Hybrid continuous touch/friction formulation ================================================== To express the balance of the structure, taking into account the contact and friction, we have two ways of proceeding: 1. The classical form of the principle of virtual work is expressed by considering contact as a surface interaction effort (it is integrated into the expression of the work of internal forces); To complete the system, we add the weak expression of the laws of touch-friction; 2. We write the energy of the system directly in the form of mixed variational inequality (without constraining the solution space by the contact/friction conditions). The expression of the criteria of optimality through the use of augmented Lagrangian (saddle point problem) makes it possible to find the system established from PTV; Spaces ------- Hilbert space :math:`{H}^{1}`, a complete vector space, usually the Sobolev space provided with an appropriate norm (the dot product), is the space of measurable square functions :math:`{L}^{2}` and whose derivative in the weak sense also belongs to :math:`{L}^{2}`. The dual space of :math:`{H}^{1}` will be marked :math:`{H}^{\mathrm{-}1}`. :math:`{\mathit{CA}}^{i}` refers to the space containing the kinematically admissible functions such as: .. math:: :label: eq-143 {\ mathit {CA}} ^ {i}\ mathrm {=}\ mathrm {=}\ left\ {{u} ^ {u} ^ {1} ({\ Omega} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i}); {u} ^ {i} :math:`{H}^{1\mathrm{/}2}(\Gamma )` is the space for function traces (belonging to :math:`{H}^{1}`) on border :math:`\Gamma` and :math:`{H}^{\mathrm{-}1\mathrm{/}2}(\Gamma )` is its dual space. To simplify, we will note :math:`H\mathrm{=}{H}^{\mathrm{-}1\mathrm{/}2}(\Gamma )` and :math:`H\mathrm{=}{({H}^{\mathrm{-}1\mathrm{/}2}(\Gamma ))}^{2}`, its two-dimensional counterpart (for friction). .. _RefNumPara__12971_1910384451: Principle of virtual work ----------------------------- The equations for body balance :math:`{B}^{i}` are: .. math:: : label: eq-144 \ mathrm {\ {}\ begin {array} {ccc} {\ text {div}} {\ text {div}}\ underline {\ pi}}} ^ {i} + {f} _ {v} ^ {i}\ mathrm {=} {i}\\}\\}\\}}\ {i}\\ {i}\\\ underline {i}\\\\ underline {i}\\\\\\ underline {i}\\\\\\ underline {\}\\ 0&\ text {in} {\ text {in} {\ omega} _ {0} ^ {i} & (a)\\ {\ underline {\ underline {\ pi} & (a)\\ {\ underline {\ underline {\ Pi}}}} ^ {i} {N} ^ {i}\ mathrm {=} {i}\ mathrm {=} {f} _ {i} &\ text {on} {\ Gamma} _ {g} ^ {i} & (b)\\ {i} & (b)\ {u} {i} & (c)} ^ {i} & (c)\ ^ {i} & (c)\ {i} {\ underline {\ underline {\ Pi}}}} ^ {i} {i} {N} {N} {N} ^ {i}\ mathrm {=} {\ lambda} ^ {i} &\ text {on} {\ Gamma}} {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {c} ^ {i} & (d)\ end {array} Where :math:`{\underline{\underline{P}}}^{i}` refers to the first *Piola-Kirchhoff* stress tensor* (non-symmetric), :math:`{u}^{i}` the displacement field and :math:`{\lambda }^{i}` the force density due to the friction contact interactions between the two solids. .. _RefNumPara__45803_622701088: Virtual work of internal forces ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The principle of virtual work (PTV) makes it possible to express the bilinear form of internal forces: .. math:: : label: eq-145 {G} _ {\ text {int}} ^ {i}} ^ {i}\ mathrm {=} {\ mathrm {\ int}} _ {{\ omega} _ {0} ^ {i}} ({\ underline {\ underline {i}}}} ^ {i}} ^ {i}} ^ {i}\ mathrm {:} {\ underline {F}}}} ^ {i}\ mathrm {:} {\ underline {F}}}} ^ {i}\ mathrm {:} {\ underline {F}}}} ^ {i}\ mathrm {:} {\ underline {F}}}} ^ {i}\ mathrm {:} {\ underline {F}}}} {\ underline {F}}} Underline {\ mathit {grad}}}}\ tilde {u}} ^ {i}) by\ Omega Where :math:`{\stackrel{̃}{u}}^{i}` refers to the field of virtual travel. :math:`{\underline{\underline{S}}}^{i}` is the second Piola-Kirchhoff stress tensor, symmetric and purely Lagrangian, linked to the first tensor :math:`{\underline{\underline{\Pi }}}^{i}` by the relationship: .. math:: :label: eq-146 {\ underline {\ underline {S}}}} ^ {i}\ mathrm {=} {({\ underline {\ underline {F}}}} ^ {i})}} ^ {\ mathrm {-}} 1} {\ mathrm {-} 1}} {\ mathrm {-} 1} {\ underline {=}} {\ underline {\ pi}}} ^ {i})} ^ {i} .. _RefNumPara__45805_622701088: Virtual work of external efforts ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The density of the volume forces :math:`{f}_{v}^{i}` and surface forces :math:`{f}_{s}^{i}` applied to the two solids produce a work :math:`{G}_{\text{ext}}^{i}`: .. math:: : label: eq-147 {G} _ {\ text {ext}}} ^ {i}\ mathrm {=} {\ mathrm {\ int}} _ {{\ omega} _ {0} ^ {i}} {f} _ {v} _ {v}} ^ {v} ^ {i} ^ {i}) {u} ({u} ^ {i}}) {\ tilde {u}}} ^ {i} _ {{\ Gamma} _ {g} ^ {i}} {f}} {f} _ {s} ^ {i} ({u} ^ {i}) {\ tilde {u}}} ^ {i}} ^ {i} d\ Gamma Virtual work of touch-friction forces ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ :math:`{G}_{\text{cf}}^{i}` is the work of the touch-friction forces on the solid :math:`{B}^{i}`: .. math:: : label: eq-148 {G} _ {\ text {cf}} ^ {i}} ^ {i}\ mathrm {=} {\ mathrm {\ int}} _ {{\ Gamma} _ {c} ^ {i}} {\ lambda}} {\ lambda}} ^ {i}}} {\ lambda}} ^ {i}}} {\ lambda} ^ {i}}} {\ lambda} ^ {i}}} {\ lambda} ^ {i}} ^ {i}} {i}} {\ lambda} ^ {i}} ^ {i}} {i} d\ Gamma :math:`{\lambda }^{i}` is the density of the touch-friction forces experienced by the solid :math:`{B}^{i}`. Using the matching procedure, described in sub-paragraph § :ref:`2.2.1 `, the *principle of action and reaction* is written locally, on the initial configurations in the following form: .. math:: : label: eq-149 {\ lambda} ^ {1} ({p} ^ {1}, t) d {\ Gamma} _ {c} ^ {1} + {\ lambda} ^ {2} (\ stackrel {} {p} {p}, t) d {\ Gamma} _ {c} ^ {2}\ mathrm {=} 0 We set :math:`{\Gamma }_{c}\mathrm{=}{\Gamma }_{c}^{1}` and :math:`{\lambda }^{1}\mathrm{=}\lambda` and we can therefore take: .. math:: :label: eq-150 {\ lambda} ^ {1} d {\ Gamma} _ {c}} _ {c} ^ {1}\ mathrm {=}\ mathrm {-} {\ lambda} ^ {2} d {\ Gamma} _ {c} _ {c} _ {c} _ {c} This gives us a new form of the contact reaction term: .. math:: : label: eq-151 {G} _ {\ text {cf}} ^ {i}}\ to {G}}\ to {G} _ {\ text {cf}}\ mathrm {\ int}} _ {{\ Gamma} _ {c}}}\ lambda ({\ tilde {u}}}}\ lambda ({\ tilde {u}}} {c}}}}\ lambda ({\ tilde {u}}} {c}}}}\ lambda ({\ tilde {u}}}}}\ lambda ({\ tilde {u}}}} {c}}}\ lambda ({\ tilde {u}}}}\ lambda ({\ tilde {u}}}} {c}}}\ lambda ({\ tilde {u}}}}\ Gamma} _ {c}\ mathrm {=} {\ mathrm {\ int}}} _ {{\ Gamma} _ {c}}\ lambda\ mathrm {}\ tilde {u}\ tilde {u}\ mathrm {u}\ mathrm {\ int}}\ mathrm {\ int}}} _ {c} :math:`{\tilde{u}}^{i}` are the virtual displacement fields, kinematically admissible on each of the solids. In addition, we asked :math:`{\Gamma }_{c}={\Gamma }_{c}^{1}` and :math:`{\lambda }^{1}\mathrm{=}\lambda`. It is reported that the force density :math:`{\lambda }^{2}` is extended by zero at the points of :math:`{\Gamma }_{c}^{2}` without opposite to :math:`{\Gamma }_{c}^{1}`. The contact force density :math:`\lambda` is decomposed into a normal part :math:`{\lambda }_{n}\mathrm{.}n` and a tangential part :math:`{\lambda }_{\tau }` such that: .. math:: :label: eq-152 \ lambda\ mathrm {=} {\ lambda} _ {\ tau} + {\ lambda} _ {n} n By projecting onto the unit ball (see () in § :ref:`3.2.4 `), we have: .. math:: : label: eq-153 {\ lambda} _ {\ tau}\ mathrm {=}\ mu {\ lambda} _ {n}\ Lambda The reaction is discontinuous and takes two values (contact or not). We use the :math:`{S}_{u}^{{g}_{n}}` sign field introduced in § :ref:`3.1.4 `: .. math:: : label: eq-154 \ lambda\ mathrm {=} {S} _ {u} _ {u} ^ {{g} _ {n}}} ({\ lambda} _ {n} n+\ mu {\ lambda} _ {n}\ Lambda) The friction semi-multiplier is also discontinuous and takes two values (sliding or adherent). Using the :math:`{S}_{f}^{{h}_{\tau }}` sign field introduced in § :ref:`3.2.5 `: .. math:: : label: eq-155 \ Lambda\ mathrm {=} {S} _ {f} _ {f} ^ {{h}} _ {\ tau}}\ Lambda + (1\ mathrm {-} {S} _ {f} _ {{h}} _ {\ tau}})\ tau By putting the equation back into the equation: .. math:: : label: eq-156 \ lambda\ mathrm {=} {S} _ {u} _ {u} ^ {{g} _ {n}}}\ left\ {{\ lambda} _ {n} n+\ mu {\ lambda} _ {n} _ {n}} ({S}} _ {f}} ^ {f}} ^ {f} ^ {{f}} ^ {{f}} ^ {{tau}}}\ Lambda + (1\ mathrm {-} {} n+\ mu {\ lambda}} _ {n}} _ {n}} {n} _ {n}} {n}} (n) {n}} _ {n}} (n)} {n} _ {n}} (n)} {n}} ({n}}} {n}} h} _ {\ tau}})\ tau)\ right\} We are going to add here the term called "stabilization", which will make it possible to find the behavior of a standard augmented Lagrangian. The idea is as follows, in case of contact with adhesion (:math:`{S}_{u}^{{g}_{n}}\mathrm{=}1` and :math:`{S}_{f}^{{h}_{\tau }}\mathrm{=}1`), the contact friction reaction () is written: .. math:: : label: eq-157 \ lambda\ mathrm {=} {\ lambda} _ {n} n+\ mu {\ lambda} _ {n}\ Lambda We can modify the writing of the contact/friction reaction: .. math:: : label: eq-158 \ lambda\ mathrm {=} {S} _ {u} _ {u} ^ {{g} _ {n}}}\ left\ {({\ lambda} _ {n}\ mathrm {-} {\ rho} _ {\ rho} _ {n}} _ {n}) _ {n}) n+\ mu {\ g} _ {n}}}\ left\ {{{S} _ {f} {\ rho} _ {f} {\ rho} _ {f} {n} _ {f} {d} _ {h} _ {n}) {\ tau}} {h} _ {\ tau} + (1\ mathrm {-} {S} _ {f} ^ {{h} _ {\ tau}})\ tau\ tau}})\ tau\ tau\ tau}}\ tau\ tau\ tau} This addition does not change anything in the case of the adherent contact, because, in this case, :math:`{d}_{n}\mathrm{=}0` and :math:`{v}_{\tau }\mathrm{=}0`, which makes it possible to find (). On the other hand, in the other cases (no contact or sliding