Dynamic Contact Problem for Viscoelastic von Kármán-Donnell Shells

Igor Bock, Jiří Jarušek

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Igor Bock, Jiří Jarušek. Dynamic Contact Problem for Viscoelastic von Kármán-Donnell Shells. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. pp.70-75, 10.1007/978-3-642-36062-6_7. hal-01347524

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# Dynamic Contact Problem
for Viscoelastic von Kármán-Donnell Shells

Igor Bock and Jiří Jarušek

Institute of Computer Science and Mathematics FEI,
Slovak University of Technology, 812 19 Bratislava 1, Slovakia,

igor.bock@stuba.sk

Institute of Mathematics, Academy of Sciences of the Czech Republic,

Žitná 25, 115 67 Praha 1, Czech Republic,

jarusek@math.cas.cz

**Abstract.** We deal with initial-boundary value problems describing vertical vibrations of viscoelastic von Kármán-Donnell shells with a rigid inner obstacle. The short memory (Kelvin-Voigt) material is considered. A weak formulation of the problem is in the form of the hyperbolic variational inequality. We solve the problem using the penalization method.

**Keywords:** Von Kármán-Donnell shell, unilateral dynamic contact, viscoelasticity, solvability, penalty approximation

## 1 Introduction

Contact problems represent an important but complex topic of applied mathematics. Its complexity profounds if the dynamic character of the problem is respected. For elastic problems there is only a very limited amount of results available (cf. [3] and there cited literature). Viscosity makes possible to prove the existence of solutions for a broader set of problems for membranes, bodies as well as for linear models of plates. The presented results extend the research made in [2], where the problem for a viscoelastic short memory von Kármán plate in a dynamic contact with a rigid obstacle was considered. Our results also extend the research made for the quasistatic contact problems for viscoelastic shells (cf. [1]). A thin isotropic shallow shell occupies the domain

$$G = \{(x, z) \in \mathbb{R}^3 : x = (x_1, x_2) \in \Omega, |z - Z| < h/2\},$$

where $h > 0$ is the thickness of the shell, $\Omega \subset \mathbb{R}^2$ is a bounded simply connected domain in $\mathbb{R}$ with a sufficiently smooth boundary $\Gamma$. We set $I = (0, T)$ a bounded time interval, $Q = I \times \Omega$, $S = I \times \Gamma$. The unit outer normal vector is denoted by $\mathbf{n} = (n_1, n_2)$, $\tau = (-n_2, n_1)$ is the unit tangent vector. The displacement is denoted by $\mathbf{u} = (u_i)$. The strain tensor is defined as

$$\varepsilon_{ij}(\mathbf{u}) = \frac{1}{2}(\partial_i u_j + \partial_j u_i + \partial_i u_3 \partial_j u_3) - k_{ij}u_3 - x_3 \partial_{ij} u_3, \quad i, j = 1, 2$$
with $k_{12} = k_{21} = 0$ and the curvatures $k_{ii} > 0$, $i = 1, 2$.

Further, we set

$$[u, v] \equiv \partial_{11}u\partial_{22}v + \partial_{22}u\partial_{11}v - 2\partial_{12}u\partial_{12}v.$$

In the sequel, we denote by $W_p^k(M)$, $k \ge 0$, $p \in [1, \infty]$ the Sobolev spaces defined on a domain or an appropriate manifold $M$. By $\dot{W}_p^k(M)$ the spaces with zero traces are denoted. If $p = 2$ we use the notation $H^k(M)$, $\dot{H}^k(M)$. The duals to $\dot{H}^k(M)$ are denoted by $H^{-k}(M)$. For the anisotropic spaces $W_p^k(M)$, $k = (k_1, k_2) \in \mathbb{R}_+^2$, $k_1$ is related with the time variable while $k_2$ with the space variables. We shall use also the Bochner-type spaces $W_p^k(I; X)$ for a time interval $I$ and a Banach space $X$. Let us remark that for $k \in (0, 1)$ their norm is defined by the relation

$$\|w\|_{W_p^k(I;X)}^p = \int_I \|w(t)\|_X^p dt + \int_I \int_I \frac{\|w(t) - w(s)\|_X^p}{|s - t|^{1+kp}} ds dt.$$

