13. 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.

13.1. Preliminary matrix quantities

We start by describing the discretized form of a certain number of quantities. Discretizing management \({\stackrel{ˆ}{\tau }}_{k}\) will give us:

\[\]

: 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:

(13.1)\[ {\ underline {\ underline {\ stackrel {} {} {P}}}}}} _ {k} ^ {B (\ mathrm {0.1})}\ stackrel {\ text {Discretization}}} {\ to}}\ left [{\ stackrel {} {P}}} _ {k} ^ {B}\ right]\]

The process of linearizing the quantity relative to the unit ball (§ 3.2.4, see ()) gives us:

\[\]

: 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:

\[\]

: 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:

\[\]

: 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 [1right]mathrm {-}left{{stackrel {} {} {stackrel {} {tau}} {tau}} {tau}} _ {tau}}} _ {tau}} _ {tau}}} _ {tau}}} _ {tau}}} _ {tau}}}rangle)

13.2. Quantities for balance

We start by considering the matrix terms resulting from the linearization of the equilibrium equation. The discretization of Jacobian women () and ():

\[\]

: 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 ():

\[\]

: 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:

\[\]

: 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 {-}mathrmrangle} {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}}rightmathrm {{} {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 \(\left[{P}^{n}\right]\mathrm{=}\mathrm{\{}{n}_{h}\mathrm{\}}\mathrm{\langle }{n}_{h}\mathrm{\rangle }\). Finally:

\[\]

: 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 ():

(13.2)\[\begin{split} \ 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}\end{split}\]

Let’s consider the first integral, by discretizing, we get:

\[\]

: 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:

\[\]

: 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 [Gright]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 [Gright]leftleft [{stackrel {stackrel {} {N}}} ^ {m}right]

With the \(\left[G\right]\) matrix such as:

\[\]

: label: eq-400

begin {array} {c}left [Gright]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:

\[\]

: 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:

\[\]

: 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 [Hright]left [{Hright]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 [Hright]leftleft [{stackrel {stackrel {} {N}}} ^ {m}right]

With the \(\left[H\right]\) matrix such as:

(13.3)\[ \ left [H\ right]\ mathrm {=} {m} {m} _ {m} _ {11} _ {11} {11} {1} ^ {m}\ right]} ^ {T}\ mathrm {\ {T}\ mathrm {\ {m}}\ mathrm {\} {n} _ {h} {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {m}}\ mathrm {\ langle} {x} _ {h} ^ {m} {m}}\ mathrm {\ rangle} {\ left [{\ stackrel {} {B}}} _ {1} ^ {m}\ right]} ^ {T} + {m} _ {21} {\ left [{\ stackrel {} {n} {n} {B}}} _ {2} ^ {m}\ right]} ^ {T}\ mathrm {\ {} {n} {n} _ {n} _ {h}\ mathrm {\}}\ mathrm {\ langle} {x} {x} _ {h} ^ {m}\ mathrm {\ rangle} {\ left [{\ stackrel {} {B} {B}}}} _ {B}}} _ {B}}} _ {12} {\ left [{\ stackrel {} {B}} {B}}} _ {1} ^ {m}\ right]} ^ {T}} ^ {T}\ mathrm {\ {} {n} _ {h}\ mathrm {\ langle} {x} _ {h} _ {h} _ {h} ^ {m} ^ {m} ^ {m}}\ m} ^ {m}}\ right}\ mathrm {\ rangle} {\ left [{\ stackrel {} {B}}}} _ {2} ^ {m}\ right]} ^ {T} + {m} _ {22} {\ left [{\ stackrel {} {B}}} _ {2} ^ {m}\ right]} ^ {T}\ mathrm {\ {} {t}\ mathrm {\}}\ mathrm {\ langle} {x} _ {h} _ {h} ^ {m} ^ {m}\ m}\ mathrm {\ rangle} {\ m} {m}} ^ {m}\ right} {m} {m} {m} {m} {m} {m} {m} {m} {m}} {m} {m} {m} {m} {m} {m} {m} {m} {m}} {m} {m} {m}} {m} {m} {m}} {m} {m} {m}} {m} {m} {m} {m} {m} {^ {T}\]

