Abstract
We deal with the Signorini contact problem between two Timoshenko beams. In this work we use the theory of semigroups to show the existence of solutions that decay uniformly to zero. This method is new and more effective than the widely used energy method. This is because in particular we obtain uniform decay of the solutions to zero for any boundary condition. A second important point is that we can take advantage of stabilization results of others linear dynamic systems with different dissipative mechanisms and apply them through our method for Contact Problems (see Sect. 4). Finally, thanks to Lipschitzian perturbations we can generalize the Signorini problem to more general semi linear problems in a simple way (see Sect. 4.3).
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1 Introduction
This work is focused on the mechanical evolution of two dissipative Timoshenko beams in unilateral contact across a joint with clearance. The area-centers of gravity of beams in their (stress free and isothermal) reference configurations are given by the intervals \(I_{1}:=(0,\ell _{*})\) and \(I_{2}:=(\ell _{*},\ell )\), respectively. Let \(0 < T \leq \infty \). We denote by \(\varphi _{1} = \varphi _{1}(x, t) : I_{1} \times (0, T) \to \mathbb{R}\) the transverse displacement (vertical deflection) of the cross section at \(x\in I_{1}\) and at time \(t\in (0,T)\), by \(\varphi _{2} = \varphi _{2}(x, t) : I_{2} \times (0, T) \to \mathbb{R}\) the transverse displacement (vertical deflection) of the cross section at \(x\in I_{2}\) and at time \(t\in (0,T)\). Supposing that plane cross sections remain plane, the angles of rotation of a cross section are defined respectively by \(\psi _{1} = \psi _{1}(x, t) : I_{1}\times (0, T) \to \mathbb{R}\) and \(\psi _{2} = \psi _{2}(x, t) : I_{2} \times (0, T) \to \mathbb{R}\). The physical setting is represented by Fig. 1.
We describe the evolution of the system, under consideration, by the following equations (for details, see e.g. [9, 16, 18]),
where \(\gamma _{1,i}\) and \(\gamma _{2,i}\) are real non negative functions defined in \(I_{i}\) with \(i=1,2\). Here the coefficients represent: \(\rho _{1}=\rho A\) the mass density, \(\rho _{2}=\rho I\) the moment of mass inertia, \(k=\kappa G A\) the shear modulus of elasticity, \(b= EI\) the rigidity coefficient of cross-section, where \(E\) is the Young modulus, \(G\) is the modulus of rigidity, \(\kappa \) is the transversal shear factor, and \(I\) is the moment of inertia. Functions \(S\) and \(M\) stand for the shear force and the bending moment, respectively. Subscripts \(x\) and \(t\) represent partial derivatives with respect to \(x\) and \(t\). Henceforth, unless stated otherwise, the index \(i\), as a subscript of the variables involved, will always take the values 1 and 2. The initial conditions are given by
for some given functions \(\varphi _{i0}, \varphi _{i1},\psi _{i0}, \psi _{i1}:I_{i}\to \mathbb{R}\). In addition, we suppose that, at \(x=0\), \(x=\ell _{*}\) and \(x=\ell \),
The joint at \(x=\ell _{*}\) is modeled with the Signorini non penetration condition (see, e.g., [11]). In particular, the joint with gap \(g\) is asymmetrical so that \(g=g_{1}+g_{2}\), where \(g_{1}>0\) and \(g_{2}>0\) are, respectively, the upper and lower clearance, when the system is at rest. Then, at the point \(\ell _{*}\) of the beam is assumed to move vertically only between two stops, namely
This condition assures that the displacement at \(x=\ell _{*}\) is constrained between the stops \(g_{1}\) and \(g_{2}\). The stress at this point if given by
Moreover, we prescrive the condition
where \(\partial \chi _{v}(u)\) denotes the subdifferential of the indicator function \(\chi _{v}(u)\),
namely
Let us spend a few words on the condition expressed by the condition above. When
is verified, there is no contact, the ends at \(x=\ell _{*}\) are free, and \(S(\ell _{*},t)=0\). On the other hand, when
the ends at \(x=\ell _{*}\) are in contact. More precisely, when the contact occurs at the lower end, relations \(\varphi _{2}(\ell _{*},t) - g_{2}=\varphi _{1}(\ell _{*},t)\) and \(S(\ell _{*},t)\geq 0\) hold; when the contact takes place at the upper end, relations \(\varphi _{2}(\ell _{*},t)=\varphi _{1}(\ell _{*},t)-g_{1}\) and \(S(\ell _{*},t)\leq 0\) are verified.
This manuscript engages and develops within the study of the existence and asymptotic behavior of solutions associated with contact problems between two beams. The system specified by (1.1)–(1.5), and the questions related, can be regarded as an extension to the viscoelastic case of the problem studied in [10]. Dynamic models for vibrations transmission across joints are of considerable interest in various industrial settings and in many applications. In most articles currently present in the literature, the Signorini contact problem has been analyzed in a weak sense, namely by considering an approximate version of the Signorini problem via the introduction of a normal compliance condition as regularization of the Signorini condition. The exponential stability of a solution as time goes to infinity is obtained in the approximate framework: the exponential decay for the approximate solution is found by introducing a suitable Lyapunov functional and by using the multiplier method. Then, by weak lower semicontinuity arguments, the exponential decay is achieved for a solution to the original problem.