By $C(M)$ we denote the spaces of continuous functions on a (possibly relatively) compact manifold $M$. They are equipped with the max-norm. Analogously the spaces $C(M; X)$, are introduced for a Banach space $X$. The following generalization of the Aubin's compactness lemma verified in [4] Theorem 3.1 will be essentially used:

**Lemma 1** Let $B_0 \hookrightarrow B \hookrightarrow B_1$ be Banach spaces, the first reflexive and separable. Let $1 < p < \infty$, $1 \le r < \infty$. Then

$$W \equiv \{v; v \in L_p(I; B_0), \dot{v} \in L_r(I, B_1)\} \hookrightarrow L_p(I; B).$$

# 2 Short memory material

## 2.1 Problem formulation

Employing the Einstein summation, the constitutional law has the form

$$\sigma_{ij}(\mathbf{u}) = \frac{E_1}{1 - \mu^2} \partial_t ((1 - \mu)\varepsilon_{ij}(\mathbf{u}) + \mu\delta_{ij}\varepsilon_{kk}(\mathbf{u})) + \frac{E_0}{1 - \mu^2} ((1 - \mu)\varepsilon_{ij}(\mathbf{u}) + \mu\delta_{ij}\varepsilon_{kk}(\mathbf{u})).$$

The constants $E_0$, $E_1 > 0$ are the Young modulus of elasticity and the modulus of viscosity, respectively. We shall use the abbreviation $b = h^2/(12\rho(1 - \mu^2))$, where $h > 0$ is the shell thickness and $\rho$ is the density of the material. We involve the rotation inertia expressed by the term $a\Delta\ddot{u}$ in the first equation of the considered system with $a = \frac{h^2}{12}$. It will play the crucial role in the deriving a strong convergence of the sequence of velocities $\{\dot{u}_m\}$ in the appropriate space. We assume the shell clamped on the boundary. We generalize the dynamic elastic model due to the von Kármán-Donnell theory mentioned in [6]. The classical formulation for the deflection $u_3 \equiv u$ and the Airy stress function $v$ is then the
initial-value problem

$$
\left.
\begin{array}{l}
\ddot{u} + a\Delta\ddot{u} + b(E_1\Delta^2\dot{u} + E_0\Delta^2u) - [u, v] - \Delta_k * v = f + g, \\
u - \Psi \ge 0, \ g \ge 0, \ (u - \Psi)g = 0, \\
\Delta^2v + E_1\partial_t(\frac{1}{2}[u, u] + k_{11}\partial_{22}u + k_{22}\partial_{11}u) \\
\qquad + E_0(\frac{1}{2}[u, u] + \Delta_k u) = 0
\end{array}
\right\}
\quad \text{on } Q,
\qquad (1)
$$

$$
u = \partial_n u = v = \partial_n v = 0 \text{ on } S, \qquad (2)
$$

$$
u(0, \cdot) = u_0, \dot{u}(0, \cdot) = u_1 \text{ on } \Omega. \qquad (3)
$$

The obstacle function $\Psi \in L_{\infty}(\Omega)$ is fulfilling $0 < U_0 \le u_0 - \Psi$ in $\Omega$ and

$$
\Delta_k u \equiv \partial_{11}(k_{22}u) + \partial_{22}(k_{11}u), \qquad (4)
$$

$$
\Delta_k^* v \equiv k_{22} \partial_{11} v + k_{11} \partial_{22} v. \qquad (5)
$$