Finally, the last contribution:

\[\]

: 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:

\[\]

: 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 [Lright]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 [Lright]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 [Lright]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 [Lright]left [Lright]left [{N} ^ {e}right]

With the \(\left[L\right]\) matrix such as:

(13.4)\[ \ left [H\ right]\ mathrm {=} {m} _ {m} _ {11} {\ kappa} _ {11} {m} _ {11} {\ left [{\ stackrel {} {B} {B}}}} _ {1} {B} {B}}}} _ {1} {B}}}} _ {1} ^ {m}\ right] + {m} _ {21} {\ kappa} _ {21} {21} {m} {m} _ {21} {\ left [{\ stackrel {} {B}}} _ {2} ^ {m}\ right]}}} ^ {t}} ^ {T}}\ mathrm {\}}} ^ {right]}} ^ {t}} ^ {T}}\ mathrm {\} _ {h} ^ {m}\ mathrm {\ rangle}\ left [{\ stackrel {} {B}}} _ {1} ^ {m}\ right] + {m} _ {12} {\ kappa} _ {12} {m}} _ {12} _ {12} {12} {\ left [{\ stackrel {}} {B}}} _ {1} ^ {m}\ right]} ^ {m}\ right]} ^ {T}\ mathrm {\ {} {n} _ {h}\ mathrm {\}}\ mathrm {\ langle} {x} _ {h} ^ {m}\ mathrm {\ rangle}\ left [{\ stackrel {}}\ left [{\ stackrel {} {B} {B}}}} _ {2} ^ {m}\ right] + {m} _ {22} {\ kappa}}\ left [{\ stackrel {} {B} {B}}}} _ {22} {\ kappa} _ {22} {\ kappa} _ {22}\ left [{\ stackrel {} {B} {B}}}} {m} _ {22} {\ left [{\ stackrel {} {B}}} _ {2} ^ {m}\ right]} ^ {T}\ mathrm {\ {} {n} {n} _ {h}\ mathrm {\}}\ mathrm {\ langle} {x} _ {h} ^ {m}\ mathrm {\ rangle}}\ left [{\ stackrel {\ rangle}\ left [{\ stackrel {}} {B}}} _ {2} ^ {m} ^ {m} {m}\ m}\ m}\ m}\ right]\]

We now consider the discretization of Jacobian women () and ():

\[\]

: 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 (\({\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1\) and \({\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}1\)):

\[\]

: 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 \({\underline{\underline{P}}}^{\tau }\) and travel gives us:

(13.5)\[ \ left [{K} _ {\ mathit {ee}}} ^ {\ text {ee}} ^ {\ text {f}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}} _ {J} _ {J} _ {c} _ {c}\ mu {\ stackrel {}} {\ rho}} _ {\ stackrel {\ omega} _ {\ omega} _ {c} {J} _ {J} _ {J} _ {c} _ {c} _ {c}} _ {\ stackrel {\ lambda}} _ {k} {\ left [{N} ^ {e}\ right]} ^ {e}\ right]} ^ {T}\ left [{P}\ right]\ left [{P} ^ {\ tau} ^ {\ tau}\ right]\ right]\ right]\ left [{N} ^ {e}\ right] \ left [{K} _ {\ mathit {mm}}} ^ {\ text {mm}} ^ {\ text {f}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}} _ {J}} _ {c} _ {c}\ mu {\ stackrel {\ rho}}} _ {\ stackrel {\ omega} _ {\ omega} _ {c} {J}} _ {c} {J} _ {c}} _ {c}} _ {\ stackrel {\ lambda}} _ {k} {\ left [{N} ^ {m}\ right]} ^ {m}\ right]} ^ {T}\ left [{P}\ right]\ left [{P} ^ {\ tau} ^ {\ tau}\ right]\ right]\ right]\ left [{N} ^ {m}\ right] \ left [{K} _ {\ mathit {em}}} ^ {\ text {em}} ^ {\ text {f}}\ right]\ mathrm {=} + {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c} _ {c}\ c}\ c} _ {k}\ mu {\ stackrel {}} {\ rho}} _ {k}\ left [{N} ^ {e}\ right]} ^ {T}}\ left [{P} ^ {\ tau}\ right]\ left [{P} ^ {\ tau}\ right]\ right]\ left [{N} ^ {m}\ right] \ left [{K} _ {\ mathit {me}}} ^ {\ text {me}} ^ {\ text {f}}\ right]\ mathrm {=} + {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c} _ {c}\ c}\ c} _ {k}\ mu {\ stackrel {}} {\ rho}} _ {k}\ left [{N} ^ {m}\ right]} ^ {T}} ^ {T}\ left [{P} ^ {\ tau}\ right]\ left [{P} ^ {\ tau}\ right]\ right]\ left [{N} ^ {e}\ right]\]

