11. Appendix A: Calculation of Jacobian terms#
11.1. Equilibrium equation#
For the equilibrium equation, the virtual quantities are total variations with respect to the movements. The notation \(\tilde{\delta }\) will therefore be used. The first row of the Jacobian matrix \(\left[J\right]\) consists of three terms:
: label: eq-302
{J} _ {uu}mathrm {=} {J} _ {J} _ {J}} _ {uu} ^ {text {int}}} ^ {text {int}}} + {J}} _ {J} _ {int}}} + {J} _ {int}} ^ {f} + {J} _ {uu}} ^ {f} {J} _ {uc}mathrm {=} {J} _ {uc} ^ {c} + {J} _ {uc}} ^ {f} {J} _ {uf}mathrm {=} {J} _ {J} _ {uf} ^ {c} + {J} _ {uf} _ {uf} ^ {f}
We already know that the internal and external forces of bodies will not depend on the contact Lagrangian, nor on the frictional Lagrangian:
: label: eq-303
{J} _ {uc} ^ {text {int}}}mathrm {=} {J} _ {uc} ^ {text {ext}}}mathrm {=} 0
But also that the contact pressure does not depend on the friction pressure:
: label: eq-304
{J} _ {uf} ^ {text {c}}mathrm {=} 0
While the friction pressure depends on the contact pressure (see § 5.4.1):
: label: eq-305
{J} _ {uc} ^ {text {f}}mathrm {ne} 0
These are not simplifying hypotheses, but the logical consequence of writing the physical model.
11.1.1. Second variation with respect to travel#
In this paragraph, the second variation will be understood as total variation with respect to trips \({u}^{i}\). It will be rated \({\Delta }_{{u}^{i}}\mathrm{=}\tilde{\Delta }\). We are not going to go back in detail to the calculation of the nonlinear terms \({J}_{uu}^{\text{int}}\) and \({J}_{uu}^{\text{ext}}\) corresponding to the internal forces \({G}_{\text{int}}^{i}\) and external forces \({G}_{\text{ext}}^{i}\) since they concern the behavior and the kinematics of solids without frictional contact. For example, the variation in the work of internal forces introduces tangent matrices \({\underline{\underline{K}}}_{t}^{i}\) such as:
: label: eq-306
{J} _ {uu} ^ {text {int}}mathrm {int}}mathrm {=}tilde {Delta} ({G} _ {text {int}} ^ {i})mathrm {=} {=} {delta u}} {delta u}} {delta u}} {delta u}} {delta u} ^ {i}} {delta u} ^ {i}}
We start by evaluating \({J}_{uu}^{\text{c}}\) written with the normal game variation:
Trivially:
Expanding on the expression:
: label: eq-309
{J} _ {uu} ^ {text {c}}}mathrm {=}mathrm {-} {mathrm {int}} _ {Gamma} _ {c}} {S}} {S}} {S} _ {u} _ {u}} ^ {u} ^ {u} ^ {u} ^ {u} ^ {u} ^ {g} _ {g} _ {g} _ {g}} {g} {g}} {g} {g}} {g} {g} {g}} {g} {g}} {g} _ {g}} {g} _ {g}} {g} _ {g}} {g} {g}} {g} {g}} {g} {g}} {g} {g}} d {Gamma} _ {c} + {mathrm {int}}} _ {{Gamma} _ {c}} {S} _ {u} ^ {{g} _ {n}} {rho}} _ {rho} _ {n} _ {n} _ {n} d {rho} _ {n}} _ {c} _ {c}
Let’s go back to the expressions established in § 2.3. The first total variation of the normal game is (see ()):
: label: eq-310
tilde {delta} {d} _ {n}mathrm {=} ({delta} _ {t} xmathrm {-} {delta} _ {t}overline {x})mathrm {cdot} n
And so, by applying ():
: label: eq-311
tilde {delta} {d} _ {n}mathrm {=} ({delta} _ {t} umathrm {-} {delta} _ {t}overline {u})mathrm {cdot} n
The second variation is much more complex (see ()):
: label: eq-312
begin {array} {cc}tilde {Delta}tilde {Delta}tilde {delta} {d} _ {n}delta} _ {delta} _ {t} {d} _ {d} _ {d} _ {d} _ {d} _ {n} _ {n}mathrm {=}} &mathrm {-} nmathrm {-} nmathrm {cdot} (Delta {zeta}}} _ {d} _ {d} _ {d} _ {d} _ {d} _ {d} _ {n}mathrm {=}} alpha}frac {mathrm {partial} ({delta} _ {t}overline {x})} {mathrm {partial} {zeta} _ {alpha}}} +frac {mathrm {partial} ({mathrm {partial}} ({mathrm {partial}} {zeta})} {mathrm {partial}} {zeta} _ {alpha}}delta {zeta} _ {alpha} _ {alpha})\ &mathrm {-}delta {zeta} _ {alpha} {kappa} _ {alphabeta}}\ beta}\ zeta} _ {zeta} _ {beta} _ {n} (nmathrm {cdot}}frm ac {mathrm {partial} ({delta} _ {t}overline {x})} {mathrm {partial} {partial} {partial} {zeta} _ {alpha}} + {kappa}} _ {alphagamma} _ {alpha}} _ {alpha}} {alphabeta} (nmathrm {cdot}frac {alphadot}frac {alphagamma}}frac {alphagamma}}frac {alphagamma}} {alphabeta} (nmathrm {cdot}}frac {alphagamma}} {alphabeta} (nmathrm {cdot}}frac {alphagamma}}frac {alphagamma}} {alphabeta} (nmathrm {cdot}}frac {alphagamma}} {alphabeta}zeta} _ {beta}} + {kappa} _ {betasigma}delta {zeta} ^ {sigma})end {array})end {array}