We define the operators $L: H^2(\Omega) \to \hat{H}^2(\Omega)$, $\Phi: H^2(\Omega) \times H^2(\Omega) \to \hat{H}^2(\Omega)$
by uniquely solved equations

$$
(\Delta L u, \Delta w) \equiv (\Delta_k u, w) \quad \forall w \in \hat{H}^2(\Omega), \qquad (6)
$$

$$
(\Delta\Phi(u, v), \Delta w) \equiv ([u, v], w) \quad \forall w \in \hat{H}^2(\Omega). \qquad (7)
$$

with the inner product $(\cdot, \cdot)$ in the space $L_2(\Omega)$. The operator $L$ is linear and compact. The bilinear operator $\Phi$ is symmetric and compact. Moreover due to Lemma 1 from [5] $\Phi: H^2(\Omega)^2 \to W_p^2(\Omega)$, $2 < p < \infty$ and

$$
\|\Phi(u, v)\|_{W_p^2(\Omega)} \le c \|u\|_{H^2(\Omega)} \|v\|_{W_p^1(\Omega)} \quad \forall u \in H^2(\Omega), v \in W_p^1(\Omega). \quad (8)
$$

We have also $L: H^2(\Omega) \mapsto W_p^2(\Omega)$, $2 < p < \infty$ and

$$
\|Lu\|_{W_p^2(\Omega)} \le c \|u\|_{H^2(\Omega)} \quad \forall u \in H^2(\Omega). \qquad (9)
$$

For $u, y \in L_2(I; H^2(\Omega))$ we define the bilinear form $A$ by

$$
A(u, y) := b(\partial_{kk}u \partial_{kk}y + \mu(\partial_{11}u \partial_{22}y + \partial_{22}u \partial_{11}y) + 2(1-\mu)\partial_{12}u \partial_{12}y).
$$

We introduce shifted cone $\mathcal{K}$ by

$$
\mathcal{K} := \{ y \in H^{1,2}(Q) ; \dot{y} \in L_2(I, \hat{H}^1(\Omega)) ; y \ge \Psi \}. \quad (10)
$$

Then the variational formulation of the problem (1-3) has the form of

**Problem P.** Find $u \in \mathcal{K}$ such that $\dot{u} \in L_2(I; \hat{H}^2(\Omega))$ and

$$
\begin{align}
& \int_Q (E_1 A(\dot{u}, y-u) + E_0 A(u, y-u)) \,dx \,dt \nonumber \\
& + \int_Q [u, E_1 \partial_t(\frac{1}{2}\Phi(u, u) + Lu) + E_0(\frac{1}{2}\Phi(u, u) + Lu)](y-u) \,dx \,dt \nonumber \\
& + \int_Q \Delta_k (E_1 \partial_t(\frac{1}{2}\Phi(u, u) + Lu) + E_0(\frac{1}{2}\Phi(u, u) + Lu))(y-u) \,dx \,dt \nonumber \\
& - \int_Q (a \nabla \dot{u} \cdot \nabla (\dot{y} - \dot{u}) + \dot{u} (\dot{y} - \dot{u})) \,dx \,dt \nonumber \\
& + \int_Q (a \nabla \dot{u} \cdot \nabla (y-u) + \dot{u} (y-u)) (T, \cdot) \,dx \nonumber \\
& \ge \int_{\Omega} (a \nabla u_1 \cdot \nabla (y(0, \cdot) - u_0) + u_1(y(0, \cdot) - u_0)) \,dx \nonumber \\
& + \int_Q f(y_1 - u) \,dx \,dt \quad \forall y \in \mathcal{K}. \tag{11}
\end{align}
$$
## 2.2 The penalization

For any $\eta > 0$ we define the penalized problem

**Problem P**<sub>η</sub>. Find $u \in H^{1,2}(Q)$ such that $\dot{u} \in L_2(I; \hat{H}^2(\Omega))$, $\ddot{u} \in L_2(I; \hat{H}^1(\Omega))$,

$$ 
\begin{aligned}
& \int_Q (\ddot{u}z + a\nabla\ddot{u} \cdot \nabla z + E_1 A(\dot{u}, z) + E_0 A(u, z)) \,dx \,dt \\
& + \int_Q [u, E_1 \partial_t(\frac{1}{2}\Phi(u, u) + Lu) + E_0(\frac{1}{2}\Phi(u, u) + Lu)]z \,dx \,dt \\
& + \int_Q \Delta_k (E_1 \partial_t(\frac{1}{2}\Phi(u, u) + Lu) + E_0(\frac{1}{2}\Phi(u, u) + Lu)) \,z \,dx \,dt \\
& = \int_Q (f + \eta^{-1}(u - \Psi)^{-})z \,dx \,dt \quad \forall z \in L_2(I; H^2(\Omega))
\end{aligned}
\quad (12) $$

and the conditions (3) remain valid.