The second part concerns the slippery case (\({\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1\) and \({\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}0\)):

\[\]

: 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 ():

\[\]

: 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 ():

\[\]

: 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}Deltalambda d {gamma} _ {m})mathrm {cdot} {n} {n}} _ {c}

We must discretize:

(13.6)\[ {J} _ {uc} ^ {\ text {c}}}\ stackrel {\ text {c}}}\ stackrel {\ text {c}}}\ stackrel {\ text {c}}}\ right],\ left [{K}}}\ right],\ left [{K}} _ {\ mathit {mc}}} ^ {\ text {c}}} ^ {\ text {c}}} ^ {\ text {c}}\ right]\]

This gives us:

(13.7)\[ \ left [{K} _ {\ mathit {ec}}} ^ {\ text {c}}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c}} {c} {c}} {c} {c} _ {c} _ {e} _ {e} _ {n}} _ {n}} _ {n}} _ {n}} _ {n}} _ {n}} _ {N}} _ {N} {e}\ right]} ^ {T}\ mathrm {\ {} {n} {n} _ {h}\ mathrm {\}}\ langle\ psi\ rangle \ left [{K} _ {\ mathit {mc}}} ^ {\ text {c}}}\ right]\ mathrm {=} + {\ omega} _ {J} _ {c} {c} {\ stackrel {} {mc}}} {\ stackrel {} {mc}}} {\ stackrel {} {mc}} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {c} {\ stackrel {}\ mathrm {\ {} {n} _ {h}\ mathrm {\}}\ langle\ psi\ rangle\]

From the friction reaction, that is to say Jacobians () and ():

\[\]

: 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 (\({\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1\) and \({\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}1\)):

\[\]

: label: eq-416

The discretization of \({\underline{\underline{P}}}^{\tau }\), \(u\) travel, and \(\lambda\) contact pressure gives us:

(13.8)\[ \ left [{K} _ {\ mathit {ec}}} ^ {\ text {ec}} ^ {\ text {f}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}}} _ {J}} _ {J} _ {J} _ {J} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} right]\ left\ {\ psi\ right\} \ left [{K} _ {\ mathit {mc}}} ^ {\ text {mc}} ^ {\ text {f}}\ right]\ mathrm {=} + {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _\ psi\ right\}\]

The second part concerns the slippery case (\({\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1\) and \({\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}0\)):

\[\]

: label: eq-418

{J} _ {uc} ^ {text {f}mathrm {2}}mathrm {,2}}mathrm {=}mathrm {int}} _ {{Gamma} _ {c} _ {c}}muc}}}mathrm {,2}}}mathrm {2}}}mathrm {=}}mathrm {-} {int}} _ {Gamma} _ {c} _ {c}} {c}}muleft}muleft [{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}deltalambda d {gamma} {m})right]mathrm {cdot} {dot} {dot} {dot} {dot} {stackrel {} {tau}} _ {k}deltalambda d {Gamma} _ {c}

This is written, in particular using the discretization of the projection on the unit ball ():

\[\]

: 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{psiright} 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{psiright}