We use the metric tensor \(\underline{\underline{m}}\) and the tensor of the second fundamental form \(\underline{\underline{\kappa }}\). All these expressions only occur in case of contact (\({S}_{u}^{{g}_{n}}\mathrm{=}1\)) and therefore when \({d}_{n}\mathrm{=}0\). This assumption simplifies the expression a lot. For (), removing the term in \({d}_{n}\) and with ():
: label: eq-313
tilde {Delta}tilde {delta} {d} {d} _ {d} _ {n}mathrm {=} {delta} _ {t} {d} _ {d} _ {n} _ {n}mathrm {=}mathrm {-} nmathrm {.} (Delta {zeta} _ {alpha}frac {mathrm {partial} ({delta} _ {t}overline {u})} {mathrm {partial} {zeta} {zeta}} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta} _ {thrm {partial} {zeta} _ {alpha}}}delta {zeta}} _ {alpha})mathrm {-}delta {zeta} _ {alpha} {kappa} _ {kappa} _ {kappa} _ {kappa} _ {kappa} _ {kappa} _ {kappa} _ {kappa} _ {kappa} _ {kappa} _ {kappa} _ {kappa} _ {kappa} _ {kappa} _ {kappa}} _ {kappa}
To find \(\delta {\zeta }_{\alpha }\) and \(\Delta {\zeta }_{\alpha }\), you need to solve the following system (contravariant version of ()):
: label: eq-314
{m} ^ {alphabeta}delta {zeta} _ {zeta} _ {zeta} _ {zeta} _ {alphabeta}} {mathrm {partial} {partial} {partial} {zeta}} {zeta}} ^ {alpha}}mathrm {beta}}mathrm {cdot} ({delta}} _ {t} xmathrm {-} {delta} _ {t}overline {x})todelta {zeta})todelta {zeta} _ {alphabeta} ({delta} _ {t})umathrm {-} umathrm {-} {-} {delta} _ {delta} _ {t}overline {u})mathrm {cdot}frac {mathrm {partial}overline {x}} {mathrm {partial} {zeta} ^ {beta}} {m} ^ {alphabeta}Delta {zeta} _ {zeta} _ {zeta} _ {zeta} _ {zeta}} {mathrm {partial} {partial} {partial} {zeta}} {zeta}} ^ {alpha}}mathrm {beta}}mathrm {cdot}} ({delta} _ {t} xmathrm {-} {Delta} _ {t}overline {x})toDelta {zeta})toDelta {zeta} _ {alphabeta} ({Delta} _ {t}) umathrm {-} umathrm {-} {-} {delta} _ {delta} _ {t}overline {u})mathrm {cdot}frac {mathrm {partial}overline {x}} {mathrm {partial} {zeta} ^ {beta}}
We took \({d}_{n}\mathrm{=}0\), which simplifies the expressions for (). Finally, we have:
: label: eq-315
{J} _ {uu} ^ {text {c}}mathrm {=}underset {{J} _ {uu} ^ {text {c}mathrm {,1}}}}} {underset {underbrace {}}} {underbrace {}}} {underbrace {}}} {}} {S}} {S} _ {u} ^ {{g} _ {n}} {n}} {g} _ {g} _ {n}tilde {Delta} {d} _ {Gamma} _ {c}}}}underset {{c}}}}}}underset {{c}}}}underset {c}}} {underset {c}}}}}underset {c}}}}}underset {{c}}}}}underset {{c}}}}}underset {{c}}}}}underset {{c}}}}}underset {{c}}}}}underset {{c}}}}}underset {{c}}}}}underset {{c}}}} {mathrm {int}} _ {{Gamma} _ {Gamma} _ {c}} _ {c}}} {c}} {c}} _ {n}tilde {Delta} {d} {d} _ {d} _ {n}tilde {delta} {d}} _ {n} d {gamma} _ {c}}}
With:
: label: eq-316
begin {array} {cc} {J} _ {uu} _ {uu} ^ {text {c} ^ {text {c}}mathrm {=} & {mathrm {int}}} _ {{Gamma} _ {Gamma} _ {Gamma} _ {\ Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {c}}} _ {c}}} _ {c}}} _ {c}}} _ {c}}} {c}} {c}} {s}} {S} _ {cdot} _ {n} nmathrm {cdot} (({delta} _ {t} umathrm {-} {delta}} {delta} _ {t}overline {u})mathrm {cdot}frac {mathrm {partial}overline {partial}overline {x}}} {overline {x}}} {delta}} {partial}} {partial}overline {x}}} {mathrm {x}}} {mathrm {x}}} {mathrm {partial}}} {frac {beta}}frac {beta}}frac {beta}}frac {beta}} mathrm {partial} ({Delta} _ {t}overline {u})} {mathrm {partial} {zeta} _ {alpha}}) d {gamma} _ {c}{c}\\ & {mathrm {int}}} _ {mathrm {int}}} _ {gamma}} {s} _ {u} ^ {{g} _ {n}}} {g} _ {n} nmathrm {cdot} (frac {mathrm {partial} ({delta} _ {t}overline {u})} {mathrm {partial} {partial} {partial} {zeta}} _ {alpha}}frac { mathrm {partial}overline {x}}} {mathrm {partial} {partial} {zeta} ^ {beta}} {m} _ {alphabeta}mathrm {cdot} ({Delta}}} ({Delta}} _ {t}overline {u})) d {Gamma} _ {c} ({dot}} ({dot}} ({dot}}) ({Delta} _ {t}) d {Gamma} _ {c} ({dot}} ({dot}}) ({Delta} _ {t})}\ & {mathrm {int}}} _ {{Gamma}} _ {Gamma} _ {c}} {s} _ {g} _ {n} _ {n} ({delta} _ {n} ({delta} _ {t} _ {t}delta} _ {t}overline {u})mathrm {cdot} _ {n} ({delta} _ {n}} ({delta} _ {n}) ({delta} _ {n}} ({delta} _ {n}) ({delta} _ {n}} ({delta} _ {n}) ({delta} _ {n}) ({delta} _ {n} _ {n}} ({delta} _ {n} mathrm {partial}overline {x}} {mathrm {partial} {partial} {zeta} ^ {beta}} {m} _ {alphabeta} {kappa} _ {alphabeta} {m}} {m} {m}} _ {alphabeta} _ {alphabeta} _ {alphabeta} {m} _ {alphabeta} {m} _ {alphabeta} {m} _ {alphabeta} {m} _ {alphabeta} {m} _ {alphabeta} {m} _ {alphabeta} {m} _ {alphabeta} {m}} _ {alphabeta} {m}} _ {alphabeta} {m}} {m}zeta} ^ {alpha}}mathrm {cdot} ({delta} _ {t} umathrm {-} {delta}} _ {t}overline { u}) d {Gamma} _ {c}end {array}
And:
: label: eq-317
{J} _ {uu} ^ {text {c}}mathrm {c}mathrm {2}}}mathrm {=} {mathrm}} _ {c}} {S}} {S}} {S} _ {u} _ {u}} ^ {u} ^ {g} _ {g} _ {n}}} {rho}} {rho} _ {n}left{({delta}} _ {delta} _ {c}}} {c}}} {s}} {s}} {s}} {s}} {s}} {s}} {s}} {s}} {s}} {s}} {u} umathrm {-} {delta} _ {t}overline {u})mathrm {cdot} nright}left{nmathrm {cdot} ({Delta} _ {t} _ {t} umathrm {-} umathrm {-} umathrm {-} umathrm {-} {-} {Delta})mathrm {-} {Delta})right} d {Gamma} _ {c}
We will now consider the equilibrium term \({J}_{uu}^{\text{f}}\) corresponds to the reaction for friction:
: label: eq-318
{J} _ {uu} ^ {text {f}}mathrm {=}mathrm {=}mathrm {-} {tilde {Delta}} _ {{u} ^ {i}}} ({G} _ {text {f}}})mathrm {=}}mathrm {-} {tilde {Delta}} _ {{u} ^ {i}}} ({G} _ {text {f}}})mathrm {f}}})mathrm {=}}mathrm {-} {tilde {Delta}} _ {{u} ^ {i}}} ({G} _ {text {f}}})mathrm {f}}} ({mathrm {int}}} _ {{Gamma} _ {c}}}mu {lambda}} _ {n} _ {u} ^ {{g} _ {n}} {S}} {S}} _ {f} _ {f} _ {{f}} ^ {h} _ {tau}} {h} _ {tau}} {h} _ {tau} _ {tau} _ {tau} _ {cdot} {n}}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {_ {tau} d {Gamma} _ {c} _ {c} + {mathrm {int}}} _ {gamma} _ {lambda} _ {n} {S} _ {u} _ {u} _ {u}} ^ {u}} ^ {u}} ^ {tau} _ {u} _ {u}} _ {u}} _ {u}} ^ {tau}} _ {u}} _ {u}} ^ {tau}} _ {u}} ^ {tau}} _ {u}} ^ {tau}} _ {u}} ^ {tau}} _ {u}} ^ {tau}} _ {u}} ^ {tau}} _ {u}} ^ {tau}} _ {u}} ^ {mathrm {cdot} {tilde {delta} d} _ {tau} d {Gamma} _ {c})
Equation () uses the total variation of the tangent game \({\tilde{\delta }d}_{\tau }\) with respect to the displacements. We are going to make simplification hypotheses. We will overlook the second variation of the tangent game, that is to say:
: label: eq-319
tilde {Delta} {tilde {delta} d} _ {tau}mathrm {=} 0
This gives us:
: label: eq-320
{J} _ {uu} ^ {text {f}}mathrm {f}}mathrm {=}mathrm {-} {mathrm {int}} _ {c}}mu {lambda}}mu {lambda}}}mu {lambda}} _ {lambda} _ {lambda} _ {lambda} _ {lambda} _ {lambda} _ {lambda} _ {tau}}tilde {Delta} {h} _ {tau}mathrm {tau}mathrm {cdot} {dot} {delta} d {Gamma} _ {c}mathrm {-} {mathrm {-}} {mathrm {tau}}mathrm {tau} {-}mathrm {-} -} {mathrm {int}}} _ {int}} _ {u} ^ {{g} _ {n}} (1mathrm {-} {S}} _ {f} ^ {{h} _ {tau}})tilde {delta}taumathrm {mathrm {cdot} {cdot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot
We write the variation of the Lagrange friction semi-multiplier \({h}_{\tau }\). Using the kinematic hypothesis ():
: label: eq-321
{h} _ {tau}mathrm {=}}Lambda + {rho} _ {t} {v} _ {tau}mathrm {=}Lambda + {stackrel {} {}} {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}}
And so:
: label: eq-322
tilde {delta} {h} _ {tau}mathrm {=}tilde {delta}Lambda + {stackrel {} {rho}} _ {rho}} _ {t} {mathrm {}}mathrm {}} _ {tau}mathrm {=} {mathrm {}} {mathrm {}} _ {stackrel {=} {mathrm {}} {stackrel {}} {rho}} _ {t} {mathrm {}tilde {delta} umathrm {}}} _ {tau}
So:
: label: eq-323
tilde {delta} {h} _ {tau}mathrm {=} {stackrel {} {rho}} _ {t} {underline {underline {P}}}}}} ^ {tau}}}} ^ {tau}}}} ^ {tau} ({tau}} ({tau}) ({tau} ({tau}) ({delta} _ {t})
In an analogous manner:
: label: eq-324
tilde {delta} {d} _ {tau}mathrm {=} {underline {underline {P}}}} ^ {tau} ({delta} _ {t} umathrm {-} {-} {delta} _ {t}overline {u})