**Lemma 2** Let $f \in L_2(Q)$, $u_0 \in \hat{H}^2(\Omega)$, and $u_1 \in \hat{H}^1(\Omega)$. Then there exists a solution $u$ of the problem $\mathcal{P}_\eta$.

*Proof.* Let us denote by $\{w_i \in \hat{H}^2(\Omega); i = 1, 2, ...\}$ a basis of $\hat{H}^2(\Omega)$ orthonormal in $H^1(\Omega)$ with respect to the inner product

$$ (u, v)_a = \int_{\Omega} (uv + a\nabla u \cdot \nabla v) \,dx, \quad u, v \in H^1(\Omega). $$

We construct the Galerkin approximation $u_m$ of a solution in a form

$$ u_m(t) = \sum_{i=1}^{m} \alpha_i(t)w_i, \quad \alpha_i(t) \in \mathbb{R}, \quad i = 1, \dots, m, \quad m \in \mathbb{N}, \quad (13) $$

$$ 
\begin{aligned}
& (\ddot{u}_m(t), w_i)_a + \int_{\Omega} (E_1 A(\dot{u}_m(t), w_i) + E_0 A(u_m(t), w_i)) \,dx + \\
& \int_{\Omega} \Delta (E_1 \partial_t(\frac{1}{2}\Phi(u_m, u_m) + Lu_m) + E_0(\frac{1}{2}\Phi(u_m, u_m) + Lu_m)) \\
& \times \Delta(\Phi(u_m, w_i) + Lw_i) \,dx \\
& = \int_{\Omega} (f(t) + \eta^{-1}(u_m(t) - \Psi)^{-})w_i \,dx, \quad i = 1, \dots, m,
\end{aligned}
\quad (14) $$

$$ u_m(0) = u_{0m}, \quad \dot{u}_m(0) = u_{1m}, \quad u_{0m} \to u_0 \quad \text{in } \hat{H}^2(\Omega), \quad u_{1m} \to u_1 \quad \text{in } \hat{H}^1(\Omega). \quad (15) $$

After multiplying the equation (14) by $\dot{\alpha}_i(t)$, summing up with respect to $i$, taking in mind the definitions of the operators $\Phi, L$ and integrating we obtain the a priori estimates not depending on $m$:

$$ 
\begin{aligned}
& \|\dot{u}_m\|_{L_2(I; \hat{H}^2(\Omega))}^2 + \|\dot{u}_m\|_{L_\infty(I; \hat{H}^1(\Omega))}^2 + \|u_m\|_{L_\infty(I; \hat{H}^2(\Omega))}^2 \\
& + \|\partial_t \Phi(u_m, u_m)\|_{L_2(I; \hat{H}^2(\Omega))}^2 + \|\partial_t Lu_m\|_{L_2(I; \hat{H}^2(\Omega))}^2 \\
& + \eta^{-1} \|(u_m - \Psi)^{-}\|_{L_\infty(I; L_2(\Omega))} \le c \equiv c(f, u_0, u_1).
\end{aligned}
\quad (16) $$

Moreover the estimates (8), (9) imply

$$ \|\partial_t \Phi(u_m, u_m)\|_{L_2(I; W_p^2(\Omega))} + \|\partial_t Lu_m\|_{L_2(I; W_p^2(\Omega))} \le c_p \quad \forall p > 2. \quad (17) $$