We will now discretize the quantities concerning the third column, relating to the friction pressure. From the friction reaction and the Jacobians () and ():

\[\]

: 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 (\({\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1\) and \({\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}1\)):

\[\]

: 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}DeltaLambda d {gamma} _ {u}}

The discretization of \({\underline{\underline{P}}}^{\tau }\), the movements \(u\) and the friction pressure \(\Lambda\) gives us:

(13.9)\[ \ left [{K} _ {\ mathit {ef}}} ^ {\ text {f}} ^ {\ text {f}}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J}} _ {J} _ {J} _ {c} _ {c}\ c} _ {\ stackrel}} {\ lambda}} _ {\ lambda}} _ {\ left [{N} {\ omega} _ {n} {\ omega} _ {c} {J} _ {J} _ {J} _ {c}} _ {c}} _ {\ stackrel}}\ right]} ^ {T}\ left [{P} ^ {\ tau} ^ {\ tau}\ right]\ left [T\ right]\ left [\ psi\ right] \ left [{K} _ {\ mathit {mf}}} ^ {\ text {f}} ^ {\ text {f}}\ right]\ mathrm {=} + {\ omega} _ {c} {J} _ {J} _ {c} _ {c} _ {c}\ c}\ c} _ {\ stackrel {f}} {c} _ {\ stackrel {f}} {c} _ {\ stackrel {f}} {c} _ {c} {J} _ {c} _ {c}} _ {\ stackrel {f}} {c} _ {c}} _ {\ stackrel {f}} {c} _ {c} {J} _ {c} _ {c}} _ {\ stackrel {f}}\ right]\ left [{P} ^ {\ tau}\ right]\ right]\ left [T\ right]\ left [\ psi\ right]\]

The second part concerns the slippery case (\({\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1\) and \({\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}0\)):

(13.10)\[\begin{split} {J} _ {uf} ^ {\ text {f}\ mathrm {f}\ mathrm {2}}\ mathrm {=}\ mathrm {\ int}} _ {{\ Gamma} _ {c} _ {c}}}\ mu {\ stackrel {2}}}\\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {2}}}\ mu {\ stackrel {\ tau} ({\ delta} _ {t} {t} {\ stackrel {} {u}}} _ {k} ^ {e}\ mathrm {-} {\ delta} _ {t} {\ stackrel {} {u}} {u}}} {u}}}} _ {k} ^ {m}}}} _ {k} ^ {m})\ right]\ mathrm {\ cdot}\ left [{\ underline {\ underline {\ underline {\ stackrel {} {u}}}} _ {k} ^ {m}})\ right]\ mathrm {\ cdot}\ left [{\ underline {\ underline {\ underline {\ stackrel {} {u}}} {u}}} ackrel {} {P}}}}} _ {k}} ^ {B (\ mathrm {0.1})}\ Delta\ Lambda\ right] d {\ Gamma} _ {c}\end{split}\]

This is written, in particular using the discretization of the projection on the unit ball ():

\[\]

: 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 [Tright]left [psiright] 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 [Tright]left [right]left [psiright]

13.3. 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:

\[\]

: label: eq-425

{J} _ {cu}stackrel {text {Discretization}} {text {Discretization}}} {to}left [{K}}right],left [{K} _ {mathit {cm}}right]

Starting with the Jacobian:

\[\]

: 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:

\[\]

: label: eq-427

left [{K} _ {mathit {ce}}right]mathrm {=}mathrm {=}mathrm {-} {omega} _ {c} {stackrel {} {S}} {S}}}right]mathrm {{S}}}{u, k}} _ {g} _ {n}}left{psiright}}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:

(13.11)\[ {J} _ {cc}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {cc}}}\ right]\]

Starting with the Jacobian:

\[\]

: label: eq-429

{J} _ {cc}mathrm {=}mathrm {-}mathrm {-}frac {1} {rho} _ {int}} {mathrm {int}} _ {Gamma} _ {c}} _ {{Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {c} _ {c}}}}DeltaLambda}}}DeltaLambda (1mathrm {-}} {stackrel {-} {S}}} _ {u, k} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {c} _ {c}}} _ {n}})deltalambda d {Gamma} _ {c}