For the variation of the standard semi-multiplier \(\tilde{\Delta }\tau\), which corresponds to the projection on the unit ball, we only consider the sliding variation (\({S}_{f}^{{h}_{\tau }}\mathrm{=}0\)) since the term is cancelled out in the adherent case. We have (see ()):
: label: eq-325
{Delta} _ {{h} _ {tau}} (tau) (tau)mathrm {=} {underline {underline {P}}}} ^ {B (mathrm {0,} 1)}} ^ {B (mathrm {0,} 1)}
By applying the variation of a compound function:
: label: eq-326
tilde {Delta}taumathrm {=} {delta} {=} {delta} _ {tau}} (tau)tilde {delta} {h} _ {tau}mathrm {=} {=} {stackrel {delta}} _ {stackrel {} {rho}}} _ {underline {delta} {h}} _ {underline {P}}} _ {tau}}} ^ {tau}mathrm {=} {=} {stackrel {} {rho}} _ {underline {P}}}} ^ {tau}}} ^ {tau} {tau}}} ^ {tau}mathrm {=} {=} underline {underline {P}}}} ^ {B (mathrm {0,} 1)} ({Delta} _ {t} umathrm {-} {Delta} _ {Delta} _ {t}overline {u})
This gives us:
: label: eq-327
{J} _ {uu} ^ {text {f}} {text {f}}mathrm {=}underset {{J} _ {uu} ^ {text {f}mathrm {,1}}}} {underset {underbrace {}}}} {underset {underbrace {}}}}} {underset {underbrace {}}}} {underset {underbrace {}}} {}} {underbrace {}}} {}} {underbrace {}}} {} {underbrace {}}} {}} {underbrace {}}} {} {underbrace {}}} {}} {underbrace {}}} {underbrace {lambda} _ {n} {S} _ {u} ^ {{g}} ^ {{g} _ {n}}} {S} _ {{h} _ {tau}}}tilde {delta} {h} {h} _ {tau} {h}} _ {tau} {h}}}tilde {delta} {h}}}tilde {delta} {h}}}tilde {delta} {h}}}tilde {delta} {h}}}tilde {delta} {h}}}tilde {delta} {h}}}tilde {Delta} {h}}}tilde {Delta} {h}}}tilde {Delta} {h}}}tilde {{{J} _ {uu} ^ {text {f}}mathrm {2}}}} {underset {underbrace {}} {mathrm {-} {mathrm {int}}} _ {int}}} _ {gamma}} _ {gamma}} _ {c}}}mu {lambda}} _ {n}} (1mathrm {-} {S} _ {f} _ {f} ^ {{h}} _ {tau}})tilde {delta}taumathrm {cdot} {tilde {delta} d} {delta} d {tau}}}
With:
: label: eq-328
{J} _ {uu} ^ {text {f}mathrm {f}mathrm {,1}}mathrm {=}mathrm {int}} _ {{Gamma} _ {c} _ {c}}mu {c}}}mu {lambda}}mu {lambda}}mu {lambda}} _ {lambda} _ {u} ^ {{g} _ {n}} {S} _ {f} _ {f} ^ {{f} ^ {{h}} _ {tau}}left [{underline {P}}}} ^ {tau} ({delta} _ {t} _ {t} umathrm {t}delta}}left [{underline {underline {P}}}} ^ {tau} ({tau} ({delta} _ {t} umathrm {-} {Delta} _ {t}overline {u})right] d {gamma} _ {c}
And:
: label: eq-329
{J} _ {uu} ^ {text {f}mathrm {f}mathrm {2}}mathrm {=}mathrm {int}}} _ {{Gamma} _ {c} _ {c}}mu {c}}}mu {lambda}}mu {lambda}}mu {lambda}} _ {lambda} _ {u} ^ {{g} _ {n}} (1mathrm {-} {S}} {S} _ {f} ^ {{h} _ {tau}})left [{underline {P}}}}} ^ {tau}}}} ^ {tau}}}}} ^ {tau}}}}}} ^ {tau}}}}}} ^ {tau}}}}}} ^ {tau}}}}} ^ {tau}}}}} ^ {tau}}}}} ^ {tau}}}}} ^ {tau}}}}} ^ {tau}}}}} ^ {tau}}}}} ^ {tau}}}}} ^ {tau}}}}} ^ {tau}}}} thrm {cdot}left [{underline {underline {P}}}}} ^ {tau} {underline {underline {P}}} ^ {B (mathrm {0,} 1)}} ({Delta} 1)}} ({Delta} _ {t}overline {u})} (mathrm {0,} 1)}} ({Delta}} 1)} ({Delta}} 1)} ({Delta} _ {t} umathrm {-} {Delta}} _ {t} _ {t}overline {u})right] d {Gamma} _ {c}
11.1.2. Second variation with respect to contact pressure#
Now, the second variation is understood as a change from \({\lambda }_{n}\). We start by evaluating \({J}_{uc}^{\text{c}}\) such as:
: label: eq-330
{J} _ {uc} ^ {text {c}}}mathrm {=}mathrm {=}mathrm {-} {Delta} _ {n}} ({G} _ {text {c}}})mathrm {c}})mathrm {=}mathrm {-} {delta} _ {n}} ({mathrm}} _ {n}} ({mathrm}}int}} _ {{Gamma} _ {c}} {c}} {s}} {S}} {S} _ {u} ^ {n}} {g} _ {n} _ {d} _ {d} _ {n}} d {Gamma} _ {c})
Remember that the variation of the normal game is equal to ():
: label: eq-331
tilde {delta} {d} _ {n}mathrm {=} ({delta} _ {t} umathrm {-} {delta} _ {t}overline {u})mathrm {cdot} n
And:
: label: eq-332
{Delta} _ {{lambda} _ {n}} ({lambda} _ {n})mathrm {=}Delta {lambda} _ {n}
There is left:
: label: eq-333
{J} _ {uc} ^ {text {c}}}mathrm {=}mathrm {-} {mathrm {int}} _ {Gamma} _ {c}} {S}} {S}} _ {u} _ {u} _ {u} ^ {g} _ {n}}} ({delta} _ {n}}} ({delta} _ {t}} umathrm {-} {delta}} {delta}} {delta} _ {t}overline {u})mathrm {cdot} nDelta {lambda} _ {n} d {Gamma} _ {c}