After multiplying the equation (14) by $\ddot{\alpha}_i(t)$, summing up and integrating we obtain the estimate of $\ddot{u}_m$

$$ \|\ddot{u}_m\|_{L_2(I; H^1(\Omega))} \le c_\eta, \quad m \in \mathbb{N}. \quad (18) $$
Applying the estimates (16)-(18), the compact imbedding theorem and the in-
terpolation, we obtain for any $p \in [1, \infty)$, a subsequence of $\{u_m\}$ (denoted again
by $\{u_m\}$), a function $u$ and the convergences

$$
\begin{align}
\ddot{u}_m \to \ddot{u} \text{ in } L_2(I; H^1(\Omega)), \nonumber \\
\dot{u}_m \to^* \dot{u} \text{ in } L_\infty(I; \dot{H}^1(\Omega)), \nonumber \\
\dot{u}_m \to \dot{u} \text{ in } L_2(I; \dot{H}^2(\Omega)), \nonumber \\
\dot{u}_m \to \dot{u} \text{ in } L_p(I; \dot{H}^p(\Omega)) \cap L_\infty(I; H^{2-\varepsilon}(\Omega)) \quad \forall \varepsilon > 0, \tag{19} \\
u_m \to u \text{ in } C(\bar{I}; W_p^1(\Omega)), \nonumber \\
\partial_t \left( \frac{1}{2} \Phi(u_m, u_m) + Lu_m \right) \to \partial_t \left( \frac{1}{2} \Phi(u, u) + Lu \right) \text{ in } L_2(I; W_p^2(\Omega)) \nonumber
\end{align}
$$

implying that a function $u$ fulfils the identity (12). The initial conditions (3)
follow due to (15) and the proof of the existence of a solution is complete.

2.3 Solving the original problem

We verify the existence theorem

**Theorem 1** Let $f \in L_2(Q)$, $u_i \in \dot{H}^2(\Omega)$, $i = 0, 1$, $0 < U_0 \le u_0 - \Psi$. Then there exists a solution of the Problem P.

*Proof.* We perform the limit process for $\eta \to 0$. We write $u_\eta$ for the solution of the problem $\mathcal{P}_{1,\eta}$. The *a priori* estimates (16) imply the estimates

$$
\begin{align}
& \|\dot{u}_{\eta}\|_{L_2(I; \dot{H}^2(\Omega))}^2 + \|\dot{u}_{\eta}\|_{L_{\infty}(I; \dot{H}^1(\Omega))}^2 + \|u_{\eta}\|_{L_{\infty}(I; \dot{H}^2(\Omega))}^2 \nonumber \\
& + \|\partial_t \Phi(u_{\eta}, u_{\eta})\|_{L_2(I; W_p^2(\Omega))}^2 + \|\partial_t Lu_{\eta}\|_{L_2(I; W_p^2(\Omega))}^2 \tag{20} \\
& \qquad + \eta^{-1} \|(u_{\eta} - \Psi)^{-}\|_{L_{\infty}(I; L_2(\Omega))} \le c_p, \quad p > 2. \nonumber
\end{align}
$$

To get the crucial estimate for the penalty, we put $z = u_0 - u_\eta(t, \cdot)$ in (12) and
obtain the estimate

$$
\begin{align*}
0 \le U_0 \int_Q \eta^{-1} (u_\eta - \Psi)^{-} dx \, dt &\le \int_Q \|\eta^{-1} (u_\eta - \Psi)^{-} (u_0 - \Psi) dx \, dt \\
&\le \int_Q \|\eta^{-1} (u_\eta - \Psi)^{-} (u_0 - u_\eta) dx \, dt \\
&= \int_Q (\dot{u}_\eta^2 + a |\nabla \dot{u}_\eta|^2 + A((E_1 \partial_t u_\eta + E_0 u_\eta), u_0 - u_\eta) \\
&\quad + E_1 \partial_t (\Delta(L u_\eta + \frac{1}{2} \Phi(u_\eta, u_\eta))) \Delta(L(u_0 - u_\eta) + \Phi(u_\eta, u_0 - u_\eta)) \\
&\quad + E_0 \Delta(L u_\eta + \frac{1}{2} \Phi(u_\eta, u_\eta)) \Delta(L(u_0 - u_\eta) + \Phi(u_\eta, u_0 - u_\eta))) dx \, dt \\
&\quad - \int_Q f(u_0 - u_\eta) dx \, dt + \int_\Omega ((\dot{u}_\eta(u_0 - u_\eta) + a \nabla \dot{u}_\eta \cdot \nabla(u_0 - u_\eta))(T, \cdot)) dx.
\end{align*}
$$