We get:

\[\]

: 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{psiright}mathrm {langle}psimathrm {rangle}

13.4. 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:

\[\]

: 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 (\({\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1\) and \({\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}1\)):

(13.12)\[ {J} _ {fu} ^ {1}\ mathrm {=}\ mathrm {=}\ mathrm {-} {\ mathrm {\ int}} _ {\ Gamma} _ {c}}\ mu {\ stackrel {}}}\ mathrm {\ stackrel {}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}} {}} {}}\ mu {\ stackrel {}}}\ mu {\ stackrel {}}} {\ stackrel {}} {} ^ {\ tau} ({\ Delta} _ {t} {m} _ {m} _ {m})\ mathrm {-} {\ Delta} _ {t} {u} ^ {m})\ right] d {\ Gamma} {u} {u} {u} {u} {u} {u} {u} {m})\ right] d {\ Gamma} _ {c}\]

The discretization of \({\underline{\underline{P}}}^{\tau }\), the movements \(u\) and the friction pressure \(\Lambda\) gives us:

(13.13)\[ \ left [{K} _ {\ mathit {fe}}} ^ {1}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J} _ {c}\ mu {\ c}\ mu {\ stackrel {}}\ mu {\ stackrel {}}\ mu {\ stackrel {}} {\ stackrel {}} {\ stackrel {}} {\ stackrel {}} {\ stackrel {}} {\ lambda}} {\ lambda}} _ {\ left [T\ right]} ^ {T} {\ left [T\ right]} ^ {T} {\ left [T\ right]} ^ {T} {\ left [T\ right]} ^ {{T}\ left [{P} ^ {\ tau}\ right]\ left [{N} ^ {e}\ right] \ left [{K} _ {\ mathit {fm}}} ^ {1}\ right] ^ {1}\ right]\ mathrm {=} + {\ omega} _ {c}\ mu {\ stackrel {} {\ stackrel {}} {\ lambda}} {\ lambda}} {\ lambda}}} _ {\ left [{P} ^ {\ tau}\ right]\ right]\ left [{N} ^ {m}\ right]\]

The second part concerns the slippery case (\({\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1\) and \({\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}0\)):

(13.14)\[ {J} _ {fu} ^ {2} =- {\ int} =- {\ int} _ {{\ mathrm {\ Gamma}} _ {c}}\ mathrm {\ mu} {\ widehat {\ mathrm {\ lambda}}}} _ {\ int}} _ {\ int}} _ {k}\ mathrm {\ delta}\ mathrm {\ Lambda}}\ mathrm {\ Lambda}} {\ widehat {\ mathrm {\ lambda}} {\ mathrm {\ lambda}} {\ widehat {\ mathrm {\ lambda}} {\ mathrm {\ lambda}} {\ underline {\ underline}} P}}} ^ {\ mathrm {\ tau}}} {\ underline {\ underline {\ underline {\ widehat {P}}}}} _ {k} ^ {B (\ mathrm {0.1})}\ left ({\ mathrm {\ tau}}}}}\ left ({\ mathrm {\ tau}})}\ left ({\ mathrm {\ tau}}}}\ left ({\ mathrm {\ tau}}}}\ left ({\ mathrm {\ tau}}}}\ left ({\ mathrm {\ tau}}}}\ left ({\ mathrm {\ tau}}}}\ left ({\ mathrm {\ tau}}}}\ left ({\ mathrm {\ tau}} m}\ right)\ right] d {\ mathrm {\ Gamma}} _ {c}\]

This is written, in particular using the discretization of the projection on the unit ball ():