In this paper we follow a new and different approach. We consider the linear Timoshenko model coupled to a dynamic boundary condition defined by an ordinary differential equation (hybrid system), the coupling is defined through a parameter \(\epsilon \) which we will then approximate to zero, see the system (2.3) below. We use semigroup theory to show the well-posedness of the problem, as well as the exponential stability of the corresponding model. We arrive at the problem of contact with normal compliance condition through a Lipschitzian perturbation.
Finally, by setting \(\epsilon \to 0\) the dynamic boundary condition becomes static and due to the characteristics of the chosen Liptchitzian perturbation (see the last equation in system (3.7)), we arrive at the Signorini conditions which proves the existence of solution to problem (1.1)–(1.5). This procedure is possible thanks to the observability inequalities that Timoshenko model possesses. We believe that this method is more efficient than the usual penalty method (see [4, 11, 15] and the references contained therein) because in this way we obtain more general results about the asymptotic behavior of the solution. In particular, we show that the boundary conditions of the model do not play any important role in the test of asymptotic behavior. This means that the decay result can be proved for any boundary condition, different from the results obtained in [4, 6, 7, 15] where boundary conditions played an important role in the proof of exponential decay.
The remaining part of this paper is organized as follows. In Sect. 2 we show the well possedness of the linear hybrid model. In Sect. 3 we find the main result of this paper: the existence of a solution to Signorini problem, which decays exponentially as the linear semigroup. Finally, in Sect. 4 we give some applications of our result.
2 The Hybrid Linear Model as a Compact Perturbation
To fix ideas, we consider the viscoelastic constitutive law of Kelvin-Voigt type
with \(\widetilde{\kappa}_{i} \) and \(\widetilde{b}_{i}\) nonnegative functions characterizing the viscosity of the beam. Moreover, we assume that both have the same support, namely
In general, we assume that the viscous component is effective over sub intervals of \(I_{i}\). We will denote as \(I_{E}\) any subinterval of \(I_{i}\) where the viscosity is not efective, that is \(\widetilde{\kappa}_{i}=\widetilde{b}_{i}=0\) over \(I_{E}\).
To apply the semigroup theory to study the Signorini problem, let’s start by considering two uncoupled linear hybrid models, one defined over the interval \(I_{1}:=(0,\ell _{*})\) and the other one over the interval \(I_{2}:=(\ell _{*},\ell )\), approaching the penalized problem associated to (1.1)–(1.5), and given by
satisfying the boundary conditions over \(I_{1}=]0,\ell _{*}[\) and \(I_{2}=]\ell _{*},\ell [\)
and verifying the initial conditions (1.2) together with
for some given numbers \(v_{i,0}\) and \(v_{i,1} \in \mathbb{C}\).
The dynamic boundary condition (2.3)3 can be interpreted as a beam rigidly attached at the end \(x=\ell _{*}\) to a tip body that models a sealed container with a granular material, for example sand. This granular material dampens the movement of the system by internal friction (for details, see [2, 3, 14]).
The phase space of our problem is
where
Denoting by \(U_{i}=(\varphi _{i},\Phi _{i},\psi _{i},\Psi _{i},v_{i},V_{i})^{\top}\), we define the norm
Equations (2.3) are uncoupled and independent one to the other for \(i=1,2\). For sake of simplicity, in what follows we remove the subindex \(i\) from the variables. Denoting by \(B^{\top}\) the transpose of a matrix \(B\) and \(\Phi =\varphi _{t}\), \(\Psi =\psi _{t}\) and \(V=v_{t}\) we have
where \(\mathcal{U}:=\left (\varphi (t),\Phi (t),\psi (t),\Psi (t)\right )^{ \top}\) and \(\mathcal{V}:=\left (v(t),V(t)\right )^{\top}\). Hence, system (2.3) can be written as a linear ODE in \(\mathcal{H}_{i}\) of the form
where the domain \(\mathcal{D}(\mathcal{A}_{i})\) of the linear operator \(\mathcal{A}_{i}:D(\mathcal{A}_{i})\subset \mathcal{H}_{i} \to \mathcal{H}_{i}\) is given by
and
According to Lumer-Phillips Theorem (see, e.g., [12, Theorem 1.2.4] or [17, Theorem 1.4.3]), the operator \(\mathcal{A}_{i}\) is the infinitesimal generator of a contraction semigroup
In particular, \(\mathcal{A}_{i}\) is dissipative. Indeed, for every \(U \in \mathcal{D}(\mathcal{A}_{1})\),
Here we used that \(V=\Phi (\ell _{*})\). For the constitutive law (2.1), we get
Similarly for \(\mathcal{A}_{2}\). Hence we have
For any initial datum \(U_{0}=\left (\varphi _{0},\varphi _{1},\psi _{0},\psi _{1},v_{0},v_{1} \right )^{\top }\in \mathcal{H}_{i}\) the solution to (2.6) is denoted by