We are now considering evaluating \({J}_{uc}^{\text{f}}\) as:
: label: eq-334
{J} _ {uc} ^ {text {f}}}mathrm {=}mathrm {=}mathrm {-} {Delta} _ {n}} ({G} _ {text {f}}})mathrm {f}})mathrm {=}mathrm {-} {delta} _ {n}} ({mathrm}}} ({mathrm}}int}} _ {{Gamma} _ {c}}}mu {lambda}}mu {lambda} _ {n}} {n}} {S} _ {f} ^ {{h} _ {{h} _ {{h} _ {tau}} _ {tau}} {h} _ {tau}} {h} _ {tau}} {h} _ {tau}} {h} _ {tau}} {h} _ {tau}} {h} _ {tau}} {h} _ {tau}} {h} _ {tau}} {h} _ {tau}} {h} _ {tau}} {h} _ {tau}} {h} _ {tau}} {h} _ {tau}Gamma} _ {c} + {mathrm {int}}} _ {{gamma} _ {c}}mu {lambda} _ {n} {S} _ {u} ^ {{g} _ {n}}}} (1mathrm {int}}}} (1mathrm {int}}}} (1mathrm {int}}})taumathrm {n}}} (1mathrm {int}}}} (1mathrm {int}}}} (1mathrm {int}}}} (1mathrm {int}}}} (1mathrm {int}}}} (1mathrm {- int}}}})} {tilde {delta} d} _ {tau} d {gamma} _ {c})
We assumed (variant 5, see § 5.4.1):
: label: eq-335
{Delta} _ {{lambda} _ {n}} (tau)mathrm {=} 0
And we have:
: label: eq-336
{Delta} _ {{lambda} _ {n}}} {tilde {delta} d} _ {tau}mathrm {=} 0
There is left:
: label: eq-337
{J} _ {uc} ^ {text {f}} {text {f}}mathrm {=}}underset {{J} _ {uc} ^ {text {f}mathrm {,1}}} {underset {underbrace {}}} {underbrace {}}}underbrace {}}} {underbrace {}}}underbrace {}}} {underbrace {}}} {underbrace {}}} {underbrace {}}} {underbrace {}}} {underbrace {}}} {underbrace {}}} {underbrace {}}} {underbrace {}}} {underbrace {}}} {underbrace {_ {u} ^ {{g} _ {n}} {s}} {S} _ {f}} ^ {{h} _ {tau} _ {tau}mathrm {cdot} {tilde {delta} d} {tilde {delta} d}} {tilde {delta} d {delta} d {dot}} {dot} {dot} {dot} {dot} {tilde {delta} d} d {cdot} {dot} {dot} {tilde {delta} d} d {delta} d {c}}} {tilde {delta} d} {dot} {dot} {tilde {delta} d} d {delta} d {c}}} {uc} ^ {text {f}mathrm {,2}}}} {underset {underbrace {}} {mathrm {-} {mathrm {int}} _ {{Gamma} _ {c} _ {c}}}}mu {S}}}}mu {S}}}}mu {S}}} {c}}}mu {S}}}mu {S}}}mu {S}}}mu {S}}}mu {S}}}mu {S}}}mu {S}}}mu {S}}}mu {S}}}mu {S}}}mu {S}}}mu {S}}}mu {S}}}mu {S} {{h} _ {tau}})taumathrm {cdot} {tilde {delta} d} _ {tau}Delta {lambda} _ {lambda} _ {lambda} _ {lambda}} _ {lambda}} _ {n} d {gamma} _ {c}}}
With:
: label: eq-338
{J} _ {uc} ^ {text {f}mathrm {f}mathrm {,1}}mathrm {=}mathrm {int}} _ {{Gamma} _ {c} _ {c}}}mu {S}}}}mu {S}} _ {gamma} _ {c} _ {c}}}left [{underline {underline {P}}}} ^ {tau}} ({delta}} _ {t} umathrm {-} {delta} _ {t}overline {u})right]mathrm {cdot} {h} {h} {h} {h} _ {h} _ {h} _ {h} _ {c}
And:
: label: eq-339
{J} _ {uc} ^ {text {f}mathrm {f}mathrm {2}}}mathrm {=}mathrm {int}} _ {{Gamma} _ {c} _ {c}}}mu {S}}}}mathrm {S} _ {c}}}}mu {S} _ {c}}}}mu {S} _ {c}}}mu {S}}} {c}}}mu {S}}} {c}}}mu {S}}} {c}}}mu {S}}} {c}}}mu {S}}} {c}}}mu {S}}} {c}}}mu {S}}} {c}} h} _ {tau}})left [{underline {underline {underline {P}}}} ^ {tau} ({delta} _ {t} umathrm {-} {delta} _ {t}overline {u}){t}overline {u})right]right]mathrm {cdot}tauDelta {lambda} _ {c} _ {c} _ {c}
11.1.3. Second variation with respect to the friction pressure#
Finally, the second variation will be understood as a variation with respect to \(\Lambda\). The only non-null term is \({J}_{uf}^{\text{f}}\), such as:
: label: eq-340
{J} _ {uf} ^ {text {f}}mathrm {f}}mathrm {=}mathrm {-} {Delta} _ ({G} _ {text {f}})mathrm {=}}mathrm {=}}mathrm {=}}mathrm {-} {-} {Delta} _ {Lambda} ({mathrm {int}}})mathrm {=}}mathrm {=}}mathrm {-} {-} {Delta} _ {Lambda} ({mathrm {int}}}} _ {Gamma} _ {c}}mu {lambda} _ {n} {s} _ {s} _ {u} _ {u} ^ {{g} _ {f} ^ {{h} _ {tau}} {h}} {h} _ {h}} {h} _ {tau} {h}} {h}} {h}} {h}} {h} _ {h} _ {h} _ {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {h}} {thrm {int}} _ {{Gamma} _ {c}}}mu {lambda}}mu {lambda} _ {s} _ {n}} (1mathrm {-} {n}}} (1mathrm {-} {-} {S}} _ {c}})taumathrm {cdot} {n}} (1mathrm {-} {n}}} (1mathrm {-} {-} {S}} {S}} (1mathrm {-} {S}} {S} _ {n}}} (1mathrm {-} {-} {S} _ {S}} _ {delta}} (1mathrm {-} {S}} _ {tau} d {Gamma} _ {c})
Variations of \({h}_{\tau }\) with respect to \(\Lambda\) are as follows:
: label: eq-341
{Delta} _ {Lambda} ({h} _ {tau})mathrm {tau})mathrm {=} {tau} (Lambda + {stackrel {} {rho}} {rho}} _ {rho}} _ {rho}} _ {tau})mathrm {=} {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}}
We have:
: label: eq-342
{Delta} _ {Lambda}taumathrm {=} {mathrm {=} {delta}} _ {tau}} (tau) {Delta} _ {Lambda} ({h} _ {tau})mathrm {tau})mathrm {=} {=} {underline {delta}}}} (tau) {delta} _ {Lambda} ({h} _ {tau}} ({h} _ {tau}) ({h} _ {tau}} ({h} _ {tau}) ({h} _ {tau}) ({h} _ {tau}} ({h} _ {tau}) ({h} _ {tau}) ({Lambda
Likewise, trivially:
: label: eq-343
{Delta} _ {Lambda} {tilde {delta} d} _ {tau}mathrm {=} 0
This gives us:
: label: eq-344
{J} _ {uf} ^ {text {f}} {text {f}}mathrm {=}underset {{J} _ {uf} ^ {text {f}mathrm {,1}}}} {underset {underbrace {}}}} {underset {underbrace {}}}}} {underset {underbrace {}}}} {underset {underbrace {}}} {}} {underbrace {}}}} {underset {underbrace {}}}} {underbrace {}}} {}} {underbrace {}}} {}} {underbrace {}}} {}} {underbrace {}}} {}} {lambda} _ {n} {S} _ {u} ^ {{g}} ^ {{g} _ {n}}} {S} _ {{h} _ {tau}}DeltaLambdamathrm {cdot} {dot} {dot} {dot}} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {tilde {delta} d} _ {uf} _ {uf} _ {uf} _ {uf} ^ {text {f}mathrm {,2}}}} {underset {underbrace {}} {mathrm {-} {mathrm {int}} _ {Gamma} _ {c} _ {c}}}mu {lambda}}}mu {lambda}}}mu {lambda}}}mu {lambda}}mu {lambda}}mu {lambda} _ {lambda} _ {lambda} _ {lambda} _ {lambda}}mu {lambda}} _ {lambda} _ {lambda} _ {lambda}}mu {lambda}} _ {lambda} _ {lambda} _ {lambda} S} _ {f} ^ {{h} _ {tau}}) {tau}}) {underline {tau}}) {underline {P}}} ^ {B (mathrm {0,} 1)}}mathrm {cdot}} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {dot} {tilde {delta} d} _ {delta} d} _ {delta} d} _ {tau}DeltaLambda d {gamma}
With:
: label: eq-345
{J} _ {uf} ^ {text {f}mathrm {f}mathrm {,1}}mathrm {=}mathrm {int}} _ {{Gamma} _ {c} _ {c}}}mu {c}}}mu {lambda}}}mu {lambda}}}mu {lambda}}mu {lambda}} _ {lambda} _ {lambda} _ {f}}}mu {lambda} _ {lambda} _ {f}} {lambda} _ {f}} {lambda} _ {f}} {{h} _ {tau}}DeltaLambdamathrm {cdot}left [{underline {underline {P}}}} ^ {tau} ({delta} _ {t} umathrm {-} umathrm {-}mathrm {-}cdot}dot}left [{t}underline {P}}}}} ^ {tau} ({tau}) ({tau} ({delta} _ {t}} umathrm {-}) umathrm {-} {-} {delta} _ {c}
And:
11.2. Contact law#
For the law of contact (in weak form) the virtual quantities are variations with respect to the contact pressure \({\lambda }_{n}\). The notation \(\delta\) will therefore be used in this sense. The second row of the Jacobian matrix \(\left[J\right]\) consists of three terms:
: label: eq-347
{J} _ {cu} {J} _ {cc}
As the contact pressure does not depend on the friction pressure, we already know that:
11.2.1. Second variation with respect to trips#
In this paragraph, the second variation will always be understood as total variation with respect to movements \({u}^{i}\). It will be rated \({\Delta }_{{u}^{i}}\mathrm{=}\tilde{\Delta }\). We start by evaluating \({J}_{cu}\) such as:
: label: eq-349
{J} _ {cu}mathrm {=} {Delta} _ {{u} ^ {i}}} ({tilde {G}}} _ {text {c}})mathrm {=}tilde {Delta}}tilde {Delta}} (tilde {Delta}} (mathrm {Delta}} (mathrm {-}}frac {-}}frac {1} {rho} _ {n}}} {text {c}})mathrm {=}}tilde {Delta}}tilde {Delta}} (mathrm {Delta}} (mathrm {-}}frac {-}} {rho} _ {n}} _ {{Gamma} _ {c}}left{{lambda}} _ {n}mathrm {-} {S} _ {u} ^ {{g} _ {n}} {g}} {g} _ {g} _ {g} _ {g}} _ {g}} _ {g}} _ {g}} _ {c} _ {c})
We have \(\tilde{\Delta }{\delta \lambda }_{n}\mathrm{=}0\). So all that’s left is the term in \(\tilde{\Delta }{d}_{n}\) whose expression is given by ():
: label: eq-350
{J} _ {cu}mathrm {=}mathrm {-}mathrm {-} {mathrm {int}} _ {c}} {S} _ {u} ^ {{g} _ {g} _ {n} _ {n}}} {n}} {deltalambda}} {deltalambda} _ {n} nmathrm {cdot} ({delta}} _ {t} umathrm {-} {Delta} _ {t}overline {u}) d {Gamma} _ {c}
It is immediately observed that \({J}_{cu}\mathrm{\ne }{J}_{uc}\) because of (). In friction, the matrix is not symmetric.