contact), the parameters :math:`{\rho }_{n}` and :math:`{\rho }_{t}` make it possible to avoid zero on the diagonal of the matrix. Finally, the contact friction reaction term in PTV can be broken down into a component for contact: .. math:: : label: eq-159 {G} _ {\ text {c}}\ mathrm {=} {=} {\ mathrm {\ int}} _ {\ Gamma} _ {c}} {S} _ {u} ^ {{g} _ {n} _ {n}} {n}} {g}} {n} {g}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {n}} {g}} {n}} {n}} {n}} {n}} {g}} {n}} {n}} {n}} {n}} {g}} {n}} {n}} {n}} {{c} And a component for friction: .. math:: :label: eq-160 \ begin {array} {cc} {G} _ {\ text {f}} _ {\ text {f}}}\ mathrm {=} & {\ mathrm {\ int}} _ {c}}\ mu {\ lambda}}\ mu {\ lambda} _ {n}} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lamb da} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ tau}} _ {\ tau}} h} _ {\ tau}\ mathrm {\ cdot} {\ mathrm {\ cdot} {\ dot} {\ dot}} {\ mathrm {\ tau} d {\ gamma} _ {c} +\\ & {\ cdot} +\\ & {\ mathrm {\ cdot}} {\ mathrm {\ dot}} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} ^ {{g} _ {n}} (1\ mathrm {-} {S}} _ {f} ^ {{h} _ {\ tau}})\ tau\ mathrm {\ cdot} {\ mathrm {} {\ mathrm {}}\ tilde {}\ tilde {u}\ tilde {u}\ tilde {u}\ mathrm {}} _ {\ tau} _ {c}\ end {array}\ end {array} Weak formulation of the law of contact ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We are introducing the weak form of the law of contact. To do this, simply repeat equation (a) by multiplying it by the contact Lagrange test field: .. math:: : label: eq-161 {\ tilde {G}} _ {\ text {c}}\ mathrm {=}\ mathrm {-}\ frac {1} {\ rho} _ {n}} {\ mathrm {\ int}}} {\ mathrm {\ int}}} _ {int}} _ {int}} _ {u} _ {u} _ {u} ^ {{g} _ {n}} {g}} {g} _ {n}\ right\} {\ tilde {\ lambda}} _ {n} d {\ Gamma}} _ {c} Weak formulation of the law of friction ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We introduce the weak form of the law of friction. To do this, simply repeat equation (b) by multiplying it by the Lagrange friction test field: .. math:: : label: eq-162 \ begin {array} {cc} {\ tilde {G}}} _ {\ text {G}} _ {\ text {f}}}\ mathrm {=} &\ frac {1} {{\ rho} _ {t}}} {\ mathrm {\ int}}} _ {\ mathrm {\ int}}} _ {\ int}}} _ {\ mathrm {\ int}}} _ {\ int}} _ {\ int}} _ {\ int}} _ {{g}} _ {{g}} _ {n}}\ Lambda\ mathrm {\ cdot}\ tilde {\ Lambda}\ tilde {\ Lambda} d {\ Gamma} _ {c}\ mathrm {-}\\ &\ frac {1} {{\ rho}} _ {\ rho} _ {\ rho} _ {t}} _ {t}}} {t}}} {\ mathrm {\ int}}} {\ mathrm {\ int}}} _ {\ Gamma} _ {c}}\ mu {\ lambda} _ {\ lambda} _ {n} {\ rho} _ {n} {\ rho} _ {t}} _ {t}}} {\ mathrm {\ int}}} {\ mathrm {\ int}} {u} ^ {{g} _ {n}} {S}} {S} _ {f} ^ {{h} _ {\ tau}} {h} _ {\ tau}\ mathrm {\ cdot}\ tilde {\ Lambda}\ d {\ Lambda} d {\ Lambda} d {\ Gamma} d {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {c} _ {c}\ mathrm {-}\ frac {1} {{\ rho}} _ {\ rho} _ {\ Lambda} d {\ Lambda} d {\ Gamma} d {\ Gamma} _ {\ Gamma} _ thrm {\ int}} _ {{\ Gamma} _ {c}} _ {c}}\ mu {\ lambda}} _ {n}} _ {n}} (1\ mathrm {-} {n}}} (1\ mathrm {-} {S}} _ {c}})\ mathrm {\ cdot}} (1\ mathrm {-} {n}}} (1\ mathrm {-} {n}}} (1\ mathrm {-} {S}} {S} _ {n}}} (1\ mathrm {-} {s}})\ mathrm {-} {n}}} (1\ mathrm {-} {s}}} (1\ mathrm {-} {S}} {S}\ Gamma} _ {c} +\\ & {\ mathrm {\ int}} _ {{\ Gamma} _ {c}}} (1\ mathrm {-} {S} _ {u} ^ {{g} _ {n}})\ Lambda\ mathrm {\ cdot}\ tilde {\ cdot}\ tilde {\ Lambda} d {\ Lambda} d {\ Gamma} _ {c}\ end {array}})\ Lambda\ mathrm {\ cdot}\ cdot}\ tilde {\ Lambda} d {\ Lambda} d {\ Gamma} _ {c}}\ end {array} Application of the principle of virtual work ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If we consider the semi-static problem, we seek to solve the balance in weak form by applying the principle of virtual works: .. math:: : label: eq-163 \ textrm {Find fields} ({u} ^ {1}, {u} ^ {2}, {\ lambda}} _ {n},\ Lambda)\ mathrm {\ in} {\ text {CA}} ^ {1}\ mathrm {\ times}\ mathrm {\ times}}\ mathrm {\ times}} ^ {2}\ mathrm {\ times} H\ mathrm {\ times} H\ mathrm {\ times} H \ mathrm {\ {}\ begin {array} {cc}\ mathrm {\ sum} _ {i\ mathrm {=} 1} ^ {2}\ left [{G} _ {\ text {ext}}} ^ {ext}}} ^ {i}} ^ {i}} ^ {i}} ^ {i}} ^ {i}} ^ {i} (W,\ tilde {W}) (W,\ tilde {W}) (W,\ tilde {W}) _ {\ text {int}}} ^ {i}} ^ {i}} ^ {i}} ^ {i}} ^ {i} (W,\ tilde {W}} (W,\ tilde {W}}) de {W})\ right]\ mathrm {-} {G} _ {\ text {c}} _ {\ text {c}} (W,\ tilde {W})\ mathrm {-} {G} _ {\ text {f}}} (W,\ tilde {F}}} (W,\ tilde {F}}} (W,\ tilde {F}}} (W,\ tilde {W}}) (W,\ tilde {W}})\ mathrm {-} {G}} _ {\ text {f}}} (W,\ tilde {F}}} (W,\ tilde {W}}) (W,\ tilde {W}} (W,\ tilde {W}}) (W,\ tilde {W}} W,\ tilde {W})\ mathrm {=} 0& (b)\\ {\ tilde {G}} _ {\ text {f}} (W,\ tilde {W})\ mathrm {=} 0& (b)\ end {array})\ mathrm {=} 0& (b)\ end {array} \ mathrm {\ forall}\ tilde {W}\ mathrm {=}\ mathrm {=} ({\ tilde {u}}} ^ {2}, {\ tilde {\ lambda}}} _ {n}} _ {n},\ tilde {\ Lambda})\ mathrm {\ in} {\ text {CA}} ^ {1}} ^ {1}\ mathrm m {\ times} {\ text {CA}}} ^ {2}\ mathrm {\ times} H\ mathrm {\ times} H Mixed variational inequality ------------------------------ Overall, the aim is to find the solution of the following problem (mixed variational inequality): .. math:: :label: eq-164 \ textrm {Find fields} W\ mathrm {=} ({u} ^ {1}, {u} ^ {1}, {u} ^ {2}, {\ lambda},\ Lambda)\ mathrm {\ in} {\ text {CA}} {\ text {CA}}} {\ text {CA}}\ mathrm {\ times}} {\ times} H\ mathrm {\ times}} ^ {1}\ mathrm {\ times}} ^ {1}\ mathrm {\ times}} ^ {1}\ mathrm {\ times}} ^ {1}\ mathrm {\ times}} ^ {1}\ mathrm {\ times}} H\ mathrm {\ times}} H\ mathrm {\ times}} H\ mathrm {\ times} ({u} ^ {1}, {u} ^ {2}, {\ lambda}} _ {n},\ Lambda)\ mathrm {=}\ underset {({\ tilde {u}}} ^ {1}} ^ {2}}, {\ tilde {u}}} ^ {2})} {\ text {argmin}}}\ underset {({\ tilde {u}}} ^ {1}}, {1}}, {\ tilde {u}} {1}, {\ tilde {u}} {1}, {\ tilde {u}} {1}, {\ tilde {u}} {1}, {\ tilde {u}} {1}, {\ tilde {u}} {1}, {\ tilde {n},\ tilde {\ Lambda})}} {\ text {argmax}}}\ left\ {\ mathrm {\ sum} _ {i\ mathrm {=} 1} ^ {2}\ left [{W}} _ {\ text {W} _ {\ text {W}} _ {\ text {ext}} ^ {W}}\ left [{W}}\ left [{W}} _ {\ left [{W}} _ {\ left [{W}} _ {\ left [W}} _ {\ left [{W}} _ {\ left [W}} _ {\ left [{W}} _ {\ left [W}} _ {\ left [{W}} _ {\ left [W}} _ {\ left [{W}} _ {\ left m {-} {W} _ {\ text {c}}}\ mathrm {-} {W} _ {\ text {f}}\ right\} :math:`{W}_{\text{int}}^{i}` and :math:`{W}_{\text{ext}}^{i}` correspond to internal energy and external energy. We're going to write the expression for contact and friction energy. Let :math:`{l}_{n}` be the continuous and differentiable density of contact energy: .. math:: : label: eq-165 {W} _ {\ text {c}}\ mathrm {=} {\ mathrm {\ int}} _ {{\ Gamma} _ {c}} {l}} {l} _ {n} d {\ Gamma} _ {c} And :math:`{l}_{t}` the continuous and differentiable density of the frictional energy: .. math:: : label: eq-166 {W} _ {\ text {f}}\ mathrm {=} {\ mathrm {\ int}} _ {{\ Gamma} _ {c}}\ mu {\ lambda}} _ {n} {l} _ {n} {l} _ {l} _ {l} _ {l} _ {l} _ {l} _ {l} _ {l} _ {c} Compared to a conventional augmented Lagrangian formulation, variable :math:`{\lambda }_{\tau }\mathrm{=}\mu {\lambda }_{n}\Lambda` was changed (see § :ref:`3.2.4 `). For contact energy density: .. math:: : label: eq-167 {l} _ {n}\ mathrm {=}\ frac {1} {1} {2 {\ rho} _ {n}}} ({S} _ {u} ^ {g} _ {n}}} {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}} {g} Remember that the normal game :math:`{d}_{n}` is written: .. math:: : label: eq-168 {d} _ {n}\ mathrm {=} {\ mathrm {=}} {\ mathrm {}} _ {n}\ mathrm {=} (x\ mathrm {-}}\ stackrel {} {x})\ mathrm {\ cdot} n Likewise, the sign field depends on the movement fields and the contact Lagrangian: .. math:: : label: eq-169 {S} _ {u} ^ {{g} _ {n}}\ mathrm {=}\ mathrm {=}\ mathrm {\ {}\ begin {array} {cc} 1&\ text {si} {c} 1&\ text {si} {g} {g} _ {g} _ {g} _ {n} _ {n} _ {n}\ mathrm {\ le} 0\ end {array} {cc} {c}\ begin {array} {cc} 1&\ text {si} {g} 0&\ text {si} {g} _ {g} _ {g} _ {n} _ {n}}\ mathrm {\ le} 0\ end {array} {cc} {c}\ begin {array} {cc} 1&\ text {si} {} _ {n} >0\ end {array}\ end {array} For the frictional energy density: .. math:: : label: eq-170 \ begin {array} {cc} {l} _ {t}\ mathrm {=}\ mathrm {=} &\ frac {1} {2 {\ rho} _ {\ tau}}} (1\ mathrm {-} {S}} {S}} _ {t} _ {t}}\ mathrm {=}}\\ frac {1}})\ Lambda\ mathrm {\ cdot}\\ &\ frac {1} {2 {\ rho} _ {\ tau}} {S}} {S} _ {u} _ {{g} _ {n}} {S} _ {{h} _ {\ tau}}\ left [{h}}\ left [{h} _ {\ h} _ {\ tau}\ Lambda\ right] +\\ &\ frac {1} {1} {2 {\ rho} {2 {\ rho} _ {rho}} _ {n}} (1\ mathrm {-} {S}} _ {S} _ {f} _ {\ rho}} _ {\ tau}})\ tau\ mathrm {\ cdot}\ tau +\\ &\ frac {1} {2 {\ rho} _ {\ tau}} {S}} {S}} {S}} {S}} {S}} {s} _ {f} ^ {{h} _ {\ tau} _ {\ tau}}) (\ mathrm {-}}) (\ mathrm {-}}) (\ mathrm {-}} 2\ mathrm {-}} 2\ mathrm {\ parallel} {-}} {h} _ {\ tau} _ {f} ^ {{h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _ {h} _\ mathrm {\ cdot}\ Lambda)\ end {array} To find the expression of balance in the weak sense, simply apply the optimality condition () to () and (). By variation :math:`{\delta }_{u}` on contact energy :math:`{W}_{\text{c}}`, we obtain: .. math:: : label: eq-171 {\ delta} _ {{u} ^ {i}} {W}} {W} _ {\ text {c}}\ mathrm {=} {\ delta} _ {{u} ^ {i}} ({\ mathrm {\ int}}}} {W}} {W}} {W} _ {\ gamma}}} {l}} {l} _ {\ gamma} _ {c}})\ mathrm {=} {\ int}}} _ {\ gamma}} {\ mathrm {\ int}}} _ {\ gamma}} {\ mathrm {\ int}}} {W}} {W} _ {\ gamma}} {W}} {W} _ {\ gamma}} {thrm {\ int}} _ {{\ Gamma} _ {c} _ {c}} {c}} {c}} {c} _ {n}} {g} _ {n}\ delta {d} _ {d} _ {n} d {n} d {\ Gamma} _ {c} We find the expression of the virtual work of contact efforts (). Likewise, by variation :math:`{\delta }_{{u}^{i}}` on the contact energy :math:`{W}_{\text{f}}`, we obtain the expression of the virtual work of the friction forces (): .. math:: : label: eq-172 \ begin {array} {cc} {\ delta} _ {{u}} _ {{u}} ^ {i}} {W}} _ {\ text {f}} &\ mathrm {=} {\ delta} _ {\ delta} _ {\ delta} _ {\ delta} _ {\ delta} _ {i}} _ {i}} {\ u} ^ {i}}} ({\ mathrm {\ int}}} {\ mathrm {\ int}}} _ {t} _ {t} d {\ Gamma} _ {c})\\ &\ mathrm {=} {\ mathrm {\ int}} _ {{\ Gamma} _ {c}} {S} _ {u} ^ {{g} _ {g} _ {n} _ {n}}}}\ mu {n}}} {n}}}\ mu {\ lambda}}}\ mu {\ lambda}}}\ mu {\ lambda}}}\ mu {\ lambda}}\ mu {\ lambda}}\ mu {\ lambda}}\ mu {\ lambda}}\ mu {\ lambda}} _ {\ lambda} _ {\ tau}\ mathrm {\ cdot} {\ delta v} _ {\ tau} _ {\ tau} d {\ gamma} _ {c} +\\ & {\ mathrm {\ int}} _ {\ Gamma} _ {c}} {S}} {S} {S} {S} {S} {S} {S} {S} {S} _ {S} _ {S} _ {f} ^ {{h} _ {\ tau}})\ tau\ tau\ mathrm {\ cdot} {\ delta v} _ {\ tau} d {\ Gamma} _ {\ gamma} _ {c}\ end {array} To find the expression of the virtual work of the law of contact (equation), we calculate the variation of :math:`{W}_{c}` by the contact Lagrangian :math:`{\lambda }_{n}` and for the virtual work of the law of friction (equation), we express the variation of :math:`{W}_{f}` by the normalized friction lagrangian :math:`\Lambda`. Variational formulation penalized ------------------------------------ To write the penalized variational formulation, we regulate the laws of contact and friction by writing: .. math:: : label: eq-173 {\ lambda} _ {n}\ mathrm {=}\ mathrm {=}\ mathrm {-} {({\ kappa} _ {n})} ^ {\ text {+}} \ Lambda\ mathrm {=} {P} ^ {B (\ mathrm {0.1})} ({\ kappa} _ {\ tau} {v} _ {\ tau} _ {\ tau}) The two parameters :math:`{\kappa }_{n}` and :math:`{\kappa }_{\tau }` are strictly positive penalty parameters. :math:`{(\mathrm{.})}^{\text{+}}` refers to the positive part and :math:`{\mathit{Proj}}_{B(\mathrm{0,}1)}` the projection onto the unit ball (see definition in § :ref:`3.2.4 `). The weak form of the contact reaction is written as: .. math:: : label: eq-174 {G} _ {\ text {c}}\ mathrm {=} {\ mathrm {\ int}} _ {{\ Gamma} _ {c}} {\ kappa} _ {n} {d} _ {n} {d} _ {n} _ {s} _ {s} _ {u} _ {u} _ {u}} _ {n}} {\ mathrm {}} _ {n} d {\ Gamma} _ {c} For the friction reaction: .. math:: : label: eq-175 \ begin {array} {cc} {G} _ {\ text {f}} _ {\ text {f}}}\ mathrm {=} &\ mathrm {\ int}}} _ {{\ Gamma} _ {c}}\ mu {\ c}}}\ mu {\ lambda}}}\ mu {\ lambda}}}\ mu {\ lambda}}}\ mu {\ lambda}} _ {\ lambda} _ {n} {\ lambda} _ {n} {\ lambda} _ {n} {\ lambda} _ {n} {\ lambda} _ {n}} {\ tilde {S}} _ {f} ^ {{u} ^ {{h} _ {\ tau}} {v} _ {\ tau}\ mathrm {\ cdot} {\ mathrm {}}\ tilde {u}\ tilde {u}\\ u}\\ u}\\ u}\\ mathrm {-}\\ mathrm {-}\ mathrm {u}\\ mathrm {-}\ mathrm {u}\\ mathrm {\ int}} _ {{\ Gamma} _ {c}}\ mu {\ lambda}}\ mu {\ lambda}} _ {\ tilde {S}} _ {n}} (1\ mathrm {-} {-} {\ tilde {S}}} {\ tilde {S}}}}\ mu {\ lambda}} _ {s}} _ {t} _ {t}})\ frac {{\ kappa} _ {n}}} (1\ mathrm {-} -} {c}}}\ mu {\ lambda}}}\ mu {\ lambda}} _ {\ lambda}} _ {\ lambda}} _ {\ lambda}} _ {\ lambda}} _ {s}} v} _ {\ tau}} {\ mathrm {\ parallel} {\ parallel} {\ parallel} {\ parallel} {\ parallel}} {\ mathrm {\ cdot} {\ cdot} {\ mathrm {} {\ mathrm {}}\ tilde {}}\ tilde {u}\ tilde {u} {u}\ mathrm {}} _ {\ tau} d {\ parallel}}\ mathrm {\ cdot} {\ cdot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ dot} {\ mathrm {}} {\ mathrm {}} {\ mathrm {}}\ tilde {u}\ The sign fields are changed: .. math:: : label: eq-176 {\ tilde {S}} _ {u} ^ {{g} _ {n}}}\ mathrm {=}\ mathrm {\ {}\ begin {array} {c}\ begin {array} {cc} 1&\ text {c} {cc} 1&\ text {si}} _ {n}\ mathrm {\ the} 0\ text {(contact)} {cc} 1&\ text {si} _ {n}\ mathrm {\ the} 0\ text {(contact)} {cc} 1&\ text {si}} 1&\ text {si}}\ text {si}}\ if}\ mathrm {-} {d} _ {n}\ mathrm {\ le} 0\ text {(contact)} {cc} 1&\ text {si}}\\\ begin {array} {cc} 0&\ text {si}\ text {si}\ mathrm {-} {d} _ {n} >0\ text {(detachment)}}\ end {array}}\ end {array} {\ tilde {S}} _ {f} ^ {{h} _ {{h} _ {\ tau}}\ mathrm {=}\ mathrm {\ {}\ begin {array} {c}\ begin {array} {cc} {cc} 1&\ text {c} {cc} 1&\ text {\ h} {cc} 1&\ text {if} _ {\ parallel}\ mathrm {\ le} 1\ text {(adherence)}\ end {array}}\ end {array}\\ array} {cc} 0&\ text {si}\ mathrm {\ parallel} {\ kappa} {\ kappa} _ {t} {v}} _ {v} _ {v} _ {\ v} _ {\ v} _ {\ v} _ {\ v} _ {\ tau} _ {\ tau} _ {\ tau}\ {tau}\ mathrm {\ parallel} >1\ text {(detachment)}\ end {array}\ end {array}\ end {array}\ end {array}\ end {array}\ end {array}\ end {array} In the penalized