Dynamic contact of a thermoelastic Mindlin–Timoshenko beam with a rigid obstacle

Abstract

We concentrate on the dynamics of a thermoelastic Mindlin–Timoshenko beam striking a rigid obstacle. We state classical formulations involving complementarity conditions. Weak formulations are in the form of systems consisting of a hyperbolic variational inequality for a deflection, a hyperbolic and a parabolic equation for an angle of rotation and a thermal strain, respectively. The penalization method is applied to solve the unilateral problem. The time derivative of the function representing the deflection of the beam’s middle line is not continuous due to the hitting the obstacle. The acceleration term has the form of a vector measure.

Keywords

Thermoelastic Mindlin–Timoshenko beam dynamic contact rigid obstacle penalization vector measure

1. Introduction and notation

Dynamic contact problems are not frequently solved in the framework of variational inequalities. For elastic problems there is only a very limited amount of results available (cf. the work by Eck et al. [1] and there cited literature). We have solved these problems for geometrically nonlinear plates and shells in the work by Bock and Jarušek [2, 3] respectively. In the work by Bock et. al. [4] we concentrated not only on purely mechanical impact to the plate being under some load and possibly contacting an rigid obstacle, but also have taken in mind the heat balance of this process. We consider here a thermoelastic Mindlin–Timoshenko beam in a dynamic contact with a rigid obstacle. We shall use a similar model in the work by Berti et al. [5]. The acceleration second time derivative term poses the greatest difficulties in solving dynamic contact problems. In contrast to the work by Berti et al. [5], where the boundary contact has been considered and the acceleration term does not appear explicitly, we express it here in the form of a vector measure in a similar way as in the work by Bock et al. [4], where the geometrically nonlinear plate model with a thermal effect has been investigated. We deal here with a one-dimensional beam model, because of a second order equation with respect to the deflection and the angle of the rotation. The second order two dimensional dynamic contact problem with the inner obstacle has still not been solved. The only known result for the case of a boundary contact has been received by Kim [6].

We study the mechanical behaviour of a beam of the length $L > 0$ hitting the rigid obstacle and assume its following material, thermal and mechanical characteristics

$κ > 0$ – the shear modulus of elasticity;

$a > 0$ – the moment of mass inertia;

$b > 0$ – the rigidity coefficient of cross section;

$ν \in [0, \frac{1}{2})$ – Poisson ratio;

$d > 0$ – the thermal conductivity;

$e > 0$ – the material dependent coupling coefficient.

The beam is acting under the perpendicular load $f : Q \to ℝ$ , surface load $p : Q \to ℝ$ and a heat source $q : Q \to ℝ$ , where

Q = (0, T] \times (0, L)

is the time-length domain.

The beam is assumed to be simply supported on both its ends. We denote by $(u, φ, θ) : Q \mapsto ℝ^{3}$ as a function representing a vertical displacement of the longitude axis, an angle of a rotation and a thermal strain resultant, respectively, by $(u_{0}, φ_{0}, θ_{0}) : (0, L) \mapsto ℝ^{3}$ the initial values and by $(v_{0}, ω_{0}) : (0, L) \mapsto ℝ^{2}$ the initial velocities.

2. Classical formulation and its penalization

For simplicity, the rigid obstacle of the form y=0 is considered. We denote by g the unknown contact force between the beam and the obstacle. The first and the second-time derivatives of any function w are denoted by $ẇ, ẅ$ .

Using the complementary principle we arrive at the following initial-boundary value problem

\begin{matrix} ü - κ {[(u_{x} + φ)]}_{x} = f + g, \end{matrix}

(1)

\begin{matrix} u \geq 0, g \geq 0, u g = 0, \end{matrix}

(2)

\begin{matrix} a \overset{\cdot\cdot}{φ} + κ (u_{x} + φ) - b (φ_{xx} + \frac{1 + ν}{2} θ_{x}) = p, \end{matrix}

(3)

\begin{matrix} \overset{\cdot}{θ} - d θ_{xx} - e {\overset{\cdot}{φ}}_{x} = q on Q, \end{matrix}