Considering the resolvent equation
and taking inner product with \(U\) over the phase space \(\mathcal{H}_{i}\), we get
In particular, we have
In this section we will make a comparison between the hybrid model and the non hybrid Timoshenko model given by
for \(i=1,2\), satisfying the boundary conditions
Let us denote the infinitesimal generator of system (2.14)–(2.16) by \(\mathcal{A}_{i,T}\) where
The phase space we consider for the above model is
Hence the domain \(\mathcal{D}(\mathcal{A}_{i,T})\) of the linear operator \(\mathcal{A}_{i,T}:D(\mathcal{A}_{i,T})\subset \mathbf{H}_{i} \to \mathbf{H}_{i}\) is given by
Similarly as the hybrid model, we have
with \(\gamma _{1},\gamma _{2}\geq 0\). Under this notations we get that system (2.14)–(2.16) can be written as
Let us denote by \(\mathbf{T}_{i}=e^{t\mathcal{A}_{i,T}}\) the semigroup associated to system (2.14). The main objective of this section is to show that the semigroup \(\mathbf{T}_{i}\) is exponentially stable if and only the semigroup \(\mathcal{T}_{i}\) is also exponentially stable. This means that the dissipation produced by the ODE in (2.3)3, of the hybrid model, is not relevant. Let us introduce the space
intended as the extended phase space. Let us denote by \(\Pi _{i}\) the projection of \(\mathcal{H}_{i}\) onto \(\widetilde{\mathbf{H}}_{i}\):
Let us decompose the infinitesimal generator \(\mathcal{A}_{i}\) in the following way
with
where \(\boldsymbol{\gamma}_{1}\varphi =\varphi _{x}(\ell _{*})\) and \(\boldsymbol{\gamma}_{0}\psi =\psi (\ell _{*})\). Hence, recalling that \(U:=(\mathcal{U}, \mathcal{V})^{\top}\), where \(\mathcal{U}:=(\varphi ,\Phi ,\psi ,\Psi )\) and \(\mathcal{V}:=(v,V)\), we get
Under the above conditions we can state the following Lemma:
Lemma 2.1
The difference \(\mathcal{T}_{i}(t)-\mathbf{T}_{i}(t)\Pi \) is a compact operator over \(\mathcal{H}_{i}\). Hence the corresponding essential types \(\omega _{\textit{ess}}(\mathcal{T}_{i})\) and \(\omega _{\textit{ess}}(\mathbf{T}_{i}(t)\Pi )\) are equal.
Proof
Note that the solution of \(U_{t}-\mathcal{A}_{i}U=0\), \(U(0)=U_{0}\) can be written as
with \(U_{0}=(\mathcal{U}_{0}, \mathcal{V}_{0})^{\top}\) which implies that
Therefore
Note that the right hand side of the above equation is a compact operator, therefore
is a compact operator. So our conclusion follows. □
Remark 2.1
Let us denote by \(\mathbf{A}\) the operator \(\mathcal{A}_{i}\) or \(\mathcal{A}_{i,T}\), then resolvent operator \((\mu I-\mathbf{A})^{-1}\) is not compact.
Here we consider \(\gamma _{1,i}=0\) and \(\gamma _{2,i}=0\). We will show that the spectrum of \(\mathbf{A}\), \(\sigma (\mathbf{A})\), does not contain only eigenvalues. In fact, let us consider \(\mu \in \mathbb{R}\), then the resolvent equation can be written as
where we assume that \(\kappa \), \(\widetilde{\kappa}\), \(b\) and \(\widetilde{b}\) are positive constant. Taking \(f_{1}=f_{3}=0\), the above system can be written as
Let us consider the numbers \(\mu _{1}=-\kappa /\widetilde{\kappa}\) and \(\mu _{2}=-b/\widetilde{b}\) none of them belong to the resolvent set of \(\mathbf{A}\). In fact from (2.21) we get
On the other hand, if \(\mu _{1}\in \rho (\mathbf{A})\) the corresponding solution \(U\) must satisfy \(U\in D(\mathbf{A})\) which in particular means that \(\varphi \in H^{1}(0,\ell _{*})\), which is contradictory to (2.23).
Similarly, if \(\mu _{2}\in \rho (\mathbf{A})\) using (2.21) and (2.22) we get
But \(U\) must be in \(D(\mathbf{A})\) which in particular means that \(\psi \in H_{0}^{1}(0,\ell _{*})\). This is contradictory to (2.24). Hence \(\mu _{2}\notin \rho (\mathbf{A})\).
Finally we will show that one of them is not an eigenvalue. Here \(f_{2}=f_{4}=0\), let us suppose that \(\mu _{2}<\mu _{1}\). We will show that \(\mu _{1}\) is not an eigenvalue. Note that (2.21) implies that \(\varphi =0\) and (2.22) can be written as
Multiplying the above equation by \(\psi \), integrating by parts we get
So we have that \(\psi =0\) which implies that \(U=0\) that is a contradiction.
If \(\mu _{1}<\mu _{2}\), then we have that \(\mu _{2}\) is not an eigenfunction. This because system (2.21)–(2.22) can be written as
From the above equations we get
So we have that \(U=0\) which is a contradiction. The same result holds when \(\mu _{1}=\mu _{2}\). Then our conclusion follows.
Lemma 2.2
Let us denote by \(\mathbf{A}\) the operator \(\mathcal{A}_{i}\) or \(\mathcal{A}_{i,T}\). If \(i\mathbb{R}\not \subset \varrho (\mathbf{A})\) then there exists \(0\ne \sigma \in \mathbb{R}\) such that \(i\sigma U -\mathbf{A} U= 0\).