11.2.2. Second variation with respect to contact pressure#
The second variation will be understood as a variation with respect to \({\lambda }_{n}\). We calculate \({J}_{cc}\) such that:
: label: eq-351
{J} _ {cc}mathrm {=} {Delta} _ {delta} _ {lambda} _ {n}} ({tilde {G}}} _ {text {c}})mathrm {=} {delta}} {delta} _ {delta} _ {delta} _ {n} _ {n} _ {n}} (mathrm {-}}frac {1} {rho} _ {n}}} {mathrm {int}} _ {{Gamma} _ {c}}}left{{lambda} _ {n}mathrm {-} {S} _ {u} ^ {{g} _ {g} _ {n} _ {n}}} ({lambda}}} ({lambda} _ {n}})right} {deltalambda} _ {n} d {Gamma} _ {c})
The second variation \({\Delta }_{{\lambda }_{n}}{\delta \lambda }_{n}\mathrm{=}0\) is zero and \({\Delta }_{{\lambda }_{n}}({\lambda }_{n})\mathrm{=}\Delta {\lambda }_{n}\), so:
Note that this term only occurs when there is no contact (\({S}_{u}^{{g}_{n}}\mathrm{=}0\)).
11.2.3. Second variation with respect to the friction pressure#
The second variation will be understood as a variation with respect to \(\Lambda\). The term \({J}_{cf}\) sucks:
: label: eq-353
{J} _ {cf} = {mathrm {Delta}}} _ {mathrm {Lambda}}left ({stackrel {~} {G}}}} _ {text {c}}right) =0
11.3. Law of friction#
For the law of friction (in weak form) the virtual quantities are variations with respect to the friction pressure \(\Lambda\). The notation \(\delta\) will therefore be used in this sense. The third row of the Jacobian matrix \(\left[J\right]\) consists of three terms:
: label: eq-354
{J} _ {fu} {J} _ {fc}
11.3.1. Second variation with respect to trips#
In this paragraph, the second variation will always be understood as total variation with respect to movements \({u}^{i}\). It will be rated \({\Delta }_{{u}^{i}}\mathrm{=}\tilde{\Delta }\). We start by evaluating \({J}_{fu}\) such as:
: label: eq-355
begin {array} {cc} {J} _ {fu}mathrm {=} {Delta} _ {{u} ^ {i}}} ({tilde {G}} _ {text {f}})mathrm {f}})mathrm {=}}mathrm {=}} & +tilde {=}} & +tilde {Delta}} (frac {1}} {stackrel {}} {rho}})mathrm {=}} & +tilde {Delta} (frac {1}} {stackrel {}} {rho}})mathrm {=}} & +tilde {Delta}} (frac {1}} {}} {mathrm {int}}} _ {{Gamma} _ {Gamma} _ {c}}mu {lambda} _ {s} _ {n}}}Lambdamathrm {cdot}Lambdamathrm {cdot}\mathrm {cdot}\mathrm {cdot}\mathrm {cdot}\ delta}deltaLambda d {lambda} _ {c})\ &mathrm {-}}tilde {Delta} (frac {1} {{stackrel {} {} {rho}}} _ {t}} {mathrm {int}} _ {c}}mu {lambda}}mu {lambda} _ {n}} {lambda} _ {lambda} _ {lambda} _ {lambda} _ {lambda} _ {lambda} _ {lambda} _ {lambda} _ {lambda} _ {tau} _ {tau}} h} _ {tau}mathrm {cdot}deltaLambda d {Gamma} _ {c})\ &mathrm {-}tilde {Delta} (frac {1} {frac {1}} {{stackrel {1}} {{stackrel {}} {stackrel {}} {stackrel {} {} {rho}}} _ {rho}} _ {rho}} _ {c}} mu {lambda} _ {n} {S} _ {u} _ {u}} ^ {{g} _ {n}}} (1mathrm {-} {S} _ {{h} _ {tau}})taumathrm {tau}}})taumathrm {tau}}})taumathrm {-}})taumathrm {-}}tilde {Delta} ({mathrm {int}}} _ {{Gamma}} _ {c}} (1mathrm {-} {S} _ {g} _ {n}})Lambdamathrm {int}})Lambdamathrm {int}}} _ {n}})Lambdamathrm {cdot}deltaLambdaLambdaLambda d {Gamma} _ {c})end {array}}
We reuse ():
: label: eq-356
tilde {Delta}taumathrm {=} {stackrel {} {rho}} _ {t} {underline {underline {P}}} ^ {tau} {tau} {tau} {underline {=} {=} {underline {P}}}} ^ {B (mathrm {0,} 1)}} ^ {delta} _ {t} umathrm {-} {Delta} _ {t}overline {u})
We also have \(\tilde{\Delta }\delta \Lambda \mathrm{=}0\) and \(\tilde{\Delta }\Lambda \mathrm{=}0\). So:
With:
: label: eq-358
{J} _ {fu} ^ {1}mathrm {=}mathrm {=}mathrm {-} {mathrm {int}} _ {gamma} _ {lambda}}mu {lambda} _ {n}} _ {n}} {gamma} _ {c}}}lambda} _ {lambda}} _ {lambda} _ {lambda} _ {lambda} _ {n}} _ {tau}}deltaLambdamathrm {cdot}left [{underline {underline {P}}}} ^ {tau} ({Delta} _ {t} umathrm {-} {Delta}} {Delta} _ {Delta} _ {Delta} _ {Delta} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c} _ {c}
And:
: label: eq-359
{J} _ {fu} ^ {2}mathrm {=}mathrm {=}mathrm {-} {mathrm {int}} _ {gamma} _ {lambda}}mu {lambda} _ {2} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f}} _ {f} _ {f}} _ {f}} _ {f} {s}} _ {f}} _ {f} {s}} _ {f}} _ {f} {s}} _ {f}} _ {f} {s}} _ {f} {f} {s}} _ {f} {s}} _ {f} {s}} _ {f}} _ {f} {s}} _ {tau}})deltaLambdamathrm {cdot}left [{underline {underline {P}}}} ^ {tau} {underline {underline {P}}}}} ^ {underline {P}}}}} ^ {underline {P}}}} ^ {underline {P}}}}} ^ {underline {P}}}} ^ {underline {P}}}} ^ {underline {P}}}} ^ {underline {P}}}} ^ {underline {P}}}} ^ {underline {P}}}} ^ {underline {P}}}} ^ {underline {P}}}} ^ {underline {P}}overline {u})right] d {Gamma} _ {c}
11.3.2. Second variation with respect to contact pressure#
The second variation will be understood as a variation with respect to \({\lambda }_{n}\). We start by evaluating \({J}_{fc}\) such as:
: label: eq-360
{J} _ {fc}mathrm {=} {Delta} _ {lambda} _ {n}} ({tilde {G}}} _ {text {f}})
Using variant 5 (see § 5.4.1), we have \({\Delta }_{{\lambda }_{n}}({\tilde{G}}_{f})\mathrm{=}0\), which implies \({J}_{fc}\mathrm{=}0\).