(4)

\begin{matrix} u (t, 0) = w (0), u (t, L) = w (L), \end{matrix}

(5)

\begin{matrix} φ_{x} (t, 0) = φ_{x} (t, L) = 0, \end{matrix}

(6)

\begin{matrix} θ (t, 0) = θ (t, L) = 0, t \in (0, T], \end{matrix}

(7)

\begin{matrix} u (0, x) = u_{0} (x), \overset{\cdot}{u} (0, x) = v_{0} (x), \end{matrix}

(8)

\begin{matrix} φ (0, x) = φ_{0} (x), \overset{\cdot}{φ} (0, x) = ω_{0} (x), \end{matrix}

(9)

\begin{matrix} θ (0, x) = θ_{0} (x), x \in (0, L) . \end{matrix}

(10)

Let $u^{-} = max {0, - u}$ be the negative part of u. For any $η > 0$ we formulate the penalized problem consisting of the equation

ü - κ {[(u_{x} + φ)]}_{x} = f + η^{- 1} u^{-},

(11)

and equations (3) to (10).

3. Solving the penalized variational problem

Before coming to a variational formulation of the problem given by equations (11) and (3) to (10) we introduce some spaces of functions and state the assumptions on data.

Hilbert spaces

\begin{matrix} H \equiv L_{2} (0, L) = {y : (0, L) \mapsto ℝ : \int_{0}^{L} y^{2} dx < \infty}, \\ H^{1} (0, L) = {y \in L_{2} (0, L) : y^{'} \in L_{2} (0, L)}, \\ V = {y \in H^{1} (0, L) : y (0) = y (L) = 0}, \\ L_{2} (Q) = {v : Q \mapsto ℝ : \int_{Q} v^{2} dt dx < \infty} \end{matrix}

with the inner products and the norms

\begin{matrix} (y, z) = \int_{0}^{L} y (x) z (x) dx, | y |_{0} = {(y, y)}^{1 ∕ 2}, y, z \in H, \\ {(y, z)}_{1} = \int_{0}^{L} [y (x) z (x) + y^{'} (x) z^{'} (x)] dx, \\ ∥ y ∥_{1} = {(y, y)}_{1}^{1 ∕ 2}, y, z \in H^{1} (0, L), \\ ((y, z)) = (y^{'}, z^{'}), ∥ y ∥ = {((y, y))}^{1 ∕ 2}, y, z \in V, \\ {(u, v)}_{Q} = \int_{Q} u (t, x) v (t, x) dt dx, | v |_{Q} = {(v, v)}^{1 ∕ 2}, \\ y, z \in L_{2} (Q) . \end{matrix}

Vector-valued function spaces for a Banach space X,

$I = (0, T)$ and $p \in [1, + \infty]$

L_{p} (I; X) = {y : (0, T) \mapsto X : ∥ y (\cdot) ∥_{X} \in L_{p} (0, T)} .

$C (Ī; X)$ and $C_{w} (Ī; X)$ are the spaces of continuous and weakly continuous functions $y : Ī \mapsto X, Ī = [0, T]$ . If $X *$ is the dual space of linear continuous functionals over $X$ , then

y \in C_{w} (Ī; X) \Leftrightarrow {⟨ F, y (\cdot) ⟩}_{X^{*} \times X} \in C (Ī; ℝ) \forall F \in X^{*} .

We assume the right-hand sides

f, p, q \in L_{2} (Q),

(12)

the initial data

u_{0}, φ_{0} \in H^{1} (0, L), v_{0}, ω_{0}, θ_{0} \in L_{2} (0, L),

(13)

and the Dirichlet boundary function

\begin{matrix} \begin{matrix} w \in H^{1} (0, L), w \geq w_{0} > 0 on [0, L]; \\ w (0) = u_{0} (0), w (L) = u_{0} (L) . \end{matrix} \end{matrix}

(14)

Let $⟨ \cdot, \cdot ⟩ = {⟨ \cdot, \cdot ⟩}_{V^{*} \times V}, {⟨ \cdot, \cdot ⟩}_{- 1} = {⟨ \cdot, \cdot ⟩}_{H^{1} {(0, L)}^{*}, \times H^{1} (0, L)}$ .

