The semi-smooth Newton-PDAS methods appear to be one of the most relevant methods for solving friction contact problems (cf. [27, 29, 30]). These methods are based on the following principle: the conditions of contact and friction are reformulated in terms of nonlinear complementarity functions whose solution is provided by the semi-smooth iterative method of Newton [26, 27]. To this end, we need the generalised derivative of complementary functions for contact and friction. In practice, the conditions of contact with Coulomb’s friction can be formulated in terms of a fixed point problem related to a quasi-optimisation one. From a purely algorithmic point of view, the main goal of these methods is to separate the nodes potentially in contact into two subsets (active and inactive) and to find the correct subset of all the nodes actually in active contact (subset (mathcal{A })), as opposed to those that are inactive (subset (mathcal{I })). In practice, the semi-smooth Newton-PDAS methods do not require the use of Lagrange multipliers. In fact, the boundary conditions on the subsets (mathcal{A }) and (mathcal{I }) are directly imposed thanks to a semi-smooth Newton method, and consequently, their implementation could be achieved without much effort.
In order to generalise the frictional contact conditions for deformable and rigid problems, we will make a change of variables. Let us take the following sthenic and kinematic variables:
$$begin{aligned} & – text{For deformable problems:} quad phi _{n}=lambda ^{alpha}_{n}, boldsymbol{phi }_{t}=boldsymbol{lambda }^{alpha}_{t}, w_{n}=u^{ alpha}_{n}, boldsymbol{w}_{t}= dot{boldsymbol{u}}^{alpha}_{t}, end{aligned}$$
(3.1)
$$begin{aligned} & – text{For rigid problems:} quad phi _{n}=p^{alpha}_{n}, boldsymbol{phi }_{t}=boldsymbol{p}^{alpha}_{t}, w_{n}=v_{n}, boldsymbol{w}_{t}={ boldsymbol{v}}^{alpha}_{t}. end{aligned}$$
(3.2)
Complementary function for frictional contact
Signorini’s conditions (2.2) or (2.15) can be formulated from the following nonlinear complementary function ({mathcal{C}}_{n}^{phi}(w_{n},phi _{n})):
$$begin{aligned} {mathcal{C}}_{n}^{phi}(w_{n},phi _{n})=phi _{n} – [phi _{n} – gamma _{n} w_{n}]_{+} quad forall alpha in {mathcal{S}}, end{aligned}$$
(3.3)
where ({mathcal{S}}) is the set of all potential contact nodes and (gamma _{n}>0) is the normal active set parameter. Let us state a first result.
Proposition 3.1
The unilateral contact conditions (2.2) or (2.15) for each potential contact are equivalent to ({mathcal{C}}_{n}^{phi}(w_{n},phi _{n})=0), where (phi _{n}) is the normal reaction (or impulse) force.
For the proof, see [35] for deformable problems and see [34] for rigid problems.
Frictional Coulomb’s conditions (2.3) or (2.17) can be formulated from the following nonlinear complementary function ({mathcal{C}}_{t}^{phi}(w_{n},boldsymbol{phi }_{t},phi _{n}, boldsymbol{phi }_{t})):
$$begin{aligned} {mathcal{C}}_{t}^{phi}(w_{n}, boldsymbol{w}_{t},phi _{n}, boldsymbol{phi }_{t})=max{bigl(mu phi _{n}, Vert boldsymbol{ phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert bigr)}boldsymbol{phi }_{t} – mu phi _{n}( boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t}) quad forall alpha in {mathcal{S}}, end{aligned}$$
(3.4)
where (gamma _{t}>0) is the tangential active set parameter. Then, let us present this result.
Proposition 3.2
The frictional contact conditions (2.3) or (2.17) for each contact are equivalent to ({mathcal{C}}_{t}^{phi}(w_{n},boldsymbol{w}_{t},phi _{n}, boldsymbol{phi }_{t})=0), where (boldsymbol{phi }_{t}) is the tangential reaction (or impulse) force between two particles in contact.
For the proof, see [34] for deformable problems and see [35] for rigid problems.
Generalised derivative of complementary functions
Now, we provide the generalised derivative of the complementary functions in the gap, stick and slip cases.
(underline{bullet text{ Gap case: }phi _{n} – gamma _{n} w_{n}le 0})
According to the complementary functions ({mathcal{C}}_{n}^{phi}(w_{n},phi _{n})=phi _{n}) and ({mathcal{C}}_{t}^{phi}(w_{n},boldsymbol{w}_{t},phi _{n},boldsymbol{phi }_{t})=| boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t}|boldsymbol{phi }_{t}), we have the following derivatives:
$$begin{aligned} &d_{w_{n}} {mathcal{C}}_{n}^{phi}=0 , end{aligned}$$
(3.5)
$$begin{aligned} &d_{phi _{n}} {mathcal{C}}_{n}^{phi}=d{phi _{n}} , end{aligned}$$
(3.6)
$$begin{aligned} &d_{w_{n}} {mathcal{C}}_{t}^{phi}=0, end{aligned}$$
(3.7)
$$begin{aligned} &d_{boldsymbol{w}_{t}} {mathcal{C}}_{t}^{phi}=-gamma _{t} boldsymbol{phi }_{t} frac{(boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t})^{T}}{ Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert },d{ boldsymbol{w}_{t}}=0, end{aligned}$$
(3.8)
$$begin{aligned} &d_{phi _{n}} {mathcal{C}}_{t}^{phi}=0, end{aligned}$$
(3.9)
$$begin{aligned} &d_{boldsymbol{phi }_{t}} {mathcal{C}}_{t}^{phi}= biggl( boldsymbol{phi }_{t} frac{(boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t})^{T}}{ Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert }+ Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert boldsymbol{I}_{2} biggr),d{boldsymbol{phi }_{t}}. end{aligned}$$
(3.10)
(underline{bullet text{ Stick case: }mu phi _{n}ge |boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t}|> 0})
Given the complementary functions ({mathcal{C}}_{n}^{phi}(w_{n},phi _{n})=gamma _{n} w_{n}) and ({mathcal{C}}_{t}^{phi}(w_{n},boldsymbol{w}_{t},phi _{n},boldsymbol{phi }_{t})= mu gamma _{t} phi _{n}boldsymbol{w}_{t}), we have
$$begin{aligned} &d_{w_{n}} {mathcal{C}}_{n}^{phi}=gamma _{n} ,d{w_{n}} , end{aligned}$$
(3.11)
$$begin{aligned} &d_{phi _{n}} {mathcal{C}}_{n}^{phi}=0 , end{aligned}$$
(3.12)
$$begin{aligned} &d_{w_{n}} {mathcal{C}}_{t}^{phi}=0, end{aligned}$$
(3.13)
$$begin{aligned} &d_{boldsymbol{w}_{t}} {mathcal{C}}_{t}^{phi}=mu gamma _{t} phi _{n} ,d{ boldsymbol{w}_{t}}, end{aligned}$$
(3.14)
$$begin{aligned} &d_{phi _{n}} {mathcal{C}}_{t}^{phi}=mu gamma _{t} boldsymbol{w}_{t} ,d{ phi _{n}}, end{aligned}$$
(3.15)
$$begin{aligned} &d_{boldsymbol{phi }_{t}} {mathcal{C}}_{t}^{phi}=0 . end{aligned}$$
(3.16)
(underline{bullet text{ Slip case: }|boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t}|> mu phi _{n}> 0})
From ({mathcal{C}}_{n}^{phi}(w_{n},phi _{n})=gamma _{n} w_{n}) and ({mathcal{C}}_{t}^{phi}(w_{n},boldsymbol{w}_{t},phi _{n},boldsymbol{phi }_{t})=| boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t}|boldsymbol{phi }_{t}- mu phi _{n} ( boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t})), it comes
$$begin{aligned} &d_{w_{n}} {mathcal{C}}_{n}^{phi}=gamma _{n} ,d{w_{n}} , end{aligned}$$
(3.17)
$$begin{aligned} &d_{phi _{n}} {mathcal{C}}_{n}^{phi}=0 , end{aligned}$$
(3.18)
$$begin{aligned} &d_{w_{n}} {mathcal{C}}_{t}^{phi}=0, end{aligned}$$
(3.19)
$$begin{aligned} &d_{boldsymbol{w}_{t}} {mathcal{C}}_{t}^{phi}= biggl(- gamma _{t} boldsymbol{phi }_{t} frac{(boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t})^{T}}{ Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert }+ mu gamma _{t} phi _{n}boldsymbol{I}_{2} biggr),d{ boldsymbol{w}_{t}}, end{aligned}$$
(3.20)
$$begin{aligned} &d_{phi _{n}} {mathcal{C}}_{t}^{phi}=- mu (boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t}),d phi _{n}, end{aligned}$$
(3.21)
$$begin{aligned} &d_{boldsymbol{phi }_{t}} {mathcal{C}}_{t}^{phi}= biggl( boldsymbol{phi }_{t} frac{(boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t})^{T}}{ Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert } + Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert boldsymbol{I}_{2} – mu phi _{n}boldsymbol{I}_{2} biggr),d{boldsymbol{phi }_{t}} . end{aligned}$$
(3.22)
By combining (3.5)–(3.22) with ({mathcal{D}}_{{mathcal{C}}_{n}^{phi}}) and ({mathcal{D}}_{{mathcal{C}}_{t}^{phi}}) the generalised derivatives of ({mathcal{C}}_{n}^{phi}) and ({mathcal{C}}_{t}^{phi}), respectively, we obtain
$$begin{aligned} &{mathcal{D}}_{{mathcal{C}}_{n}^{phi}}(w_{n},phi _{n}) ( delta w_{n},delta phi _{n})= gamma _{n}({1}_{mathrm{Stick}}+ {1}_{mathrm{Slip}})delta w_{n} + {1}_{mathrm{Gap}} delta phi _{n}, end{aligned}$$
(3.23)
$$begin{aligned} &{mathcal{D}}_{{mathcal{C}}_{t}^{phi}}(w_{n},boldsymbol{w}_{t}, phi _{n}, boldsymbol{phi }_{t}) (delta w_{n},delta boldsymbol{w}_{t},delta phi _{n}, delta phi _{t}) \ &quad = {1}_{mathrm{Gap}} Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert delta boldsymbol{phi }_{t} + {1}_{mathrm{Stick}} ( mu gamma _{t} phi _{n} delta{ boldsymbol{w}_{t}} + mu gamma _{t} boldsymbol{w}_{t}delta phi _{n} ) \ &qquad {} + {1}_{mathrm{Slip}} biggl( biggl(- gamma _{t} boldsymbol{phi }_{t} frac{(boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t})^{T}}{ Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert }+ mu gamma _{t} phi _{n}boldsymbol{I}_{2} biggr)delta{ boldsymbol{w}_{t}} – mu (boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t})delta phi _{n} \ &qquad {} + biggl( boldsymbol{phi }_{t} frac{(boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t})^{T}}{ Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert } + Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert boldsymbol{I}_{2} – mu phi _{n} boldsymbol{I}_{2} biggr)delta{boldsymbol{phi }_{t}} biggr), end{aligned}$$
(3.24)
where
$$begin{aligned} &{1}_{mathrm{Gap}} =1, qquad {1}_{mathrm{Stick}} = 0,qquad {1}_{mathrm{Slip}} = 0quad {mathrm{if}} phi _{n} – gamma _{n} w_{n}le 0, \ &{1}_{mathrm{Gap}} =0,qquad {1}_{mathrm{Stick}} = 1,qquad {1}_{mathrm{Slip}} = 0quad {mathrm{if}} mu phi _{n} ge Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert > 0, \ &{1}_{mathrm{Gap}} =0,qquad {1}_{mathrm{Stick}} = 0,qquad {1}_{mathrm{Slip}} = 1quad {mathrm{if}} Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}_{t} Vert > mu phi _{n}> 0. end{aligned}$$
Fixed point conditions from Newton’s semi-smooth approach
Using now the semi-smooth Newton formalism (indexed by the subscript k) at the current fixed point ((w^{k}_{n},boldsymbol{w}^{k}_{t},phi ^{k}_{n},phi ^{k}_{t})) of the complementary functions ({mathcal{C}}_{n}^{phi}) and ({mathcal{C}}_{t}^{phi}), one can derive the new iterate ((w^{k+1}_{n},boldsymbol{w}^{k+1}_{t},phi ^{k+1}_{n},phi ^{k+1}_{t}))
$$begin{aligned} &{mathcal{D}}_{{mathcal{C}}_{n}^{phi}}bigl(w^{k}_{n},phi ^{k}_{n}bigr) bigl(delta w^{k+1}_{n}, delta phi ^{k+1}_{n}bigr)= – {mathcal{C}}_{n}^{phi} bigl(w^{k}_{n},phi ^{k}_{n} bigr), end{aligned}$$
(3.25)
$$begin{aligned} &{mathcal{D}}_{{mathcal{C}}_{t}^{phi}}bigl(w^{k}_{n}, boldsymbol{w}^{k}_{t}, phi ^{k}_{n}, boldsymbol{phi }^{k}_{t}bigr) bigl(delta w^{k+1}_{n}, delta boldsymbol{w}^{k+1}_{t}, delta phi ^{k+1}_{n},delta boldsymbol{phi }^{k+1}_{t} bigr) \ &quad = – {mathcal{C}}_{t}^{phi} bigl(w^{k}_{n}, boldsymbol{w}^{k}_{t}, phi ^{k}_{n}, boldsymbol{phi }^{k}_{t}bigr) , end{aligned}$$
(3.26)
$$begin{aligned} & bigl(w^{k+1}_{n},boldsymbol{w}^{k+1}_{t}, phi ^{k+1}_{n},boldsymbol{phi }^{k+1}_{t} bigr) \ &quad =bigl(w^{k}_{n},boldsymbol{w}^{k}_{t}, phi ^{k}_{n},boldsymbol{phi }^{k}_{t} bigr) +bigl(delta w^{k+1}_{n},delta boldsymbol{w}^{k+1}_{t},delta phi ^{k+1}_{n}, delta boldsymbol{phi }^{k+1}_{t}bigr) . end{aligned}$$
(3.27)
(underline{bullet text{ Gap case: }{1}_{mathrm{Gap}} =1, {1}_{mathrm{Stick}} = 0, {1}_{mathrm{Slip}} = 0})
From equations (3.25) and (3.26) we have
$$begin{aligned} &phi ^{k+1}_{n}-phi ^{k}_{n}=- phi ^{k}_{n}, end{aligned}$$
(3.28)
$$begin{aligned} & biglVert boldsymbol{phi }^{k}_{t}- gamma _{t} boldsymbol{w}^{k}_{t} bigrVert bigl(boldsymbol{phi } ^{k+1}_{t}- boldsymbol{phi }^{k}_{t}bigr) = – biglVert boldsymbol{phi } ^{k}_{t}- gamma _{t} boldsymbol{w}^{k}_{t} bigrVert boldsymbol{phi }^{k}_{t}. end{aligned}$$
(3.29)
Next, the gap conditions of the semi-smooth Newton formalism are as follows:
$$begin{aligned} &phi ^{k+1}_{n}=0, end{aligned}$$
(3.30)
$$begin{aligned} &boldsymbol{phi }^{k+1}_{t}=boldsymbol{0}, end{aligned}$$
(3.31)
since (|phi ^{k}_{t}- gamma _{t} boldsymbol{w}^{k}_{t}|>0).
(underline{bullet text{ Stick case: }{1}_{mathrm{Gap}} =0, {1}_{mathrm{Stick}} = 1, {1}_{mathrm{Slip}} = 0})
From equations (3.25) and (3.26) we have
$$begin{aligned} &gamma _{n}bigl(w^{k+1}_{n}-w^{k}_{n} bigr)=-gamma _{n}w^{k}_{n}, end{aligned}$$
(3.32)
$$begin{aligned} & mu gamma _{t} phi ^{k}_{n}bigl( boldsymbol{w}^{k+1}_{t}- boldsymbol{w}^{k}_{t} bigr)+mu gamma _{t} boldsymbol{w}^{k}_{t} bigl(phi ^{k+1}_{n}- phi ^{k}_{n} bigr) = -mu gamma _{t} phi ^{k}_{n} boldsymbol{w}^{k}_{t}. end{aligned}$$
(3.33)
Next,
$$begin{aligned} &w^{k+1}_{n}=0, end{aligned}$$
(3.34)
$$begin{aligned} & boldsymbol{w}^{k+1}_{t} – boldsymbol{w}^{k}_{t} = – boldsymbol{w}^{k}_{t} frac{phi ^{k+1}_{n}}{phi ^{k}_{n}}. end{aligned}$$
(3.35)
For dynamics of rigid bodies and for a given contact node α, the fundamental principle of dynamics can be written as follows:
$$begin{aligned} &boldsymbol{w}^{alpha} = boldsymbol{w}^{alpha ,mathrm{free}} + { mathcal{W}}^{ alpha alpha} boldsymbol{phi } ^{alpha} + sum _{beta neq alpha} {mathcal{W}}^{ beta alpha} boldsymbol{phi } ^{beta}. end{aligned}$$
(3.36)
Finally, the stick conditions for rigid bodies of the semi-smooth Newton formalism are as follows:
$$begin{aligned} &w^{k+1}_{n}=0, end{aligned}$$
(3.37)
$$begin{aligned} &boldsymbol{phi }_{t}^{k+1} + frac{frac{phi ^{k+1}_{n}}{phi ^{k}_{n}}boldsymbol{w}^{k}_{t}}{{mathcal{W}}^{alpha alpha}_{tt}} = boldsymbol{phi } _{t}^{k}. end{aligned}$$
(3.38)
(underline{bullet text{ Slip case: }{1}_{mathrm{Gap}} =0, {1}_{mathrm{Stick}} = 0, {1}_{mathrm{Slip}} = 1})
For ({mathcal{C}}_{n}^{phi}), we get once again
$$begin{aligned} &w^{k+1}_{n}=0. end{aligned}$$
(3.39)
For ({mathcal{C}}_{t}^{phi}), we obtain
$$begin{aligned} & biggl(- gamma _{t} boldsymbol{phi }^{k}_{t} frac{(boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t})^{T}}{ Vert boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t} Vert }+ mu gamma _{t} phi ^{k}_{n} boldsymbol{I}_{2} biggr) bigl(boldsymbol{w}^{k+1}_{t}- boldsymbol{w}^{k}_{t}bigr) \ &qquad {} – mu bigl(boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t}bigr) bigl( phi ^{k+1}_{n}-phi ^{k}_{n}bigr) \ &qquad {} + biggl( boldsymbol{phi }^{k}_{t} frac{(boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t})^{T}}{ Vert boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t} Vert } + biglVert boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t} bigrVert boldsymbol{I}_{2} – mu phi ^{k}_{n} boldsymbol{I}_{2} biggr) bigl(boldsymbol{phi }^{k+1}_{t}- boldsymbol{phi }^{k}_{t}bigr) \ & quad = – biglVert boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t} bigrVert boldsymbol{phi }^{k}_{t}+ mu phi ^{k}_{n} bigl(boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t}bigr) . end{aligned}$$
(3.40)
Let us introduce
$$ boldsymbol{F}^{(k)}= boldsymbol{phi }^{k}_{t} frac{(boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t})^{T}}{ Vert boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t} Vert }, qquad E^{(k)}= frac{1}{ Vert boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t} Vert }. $$
(3.41)
Therefore, after an elementary computation
$$begin{aligned} &- gamma _{t} E^{(k)} bigl( boldsymbol{F}^{(k)}- mu phi ^{k}_{n} boldsymbol{I}_{2} bigr) bigl(boldsymbol{w}^{k+1}_{t}-boldsymbol{w}^{k}_{t} bigr) – mu E^{(k)} bigl(boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t}bigr) phi ^{k+1}_{n} \ &qquad {} + bigl( E^{(k)}bigl(boldsymbol{F}^{(k)}- mu phi ^{k}_{n} boldsymbol{I}_{2} bigr)+ boldsymbol{I}_{2} bigr)boldsymbol{phi }^{k+1}_{t} -E^{(k)} bigl( boldsymbol{F}^{(k)} – mu phi ^{k}_{n}boldsymbol{I}_{2} bigr)boldsymbol{phi } ^{k}_{t}= 0. end{aligned}$$
Now, let us introduce the following operators:
$$begin{aligned} &boldsymbol{M}_{alpha}^{*(k)}=E^{(k)}bigl( boldsymbol{F}^{(k)} – mu phi ^{k}_{n} boldsymbol{I}_{2}bigr), \ &boldsymbol{h}_{alpha}^{(k)}=E^{(k)} boldsymbol{F}^{(k)} bigl(boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t} bigr)= boldsymbol{phi }^{k}_{t}. \ &boldsymbol{L}_{alpha}^{*(k)}=- gamma _{t} bigl(boldsymbol{I}_{2}+ boldsymbol{M}_{alpha}^{*(k)} bigr)^{-1}boldsymbol{M}_{alpha}^{*(k)}, \ &boldsymbol{r}_{alpha}^{*(k)}=bigl(boldsymbol{I}_{2}+ boldsymbol{M}_{ alpha}^{*(k)}bigr)^{-1} boldsymbol{h}_{alpha}^{(k)}, \ &boldsymbol{v}_{alpha}^{(k)}=mu bigl(boldsymbol{I}_{2}+ boldsymbol{M}_{alpha}^{*(k)}bigr)^{-1}E^{(k)} bigl(boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t}bigr). end{aligned}$$
And, at last after computation, we have
$$begin{aligned} &boldsymbol{L}_{alpha}^{*(k)}boldsymbol{w}^{k+1}_{t}- boldsymbol{v}_{alpha}^{(k)}phi ^{k+1}_{n}+ boldsymbol{phi }^{k+1}_{t}= boldsymbol{r}_{alpha}^{*(k)} – boldsymbol{v}_{alpha}^{(k)}phi ^{k}_{n}. end{aligned}$$