(13.15)\[ \ left [{K} _ {\ mathit {fe}}} ^ {2}\ right]\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {J} _ {c}\ mu {\ c}\ mu {\ stackrel {}}\ mu {\ stackrel {}}\ mu {\ stackrel {}} {\ stackrel {}} {\ stackrel {}} {\ stackrel {}} {\ stackrel {}} {\ lambda}} {\ lambda}} _ {\ left [T\ right]} ^ {T} {\ left [T\ right]} ^ {T} {\ left [T\ right]} ^ {T} {\ left [T\ right]} ^ {T}\ left [{P} ^ {\ tau}\ right]\ right]\ left [{\ stackrel {} {P}}} _ {k} ^ {B}\ right]\ right]\ left [{N}\ right]\ left [{N} ^ {e}\ right] \ left [{K} _ {\ mathit {fe}}} ^ {2}\ right] ^ {2}\ right]\ mathrm {=} + {\ omega} _ {c}\ mu {\ stackrel {} {\ stackrel {}} {\ lambda}} {\ lambda}} {\ lambda}}} _ {\ left [{P} ^ {\ tau}\ right]\ left [{\ stackrel {} {P}}} _ {k} ^ {B}\ right]\ left [{N} ^ {m}\ right]\]

The Jacobian discretization () corresponding to the third column of the global matrix:

(13.16)\[ {J} _ {ff}\ stackrel {\ text {Discretization}} {\ to}\ left [{K} _ {\ mathit {ff}}}\ right]\]

The Jacobian is written:

(13.17)\[\begin{split} \ begin {array} {cc} {J} _ {ff} _ {ff}\ mathrm {=}} &\ frac {1} {{\ stackrel {}} {\ rho}} _ {t}} {\ mathrm {\ int}}} {\ mathrm {\ int}}} _ {\ mathrm {\ int}}} _ {\ int}} _ {\ stackrel}} _ {\ stackrel}} _ {\ stackrel}} {} {S}} _ {u, k} ^ {{h} _ {n}} ^ {{g} _ {n}} (1\ mathrm {-} {S}}} _ {f, k} ^ {{h}} _ {{h} _ {\ h} _ {\ tau}}) (\ underline {\ tau}}) (\ underline {\ tau}}) (\ underline {\ tau}}) (\ underline {\ underline {\ tau}}) (\ underline {\ stackrel {-}} {\ underline {\ underline {\ underline {\ stackrel {1}}}\ mathrm {-} {\ underline {\ underline {\ underline {\ underline {\ stackrel}}} {P}}}} _ {k} ^ {B (\ mathrm {0.1})})\ Delta\ Lambda\ mathrm {\ cdot}\ delta\ Lambda d {\ Gamma} _ {c} _ {c}\\\ &\ mathrm {-} {-} {\ mathrm {-}}} _ {\ Gamma} _ {c}}} (1\ mathrm {-}} (1\ mathrm {-}}\ stackrel {} {S}} _ {u, k} _ {u, k} ^ {{g} _ {n}})\ Delta\ Lambda\ mathrm {\ cdot}\ delta\ Lambda d {\ Lambda d {\ Gamma} _ {\ Gamma} _ {c}\ end {array}\end{split}\]

We prefer to divide it in two. A contribution for the slippery contact case (\({\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1\) and \({\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}0\)):

(13.18)\[ {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:

(13.19)\[ {J} _ {ff} ^ {2}\ mathrm {=}\ mathrm {=}\ mathrm {-} {\ mathrm {\ int}} _ {\ Gamma} _ {c}}\ Delta\ Lambda\ {c}}\ Delta\ Lambda\ {c}}\ Delta\ Lambda d {\ Gamma} _ {c}}\ Delta\ Lambda\ {c}}\]

13.5. 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.

