: : label: eq-440 \ mathrm {\ Delta} {d} _ {\ mathrm {\ tau}} = {v} _ {\ mathrm {\ tau}}\ mathrm {\ Delta} t=\ frac {1} {\ mathrm {1} {\ mathrm {\ Delta} t} {\ mathrm {\ delta} t} {\ mathrm {\ delta} t} {\ mathrm {\ delta} t} {\ mathrm {\ delta} t} {\ mathrm {\ delta} t} {\ mathrm {\ delta} t} {\ mathrm {\ delta} t} {\ mathrm {\ delta} t} {\ mathrm {\ delta} t} {\ mathrm {\ delta} t} {\ mathrm {\ delta} Mathit {nn}} ^ {T})\ mathrm {\ Delta} u Note: in the following, you can replace :math:`\mathrm{\Delta }` by :math:`\mathrm{\delta }` if necessary. Strictly speaking, if we adopt a kinematic approach specific to the mechanics of continuous media, the relative speed at a point of contact is: .. math:: :label: eq-441 v=\ frac {\ partial [{y} _ {\ mathit {escl}}} - {\ mathit {escl}}}}} {\ partial t}\ mathrm {\ Delta} t-\ frac {\ delta} t-\ frac {\ partial {y}} _ {\ mathit {y}} _ {\ mathit {\ bait}}}} {\ partial {\ mathrm {\ zeta}}} ^ {\ mathrm {\ delta} t-\ frac {\ partial {y}} _ {\ mathit {y}} _ {\ mathit {but}}} {\ mathrm {\ zeta}}} ^ {\ mathrm {\ delta} t-\ frac {\ partial {\ delta} t-\ frac {\ partial {y}}\ frac {\ partial {\ mathrm {\ zeta}}} ^ {\ mathrm {\ alpha}}} {\ partial t}\ mathrm {\ delta} t=\ frac {\ partial [{y} _ {\ partial [{y} _ {\ mathit {escl}} _ {\ mathit {escl}}}} {\ partial t} t=\ frac {\ partial [{y} _ {\ mathit {escl}} _ {\ mathit {escl}}} - {y} _ {\ mathit {escl}}} - {y} _ {\ mathit {\ delta}}]} {\ partial t}\ mathrm {\ Delta} t-\ frac {\ partial {y} _ {\ mathit {ait}}}} {\ partial {\ mathrm {\ zeta}}} ^ {\ mathrm {\ alpha}}}}\ mathrm {\ Delta} {\ delta} {\ delta}} {\ mathrm {\ delta}} {\ mathrm {\ delta}} {\ delta}} {\ mathrm {\ delta}} {\ mathrm {\ delta}} {\ delta}} {\ mathrm {\ delta}} {\ mathrm {\ delta}} {\ delta}} {\ mathrm {\ delta}} {\ Afterwards, we have: :math:`\frac{\partial [{y}_{\mathit{escl}}-{y}_{\mathit{mait}}]}{\partial t}\mathrm{\Delta }t-\frac{\partial {y}_{\mathit{mait}}}{\partial {\mathrm{\zeta }}^{\mathrm{\alpha }}}\mathrm{\Delta }{\mathrm{\zeta }}^{\mathrm{\alpha }}=\frac{\partial {g}_{n}n}{\partial t}\mathrm{\Delta }t-\frac{\partial {y}_{\mathit{mait}}}{\partial {\mathrm{\zeta }}^{\mathrm{\alpha }}}\mathrm{\Delta }{\mathrm{\zeta }}^{\mathrm{\alpha }}=\frac{\partial {g}_{n}}{\partial t}\mathrm{\delta }tn+{g}_{n}\frac{\partial n}{\partial t}\mathrm{\Delta }t-\frac{\partial {y}_{\mathit{mait}}}{\partial {\mathrm{\zeta }}^{\mathrm{\alpha }}}\mathrm{\Delta }{\mathrm{\zeta }}^{\mathrm{\alpha }}` We are going to note: :math:`{v}_{n}=\frac{\partial {g}_{n}}{\partial t}\mathrm{\Delta }tn` and :math:`{v}_{\mathrm{\tau }}={g}_{n}\frac{\partial n}{\partial t}\mathrm{\Delta }t-\frac{\partial {y}_{\mathit{mait}}}{\partial {\mathrm{\zeta }}^{\mathrm{\alpha }}}\mathrm{\Delta }{\mathrm{\zeta }}^{\mathrm{\alpha }}` So the continuous writing of relative speed is: .. math:: :label: eq-442 v= {v} _ {n} + {v} _ {\ mathrm {\ tau}} Here we are interested in the writing of :math:`{v}_{\mathrm{\tau }}`. As a first approximation, and assuming that the point is constantly in contact, we have: .. math:: :label: eq-443 {v} _ {\ mathrm {\ tau}}}\ approx\ frac {-\ partial {y} _ {\ mathit {ait}}}} {\ partial {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ { We can refer to equation (66) to take into account the variation due to normal :math:`{g}_{n}\frac{\partial n}{\partial t}`. From a vocabulary point of view in the mechanics of continuous media, it is said that the slave contact point is described according to Lagrangian kinematics while its master projected point is described according to Eulerian kinematics. The previous equation shows that by neglecting the term :math:`{g}_{n}\frac{\partial n}{\partial t}`, all the information necessary to calculate the slip in the tangential plane is contained in the variation of the master projected point provided that the status of