formulation, contact multipliers :math:`{\lambda }_{n}` and friction semi-multipliers :math:`\Lambda` theoretically no longer have a reason to exist but they are retained for reasons of computer architecture. We therefore write a weak form corresponding to the law of contact/friction: we therefore have access to contact pressures in penalized formulation: .. math:: : label: eq-177 {\ tilde {G}} _ {\ text {c}}\ mathrm {=}\ mathrm {=}\ mathrm {-}\ frac {1} {\ kappa} _ {n}} {\ mathrm {\ int}}} {\ mathrm {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}}} _ {\ int}} _ {\ int}}} _ {\ int}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {n}} {\ kappa} _ {n} {n} {d} _ {n}) {\ tilde {\ lambda}} _ {n} d {\ Gamma} _ {c} As well as to the density of adhesive force thanks to the weak form of Coulomb's law: .. math:: : label: eq-178 \ begin {array} {cc} {\ tilde {G}}} _ {\ text {G}} _ {\ text {f}}}\ mathrm {f}} {\ tilde {G}} {\ mathrm {\ int}}} _ {\ gamma {\ int}}} _ {\ gamma {\ int}}} _ {int}}} _ {int}}} _ {\ gamma} _ {c}}}\ mu {\ lambda} _ {c}}}\ mu {\ lambda} _ {\ lambda} _ {s}}} {\ tilde {S}}} {\ tilde {S}}} {\ mathrm {\ int}}} _ {int}}} _ {int}}} _ {int}} _ {int}}} _ {{g} _ {n}}\ Lambda\ mathrm {\ cdot}\ tilde {\ Lambda} d {\ Gamma} _ {c}\ mathrm {-}\\ &\ frac {1} {1} {{\ kappa}} {{\ kappa}} _ {\ kappa} _ {t}} _ {t}}} {\ mathrm {\ int}}} _ {c}}\ mu {\ lambda}} {\ lambda} _ {n}}\ mu {\ lambda} _ {n}}} {\ tilde {S}} _ {u} ^ {{g} ^ {{g} _ {g} _ {n}}} {\ tilde {S}} _ {\ tau}} {\ kappa} _ {\ kappa} _ {t} {v} _ {v} _ {v} _ {v} _ {v} _ {v} _ {v} _ {v} _ {v} _ {\ v} _ {v} _ {\ v} _ {\ v} _ {\ v} _ {\ v} _ {\ v} _ {\ v} _ {\ v} _ {\ v} _ {\ v} _ {\ v} _ {\ v} _ {\ v} _ {v} _ {\ tau} _ {v} _ {v} _ {v} _ {v}}\\ &\ frac {1} {{\ kappa} _ {\ kappa} _ {t}} {\ mathrm {\ int}} _ {\ gamma} _ {\ lambda} _ {n} {n} {\ tilde {S}}} {\ tilde {S}}} _ {f} _ {f} _ {f} _ {f} _ {f} ^ {{h} _ {\ tau}})\ frac {{\ kappa} _ {t} {v} _ {\ tau}} {\ tau}} {\ mathrm {\ parallel} {\ kappa} _ {\ tau}\ mathrm {\ parallel}}\ mathrm {\ parallel}}\ mathrm {\ parallel}}}\ mathrm {\ cdot}}\\ dot}}\\\ dot}}\ tilde {\ Lambda} d {\ Gamma} _ {c} +\\ & {\ mathrm {\ parallel}}}\\ & {\ mathrm {\ parallel}}}\\} _ {{\ Gamma} _ {c}}} (1\ mathrm {-} {\ tilde {S}} _ {u} ^ {{g} _ {n}})\ Lambda\ mathrm {\ cdot}}\ tilde {\ cdot}\ tilde {\ Lambda} d {\ Lambda} d {\ Gamma} _ {c}\ end {array} .. _RefNumPara__15809_801959034: Dynamic — Writing in speed/impulse ----------------------------------------- The Signorin model is unsuited to contact impact problems, that is to say to dynamic contact problems. As the displacement fields are irregular (sign field), their temporal integration by classical Newmark-type finite difference schemes causes parasitic (non-physical) oscillations at the time of shocks. These oscillations are all the more important the higher the order of the diagram. Signorini-Moreau's law for contact ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Moreau rewrote the Signorin model to adapt it to the case of dynamics: .. math:: :label: eq-179 \ mathrm {\ {}\ begin {array} {ccc} {\ lambda} {\ lambda} _ {n}\ mathrm {=} 0&\ text {d} _ {n} <0& (a)\\ {\ mathrm {} {\ mathrm {}} v\ mathrm {}} v\ mathrm {}} v\ mathrm {}} v\ mathrm {}} _ {n}\ the 0\ text {and} {\ lambda} _ {n}\ the 0\ text {and} {\ lambda} _ {n} v\ mathrm {=} 0&\ text {si} {d} _ {n}\ ge 0& (b)\ end {array} With the normal speed defined by: .. math:: : label: eq-180 {\ mathrm {} v\ mathrm {}}} _ {n}\ mathrm {=}\ mathrm {} v\ mathrm {}\ mathrm {}\ mathrm {\ cdot} n Moreau's model is justified by writing that if the Signorin model is true at the initial moment :math:`{t}_{0}`, then complying with Moreau's conditions is the same as solving the Signorin problem for all :math:`t>{t}_{0}`. For :math:`t={t}_{0}`, the Signorini-Moreau laws are strictly equivalent to those introduced in § :ref:`3.1.1 `: .. math:: : label: eq-181 {\ lambda} _ {n}\ mathrm {=} {S} _ {s} _ {u} ^ {{g} _ {n}} {g}} {g} _ {n} Always with the increased contact multiplier :math:`{g}_{n}` defined by: .. math:: : label: eq-182 {g} _ {n}\ mathrm {=} {\ lambda} _ {\ lambda} _ {n}\ mathrm {-} {\ rho} _ {n} _ {d} _ {n} For :math:`t>{t}_{0}`, the laws of Signorin are written as follows: .. math:: :label: eq-183 {\ lambda} _ {n}\ mathrm {=} {=} {S} _ {s} _ {u} ^ {{d} _ {n}}} {v} ^ {{\ dot {g}}} _ {g}}} _ {g}} _ {g}} _ {n} with the following two sign fields: .. math:: : label: eq-184 {S} _ {u} ^ {{d} _ {n}}}\ mathrm {=}\ mathrm {=}\ mathrm {\ {}\ begin {array} {cc} 1&\ text {si} {si} {si} {\ mathrm {-}}}}\ mathrm {-} d} _ {n}\ mathrm {-} d} _ {n}\ mathrm {\ le} 0\ end {array} {cc} {cc} 1&\ text {si} {si} {si} {si} {\ si} {\ si} {\ mathrm {-} d} _ {n} d} _ {n}\ mathrm {\ le} 0\ end {array} {cc} 0\ end {array} {cc} {cc}\ text {if} {\ mathrm {-} d} _ {n} >0\ end {array}\ end {array}\ end {array} And the increased contact multiplier *in speed* :math:`{\dot{g}}_{n}`, defined by: .. math:: : label: eq-185 {\ dot {g}} _ {n}\ mathrm {=} {\ lambda} _ {n}\ mathrm {-} {\ rho} _ {n} {\ mathrm {}} v\ mathrm {}} v\ mathrm {}} _ {n} Note that the constraint on the game has moved partly to normal speed :math:`{\mathrm{〚}v\mathrm{〛}}_{n}`. It should also be noted the change in the nature of the increase parameter :math:`{\rho }_{n}`, which now has the dimension of a force on a speed (and no longer a force on a displacement). Coulomb's law for friction ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The Coulomb model for friction is already naturally written in speed: .. math:: : label: eq-186 (1\ mathrm {-} {S} _ {u} _ {u} ^ {{d} _ {n}})\ Lambda + {S} _ {u} ^ {{d} _ {n}}\ left\ {{{n}}}\ left\ {{(1\ mathrm {-} {(1\ mathrm {-} {(1\ mathrm {-} {(1\ mathrm {-} {\ rho})\ mathrm {-} {(1\ mathrm {-} {\ rho} _ {h}})\ Lambda\ mathrm {-} {\ rho}})\ Lambda\ mathrm {-} {\ rho} _ {\ tau} {v} _ {\ tau} {S} {S} _ {f} ^ {{h} _ {\ tau}}\ mathrm {-} (1\ mathrm {-} {S} _ {f} _ {f}} ^ {{h}} _ {h} _ {\ tau}})\ frac {{h} _ {tau}} {\ tau}} {\ mathrm {\ parallel} {h} {h} _ {tau}})\ frac {\ tau}} {\ tau}} {\ mathrm {\ parallel} {h} {h} _ {tau}} {\ tau}} {\ tau} {\ tau}} {\ mathrm {\ parallel} {h}} _ {tau}} {\ tau}} {\ tau}}\ mathrm {\ parallel}}\ right\}\ right\}\ mathrm {=} 0\ text {on} {\ Gamma} _ {c} Application of the principle of virtual work ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ For the problem in dynamics, we add the term inertia :math:`{G}_{\text{dyn}}^{i}` to the equilibrium equation. So the system to be solved is: .. math:: :label: eq-187 \ textrm {Find fields} ({u} ^ {1}, {u} ^ {2}, {\ lambda}} _ {n},\ Lambda)\ mathrm {\ in} {\ text {CA}} ^ {1}\ mathrm {\ times}\ mathrm {\ times}}\ mathrm {\ times}} ^ {2}\ mathrm {\ times} H\ mathrm {\ times} H\ mathrm {\ times} H \ mathrm {\ {}\ begin {array} {cc}\ mathrm {\ sum} _ {i\ mathrm {=} 1} ^ {2}\ left [{G} _ {\ text {dyn}}} {\ text {dyn}}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}} ^ {dyn}}\ mathrm {-} {G} _ {\ text {cf}} _ {\ text {ext}}}} ^ {i}} (W,\ tilde {W})\ right]\ mathrm {-} {G} _ {\ text {cf}}} (W,\ text {cf}}} (W,\ text {cf}}} (W) (W,\ tilde {W}}) (W,\ tilde {W}})\ mathrm {=} 0& (a)\\ {\ tilde {G}}} _ {\ text {cf}}} (W,\ tilde {F}}} (W,\ tilde {W}}) (W,\ tilde {W}}),\ tilde {W})\ mathrm {=} 0& (b)\ end {array} \ mathrm {\ forall}\ tilde {W}\ mathrm {=}\ mathrm {=} ({\ tilde {u}}} ^ {2}, {\ tilde {\ lambda}}} _ {n}} _ {n},\ tilde {\ Lambda})\ mathrm {\ in} {\ text {CA}} ^ {1}} ^ {1}\ mathrm m {\ times} {\ text {CA}}} ^ {2}\ mathrm {\ times} H\ mathrm {\ times} H :math:`{G}_{\text{dyn}}^{i}` is the work of inertia efforts: .. math:: : label: eq-188 {G} _ {\ text {dyn}}} ^ {i}\ mathrm {=} {\ mathrm {\ int}} _ {{\ omega} _ {0} ^ {i}} ({\ rho}} _ {p} _ {p} _ {p} ^ {i} _ {p} ^ {i}} _ {\ rho} _ {p} ^ {i}} _ {p} ^ {i}} _ {p} ^ {i} _ {p} ^ {i} _ {p} ^ {i} _ {p} ^ {i} _ {p} ^ {i} _ {p} ^ {i} _ {p} ^ {i} _ {p} ^ {i} _ {p} ^ {i} _ {p} ^ {i} _ {p} ^ {partial} {t} ^ {2}}\ mathrm {\ cdot} {\ tilde {u}} ^ {i}) d {\ Omega} ^ {i}) d {\ omega} ^ {i} Initial conditions should be added to the system: .. math:: : label: eq-189 {u} _ {t} ^ {i}\ mathrm {=} {u} _ {0} ^ {i}\ text {in} {\ Omega} _ {0}} ^ {i} _ {0} ^ {i} {v} _ {t} ^ {i}\ mathrm {=} {v} _ {0} ^ {i}\ text {in} {\ Omega} _ {0}} ^ {i} _ {0} ^ {i} To complete the system, it is also necessary to describe the temporal integration scheme, which relates movements to speeds: .. math:: :label: eq-190 {u} _ {t} ^ {i}\ mathrm {=} {u} _ {0} ^ {i} + {\ mathrm {\ int}} _ {{t} _ {0}}} {0}}} ^ {0}}}} ^ {0}}}} ^ {0}}}} ^ {0}}}} ^ {0}}}} ^ {0}}}} ^ {0}}}} ^ {0}}}} ^ {t}}}} ^ {0}}}} ^ {t}}}} ^ {0} Building on the work of Jean [], we will use a first-order theta schema: .. math:: : label: eq-191 {u} _ {k+1} ^ {i}\ mathrm {=} {u} _ {k} ^ {i} +\ Delta {t} _ {k}\ left [(1\ mathrm {-}\ theta) {v}\ theta) {v}\ theta} _ {i}\ right] If :math:`\theta =0`, we find a purely explicit pattern. If :math:`\theta =1`, we find a purely implicit Euler-type diagram (we indeed find the approximation of the speed carried out in a semi-static way): .. math:: : label: eq-192 {v} _ {k+1}\ mathrm {=}\ frac {1} {\ delta {t} _ {k}} ({u} _ {k+1}\ mathrm {-} {-} {u} _ {k}) The pattern is stable if: .. math:: :label: eq-193 \ theta\ ge 0\ text {and} 1\ mathrm {-}\ theta\ le\ frac {2} {{\ omega} _ {m}\ Delta {t} _ {k}} where :math:`{\omega }_{m}` is the maximum pulsation of the dynamic system. In practice, it is