Sensitivity Analysis of a Scalar Mechanical Contact Problem with Perturbation of the Tresca’s Friction Law

Abstract

This paper investigates the sensitivity analysis of a scalar mechanical contact problem described by a boundary value problem involving the Tresca’s friction law. The sensitivity analysis is performed with respect to right-hand source and boundary terms perturbations. In particular, the friction threshold involved in the Tresca’s friction law is perturbed, which constitutes the main novelty of the present work with respect to the existing literature. Hence, we introduce a parameterized Tresca friction problem and its solution is characterized by using the proximal operator associated with the corresponding perturbed nonsmooth convex Tresca friction functional. Then, by invoking the extended notion of twice epi-differentiability depending on a parameter, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving Signorini unilateral conditions. Finally, numerical simulations are provided in order to illustrate our main result.

Sufficient Conditions for the Twice Epi-Differentiability of the Parameterized Tresca Friction Functional

In this appendix, the notations and assumptions introduced in Section 3 are preserved. This appendix follows from Remark 3.22. Our aim here is to prove, in some particular cases which correspond to practical situations, that the parameterized Tresca friction functional $\Phi$ is twice epi-differentiable at $u_{0}$ for $F_{0}-u_{0}\in \partial \Phi (0,\cdot )(u_{0})$, with its second-order epi-derivative given by (3.11). From the characterization of Mosco epi-convergence (see Proposition 2.6), it is sufficient to prove that, for all $w\in \mathrm {H}^{1}_{\mathrm {D}}(\Omega )$, the two conditions

• (i) for all $(w_{t})_{t>0}\subset \mathrm {H}^{1}_{\mathrm {D}}(\Omega )$ such that $(w_{t})_{t>0}\rightharpoonup w$ in $\mathrm {H}^{1}_{\mathrm {D}}(\Omega )$, then

  $$\begin{aligned} \mathrm {lim}\inf \Delta _{t}^{2}\Phi (u_{0}|F_{0}-u_{0})(w_{t})\ge & {} \mathrm {I}_{\mathcal {K}_{u_{0},\frac{\partial _{\mathrm {n}}(F_{0}-u_{0})}{g_{0}}}}(w)\\&+\int _{\Gamma _{\mathrm {T}}}g'_{0}(s)\frac{\partial _{\mathrm {n}}(F_{0}-u_{0})(s)}{g_{0}(s)}w(s)\mathrm {d}s ; \end{aligned}$$

• (ii) there exists $(w_{t})_{t>0}\subset \mathrm {H}^{1}_{\mathrm {D}}(\Omega )$ such that $(w_{t})_{t>0}\rightarrow w$ in $\mathrm {H}^{1}_{\mathrm {D}}(\Omega )$ and

  $$\begin{aligned} \mathrm {lim}\sup \Delta _{t}^{2}\Phi (u_{0}|F_{0}-u_{0})(w_{t})\le & {} \mathrm {I}_{\mathcal {K}_{u_{0},\frac{\partial _{\mathrm {n}}(F_{0}-u_{0})}{g_{0}}}}(w)\\&+\int _{\Gamma _{\mathrm {T}}}g'_{0}(s)\frac{\partial _{\mathrm {n}} (F_{0}-u_{0})(s)}{g_{0}(s)}w(s)\mathrm {d}s ; \end{aligned}$$

are satisfied.

The condition (i) is always satisfied. Indeed, from Proposition 3.17, this condition can be rewritten as

$$\begin{aligned}&\mathrm {lim}\inf \int _{\Gamma _{\mathrm {T}}}\Delta _{t}^{2}G(s)(u(s) |\partial _{\mathrm {n}}(F_{0}-u_{0})(s))(w_{t}(s))\mathrm {d}s\\&\quad \ge \int _{\Gamma _{\mathrm {T}}}\mathrm {D}_{e}^{2}G(s)(u_{0}(s) |\partial _{\mathrm {n}}(F_{0}-u_{0})(s))(w(s))\mathrm {d}s, \end{aligned}$$

which is true thanks to the dense and compact embedding , to the twice epi-differentiability of the function G(s) for almost all $s\in \Gamma _{\mathrm {T}}$ (see Proposition 3.19) and to the classical Fatou’s lemma (see, e.g., [8, Lemma 4.1 p.90]).

The condition (ii) is obviously satisfied if $w\notin \mathcal {K}_{u_{0},\frac{\partial _{\mathrm {n}}(F_{0}-u_{0})}{g_{0}}}$. Thus, one has only to prove the following assertion:

• (ii’) for all $w\in \mathcal {K}_{u_{0},\frac{\partial _{\mathrm {n}}(F_{0}-u_{0})}{g_{0}}}$, there exists $(w_{t})_{t>0}\subset \mathrm {H}^{1}_{\mathrm {D}}(\Omega )$ such that $(w_{t})_{t>0}\rightarrow w$ in $\mathrm {H}^{1}_{\mathrm {D}}(\Omega )$ and

  $$\begin{aligned} \mathrm {lim}\sup \Delta _{t}^{2}\Phi (u_{0}|F_{0}-u_{0})(w_{t})\le \int _{\Gamma _{\mathrm {T}}}g'_{0}(s) \frac{\partial _{\mathrm {n}}(F_{0}-u_{0})(s)}{g_{0}(s)}w(s)\mathrm {d}s. \end{aligned}$$

Unfortunately, we are not able to prove this assertion in a general setting yet, that is without any additional assumption on $u_{0}$ and on $\Gamma$, and in any dimension $d\ge 1$. Nevertheless, in this appendix, we prove this assertion in some particular cases which correspond to practical situations, providing sufficient conditions. In particular, in the next sections, we consider the additional assumption

• (A) the map $t\in \mathbb {R}^{+}\mapsto g_{t}\in \mathrm {L}^{2}(\Gamma _{\mathrm {T}})$ is differentiable at $t=0$.

1.1 First Example of Sufficient Condition: $u=0$ a.e. on $\Gamma _{\mathrm {T}}$

In this first example, we assume that $u_{0}=0$ a.e. on $\Gamma _{\mathrm {T}}$, therefore $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}}}}$ has a null measure. Let $w\in \mathcal {K}_{u_{0},\frac{\partial _{\mathrm {n}}(F_{0}-u_{0})}{g_{0}}}$. Then, taking the sequence $w_{t}=w$ for all $t>0$, one gets

$$\begin{aligned}&\Delta _{t}^{2}\Phi (u_{0}|F_{0}-u_{0})(w)\\&\quad =\int _{\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S+}}}\cup \Gamma ^{u_{ 0},g_{0}}_{\mathrm {T}_{\mathrm {S-}}}}\frac{g_{t}(s)|u_{0}(s)+t w(s)|-g_{t}(s)|u_{0}(s)|+t\partial _{\mathrm {n}}(F_{0}-u_{0})(s)w(s)}{t^{2}}\mathrm {d}s \\&\quad = \int _{\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S+}}}}\frac{g_{t}(s)-g_{0}(s)}{t}w(s) \mathrm {d}s-\int _{\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S-}}}} \frac{g_{t}(s)-g_{0}(s)}{t}w(s)\mathrm {d}s\\&\qquad \longrightarrow \int _{\Gamma _{\mathrm {T}}}g'_{0}(s)\frac{\partial _{\mathrm {n}}(F_{0}-u_{0})(s)}{g_{0}(s)}w(s)\mathrm {d}s, \end{aligned}$$

when $t\rightarrow 0^{+}$ from Assumption A. Therefore, Condition (ii’) is satisfied.

1.2 Second Example of Sufficient Condition: Truncature

In this second example, we introduce two disjoint subsets of $\Gamma _{\mathrm {T}}$ given by

$$\begin{aligned} \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}+}}} :=\left\{ s\in \Gamma _{\mathrm {T}} \mid u_{0}(s)>0\right\} \qquad \text{ and } \qquad \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}-}}} :=\left\{ s\in \Gamma _{\mathrm {T}} \mid u_{0}(s)<0\right\} . \end{aligned}$$

Hence, it follows that $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}}}} =\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}+}}} \cup \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}-}}}$, $\partial _{\mathrm {n}}u_{0}=-g_{0}$ a.e. on $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}+}}}$ and that $\partial _{\mathrm {n}}u_{0}=g_{0}$ a.e. on $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}-}}}$. Now, let us assume that there exists $C>0$ such that $|u_{0}|\ge C$ on $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}+}}} \cup \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}-}}}$. Let us consider $w\in \mathcal {K}_{u_{0},\frac{\partial _{\mathrm {n}}(F_{0}-u_{0})}{g_{0}}}$ and the truncature $w_{t}\in \mathrm {H}^{1}_{\mathrm {D}}(\Omega )$ of w defined by

$$\begin{aligned} w_{t}(x):= \left\{ \begin{array}{ccll} \frac{1}{\sqrt{t}} &{} &{} \text { if } w(x)\ge \frac{1}{\sqrt{t}} , \\ w(x) &{} &{} \text { if } |w(x)| \le \frac{1}{\sqrt{t}}, \\ -\frac{1}{\sqrt{t}} &{} &{} \text { if } w(x)\le -\frac{1}{\sqrt{t}}, \end{array} \right. \end{aligned}$$

for almost all $x\in \Omega$ and for all $t>0$. One deduces from Marcus–Mizel theorem (see [21]) that $w_{t}\rightarrow w$ in $\mathrm {H}^{1}_{\mathrm {D}}(\Omega )$ when $t\rightarrow 0^{+}$. Moreover, for all $t\le C^{2}$, one gets

$$\begin{aligned}&\Delta _{t}^{2}\Phi (u_{0}|F_{0}-u_{0})(w_{t})\\&\quad =\int _{\Gamma ^{{u_{0}},g_{0}}_{\mathrm {T}_{\mathrm {S+}}}\cup \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}+}}}}\frac{g_{t}(s) -g_{0}(s)}{t}w_{t}(s)\mathrm {d}s-\int _{\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S-}}} \cup \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}-}}}} \frac{g_{t}(s)-g_{0}(s)}{t}w_{t}(s)\mathrm {d}s\\&\qquad \longrightarrow \int _{\Gamma _{\mathrm {T}}}g'_{0}(s) \frac{\partial _{\mathrm {n}}(F_{0}-u_{0})(s)}{g_{0}(s)}w(s)\mathrm {d}s, \end{aligned}$$

when $t\rightarrow 0^{+}$ from Assumption A; therefore, Condition (ii’) is satisfied.

1.3 Third Example of Sufficient Condition: Truncature and Dilatation

In this third example, we take $d=2$ and $\Gamma _{\mathrm {N}}=\emptyset$, and we assume that $u_{0}$ and $\partial _{\mathrm {n}}u_{0}$ are continuous on $\Gamma$, and that $\Gamma$ is diffeomorphic to the circle $\mathrm {S}^{1}:=\left\{ (x,y)\in \mathbb {R}^{2} \mid x^{2}+y^{2}=1\right\}$. From this last assumption, for simplicity, we assume in the sequel that $\Gamma =\mathrm {S}^{1}$. In what follows, the next hypotheses are only useful to simplify the computations. Let us assume that $\Gamma _{\mathrm {T}}=\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S+}}}\cup \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}+}}}$ (in this particular case, the hypothesis on the continuity of $\partial _{\mathrm {n}}u_{0}$ is useless, see Remark A.1) where $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}+}}}$ has already been defined in the previous example, and with the following parameterizations

$$\begin{aligned} \displaystyle \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {N}+}}= & {} \left\{ \left( \cos {\theta },\sin {\theta }\right) \in \Gamma \mid \theta \in \left] \gamma _{1},\gamma _{2}\right[ \right\} ,\\ \displaystyle \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S+}}}= & {} \left\{ \left( \cos {\theta },\sin {\theta }\right) \in \Gamma \mid \theta \in \left[ \xi _{1},\gamma _{1}\right] \cup \left[ \gamma _{2},\xi _{2}\right] \right\} , \end{aligned}$$

such that $-\pi \le \xi _{1}<\gamma _{1}<\gamma _{2}<\xi _{2}\le \pi$ (see Figure 4). From the continuity of $u_{0}$, there exists $c>0$ such that $u_{0}\ge c$ on the set $\{(\cos {\theta },\sin {\theta })\in \Gamma \text {, } \theta \in \left[ \chi _{1},\chi _{2}\right] \}\subset \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {N}+}}$, with $\gamma _{1}<\chi _{1}<\chi _{2}<\gamma _{2}$. Let us consider $\omega _{1}\in \left]\xi _{1},\gamma _{1}\right[$, $\omega _{2}\in \left]\gamma _{2},\xi _{2}\right[$, and also $\alpha _{t}$, $\beta _{t}$ defined, for $t>0$ such that $\sqrt{t}\le c$, by

$$\begin{aligned} \alpha _{t}:= & {} \inf {\left\{ \alpha \in \left[ \gamma _{1},\chi _{1}\right] \mid \forall \theta \in \left[ \alpha ,\chi _{1}\right] \text {, }u_{0}(\cos {\theta },\sin {\theta })\ge \sqrt{t} \right\} },\\ \beta _{t}:= & {} \inf {\left\{ \beta \in \left[ \chi _{2},\gamma _{2}\right] \mid \forall \theta \in \left[ \chi _{2},\beta \right] \text {, }u_{0}(\cos {\theta },\sin {\theta })\ge \sqrt{t} \right\} }. \end{aligned}$$

From the continuity of $u_{0}$, ones deduces that $\alpha _{t}\rightarrow \gamma _{1}$ and $\beta _{t}\rightarrow \gamma _{2}$ when $t\rightarrow 0^{+}$.

Fig. 4

Illustration of the boundary $\Gamma$

Let $w\in \mathcal {K}_{u_{0},\frac{\partial _{\mathrm {n}}(F_{0}-u_{0})}{g_{0}}}$, and let $y_{t}\in \mathrm {H}^{1}_{\mathrm {D}}(\Omega )$ be the truncature of w given by

$$\begin{aligned} y_{t}(x):= \left\{ \begin{array}{ccll} \frac{1}{\sqrt{t}} &{} &{} \text { if } w(x)\ge \frac{1}{\sqrt{t}} , \\ w(x) &{} &{} \text { if } |w(x)| \le \frac{1}{\sqrt{t}}, \\ -\frac{1}{\sqrt{t}} &{} &{} \text { if } w(x)\le -\frac{1}{\sqrt{t}}, \end{array} \right. \end{aligned}$$

for almost all $x\in \Omega$ and for all $t>0$. As in the previous section, one gets $y_{t}\rightarrow w$ in $\mathrm {H}^{1}_{\mathrm {D}}(\Omega )$, and thus, $y_{t|\Gamma }\rightarrow w_{|\Gamma }$ in $\mathrm {H}^{1/2}(\Gamma )$ when $t\rightarrow 0^{+}$. Let us consider, for $t>0$ sufficiently small, the dilatation $z_{t}:=y_{t|\Gamma } \circ d_{t}$ of $y_{t|\Gamma }$, with $d_{t}$ given by

$$\begin{aligned}{}\begin{array}[t]{lrcl}d_{t} :&{}\Gamma &{}\longrightarrow &{}\Gamma \\ {} &{}(x_{1},x_{2})&{} \longmapsto &{}\left\{ \begin{array}{llll} \left( x_{1},x_{2}\right) &{} &{} \text { if }\left( x_{1},x_{2}\right) \in \Gamma _{\mathrm {N}}\cup \Gamma _{\mathrm {D}} , \\ \left( x_{1},x_{2}\right) &{} &{} \text { if }\left( x_{1},x_{2}\right) \in \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S+}}}\backslash \left\{ \left( \cos {\theta },\sin \theta \right) \text {, } \theta \in \left[ \omega _{1},\omega _{2}\right] \right\} , \\ d^{\omega _{1},\alpha _{t}}(x_{1},x_{2}) &{} &{} \text { if } 2\arctan {\left( \frac{x_{2}}{x_{1}+1}\right) }\in \left[ \omega _{1},\alpha _{t}\right] , \\ d^{\alpha _{t},\chi _{1}}(x_{1},x_{2}) &{} &{} \text { if } 2\arctan {\left( \frac{x_{2}}{x_{1}+1}\right) }\in \left[ \alpha _{t},\chi _{1}\right] , \\ \left( x_{1},x_{2}\right) &{} &{} \text { if } 2\arctan {\left( \frac{x_{2}}{x_{1}+1}\right) }\in \left[ \chi _{1},\chi _{2}\right] , \\ d^{\chi _{2},\beta _{t}}(x_{1},x_{2}) &{} &{} \text { if } 2\arctan {\left( \frac{x_{2}}{x_{1}+1}\right) }\in \left[ \chi _{2},\beta _{t}\right] , \\ d^{\beta _{t},\omega _{2}}(x_{1},x_{2}) &{} &{} \text { if } 2\arctan {\left( \frac{x_{2}}{x_{1}+1}\right) }\in \left[ \beta _{t}, \omega _{2}\right] , \\ \end{array} \right. \end{array} \end{aligned}$$

where

$$\begin{aligned} d^{\omega _{1},\alpha _{t}}(x_{1},x_{2})= & {} \left( \cos {\theta ^{\omega _{1},\alpha _{t}}}, \sin {\theta ^{\omega _{1},\alpha _{t}}}\right) , \text { with }\\ \theta ^{\omega _{1},\alpha _{t}}= & {} \frac{\left( \gamma _{1}-\omega _{1}\right) 2 \arctan {\left( \frac{x_{2}}{x_{1}+1}\right) }+w_{1}\left( \alpha _{t}-\gamma _{1}\right) }{\alpha _{t}-\omega _{1}},\\ d^{\alpha _{t},\chi _{1}}(x_{1},x_{2})= & {} \left( \cos {\theta ^{\alpha _{t},\chi _{1}}}, \sin {\theta ^{\alpha _{t},\chi _{1}}}\right) , \text { with }\\ \theta ^{\alpha _{t},\chi _{1}}= & {} \frac{\left( \chi _{1}-\gamma _{1}\right) 2 \arctan {\left( \frac{x_{2}}{x_{1}+1}\right) }+\chi _{1}\left( \gamma _{1}-\alpha _{t}\right) }{\chi _{1}-\alpha _{t}}, \\ d^{\chi _{2},\beta _{t}}(x_{1},x_{2})= & {} \left( \cos {\theta ^{\chi _{2},\beta _{t}}}, \sin {\theta ^{\chi _{2},\beta _{t}}}\right) , \text { with }\\ \theta ^{\chi _{2},\beta _{t}}= & {} \frac{\left( \gamma _{2}-\chi _{2}\right) 2 \arctan {\left( \frac{x_{2}}{x_{1}+1}\right) }+\chi _{2}\left( \beta _{t}-\gamma _{2}\right) }{\beta _{t}-\chi _{2}}, \\ d^{\beta _{t},\omega _{2}}(x_{1},x_{2})= & {} \left( \cos {\theta ^{\beta _{t},\omega _{2}}}, \sin {\theta ^{\beta _{t},\omega _{2}}}\right) , \text { with }\\ \theta ^{\beta _{t},\omega _{2}}= & {} \frac{\left( \omega _{2}-\gamma _{2}\right) 2\arctan {\left( \frac{x_{2}}{x_{1}+1}\right) }+\omega _{2}\left( \gamma _{2}-\beta _{t}\right) }{\omega _{2}-\beta _{t}}. \end{aligned}$$

Note that, since $-\pi \le \xi _{1}<\omega _{1}<\omega _{2}<\xi _{2}\le \pi$ (see Remark A.2), then $d_{t}$ is a well-defined bijective Lipschitz continuous map, and its inverse is also a bijective Lipschitz continuous map. Thus, it follows that $z_{t}\in \mathrm {H}^{1/2}(\Gamma )$ and also $z_{t}\rightarrow w_{|\Gamma }$ in $\mathrm {H}^{1/2}(\Gamma )$ when $t\rightarrow 0^{+}$. Then, for $t>0$ sufficiently small, we denote by $w_{t}\in \mathrm {H}^{1}_{\mathrm {D}}(\Omega )$ a lift of $z_{t}\in \mathrm {H}^{1/2}(\Gamma )$, such that $w_{t}\rightarrow w$ in $\mathrm {H}^{1}_{\mathrm {D}}(\Omega )$ when $t\rightarrow 0^{+}$. Therefore, by denoting

$$\begin{aligned} m_{t}(s)=\frac{g_{t}(s)|u_{0}(s)+t w_{t}(s)|-g_{t}(s)|u_{0}(s)|-t\partial _{\mathrm {n}}(F_{0}-u_{0})(s)w_{t}(s)}{t^{2}}, \end{aligned}$$

for $t>0$ sufficiently small and for almost all $s\in \Gamma _{\mathrm {T}}$, it follows that

$$\begin{aligned}&\Delta _{t}^{2}\Phi (u_{0}|F_{0}-u_{0})(w_{t}) = \int _{\left\{ \left( \cos {\theta },\sin {\theta }\right) \text {, } \theta \in \left[ \xi _{1},\omega _{1}\right] \right\} }m_{t}(s)\mathrm {d}s \\&\quad +\int _{\left\{ \left( \cos {\theta },\sin {\theta }\right) \text {, } \theta \in \left[ \omega _{1},\alpha _{t}\right] \right\} }m_{t}(s)\mathrm {d}s +\int _{\left\{ \left( \cos {\theta },\sin {\theta }\right) \text {, } \theta \in \left[ \alpha _{t},\chi _{1}\right] \right\} }m_{t}(s)\mathrm {d}s \\&\quad +\int _{\left\{ \left( \cos {\theta },\sin {\theta }\right) \text {, } \theta \in \left[ \chi _{1},\chi _{2}\right] \right\} }m_{t}(s)\mathrm {d}s +\int _{\left\{ \left( \cos {\theta },\sin {\theta }\right) \text {, } \theta \in \left[ \chi _{2},\beta _{t}\right] \right\} }m_{t}(s)\mathrm {d}s \\&\quad +\int _{\left\{ \left( \cos {\theta },\sin {\theta }\right) \text {, } \theta \in \left[ \beta _{t},\omega _{2}\right] \right\} }m_{t}(s)\mathrm {d}s+\int _{\left\{ \left( \cos {\theta },\sin {\theta }\right) \text {, } \theta \in \left[ \omega _{2},\xi _{2}\right] \right\} }m_{t}(s)\mathrm {d}s. \end{aligned}$$

Then, from the definition of $d_{t}$ and Assumption A, one deduces that

$$\begin{aligned} \Delta _{t}^{2}\Phi (u_{0}|F_{0}-u_{0})(w_{t})\longrightarrow \int _{\Gamma _{\mathrm {T}}}g'_{0}(s)\frac{\partial _{\mathrm {n}}(F_{0}-u_{0})(s)}{g_{0}(s)}w(s) \mathrm {d}s, \end{aligned}$$

when $t\rightarrow 0^{+}$, and thus, Condition (ii’) is satisfied.

Remark A.1

In the case where $\Gamma _{\mathrm {T}}=\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S+}}}\cup \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}+}}}$, the hypothesis $\partial _{\mathrm {n}}u_{0}$ continuous on $\Gamma$ is useless. Nevertheless, in the general case $\Gamma _{\mathrm {T}}=\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}+}}}\cup \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}-}}}\cup \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {D}}}}\cup \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S-}}}\cup \Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S+}}}$, the hypotheses $u_{0}$ and $\partial _{\mathrm {n}}u_{0}$ continuous on $\Gamma$ is sufficient to get the twice epi-differentiability of the parameterized Tresca friction functional: a part of $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S-}}}$ (resp. $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S+}}}$, resp. $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}-}}}$) is never side to side with a part of $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}+}}}$ (resp. $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}-}}}$, resp. $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}+}}}$), and thus, using an appropriate dilatation, one can obtain the same result.

Remark A.2

The hypothesis on the angles

$$\begin{aligned} -\pi \le \xi _{1}<\omega _{1}<\gamma _{1}<\gamma _{2}<\omega _{2}<\xi _{2}\le \pi , \end{aligned}$$

avoids the problem of the definition of $d_{t}$ for the point $(x_{1},x_{2})=(-1,0)$. But, in a more general case, since $\Gamma _{\mathrm {D}}$ has a positive measure, it is always possible to translate the angles in order to overcome this difficulty and get a well-defined dilatation $d_{t}$.

Remark A.3

The assumption $\Gamma _{\mathrm {N}}=\emptyset$ can be replaced by the assumption that $\Gamma _{\mathrm {N}}$ is never side to side with $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}+}}}$ and $\Gamma ^{u_{0},g_{0}}_{\mathrm {T}_{\mathrm {S}_{\mathrm {N}-}}}$. Without one of those assumptions, the dilatation may not work.