the contact point is equal to 1 and the rest is equal to 1 and the rest during the Newton process in the time step. We are going to explain the term :math:`\mathrm{\Delta }{\mathrm{\zeta }}^{\mathrm{\alpha }}`. .. math:: :label: eq-444 \ mathrm {\ Delta} {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ stackrel {~} {A}}} - {g} _ {a}}} - {g} _ {\ zeta}}}} - {g} _ {\ mathrm {\ alpha}}} - {g} _ {\ mathrm {\ alpha}}} - {g} _ {\ mathrm {\ alpha}}} - {g} _ {\ mathrm {\ alpha}}}} - {g} _ {\ mathrm {\ alpha}}}} - {g} _ {\ mathrm {\ alpha}}}} - {g} _ {\ mathrm {\ alpha}}}} {\ beta}} ^ {-1} [\ frac {\ partial {y}} _ {\ mathit {ait}}}} {\ partial {\ mathrm {\ zeta}}} ^ {\ mathrm {\ beta}}}} (\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} {\ mathrm {\ beta}}}} ^ {\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} (\ Mait}}) + {g} _ {n} n\ mathrm {\ Delta}\ frac {\ partial {y} _ {\ mathit {but}}} {\ partial {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}}} ^ {\ mathrm {\ beta}}}] For the linearization of friction, we will also need the second variation of :math:`\mathrm{\Delta }\mathrm{\delta }{\mathrm{\zeta }}^{\mathrm{\alpha }}`. .. math:: :label: eq-445 \ mathrm {\ Delta}\ mathrm {\ delta} {\ mathrm {\ delta}} {\ mathrm {\ alpha}} = {[\ stackrel {~} {\ stackrel {~} {a}} {A}}} - {g} {A}}} - {g} _ {A}}} - {g} _ {A}}} - {g} _ {A}}} - {g} _ {A}}} - {g} _ {a}}} - {g} _ {a}}} - {g} _ {n}\ stackrel {~} {H}}}]} _ {\ mathrm {\ alpha}\ mathrm {\ beta}}} ^ {-1} {-1} [\ frac {\ partial {y}} _ {\ mathit {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ beta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} mathrm {\ delta}\ frac {\ partial {y} _ {\ mathit {ait}}}} {\ partial {\ mathrm {\ zeta}} ^ {\ mathrm {\ gamma}}}}} +\ mathrm {\ delta}}} +\ mathrm {\ delta}}} +\ mathrm {\ delta}}} +\ mathrm {\ delta}}\ frac {\ delta}}\ frac {\ partial {y}} _ {\ mathit {mai}}} {\ partial {\ mathrm {\ gamma}}}} +\ mathrm {\ delta}}} +\ mathrm {\ delta}}} +\ mathrm {\ delta}}\ mathrm {\ delta}}\ frac {\ delta}}\ {\ mathrm {\ gamma}}}}\ mathrm {\ delta} {\ mathrm {\ zeta}} ^ {\ mathrm {\ gamma}}}) +\ mathrm {\ Delta} {\ mathrm {\ zeta}}} ^ {\ mathrm {\ gamma}}} (\ frac {\ partial {y} _ {\ mathit {mai}}}} {\ partial {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ beta}}}}\ mathrm {.}} \ frac {{\ partial} ^ {2} {y} _ {\ mathit {but}}} {{\ partial} ^ {2} {\ mathrm {\ zeta}}} ^ {\ mathrm {\ beta}}}} ^ {\ mathrm {\ beta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ beta}}} ^ {\ mathrm {\ beta}}})\ mathrm {\ beta}}})\ mathrm {\ beta}}})\ mathrm {\ beta}}})\ mathrm {\ beta}}}) - {g} _ {n} n (\ mathrm {\ Delta} {\ delta}} {\ mathrm {\ zeta}} ^ {\ mathrm {\ delta}}\ frac {\ partial {y}\ frac {\ partial {y}} _ {\ partial {\ mathrm {\ delta}}\ frac {\ partial {y}} _ {\ partial {y}} _ {\ partial {y}} _ {\ partial {y}} _ {\ mathit {\ delta}} _ {\ partial {y}} _ {\ mathit {\ delta}} _ {\ partial {y}} _ {\ mathit {\ delta}} _ {\ partial {y}} _ {\ mathit {\ delta}} _ {\ partial {y}} _ {\ partial {y} mathrm {\ Delta}\ frac {\ partial {y} _ {\ mathit {ait}}}} {\ partial {\ mathrm {\ zeta}} ^ {\ mathrm {\ gamma}}}}}\ mathrm {\ gamma}}}}}}\ mathrm {\ delta}} {\ mathrm {\ delta}} {\ delta}} {\ mathrm {\ gamma}}}}}\ mathrm {\ delta}} {\ mathrm {\ delta}} {\ mathrm {\ gamma}}}}}\ mathrm {\ delta}} {\ delta} {\ delta}} {\ mathrm {\ delta}} {\ The user is offered the two sliding scripts: - or by projecting the displacement increment onto the tangential plane equation (4339) via the keyword GRAND_GLIS =' NON ', - or by adopting a continuous writing of the tangent sliding relative to the master point: GRAND_GLIS =' OUI '. In theory, in small slips, the two writings give the same result. For the implementation, the discretization technique remains the same for the kinematic unknowns as described in section 6.