We state a following variational formulation of the Penalized problem given by equations (11) and (3) to (10):

Problem $P_{η}$ . We look for

{u, φ, θ} \in (w + L_{\infty} (I; V)) \times L_{\infty} (I; H^{1} (0, L)) \times L_{2} (I; V)

such that

{\overset{\cdot}{u}, \overset{\cdot}{φ}, θ} \in C (\bar{I}; H) \times C (\bar{I}; H) \times C (\bar{I}; H),

${ü, \overset{\cdot\cdot}{φ}, \overset{\cdot}{θ}} \in L_{2} (I; V^{*}) \times L_{2} (I; H^{1} {(0, L)}^{*}) \times L_{2} (I; V^{*}),$ the equations

\begin{matrix} \begin{matrix} \int_{I} ⟨ ü, y ⟩ dt + \int_{Q} (κ (u_{x} + φ) y_{x} - η^{- 1} u^{-} y) dx dt \\ = \int_{Q} fy dx dt, \end{matrix} \end{matrix}

(15)

\begin{matrix} \begin{matrix} \int_{I} {⟨ a \overset{\cdot\cdot}{φ}, ψ ⟩}_{- 1} dt + \\ \int_{Q} (b φ_{x} ψ_{x} + κ (u_{x} + φ) ψ - c θ_{x} ψ) dx dt \\ = \int_{Q} pψ dx dt, c = \frac{1}{2} b (1 + ν), \end{matrix} \end{matrix}

(16)

\begin{matrix} \begin{matrix} \int_{I} ⟨ \overset{\cdot}{θ}, z ⟩ dt + \int_{Q} (d θ_{x} z_{x} + e \overset{\cdot}{φ} z_{x}) dx dt \\ = \int_{Q} qz dx dt \end{matrix} \end{matrix}

(17)

hold for any ${y, ψ, z} \in L_{2} (I; V) \times L_{2} (I; H^{1} (0, L)) \times L_{2} (I; V)$ and the initial conditions given by equations (8) to (10) remain.

We shall verify the existence of a solution to the variational penalized problem.

Theorem 3.1. For every $η > 0$ there exists a solution ${u, φ, θ}$ of the Problem $P_{η}$ .

Proof. We introduce

{v_{i} \in V; i \in ℕ}

a basis of V orthonormal in $L_{2} (0, L)$

{w_{i} \in H^{1} (0, L); i \in ℕ}

a basis of $H^{1} (0, L)$ orthonormal with respect to

{(u, v)}_{a} = \int_{0}^{L} a uv dx, u, v \in H^{1} (0, L) .

and construct a Galerkin approximation of a solution ${u_{η},$ $φ_{η}, θ_{η}}$ in a form

\begin{matrix} u_{m} (t) = w + \sum_{j = 1}^{m} α_{j} (t) v_{j}, φ_{m} (t) = \sum_{j = 1}^{m} β_{j} (t) w_{j}, \\ θ_{m} (t) = \sum_{j = 1}^{m} γ_{j} (t) v_{j}; \\ t \in I, {α_{j} (t), β_{j} (t), γ_{j} (t)} \in ℝ^{3}, j = 1, \dots, m; m \in ℕ \end{matrix}

to satisfy the following system of equations

\begin{matrix} \begin{matrix} \int_{0}^{L} (ü_{m} v_{i} + κ (u_{mx} + φ_{m}) v_{ix} - η^{- 1} u_{m}^{-} v_{i}) dx \\ = \int_{0}^{L} f v_{i} dx, \end{matrix} \end{matrix}

(18)

\begin{matrix} \begin{matrix} \int_{0}^{L} (a {\overset{\cdot\cdot}{φ}}_{m} w_{i} + b φ_{mx} w_{ix} + κ (u_{mx} + φ_{m}) w_{i} - c θ_{mx} w_{i}) dx \\ = \int_{0}^{L} p w_{i} dx, \end{matrix} \end{matrix}

(19)

\begin{matrix} \begin{matrix} \int_{0}^{L} ({\overset{\cdot}{θ}}_{m} v_{i} + d θ_{mx} v_{ix} + e {\overset{\cdot}{φ}}_{mx} v_{ix}) dx \\ = \int_{0}^{L} q v_{i} dx, \end{matrix} \\ i = 1, \dots, m \end{matrix}

(20)

and the initial conditions

\begin{matrix} \begin{matrix} u_{m} (0) = u_{0 m}, u_{0 m} \to u_{0} in V; \\ {\overset{\cdot}{u}}_{m} (0) = v_{0 m}, v_{0 m} \to v_{0} in H; \\ φ_{m} (0) = φ_{0 m}, φ_{0 m} \to φ_{0} in H^{1} (0, L); \\ {\overset{\cdot}{φ}}_{m} (0) = ω_{0 m}, ω_{0 m} \to ω_{0} in H; \\ θ_{m} (0) = θ_{0 m}, θ_{0 m} \to θ_{0} in H . \end{matrix} \end{matrix}

(21)

There exists a local solution ${u_{m}, φ_{m}, θ_{m}}$ of equations (18) to (21) on some interval $I_{m} \equiv [0, t_{m}], 0 < t_{m} < T$ .

We multiply equation (18) by ${\overset{\cdot}{α}}_{i} (t)$ , equation (19) by ${\overset{\cdot}{β}}_{i} (t)$ , equation (20) by $\frac{c}{e} γ_{i} (t)$ add with respect to i, integrate on $[0, t_{m}]$ and obtain for $Q_{m} : = I_{m} \times (0, L)$ the relation

\begin{matrix} \int_{Q_{m}} \frac{1}{2} \partial_{t} [{\overset{\cdot}{u}}_{m}^{2} + a {\overset{\cdot}{φ}}_{m}^{2} + b φ_{mx}^{2} + κ {(u_{mx} + φ_{m})}^{2}] dx dt \\ + \int_{Q_{m}} [\frac{c}{e} θ_{m}^{2} + η^{- 1} {(u_{m}^{-})}^{2}] dx dt + \int_{Q_{m}} \frac{cd}{e} θ_{mx}^{2} dx dt \\ = \int_{Q_{m}} (f {\overset{\cdot}{u}}_{m} + p {\overset{\cdot}{φ}}_{m} + \frac{c}{e} q θ_{m}) dx dt, \end{matrix}

leading to the estimate

\begin{matrix} \begin{matrix} ∥ {\overset{\cdot}{u}}_{m} ∥_{L_{\infty} (I; H)}^{2} + ∥ u_{m} ∥_{L_{\infty} (I; V)}^{2} + ∥ {\overset{\cdot}{φ}}_{m} ∥_{L_{\infty} (I; H)}^{2} \\ + ∥ φ_{m} ∥_{L_{\infty} (I; H^{1} (0, L))}^{2} + η^{- 1} ∥ u_{m}^{-} ∥_{L_{\infty} (I; H)}^{2} \\ + ∥ θ_{m} ∥_{L_{\infty} (I; H)}^{2} + ∥ θ_{m} ∥_{L_{2} (I; V)} \\ \leq C_{1} \equiv C_{1} (f, p, q, u_{0}, v_{0}, φ_{0}, ω_{0}, θ_{0}) . \end{matrix} \end{matrix}

(22)

The prolongation to the whole interval I is due to the fact that the original estimate for $I_{m}$ does not depend on m. From equations (18) to (20) we obtain directly the dual estimates

\begin{matrix} ∥ ü_{m} ∥_{L_{2} (I; {V_{m}}^{*})} \leq C_{2} (η, f, p, q, u_{0}, v_{0}, φ_{0}, ω_{0}, θ_{0}), \end{matrix}

(23)

\begin{matrix} \begin{matrix} ∥ {\overset{\cdot\cdot}{φ}}_{m} ∥_{L_{2} (I; {W_{m}}^{*})} \leq C_{3} (f, p, q, u_{0}, v_{0}, φ_{0}, ω_{0}, θ_{0}), \\ ∥ {\overset{\cdot}{θ}}_{m} ∥_{L_{2} (I; {V_{m}}^{*})} \leq C_{4} (f, p, q, u_{0}, v_{0}, φ_{0}, ω_{0}, θ_{0}), \\ m \in ℕ, \end{matrix} \end{matrix}