Proof
Let us denote by
It is easy to see that \(0\in \rho (\mathbf{A})\), so we have \(\mathcal{N}\neq \varnothing \). Putting \(\sigma =\sup \mathcal{N}\) we have two possibilities. First \(\sigma =+\infty \), which implies that \(i\mathbb{R}\subseteq \rho (\mathbf{A})\), and that \(0<\sigma \) finite. We will reason by contradiction. Let us suppose that \(\sigma <\infty \). Then, exists a sequence \(\{\lambda _{n}\}\subseteq \mathbb{R}\) such that \(\lambda _{n}\to \sigma < \infty \) and
Hence, there exists a sequence \(\{f_{n}\}\subseteq \mathcal{H}\) verifying \(\|f_{n}\|_{\mathcal{H}}=1\) and \(\|(i\lambda _{n}I-\mathbf{A})^{-1}f_{n}\|_{\mathcal{H}}\to \infty \). Denoting by
and \(U_{n}=\dfrac{\tilde{U}_{n}}{\|\tilde{U}_{n}\|_{\mathcal{H}}}\), \(F_{n}=\dfrac{f_{n}}{\|\tilde{U}_{n}\|_{\mathcal{H}}}\) we conclude thet \(U_{n}\) verifies \(\|U_{n}\|_{\mathcal{H}}=1\) and
Since \(\|\mathbf{A} U_{n}\|_{\mathcal{H}}\leq C\) using (2.1) or (2.18) we have that over the viscoelastic component \(I_{v}=\operatorname{supp}( \widetilde{\kappa})=\operatorname{supp}( \widetilde{b})\) the sequences \(\Phi _{n}\), \(\Psi _{n}\) converges strongly, that is
Since \(\lambda _{n}\to \sigma < \infty \) the above convergence implies that
Since \(U_{n}\) is bounded in \(D(\mathbf{A})\) we conclude that \(\varphi _{n}\), \(\psi _{n}\) are bounded in \(H^{2}(]0,\ell _{*}[\setminus I_{v})\). In particular this means that there exists a subsequence of \(\varphi _{n}\) and \(\psi _{n}\), we still denote in the same way, such that
Therefore from convergences (2.25), (2.26) and (2.27) there exists a subsequence of \(U_{n}\), we still denote in the same ways such that
Hence \(\|U\|_{\mathcal{H}}=1\). Moreover, because of \(\mathbf{A}U_{n}=i\lambda _{n}U_{n}-F_{n}\), we have \(\mathbf{A}U_{n}\) converges strongly in ℋ. Since \(\mathbf{A}\) is closed, we conclude that \(U\) verifies
from where our conclusion follows. □
To show the equivalence of the exponential stability between \(\mathcal{T}_{i}(t)\) and \(\mathbf{T}_{i}(t)\) we apply the following result
Theorem 2.1
Let \(S(t)=e^{{\mathbb{A}}t}\) be a \(C_{0}\)-semigroup of contractions on Banach space. Then, \(S(t)\) is exponentially stable if and only if
where \(\omega _{ess}(S(t))\) is the essential growth bound of the semigroup \(S(t)\).
Proof
Here we use [8, Corollary 2.11] establishing that the type \(\omega \) of the semigroup \(e^{\mathbb{A}t}\) verifies
where \(\omega _{\sigma}(\mathbb{A})\) is the upper bound of the spectrum of \(\mathbb{A}\). Moreover, for any \(c>\omega _{ess}\), the set \(\mathcal{I}_{c}:=\sigma (\mathbb{A})\cap \{\lambda \in \mathbb{C}:\; \; \operatorname{Re}\lambda \geq c\}\) is finite.
Let us suppose that (2.29) is valid. Since the essential type of the semigroup \(\omega _{ess}\) is negative, identity (2.30) states that the type of the semigroup will be negative provided \(\omega _{\sigma}(\mathbb{A})<0\).
If \(\omega _{\sigma }(\mathbb{A}) \leq \omega _{ess} \) then we have nothing to prove. Let us suppose that \(\omega _{\sigma }(\mathbb{A})> \omega _{ess} \). From (2.29) and Hille-Yosida Theorem we have \(\overline{\mathbb{C} _{+}} \subset \varrho (\mathbb{A}) \), hence \(\omega _{\sigma }(\mathbb{A})\leq 0 \). On the other hand \(\mathcal{I}_{\omega _{ess}+\delta} \) is finite for \(\delta >0\) verifying \(\omega _{ess} + \delta <0 \) and \(\omega _{ess} + \delta <\omega _{\sigma }(\mathbb{A}) \). Therefore we have
Hence, the sufficient condition follows.
Reciprocally, let us suppose that the semigroup \(S(t)\) is exponentially stable, in particular it goes to zero. Then, by [5, Theorem 1.1] we have that \(i\mathbb{R}\subset \varrho (\mathbb{A})\). Moreover, since the type \(\omega \) verifies (2.30), we have that
Then, our conclusion follows. □
Remark 2.2
The above characterization is valid for any Banach space.
Theorem 2.2
The semigroup \(\mathbf{T}_{i}\) is exponentially stable if and only if the semigroup \(\mathcal{T}_{i}(t)=e^{{\mathcal{A}_{i}}t}\) associated to the hybrid system (2.3) also is exponentially stable.