Applying the *a priori* estimates (20) we obtain

$$
\| \eta^{-1} u_{\eta}^{-} \|_{L_1(Q)} \leq c(f, u_0, u_1, \Psi). \quad (21)
$$

With respect to Dirichlet conditions we obtain from (12) and (21) the dual
estimate

$$
\| - a \Delta \ddot{u}_{\eta} + \ddot{u}_{\eta} \|_{L_1(I; H^{-2}(\Omega))} \leq c. \quad (22)
$$
We take the sequence {$u_k$} $\equiv \{u_{\eta_k}\}$, $\eta_k \to 0^+$.

After applying the Lemma 1 with the spaces

$$B_0 = L_2(\Omega), \quad B = H^{-1}(\Omega), \quad B_1 = H^{-2}(\Omega)$$

we obtain the relative compactness of the sequence {$-a\Delta\dot{u}_k + \dot{u}_k$} in $L_2(I; H^{-1}(\Omega))$ and with the help of the test function $\dot{u}_k - \dot{u}$ the crucial strong convergence

$$\dot{u}_k \to \dot{u} \text{ in } L_2(I; \dot{H}^1(\Omega)). \quad (23)$$

Simultaneously we have the convergences

$$\begin{align}
\dot{u}_k &\to \dot{u} \text{ in } L_2(I; \dot{H}^2(\Omega)), \nonumber \\
\dot{u}_k &\to \dot{u} \text{ in } L_2(I; W_p^1(\Omega)), \tag{24} \\
\frac{1}{2} \partial_t \Phi(u_k, u_k) + \partial_t Lu_k &\to \frac{1}{2} \partial_t \Phi(u, u) + \partial_t Lu \text{ in } L_2(I; W_p^2(\Omega)). \nonumber
\end{align}$$

It can be verified after inserting the test function $z = y - u_k$ in (12) for $y \in \mathcal{K}$, performing the integration by parts in the terms containing $\ddot{u}$, applying the convergences (23), (24), using the definitions of the operators $L$, $\Phi$ in (6), (7) and the weak lower semicontinuity that the limit function $u$ is a solution of the original problem $\mathcal{P}$.

*Remark 1.* The existence Theorem 1 can be after some modification verified also for another types of boundary conditions.

**Acknowledgments.** The work presented here was supported by the Czech Academy of Sciences under grant P201/12/0671 and under the Institutional research plan RVD 67985840, by the Czech Ministry of Education under grant MEB 0810045, by the APVV Agency of Slovak Republic under grant 0011-09 and by Ministry of Education of the Slovak Republic under VEGA grant 1/0426/12.

## References

1. I. BOCK: On a pseudoparabolic system for a viscoelastic shallow shell. *PAMM Proc. Appl. Math. Mech.* **6** (2006), 621-622.

2. I. BOCK AND J. JARUŠEK: Unilateral dynamic contact of viscoelastic von Kármán plates. *Advances in Math. Sci. and Appl.* **16** (2006), 175-187.

3. C. ECK, J. JARUŠEK AND M. KRBEČ: *Unilateral Contact Problems in Mechanics. Variational Methods and Existence Theorems*. Monographs & Textbooks in Pure & Appl. Math. No. 270 (ISBN 1-57444-629-0). Chapman & Hall/CRC (Taylor & Francis Group), Boca Raton - London - New York - Singapore 2005.

4. J. JARUŠEK, J. MÁLEK, J. NEČAS AND V. ŠVERÁK: Variational inequality for a viscous drum vibrating in the presence of an obstacle. *Rend. Mat.*, Ser. VII, **12** (1992), 943-958.

5. H. KOCH AND A. STACHEL: Global existence of classical solutions to the dynamical von Kármán equations. *Math. Methods in Applied Sciences* **16** (1993), 581-586.

6. A.G. VOĽMIR: *Gibkije plastinky i oboločky* (in Russian), Gosizdat, Moskva, 1956.