11.3.3. Second variation with respect to the friction pressure#
The second variation will be understood as a variation with respect to \(\Lambda\). We start by evaluating \({J}_{ff}\) such as:
: label: eq-361
begin {array} {cc} {J} _ {ff} _ {ff}mathrm {=} {Delta} _ {Lambda} ({tilde {G}}} _ {text {f}})mathrm {=}} | {=} & + {Delta} & + {Delta}} & + {Delta} _ {t} _ {t}} {mathrm {int}} _ {{Gamma} _ {Gamma} _ {c}}mu {lambda} _ {s} _ {u} ^ {{g} _ {n}}}Lambdamathrm {n}}}Lambdamathrm {n}}}\ mathrm {-}}Lambdamathrm {n}}}Lambdamathrm {n}}}Lambdamathrm {n}}}Lambdamathrm {n}}}Lambdamathrm {n}}}Lambdamathrm {n}}}Lambdamathrm {n}}}Da} (frac {1} {{stackrel {} {} {rho}}} _ {t}} {mathrm {int}} _ {gamma} _ {c}}mu {lambda}}mu {lambda}} _ {lambda} _ {lambda} _ {lambda} _ {lambda} _ {lambda} _ {lambda} _ {lambda} _ {n} {S}} _ {tau}} {h} _ {tau}mathrm {cdot}deltaLambda d {Gamma} _ {c})\ &mathrm {-} {Delta} _ {Lambda} _ {Lambda} (frac {1}} (frac {1}} {frac {1}} {frac {1}} {{1}} {{stackrel {}} {stackrel {}} {rho}} _ {t}} {delta} _ {delta} _ {delta} _ {delta} _ {delta} _ {delta} _ {delta}} _ {delta} _ {delta}} _ {delta} _ {{Gamma} _ {c}}mu {lambda}} _ {n} {lambda} _ {lambda} _ {n}} (1mathrm {-} {S} _ {f} _ {f}} ^ {lambda}} ^ {{h}} _ {lambda} _ {f} ^ {h} _ {c}})taumathrm {cdot}deltaLambda d {Gamma} _ {c})&mathrm {-} {Delta} _ {Lambda} _ {Lambda} ({mathrm {int}}} _ {c}} (1mathrm {-} {S} _ {s} _ {u} _ {u} _ {n}})Lambdamathrm {cdot}deltaLambda d {Gamma} {S} _ {s} _ {c})end {array}
Variations of \({h}_{\tau }\) with respect to \(\Lambda\) are as follows ():
: label: eq-362
{Delta} _ {Lambda} ({h} _ {tau})mathrm {tau})mathrm {=} {tau} (Lambda + {stackrel {} {rho}} {rho}} _ {rho}} _ {rho}} _ {tau})mathrm {=} {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}} _ {rho}}
The contact pressure does not depend on the friction pressure, so:
We also have ():
: label: eq-364
{Delta} _ {Lambda}taumathrm {=} {underline {underline {P}}}} ^ {B (mathrm {0,} 1)}taumathrm {0,} 1)}}DeltaLambda
Finally:
: label: eq-365
begin {array} {cc} {J} _ {ff} _ {ff}mathrm {=}mathrm {=}} &frac {1} {{stackrel {}} {rho}} _ {mathrm {int}}} {mathrm {int}}} _ {mathrm {int}}} _ {int}}} _ {mathrm {int}}} _ {int}}} _ {mathrm {int}}} _ {int}}} _ {mathrm {int}}} _ {int}}} _ {mathrm {int}}} _ {int}} _ {int}} _ {int}} _ {int} n}} (1mathrm {-} {S} _ {f} ^ {{h} _ {tau}}) (underline {underline {1}}mathrm {-} {underline {underline {underline {P}}}}}} ^ {b (mathrm {0,} 1)})DeltaLambdamathrm {-}} {underline {underline {underline {P}}}}} ^ {B (mathrm {0,} 1)})DeltaLambdamathrm {-}} {underline {underline {underline {P}}}}} ^ {B (mathrm {0,} 1)})DeltaLambda\ Lambda d {Gamma} _ {c}\&mathrm {-} {mathrm {int}} _ {{Gamma} _ {c}} (1mathrm {-} {S}} {S} _ {S} _ {S} _ {S} _ {S} _ {S} _ {S} _ {S} _ {S} _ {S} _ {S} _ {s} _ {S} _ {s} _ {S} _ {s} _ {S} _ {s} _ {S} _ {S} _ {S} _ {S} _ {S} _ {S} _ {S} _ {S} _ {s} _ {c}end {array}
There is only a contribution when the contact is sliding and when the contact is inactive.