Proof
Let us suppose that \(\mathbf{T}_{i}\) is exponentially stable. First note that \(i\mathbb{R} \subset \varrho (\mathcal{A}_{i})\). In fact, let us suppose the contrary, then by Lemma 2.2 we have that there exists \(\sigma \in \mathbb{R}\) such that
Using (2.9) we find
By (2.3)3 we get
Therefore the eigenvector \(U=(\mathcal{U},\mathcal{V})^{\top}=(\mathcal{U},\mathbf{0})^{\top}\) verifies (2.15)–(2.16). Then, we have
This implies that \(i\lambda \in \sigma (\mathcal{A}_{i,T})\) which is not possible because \(\mathbf{T}_{i}\) is exponentially stable. This contradiction comes from assuming that \(i\mathbb{R} \not \subset \varrho (\mathcal{A}_{i})\). Therefore \(i\mathbb{R} \subset \varrho (\mathcal{A}_{i})\). From Lemma 2.1 we have that \(\omega _{\text{ess}}(\mathcal{T}_{i})=\omega _{\text{ess}}(\mathbf{T}_{i}(t) \Pi )<0\) then Theorem 2.1 implies the exponential stability of \(\mathcal{T}_{i}(t)=e^{{\mathcal{A}_{i}}t}\).
Finally, let us suppose that \(\mathcal{T}_{i}(t)=e^{{\mathcal{A}_{i}}t}\) is exponentially stable. In particular, we have \(i\mathbb{R} \subset \varrho (\mathcal{A}_{i})\), and as before by Theorem 2.1 it is enough to show the strong stability of \(\mathcal{A}_{i,T}\). By contradiction if \(i\mathbb{R} \not \subset \varrho (\mathcal{A}_{i,T})\), by Lemma 2.2 there exists \(\mathcal{U}\ne 0\) such that \(\mathcal{A}_{i,T}\mathcal{U}=i\sigma \, \mathcal{U}\) then the vector \(U=(\mathcal{U},0,0)\) must be an imaginary eigenvector of \(\mathcal{A}_{i}\). But this is a contradiction. Then, our conclusion follows. □
3 The Signorini Problem
In what follows we prove the well posedness of an abstract semilinear problem and we study, under suitable conditions the asymptotic behavior of the solutions. So, we introduce a local Lipschitz function ℱ defined over a Hilbert space ℋ. We suppose that for any ball \(B_{R}=\{W\in \mathcal{H}:\;\; \|W\|_{\mathcal{H}}\leq R\} \), there exists a function globally of Lipschitz \(\widetilde{\mathcal{F}_{R}}\) such that
and additionally, that there exists a positive constant \(K_{0}\) such that
Under these conditions, we present
Theorem 3.1
Let \(\{T(t)\}_{t\geq 0}\) be a \(C_{0}\) semigroup of contraction, exponentially stable semigroup with infinitesimal generator \(\mathbb{A}\) over the phase space ℋ. Let ℱ locally Lipschitz on ℋ satisfying conditions (3.1) and (3.2). Then there exists a global solution to
that decays exponentially.
Proof
By hypotheses, there exist positive constants \(c_{0}\) and \(\gamma \) such that \(\|T(t)\|\leq c_{0}e^{-\gamma t} \), and \(\widetilde{\mathcal{F}_{R}}\) globally Lipschitz with Lipschitz constant \(K_{0}\) verifying conditions (3.1) and (3.2). Let us consider the following space:
Using standard fixed point arguments we can show that there exists only one global solution to
Multiplying the above equation by \(U^{R}\) we get that
Since the semigroup is contractive, its infinitesimal generator is dissipative, therefore
Using (3.2) we get
Note that for \(R> (1+2K_{0})\|U_{0}\|_{\mathcal{H}}^{2}\), we have that
In particular,
This means that \(U^{R}\) is also solution of system (3.3) and because of the uniqueness we conclude that \(U^{R}=U\). To show the exponential stability to system (3.3), it is enough to show the exponential decay to system (3.4). To do that, we use fixed points arguments. Let us consider
Note that \(\mathcal{T}\) is invariant over \(E_{\gamma -\delta}\) for \(\delta \) small, with \(\gamma -\delta >0\). In fact, for any \(V\in E_{\gamma -\delta}\) we have
Hence \(\mathcal{T}(V)\in E_{\gamma -\delta}\). Using standard arguments we show that \(\mathcal{T}^{n}\) satisfies
with \(k_{1} \in \mathbb{R}\). Therefore we have a unique fixed point satisfying
that is \(U\) is a solution of (3.4), and since \(\mathcal{T}\) is invariant over \(E_{\gamma -\delta}\), then the solution decays exponentially. □
3.1 The Linear Model
To apply Theorem 3.1 for the linear model given by problem (2.3)–(2.5) let us introduce the functions
Let us denote by ℋ the phase space given by
Then, for any \(U_{0}\in \mathcal{H}\) let us introduce the semigroup
It is easy to verify that \(\mathcal {T}(t) \) is a contraction semigroup over ℋ. Moreover its infinitesimal generator \(\mathbb{A}\) is given by
where the domain \(\mathcal{D}(\mathbb{A})\) of the linear operator \(\mathbb{A}:\mathcal{D}(\mathbb{A})\subset \mathcal{H}\to \mathcal{H}\) is given by
Under the above conditions we have
Theorem 3.2
Let us suppose that the semigroups \(\mathbf{T}_{i}\), \(i=1,2\) are exponentially stable, then the semigroup defined in (3.5)–(3.6) is exponentially stable.
Proof
Immediate consequence of Theorem 2.2. □
3.2 The Semilinear Model
Let us consider the semilinear system
for \(i=1,2\). Note that the above system is now coupled by the dynamic boundary condition (3.7)3. The above system can be written as
where \(\mathcal{A}\) is given by (2.7) and ℱ is given by
with \(f(v_{1},v_{2})=-\dfrac{1}{\epsilon ^{2}}\left [(v_{1}^{\epsilon}-v_{2}^{ \epsilon}\!-\!g_{1})^{+}-(v_{2}^{\epsilon}-v_{1}^{\epsilon}\!-\!g_{2})^{+} \right ]\). Note that ℱ is a Lipschitz function verifying hypothesis (3.1)–(3.2) and does not depend on \(i\). In fact, \(\mathcal{F}(0)=0\). Moreover,
Remark 3.1
If the initial data verifies
then the right hand side of the (3.9) vanishes.
Theorem 3.3
The semilinear semigroup defined by system (3.7) is exponentially stable.
Proof
It is a direct consequence of Theorem 3.1. □
Next we show the energy inequality
Lemma 3.1
The solution of system (3.7) satisfies
where
and
Proof
Multiplying equation (3.7)1 by \(\widehat{\varphi}_{t}\), equation (3.7)2 by \(\widehat{\psi}_{t}\), and equation (3.7)3 by \(v_{i,t}\), summing up the product result our conclusion follows. □
Let us introduce the functional
Under the above notations we have
Lemma 3.2
Let \(]\alpha ,\beta [\) be an interval where \(\widetilde{\kappa}_{i}=\widetilde{b}_{i}=0\), then the solution of system (3.7) satisfies
Proof
Over the interval \(]\alpha ,\beta [\) where \(\widetilde{\kappa}_{i}=\widetilde{b}_{i}=0\), system (3.7) can be written as
Let us denote by \(q=x-\frac{\alpha +\beta}{2}\). Multiplying equation (3.11)1 by \(q\varphi _{i,x}\) and equation (3.11)2 by \(q\psi _{i,x}\) we get
Similarly we get
where
Summing up identities (3.12) and (3.13) performing integrations by parts and integrating overt \([0,T]\) we get
where
Using Lemma 3.1 we get
Substitution of the above inequalities into (3.14), our result follows. □
Let us introduce the convex set
With these notations we have
Theorem 3.4
For any initial data \((\varphi _{0}^{i},\varphi _{1}^{i},\psi _{0}^{i},\psi _{1}^{i})\in \mathcal{H}_{i}\) such that
there exists a weak solution to Signorini problem (1.1)–(1.4) which decays as established in Theorem 3.3.
Proof
From Theorem 3.1 we have that there exists only one solution to system (3.7). Let \(]\ell _{*}-\delta ,\ell _{*}[\) be an interval where \(\widetilde{\kappa}_{i}=\widetilde{b}_{i}=0\), using Lemma 3.1 and Lemma 3.2, we get
which means that the first order energy is uniformly bounded for any \(\epsilon >0\). This implies that \((v_{1}^{\epsilon},v_{2}^{\epsilon})\) strongly converges to \((v_{1},v_{2})\in \mathcal{K}\). Standard procedures implies that the solution of system (3.7) converges in the distributional sense to system (1.1). It remains only to show that conditions (1.4) holds. First note that
In fact, from (3.7)3 we get
Summing up the above equations we get
Then for any \(\eta \in C_{0}^{\infty}(\mathbb{R}_{+})\) we get
So we denote by \(S(\ell _{*},t)=\lim _{\epsilon \rightarrow 0}S^{\epsilon}_{1}(\ell _{*},t)= \lim _{\epsilon \rightarrow 0}S^{\epsilon}_{2}(\ell _{*},t)\). Finally, we use the observability inequality in Theorem 3.2, and we get that \(\widehat{\varphi}_{t}^{\epsilon}(\ell ,t)\) and \({S}^{\epsilon}(\ell ,t)\) are bounded in \(L^{2}(0,T)\), so is \(v_{tt}\). Using (3.7)3 we obtain
for any \(u\in \mathcal{K}\). It is no difficult to see that
In fact, from (3.7)3 \(\epsilon v_{tt}^{\epsilon}\) is bounded for any \(\epsilon >0\) (by a constant depending on \(\epsilon \)) in \(L^{2}(0,T)\). From (3.15) \(v_{i,t}^{\epsilon}\) is also uniformly bounded in \(L^{2}(0,T)\). Therefore \(v_{i,t}^{\epsilon}\) is a continuous function, uniformly bounded in \(L^{\infty}(0,T)\). Making an integration by parts we find
Hence,
Since
for all \(u\leq v_{2}^{\epsilon }+g_{1}\). Similarly we find
for all \(u\geq v_{2}^{\epsilon }-g_{2}\). Therefore, from the last two inequalities we get
for any \(u\in \mathcal{K}\) such that \(v_{2}^{\epsilon}-g_{2}\leq u\leq v_{2}^{\epsilon}+ g_{1}\). Taking the limit \(\epsilon \rightarrow 0\) we get
Using the same above procedure to equation (3.17) we get
From relations (3.18)–(3.19) and since \(S_{1}(\ell _{*},t)=S_{2}(\ell _{*},t)\), we get (1.5).