recommended to choose :math:`\theta` between :math:`0.5` and :math:`1`. Virtual work of touch-friction forces ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We will now write the virtual work of touch-friction forces in strong form. We have a component for contact: .. math:: : label: eq-194 {G} _ {\ text {c}}\ mathrm {=}\ mathrm {=}\ mathrm {-} {\ mathrm {\ int}} _ {\ gamma} _ {\ lambda}} {\ lambda} _ {n}} _ {s}}} {\ dot {g}}} {\ lambda} _ {s}}} {\ lambda} _ {n}} _ {n}} _ {n}} _ {n}} {\ dot {g}} _ {n}}} {\ mathrm {}\ tilde {u}\ mathrm {}} _ {n} d {\ Gamma} _ {c} And a component for friction: .. math:: :label: eq-195 \ begin {array} {cc} {G} _ {\ text {f}} _ {\ text {f}}}\ mathrm {=} & {\ mathrm {\ int}} _ {c}}\ mu {\ lambda}}\ mu {\ lambda} _ {n}} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} _ {\ lambda} Lambda\ mathrm {\ cdot} {\ mathrm {}\ tilde {u}\ mathrm {}} _ {\ tau} d {\ Gamma} _ {c} +\\ & {\ mathrm {\ int}}\\ mathrm {\ int}}}\ tilde {u} {\ int}}} _ {\ mathrm {\ int}}} _ {\ mathrm {\ int}}} _ {\ mathrm {\ int}}} _ {\ int}} _ {\ mathrm {\ int}}} _ {\ int}} _ {\ mathrm {\ int}}} _ {\ mathrm {\ int}}} _ {\ int}} _ {u} {\ mathrm {\ int}}} _ {n}} (1\ mathrm {-} {S} _ {f}} _ {f} ^ {{h} _ {\ tau}})\ tau\ mathrm {\ cdot} {\ mathrm {} {\}\ tilde {}}\ tilde {u}\ tilde {u}\ tilde {u}\ tilde {u}\\ mathrm {u}\\ mathrm {u}\\ mathrm {u}\ mathrm {u}\ mathrm {}}} _ {\ tau} d {\ gamma} _ {c}\ end {array}\ end {array} We don't see the sign field for velocities :math:`{S}_{v}^{{\dot{g}}_{n}}` in the expression for the strong form of the reaction due to friction, but it is present implicitly in the expression for the contact multiplier :math:`{\lambda }_{n}`. Weak formulation of the law of contact ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We now write the weak form of the contact law established by equation (): .. math:: : label: eq-196 {\ tilde {G}} _ {\ text {c}}\ mathrm {=}\ mathrm {-}\ frac {1} {\ rho} _ {n}} {\ mathrm {\ int}}} {\ int}} _ {\ int}} _ {\ int}} _ {\ int}}} _ {\ int}}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}}} _ {\ gamma}} _ {\ gamma} _ {c}}} {c}} {c}} {\ lambda}} _ {n}} {S} _ {v} ^ {{\ dot {g}} ^ {{\ dot {g}}} _ {n}\ mathrm {-} {\ rho} _ {n} {\ mathrm {} ^ {\ mathrm {}} v\ mathrm {}} v\ mathrm {}} v\ mathrm {}} v\ mathrm {}} v\ mathrm {}} v\ mathrm {}} v\ mathrm {}} _ {n}) {\ tilde {\ lambda}}} _ {n}} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {n} _ {c} Weak formulation of the law of friction ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We now write the weak form of the law of friction established by equation (): .. math:: : label: eq-197 \ begin {array} {cc} {\ tilde {G}}} _ {\ text {G}} _ {\ text {f}}}\ mathrm {=} &\ frac {1} {{\ rho} _ {t}}} {\ mathrm {\ int}}} _ {\ mathrm {\ int}}} _ {\ mathrm {\ int}}} _ {\ mathrm {\ int}}} _ {{\ int}}} _ {\ mathrm {\ int}}} _ {{\ int}} _ {\ int}} _ {{d}} _ {n}}\ Lambda\ mathrm {\ cdot}\ tilde {\ Lambda}\ tilde {\ Lambda} d {\ Gamma} _ {c}\ mathrm {-}\\ &\ frac {1} {{\ rho}} _ {\ rho} _ {\ rho} _ {t}} _ {t}}} {t}}} {\ mathrm {\ int}}} {\ mathrm {\ int}}} _ {\ Gamma} _ {c}}\ mu {\ lambda} _ {\ lambda} _ {n} {\ rho} _ {n} {\ rho} _ {t}} _ {t}}} {\ mathrm {\ int}}} {\ mathrm {\ int}} {u} ^ {{d} _ {n}} {S}} {S} _ {f}} ^ {{h} _ {\ tau}} {h} _ {\ tau}\ mathrm {\ cdot}\ tilde {\ Lambda}} d {\ Lambda} d {\ Lambda} d {\ Gamma} d {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} thrm {\ int}} _ {{\ Gamma} _ {c}}}\ mu {\ lambda}}\ mu {\ lambda} _ {s} _ {n}} (1\ mathrm {-} {n}}} (1\ mathrm {-} {S}} _ {\ S}} _ {\ tau}} (1\ mathrm {-} {s}}} (1\ mathrm {-} {S}} {S} _ {n}}} (1\ mathrm {-} {S}} {S} _ {n}}} (1\ mathrm {-} {S}} {S} _ {n}}} (1\ mathrm {-} {S}} {S} _ {n}}} (1\ mathrm} {h} _ {\ tau}\ mathrm {\ parallel}}\ mathrm {\ cdot}\ tilde {\ Lambda} d {\ Lambda} d {\ Gamma} +\\ & {\ mathrm {\ int}} _ {\ Gamma} _ {c}}}\ (1\ mathrm {\ Lambda}}}} (1\ mathrm {\ Lambda}}} (1\ mathrm {\ Lambda}}} (1\ mathrm {-}}}) (1\ mathrm {-} {\ Lambda}}} (1\ mathrm {-}}}) (1\ mathrm {-} {\ Lambda}}} (1\ mathrm {-} {\ Lambda}}} (1\ mathrm {-}}}) Gamma} _ {c}\ end {array} Energy conservation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The stabilized Lagrangian formulation for contact conserves the total system energy, linear momentum, and angular momentum. In the case of friction, energy is conserved if the friction is adherent (:math:`{S}_{u}^{{d}_{n}}{S}_{v}^{{\dot{g}}_{n}}\mathrm{=}1` and :math:`{S}_{f}^{{h}_{\tau }}\mathrm{=}1`), on the other hand it decreases (dissipation) in the case of sliding friction (:math:`{S}_{u}^{{d}_{n}}{S}_{v}^{{\dot{g}}_{n}}\mathrm{=}1` and :math:`{S}_{f}^{{h}_{\tau }}\mathrm{=}0`). .. _refnumpara__39420719: