A Three-Field Variational Formulation for a Frictional Contact Problem with Prescribed Normal Stress

: In the present work, we address a nonlinear boundary value problem that models frictional contact with prescribed normal stress between a deformable body and a foundation. The body is nonlinearly elastic, the constitutive law being a subdifferential inclusion. We deliver a three-ﬁeld variational formulation by means of a new variational approach governed by the theory of bipotentials combined with a Lagrange-multipliers technique. In this new approach, the unknown of the mechanical model is a triple consisting of the displacement ﬁeld, a Lagrange multiplier related to the friction force and the Cauchy stress tensor. We obtain existence, uniqueness, boundedness and convergence results.


Introduction
Everywhere around us we can see deformable bodies in interaction. However, even though very common, the contact phenomenon is not a trivial one; in addition, in engineering, handling the interactions between deformable bodies and obstacles is very important and requires advanced applied mathematics. The contact phenomenon can be mathematically modeled by means of boundary value problems governed by partial differential equations. Actually, the topic is very complex, involving continuum mechanics, differential equations, function spaces, calculus of variations, nonlinear analysis, control theory and numerical analysis. The importance and the abundance of the applications of contact problems in the real world has motivated a large number of scientists to investigate this kind of model. It is worth underlining that, due to their complexity, contact models do not have classical solutions. Thus, the variational methods play a crucial role in the qualitative and quantitative analysis.
In the present paper, we focus on the well-posedness and approximation results addressing a stationary frictional contact problem with prescribed normal stress, for materials governed by a multi-valued elastic operator. Using the bipotential theory, we deliver a variational formulation of the mechanical model in a form of a variational system consisting of three inequalities. Placing us in an appropriate functional setting governed by Lebesgue and Sobolev spaces for vector functions including fractional spaces on the boundary, we apply the saddle point theory and a minimization technique in order to prove the existence of at least one solution. We also pay attention to the uniqueness of the solution. Firstly, we draw attention to a partial uniqueness result related to the uniqueness in the first component. Subsequently, we discuss a global uniqueness result, not for the original problem, but for a perturbed version of it. After we investigate the boundedness of the solution of the perturbed problem, we prove a convergence result allowing an approximation of a weak solution of the contact model under consideration. The present study can be seen as a continuation of [1]; there, a two-field variational formulation for the same model was delivered; with the well-posedness of the model being studied under a more restrictive hypothesis for the prescribed normal stress. In [1], the weak solution is a pair consisting of the displacement vector and the Cauchy stress tensor. In the present study, the weak solution is a triple by considering a Lagrange multiplier related to the friction force as a component of the weak solution, in addition to the displacement vector and the Cauchy stress tensor. From the mathematical point of view, the new study is more complex. Besides the theory of bipotentials and the minimization techniques, the present study requires a saddle point technique, fractional Lebesgue and Sobolev spaces, weak topologies and a convergence of the Mosco type. It is worth emphasizing that the variational approach we propose leads to a new class of variational problems governed by Lagrange multipliers. From the mechanical point of view, the new variational approach we propose is important because it makes possible an estimation of the friction force, even a numerical computation, after passing from the qualitative study to the quantitative analysis in a future investigation.
In addition, we shall need the following theorem.
Theorem 2 (See, e.g., [35]). Let (X, · X ) be a reflexive Banach space and let K ⊂ X be a nonempty, convex, closed, unbounded subset of X. Suppose ϕ : K → R is coercive, convex and lower semicontinuous. Then, ϕ is bounded from below on K and attains its infimum in K. If ϕ is strictly convex, then ϕ has a unique minimizer.
We recall that ϕ : K → R is coercive if, for all u ∈ K, we have ϕ(u) → ∞ as u X → ∞.
Theorem 3. Let (X, (·, ·) X , · X ), (Y, (·, ·) Y , · Y ) be two Hilbert spaces and let A ⊆ X, B ⊆ Y be nonempty, closed, convex subsets. Assume that a bifunctional L : A × B → R satisfies the following conditions v → L(v, µ) is convex and lower semi-continuous for all µ ∈ B, µ → L(v, µ) is concave and upper semi-continuous for all v ∈ A.
Moreover, we assume that Then: (a) The bifunctional L(·, ·) has at least one saddle point; is strictly convex for all µ ∈ B, then A 0 contains at most one point; is strictly concave for all v ∈ A, then B 0 contains at most one point.
For a proof of (a), see [18] (p. 176). For a proof of (b), (c), (d), see [18] (p. 169). The rest of the paper has the following structure. In Section 2 we provide the functional setting we use. In Section 3 we describe the mechanical model and deliver its three-field variational formulation. In Section 4 we obtain existence, uniqueness, boundedness and convergence results. The last section provides some conclusions and final comments.
Let us introduce the vector spaces The spaces L 2 (Ω; R 3 ) and H 1 (Ω; R 3 ) are Hilbert spaces endowed with the inner products and the associated norms · L 2 (Ω;R 3 ) = (·, ·) L 2 (Ω;R 3 ) and · H 1 (Ω; respectively. The space H 1/2 (Γ; R 3 ) is also a Hilbert space endowed with the inner product and the corresponding Sobolev-Slobodeckij norm We easily observe that so The trace operator for vector functions γ : is a linear, continuous and compact operator, but it is neither an injection nor a surjection; see, e.g., [37].
is a linear, continuous operator but it is not a compact operator. Let us point out that there are some c > 0 such that Furthermore, there exists a linear, continuous operator : The operator is called the right inverse of the trace operator γ. Let S 3 be the space of second-order symmetric tensors on R 3 . Every field in S 3 is typeset in boldface. By : and · S 3 we denote the inner product and the Euclidean norm on S 3 .
We introduce now two tensor Lebesgue spaces, as follows.
Next, we introduce the following space, is the linear operator the index that follows a comma indicates a weak partial derivative with respect to the corresponding component of the independent variable. The space H 1 is a real Hilbert space endowed with the inner product The associated norm on the space H 1 is denoted by · H 1 . According to, e.g., [38], H 1 = H 1 (Ω; R 3 ) algebraically and the norms · H 1 and · H 1 (Ω;R 3 ) are equivalent.
Notice that ε is a linear and continuous operator from H 1 (Ω; R 3 ) to L 2 s (Ω; S 3 ). Let Γ 1 be a measurable part of Γ with positive surface measure. We consider the space This is a closed subspace of H 1 (Ω; Let us recall Korn's inequality: there exists c K = c K (Ω, Γ 1 ) > 0 such that see, e.g., [38,39]. Using this inequality, it can be proved that the space V is a Hilbert space endowed with the following inner product, and the corresponding norm Notice that, since the norms · H 1 and · H 1 (Ω;R 3 ) are equivalent, then there exists We proceed by introducing a closed subspace of H 1/2 (Γ; R 3 ) as follows: According to [40], the space γ(V) is a closed subspace of We note that γ( (γv)) = γv for all v ∈ V.
we can introduce an operator as follows The operator R is a linear and continuous operator. Hence, there are somec > 0 such that

The Model and Its Three-Field Variational Formulation
The physical setting is as follows: a deformable body occupies a bounded domain Ω ⊂ R 3 with smooth enough boundary Γ, partitioned in three measurable parts Γ 1 , Γ 2 and Γ 3 with positive surface measures. The body is clamped on Γ 1 , body forces of density f 0 act on Ω, surface tractions of density f 2 act on Γ 2 while on Γ 3 the body is in frictional contact with a foundation.
According to this physical setting, we state the following boundary value problem.
As usual, u = (u i ) denotes the displacement field, ε = ε(u) = (ε ij (u)) denotes the infinitesimal strain tensor, σ = (σ ij ) denotes the Cauchy stress tensor, ω is a constitutive map, ν stands for the unit outward normal to Γ, the normal and the tangential components of the displacement vector on the boundary are defined by the formulas u ν = u · ν, u τ = u − u ν ν and the normal and the tangential components of the Cauchy vector on the boundary are defined by σ ν = (σν) · ν, σ τ = σν − σ ν ν.
Due to the condition −σ ν = F, according to the engineering literature, the frictional contact model we treat is a bilateral frictional contact problem. In this context, we have to mention that the bilateral frictional contact phenomenon can be found in many components of mechanical equipment. Thus, many real-world examples can be envisaged. The mathematical and the engineering literature contains relevant applications of bilateral frictional contact models; see, e.g., [41,42].
Referring to the behavior of the materials, for significant examples of nonlinearly elastic constitutive laws described by means of subdifferential inclusions for various constitutive maps ω, see, e.g., [38] and the references therein. For the convenience of the reader, we indicate here an example of such a constitutive map: where A : S 3 → S 3 , A = (A ijkl ), A ijkl = λ 0 δ ij δ kl + µ 0 (δ ik δ jl + δ il δ jk ), 1 ≤ i, j, k, l ≤ 3, with λ 0 , µ 0 and k small enough positive material coefficients, and P K : S 3 → K denotes the projection operator on the closed and convex set K ⊂ S 3 which contains 0 S 3 . In order to study Problem 1, we make the following assumptions. Assumption 1. ω : S 3 → R is a convex and lower semicontinuous functional. In addition, there exist α, β such that 1 > β ≥ α > 0 and β ε 2 Assumption 4. The coefficient of friction satisfies k ∈ L ∞ (Γ 3 ) and k(x) ≥ 0 a.e. x ∈ Γ 3 .
Notice that the example in (14) fulfills Assumption 1. Let (u, σ) be a pair of smooth-enough functions that verify Problem 1. Using Green's formula (see, e.g., [3], (p. 145)), by taking into account (8), (10) and (11) we obtain, for all v ∈ V (σ, ε(v)) L 2 (Ω; (15) where V is the space defined in (3). As then by (15) we infer that Herein and everywhere below, v ν ( e. x ∈ Γ. By taking into consideration H3, H4, using the trace theorem for N = 3 and p = 2 and the Hölder's inequality, it follows that is a linear and continuous map, according to Riesz's representation theorem, there exists a unique element f ∈ V such that Let D be the dual of the space γ(V) defined in (6). We define λ ∈ D such that where ·, · denotes the duality pairing between D and γ(V) and Furthermore, we define a form c(·, ·) as follows, Important properties of the form c(·, ·) are given by the following lemma.
Let v ∈ V and ζ ∈ D be arbitrarily given.
We can consider M c = c c K where c > 0 appears in (2) and c K > 0 appears in (5). Next, we prove that c(·, ·) verifies the inf-sup property, or equivalently, there are some .
Using (7), we can write Thus, we can take α c = 1 c .
Let us introduce the following subset of D, see (18) for the definition of v τ . It is easy to observe that λ ∈ Λ by using (17) and (20).

Lemma 2.
The set Λ is a closed convex bounded subset of D that contains 0 D .

Proof.
It is easy to observe that 0 D ∈ Λ. In addition, the convexity can be easily obtained by means of the definition of the convex sets. Let (ζ n ) ⊂ Λ be a convergent sequence, ζ n → ζ in D as n → ∞.
We have to prove that ζ ∈ Λ. As the sequence (ζ n ) converges strongly to ζ, then ζ n ζ in D as n → ∞ and then ζ n * ζ in D as n → ∞.
Using the definition of the weak* convergence, we infer that Passing to the limit n → ∞ in the previous inequality, we conclude that ζ ∈ Λ. As a result, Λ is a closed set.
As a result,
Hence, (22) holds true in this situation too.
Proof. Let ζ ∈ Λ and let us define As c(·, ·) is a bilinear form, then the linearity of ϕ ζ is obvious. Let us prove its continuity. In order to do this, we have to prove that there exists K > 0 such that We can set K = c c K , where c is the constant in (2) and c K is the constant in (5). As a result, ϕ ζ is a linear and continuous map. Then, due to Riesz's representation theorem, there exists a unique G ζ ∈ V such that Hence, and keeping in mind the definition of the inner product on V, (4), actually we can write Let us take µ = ε( f − G ζ ) to conclude that Σ(ζ) is a nonempty subset of L 2 s (Ω; S 3 ). The convexity can be easily proved by using the definition of the convex sets.
Each solution (u, λ, σ) ∈ V × Λ × Σ(λ) of Problem 2 is called a weak solution of Problem 1. Keeping in mind (29) and (28), the bifunctional b(·, ·) can be written by means of two functionals J(·) and J * (·) as follows, Then, we observe that In consequence, Problem 2 can be equivalently written as follows.
The weak solvability of Problem 1 will be studied by means of the variational formulation stated in Problem 3.

Well-Posedness and a Convergence Result
This section is devoted to the solvability of Problem 3.
Proof. Let us introduce the following bifunctional: A pair (u, λ) ∈ V × Λ verifies (39) and (40) if and only if it is a saddle point of the bifunctional L(·, ·), i.e., Indeed, (40) is equivalent with the first inequality in the chain above. On the other hand, using the definition of L(·, ·), by a similar technique with that used in [44], we deduce that (39) is equivalent with the second inequality in (41).
Keeping in mind H1, we immediately conclude that J(·) is a convex lower semicontinuous functional such that Furthermore, according to Lemma 1, c(·, ·) is a bilinear continuous form. Therefore, L(·, ·) fulfills the conditions in Theorem 3. Moreover, according to Lemma 2, Λ is a closed convex bounded set which contains 0 D . In addition, we note that which allows us to write Therefore, applying Theorem 3, we conclude that the functional L(·, ·) has at least one saddle point (u * , λ * ) ∈ V × Λ.
Finally, if ω is, in addition, strictly convex, using Theorem 3 we immediately conclude that Problem 3 has at least one solution which is unique in its first component.
In order to obtain a global uniqueness result, one option could be to perturb Problem 3 as follows.
Let us introduce the following perturbed bifunctional.
Using similar techniques with those used in [44], it can be verified that a pair (u , λ ) ∈ V × λ verifies (42) and (43) if and only if it is a saddle point of the bifunctional L (·, ·), i.e., We observe that L (·, ·) is strictly concave in the second argument.
In order to prove the boundedness of the solution, let us set v = 0 V in (42) and ζ = 0 D in (43). With this choice, (42) and (43) lead us to since, due to H1, J(0 V ) = 0. Additionally, from this, Hence, (45) holds true. By (42), setting v = u − w with w ∈ V arbitrarily chosen, we immediately obtain As ω is Lipschitz continuous of rank L, J is Lipschitz continuous of the same rank L. Therefore, By using the inf-sup property of the form c(·, ·), we can write As a result, we obtain (46). According to (44) and keeping in mind (25) Let G λ be the unique element of V such that c(v, λ ) = (G λ , v) V for all v ∈ V.
Indeed, ε being a continuous operator, it is also a lower semicontinuous and upper semicontinuous operator. On the other hand, ε is a linear operator so it is also a convex operator. Therefore, ε is a weakly lower semicontinuous operator and a weakly upper semicontinuous operator as well. Hence, keeping in mind (50), we can write lim sup ε(G λ ) ≤ ε(G λ * ) ≤ lim inf ε(G λ ) as → 0.

Conclusions and Final Comments
In the present paper, we address a frictional contact model with prescribed normal stress. We deliver a new weak formulation that is a three-field variational formulation governed by a bipotential related to the constitutive function ω and a Lagrange multiplier related to the friction force σ τ . We establish existence, uniqueness, boundedness and convergence results. Theorem 6 indicates us that the unique solution of the perturbed Problem 4 helps us to approximate a weak solution of Problem 1. Delivering uniqueness results by omitting a perturbation technique is left open.
The advantage of the approach we propose is twofold. On the one hand, the new approach allows the inclusion of the friction force in the unknown in addition to the displacement field and the Cauchy stress tensor. On the other hand, the qualitative analysis we perform allows moving on to the quantitative analysis in order to efficiently approximate the triple weak solutions.
Funding: This research received no external funding. Data Availability Statement: Not applicable.