In case of \(\ell _{*}\in \operatorname{supp}( \widetilde{\kappa}_{i} )\) we have that \(\varphi _{t}^{\epsilon}\) is bounded in \(L^{2}(0,T;H^{1}(]\ell _{*}-\delta ,\ell _{*}[))\). So the sequences \(\varphi _{t}^{\epsilon}(\ell ^{*},t)\) converges strongly in \(C(0,T)\) as \(\epsilon \rightarrow 0\). Then, the above procedure is also valid. Hence, the proof of the existence is now complete. To show the asymptotic behavior, we get
Integrating over \([t_{1},t_{2}]\) and applying the semicontinuity of the norm, we obtain the exponential stability of a solution of the Signorini problem. □
Remark 3.2
The uniqueness of the solution to Signorini problem (1.1)–(1.4) remains an open question.
4 Applications
In this section we present some applications of exponential stability for the contact problem between two dissipative Timoshenko beams. For that, we use known results of exponential stability for linear models of Timoshenko beams and extend it first to the hybrid models, using Theorem 2.2, then we apply Theorem 3.1 and Theorem 3.2 to show that there is a solution to the Signorini contact problem between two Timoshenko beams that decay exponentially to zero.
4.1 Continuous and Discontinuous Viscoelastic Constitutive Law
In [1] the authors studied the oscillations of a beam of length \(\ell \), configurated over the interval \(]0,\,\ell [\), and splitted in three components: an elastic part \(I_{E}\), without dissipative mechanism acting over it, and two viscous parts, one of them with a continuous constitutive law we denote as \(I_{C}\) and the other, \(I_{D}\) with discontinuous (discontinuity of the first kind at the border of \(I_{D}\)) constitutive law. This components are positioned over the intervals \(I_{1}=]0, \ell _{0}[\), \(I_{2}=]\ell _{0},\,\ell _{1}[\), \(I_{3}=]\ell _{1},\ell [\). We denote by \(\widetilde{I}=I_{1}\cup I_{2}\cup I_{3}\). Under this conditions the constitutive law are given by
with \(\tilde{\kappa}\) and \(\tilde{b}\) functions of the following type:
where \(\kappa _{0}\) and \(b_{0}\) are discontinuous functions over \(]0,\ell [\) vanishing out side of \(I_{D}\). Instead \(\kappa _{1}\) and \(b_{1}\) are \(C^{1}(0,\ell )\) functions vanishing out side of \(I_{C}\), verifying
and the existence of certain positive constants \(C_{1}\), \(C_{2}\) such that
Typical examples for the function \(\tilde{\kappa}=\kappa _{0}+\kappa _{1}\) are given in Figs. 2 and 3 (\(\tilde{b}\) is similar).
Taking \(S_{1}=S\), \(M_{1}=M\) defined over \(]0,\ell ^{*}[\) and \(S_{2}=S\), \(M_{2}=M\) defined over \(]\ell ^{*}, \ell [\), under the above conditions the authors proved (see [1]).
Theorem 4.1
The semigroup \(\mathbf{T}_{i}\) associated to the linear Timoshenko system (2.14)–(2.16) is exponentially stable if the viscous discontinuous part \(I_{D}\) is not in the center of the beam, provided (4.3) and (4.4) holds.
Using our approach we extend the above result to the Signorini contact between two Timoshenko beams.
Theorem 4.2
Assume that the constitutive law of the Signorini problem (1.1)–(1.5) with \(\gamma _{1,i}=\gamma _{2,i}=0\) are given by (4.1) and (4.2). Then, for any \((\varphi _{0}^{i},\varphi _{1}^{i},\psi _{0}^{i},\psi _{1}^{i})\in \mathcal{H}_{i}\) such that
there exists a weak solution which decays exponentially, provided that conditions (4.3) and (4.4) hold and the discontinuous viscous component \(I_{D}\) is neither in the center of the first beam configured on \(]0,\ell ^{*}[\) nor in the center of the second beam configured on \(]\ell ^{*},\ell [\).
The above problem can be generalized to \(N\)-components discussed in the next subsection.
4.2 The Viscoelastic Beam with \(N\)-Components
In [13] the authors consider the transmission problem of a Timoshenko beam of length \(\ell \) composed by \(N\) components, each of them can be of three different types of materials: elastic, viscoelastic or a material with a frictional damping mechanism as illustrated in Fig. 4, for \(N= 5\).
Unlike the case studied in [1] and described in Sect. 4.1, here only viscous components with discontinuous constitutive law are considered.