In this specific problem, and for a two-dimensional case, one can obtain a simplified equivalent version of the algorithm. Let ({mathcal{D}}_{{mathcal{C}}_{t}^{phi}}^{mathrm{slip}}) be the generalised derivative of ({mathcal{C}}_{t}^{phi}) in the slip case
$$begin{aligned} {mathcal{D}}_{{mathcal{C}}_{t}^{phi}}^{mathrm{slip}}bigl( boldsymbol{w}^{alpha}_{t}, boldsymbol{phi }_{t}bigr) bigl(delta boldsymbol{w}^{alpha}_{t}, delta boldsymbol{phi }_{t}bigr)= {}&boldsymbol{phi }_{t} frac{(boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}^{alpha}_{t})^{T}}{ Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}^{alpha}_{t} Vert }bigl( delta boldsymbol{phi }_{t} – delta boldsymbol{w}^{alpha}_{t}bigr) \ & {} – mu phi _{n} bigl(delta boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}^{alpha}_{t}bigr)+ biglVert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}^{alpha}_{t} bigrVert delta boldsymbol{ phi }_{t} . end{aligned}$$
(3.42)
Denoting by t the unit slip vector, we have
$$ boldsymbol{phi }_{t}=mu phi _{n}mathbf{{t}} , qquad frac{boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}^{alpha}_{t}}{ Vert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}^{alpha}_{t} Vert }= mathbf{{t}},qquad delta boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}^{alpha}_{t}= eta {mathbf{{t}}}. $$
(3.43)
Combining (3.42)–(3.43), we get
$$begin{aligned} &{mathcal{D}}_{{mathcal{C}}_{t}^{phi}}^{mathrm{slip}}bigl(boldsymbol{w}^{alpha}_{t}, boldsymbol{phi }_{t}bigr) bigl(delta boldsymbol{w}^{alpha}_{t}, delta boldsymbol{phi }_{t}bigr)= mu phi _{n}eta bigl(mathbf{{t}}mathbf{t}^{T}- boldsymbol{I}_{2} bigr){t}+ biglVert boldsymbol{phi }_{t} – gamma _{t} boldsymbol{w}^{alpha}_{t} bigrVert delta boldsymbol{phi }_{t}. end{aligned}$$
(3.44)
Since (mathbf{{t}}mathbf{t}^{T}+mathbf{{n}}mathbf{n}^{T} = boldsymbol{I}_{2} ) in the 2D case, we have ((mathbf{{t}}mathbf{t}^{T}- boldsymbol{I}_{2})mathbf{t} = mathbf{{n}}mathbf{n}^{T} mathbf{t} = mathbf{{0}}). Using (3.26), we obtain
$$ biglVert boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t} bigrVert bigl(boldsymbol{phi } ^{k+1}_{t} – boldsymbol{phi }^{k}_{t} bigr)= – biglVert boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t} bigrVert boldsymbol{phi }^{k}_{t} + mu phi ^{k}_{n} bigl(boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t}bigr). $$
(3.45)
Therefore, (3.45) becomes
$$begin{aligned} &boldsymbol{phi }_{t}^{k+1} = mu phi ^{k}_{n} frac{boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t}}{ Vert boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t} Vert } = mu phi ^{k}_{n} mathbf{t}. end{aligned}$$
(3.46)
Finally, the slip 2D conditions of the semi-smooth Newton formalism are as follows:
$$begin{aligned} &w^{k+1}_{n}=0, end{aligned}$$
(3.47)
$$begin{aligned} &boldsymbol{phi }_{t}^{k+1} = mu phi ^{k}_{n} frac{boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t}}{ Vert boldsymbol{phi }^{k}_{t} – gamma _{t} boldsymbol{w}^{k}_{t} Vert }. end{aligned}$$
(3.48)
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