\({G}_{f}={\int }_{{\mathrm{\Gamma }}_{c}}\mathrm{\mu }{\widehat{\mathrm{\lambda }}}_{k}{\widehat{S}}_{u,k}^{{g}_{n}}\left(1-{\widehat{S}}_{f,k}^{{h}_{\mathrm{\tau }}}\right){\widehat{\mathrm{\tau }}}_{k}\cdot {\mathrm{\delta }d}_{\mathrm{\tau }}d{\mathrm{\Gamma }}_{c}\)

So far, relative slippage has been approximated as follows:

\[\]

: 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 \(\mathrm{\Delta }\) by \(\mathrm{\delta }\) 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:

(13.20)\[ 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:

\(\frac{\partial [{y}_{\mathit{escl}}-{y}_{\mathit{mait}}]}{\partial t}\mathrm{\Delta }t-\frac{\partial {y}_{\mathit{mait}}}{\partial {\mathrm{\zeta }}^{\mathrm{\alpha }}}\mathrm{\Delta }{\mathrm{\zeta }}^{\mathrm{\alpha }}=\frac{\partial {g}_{n}n}{\partial t}\mathrm{\Delta }t-\frac{\partial {y}_{\mathit{mait}}}{\partial {\mathrm{\zeta }}^{\mathrm{\alpha }}}\mathrm{\Delta }{\mathrm{\zeta }}^{\mathrm{\alpha }}=\frac{\partial {g}_{n}}{\partial t}\mathrm{\delta }tn+{g}_{n}\frac{\partial n}{\partial t}\mathrm{\Delta }t-\frac{\partial {y}_{\mathit{mait}}}{\partial {\mathrm{\zeta }}^{\mathrm{\alpha }}}\mathrm{\Delta }{\mathrm{\zeta }}^{\mathrm{\alpha }}\)

We are going to note:

\({v}_{n}=\frac{\partial {g}_{n}}{\partial t}\mathrm{\Delta }tn\) and \({v}_{\mathrm{\tau }}={g}_{n}\frac{\partial n}{\partial t}\mathrm{\Delta }t-\frac{\partial {y}_{\mathit{mait}}}{\partial {\mathrm{\zeta }}^{\mathrm{\alpha }}}\mathrm{\Delta }{\mathrm{\zeta }}^{\mathrm{\alpha }}\)

So the continuous writing of relative speed is:

(13.21)\[ v= {v} _ {n} + {v} _ {\ mathrm {\ tau}}\]

Here we are interested in the writing of \({v}_{\mathrm{\tau }}\). As a first approximation, and assuming that the point is constantly in contact, we have:

(13.22)\[ {v} _ {\ mathrm {\ tau}}}\ approx\ frac {-\ partial {y} _ {\ mathit {ait}}}} {\ partial {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\]

We can refer to equation (66) to take into account the variation due to normal \({g}_{n}\frac{\partial n}{\partial t}\). 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 \({g}_{n}\frac{\partial n}{\partial t}\), 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 \(\mathrm{\Delta }{\mathrm{\zeta }}^{\mathrm{\alpha }}\).

(13.23)\[ \ mathrm {\ Delta} {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}} ^ {\ mathrm {\ stackrel {~} {A}}} - {g} _ {a}}} - {g} _ {\ zeta}}}} - {g} _ {\ mathrm {\ alpha}}} - {g} _ {\ mathrm {\ alpha}}} - {g} _ {\ mathrm {\ alpha}}} - {g} _ {\ mathrm {\ alpha}}}} - {g} _ {\ mathrm {\ alpha}}}} - {g} _ {\ mathrm {\ alpha}}}} - {g} _ {\ mathrm {\ alpha}}}} {\ beta}} ^ {-1} [\ frac {\ partial {y}} _ {\ mathit {ait}}}} {\ partial {\ mathrm {\ zeta}}} ^ {\ mathrm {\ beta}}}} (\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} {\ mathrm {\ beta}}}} ^ {\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} (\ mathrm {\ beta}}}}} (\ Mait}}) + {g} _ {n} n\ mathrm {\ Delta}\ frac {\ partial {y} _ {\ mathit {but}}} {\ partial {\ mathrm {\ zeta}}} ^ {\ mathrm {\ zeta}}}} ^ {\ mathrm {\ beta}}}]\]

For the linearization of friction, we will also need the second variation of \(\mathrm{\Delta }\mathrm{\delta }{\mathrm{\zeta }}^{\mathrm{\alpha }}\).

(13.24)\[ \ 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.