Let us decompose the interval \(I = [0, \ell ]\) into \(N\) subintervals, \([0,\ell ]= \bigcup _{i=1}^{n} \overline{I_{i}}\) such that
Over each interval \(I_{i}\) is configured one type of material. We denote by \(I_{v}\), \(I_{e}\) or \(I_{f}\) the subinterval where the viscoelastic component, elastic component or the component with frictional mechanism is configured, respectively. In Fig. 4 the intervals \(I_{1}\) and \(I_{4}\) are of type \(I_{e}\), elastic components, \(I_{2}=]\ell _{1},\ell _{2}[\) is of viscoelastic type \(I_{v}\) and so on. Let us denote by \(\widetilde{I}\) the set
\(\widetilde{I}\) is a disconnected open set. The classical linear Timoshenko system given by
where \(\gamma _{1}\), \(\gamma _{2}\) are positive only on the intervals \(I_{f}\), vanishing over \(I_{v}\) and \(I_{e}\). Here, we consider the following Dirichlet boundary conditions:
and the initial conditions
The constitutive equations are given by
We denote by \(b_{0}\) and \(\kappa _{0}\), positive functions which characterize the viscosity over \(I_{v}\), vanishing over \(I_{e}\cup I_{f}\). Therefore the elastic coefficients are discontinuous at the points where different materials are fitted. This characterizes the transmission problem. Hence the functions \(\kappa \), \(\kappa _{0}\), \(b\), \(b_{0}\), \(\gamma _{1}\), \(\gamma _{2} :[0,\ell ]\rightarrow \mathbb{R}\) are such that its restrictions to \(I_{i}\), \(i = 1, \dots , N\), are \(C^{1}\) functions, with bounded discontinuities at the nodes \(\ell _{i}\), \(i=1,\ldots , N-1\). But even so, the stress as well as the bending moment must satisfy the laws of action and reaction at each point, therefore we have that any strong solutions of the problem must verify \(\varphi ,\; \psi ,\; S,\; M \in H^{1}(0,\ell ) \), which in particular implies the transmission conditions at the interface points \(\ell _{i}\):
for \(i = 1, \dots N-1\). A typical example of a function \(y=\kappa _{0}(x)\) is given in Fig. 5.
A similar graph would hold for function \(b_{0}\). The frictional mechanism is characterized by the functions \(y=\gamma _{i}(x)\), with \(i = 1, 2\), for the same example is given as Fig. 6.
The authors establish in [13] the following result:
Theorem 4.3
The transmission problem (4.5)–(4.10) (\(N \geqslant 2\)) is exponentially stable if and only if any elastic part of the beam is connected with at least one component with frictional damping mechanisms. Otherwise the system is polynomially stable, with a rate of decay of the order \(t^{-2}\).
Taking \(S_{1}=S\), \(M_{1}=M\) defined over \(]0,\ell ^{*}[\) and \(S_{2}=S\), \(M_{2}=M\) defined over \(]\ell ^{*}, \ell [\), under the above conditions the authors proved
Theorem 4.4
The semigroup \(\mathbf{T}_{i}\) associated to the linear Timoshenko system (4.5)–(4.10) is exponentially stable if and only if any elastic part of the beam is connected with at least one component with frictional damping mechanisms.
Therefore using our approach we extend the result in [13] to the Signorini contact problem between to Timoshenko beams (see Fig. 7).
Theorem 4.5
Assume that the constitutive law of the Signorini problem (1.1)–(1.5) with \(\gamma _{1,i}\geq 0\) and \(\gamma _{2,i}\geq 0\) are positive only on the intervals \(I_{f}\), vanishing over \(I_{v}\) and \(I_{e}\). Then for any \((\varphi _{0}^{i},\varphi _{1}^{i},\psi _{0}^{i},\psi _{1}^{i})\in \mathcal{H}_{i}\) such that
there exists a weak solution which decays exponentially, provided any elastic part of the beam is connected with at least one component with frictional damping mechanisms.
4.3 More General Semi Linear Problem
Theorem 3.4 and Theorem 4.5 can be easily extended to the semi linear Signorini problem
verifying conditions (1.2)–(1.5).
Theorem 4.6
Under the same hypothesis of Theorem 3.4or Theorem 4.5, there is at least one solution to Signorini problem (4.11) verifying conditions (1.2)–(1.5) that decays exponentially to zero.
Proof
Here we use Theorem 3.3 for the function
instead of such given in (3.8). Note that \(\mathcal{F}(0)=0\). Using the mean value theorem to \(g(s)=|s|^{\alpha }s\) we obtain the inequality
Taking the norm in ℋ and since \(\varphi ^{\epsilon}_{i}\) and \(\psi ^{\epsilon}_{i}\) belong to \(H^{1}(0,\ell )\subset L^{\infty}(0,\ell )\), then we get
Therefore, ℱ is locally Lipschtiz. Since
then
Thus, there exists a positive constant \(c_{0}\) such that
Note that, for this function, there exists the cut-off function
It is not difficult to check that
is globally Lipschtiz. Using Theorem 3.3, Theorem 4.2 and Theorem 4.5 our conclusion follows. □
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Funding
Open access funding provided by Università degli Studi di Brescia within the CRUI-CARE Agreement. J.E. Muñoz Rivera would like to thank CNPq project 307947/2022-0 for the financial support, Project Fondecyt 1230914. M.G. Naso has been partially supported by Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM).
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Muñoz Rivera, J.E., Naso, M.G. Existence and Exponential Decay for a Contact Problem Between Two Dissipative Beams. J Elast 156, 571–595 (2024). https://doi.org/10.1007/s10659-024-10064-x
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DOI: https://doi.org/10.1007/s10659-024-10064-x