(24)

where $V_{m} \subset V$ is the linear hull of ${v_{i}}_{i = 1}^{m} \subset V$ and $W_{m} \subset H^{1} (0, L)$ is the linear hull of ${w_{i}}_{i = 1}^{m} \subset H^{1} (0, L)$ . Applying the estimates given by equations (22) to (24), the compact imbedding theorem and interpolation in Sobolev spaces we obtain subsequences of ${u_{m}}, {φ_{m}}, {θ_{m}}$ and functions $u, φ, θ$ fulfilling

\begin{matrix} \begin{matrix} u_{m} ⇀^{*} u in L_{\infty} (I; V), \\ {\overset{\cdot}{u}}_{m} ⇀^{*} \overset{\cdot}{u} in L_{\infty} (I; H), \\ u_{m} \to u in C (Ī; H^{1 - ε} (0, L)) \forall ε > 0, \\ φ_{m} ⇀^{*} φ in L_{\infty} (I; H^{1} (0, L)), \\ {\overset{\cdot}{φ}}_{m} ⇀^{*} \overset{\cdot}{φ} in L_{\infty} (I; H), \\ φ_{m} \to φ in C (Ī; H^{1 - ε} (0, L)) \forall ε > 0, \\ θ_{m} ⇀ θ in L_{2} (I; V), \\ θ_{m} ⇀^{*} θ in L_{\infty} (I; H) . \end{matrix} \end{matrix}

(25)

\begin{matrix} \begin{matrix} ü_{m} ⇀ ü in L_{2} (I; Y^{*}), \\ {\overset{\cdot}{θ}}_{m} ⇀ \overset{\cdot}{θ} in L_{2} (I; Y^{*}), \\ {\overset{\cdot\cdot}{φ}}_{m} ⇀ \overset{\cdot\cdot}{φ} in L_{2} (I; W^{*}), \end{matrix} \end{matrix}

(26)

Y = ⋃_{m \in ℕ} V_{m}, \bar{Y} = V; W = ⋃_{m \in ℕ} W_{m}, \bar{W} = H^{1} (0, L) .

We continue with solving the penalized problem.

Let $D (0, T)$ be the set of functions from $C^{\infty} (0, T)$ with a compact support in $(0, T)$ , $ϕ_{i} \in D (0, T), i = 1, \dots, μ; μ \in ℕ$ and

y_{μ} = \sum_{i = 1}^{μ} ϕ_{i} (t) v_{i}, ψ_{μ} = \sum_{i = 1}^{μ} ϕ_{i} (t) w_{i}, z_{μ} = \sum_{i = 1}^{μ} ϕ_{i} (t) v_{i} .

Then

\begin{matrix} \int_{0}^{L} (\overset{\cdot\cdot}{u_{m}} y_{μ} + κ (u_{mx} + φ_{m}) y_{μx}) dx \\ = \int_{0}^{L} (f + η^{- 1} u_{m}^{-}) y_{μ} dx, \\ \int_{0}^{L} (a \overset{\cdot\cdot}{φ_{m}} ψ_{μ} + b φ_{mx} ψ_{μx} + κ (u_{mx} + φ_{m}) ψ_{μ} - c θ_{mx} ψ_{m}) dx \\ = \int_{0}^{L} p ψ_{μ} dx, \\ \int_{0}^{L} ({\overset{\cdot}{θ}}_{m} z_{μ} + d θ_{mx} z_{μx} + d θ_{mx} z_{μx} + e {\overset{\cdot}{φ}}_{m} z_{μx}) dx \\ = \int_{0}^{L} q z_{μ} dx \forall m \geq μ, t \in I . \end{matrix}

The convergences given by equations (25) and (26) imply

\begin{matrix} ⟨ ü, y_{μ} ⟩ + \int_{0}^{L} κ (u_{x} + φ) y_{μx} dx = \int_{0}^{L} (f + η^{- 1} u^{-}) y_{μ} dx, \\ a {⟨ \overset{\cdot\cdot}{φ}, ψ_{μ} ⟩}_{- 1} + \int_{0}^{L} (b φ_{x} ψ_{μx} κ (u_{x} + φ_{)} ψ_{μ} - c θ_{x} ψ_{μ}) dx \\ = \int_{0}^{L} p ψ_{μ} dx, \\ ⟨ \overset{\cdot}{θ}, z_{μ} ⟩ + \int_{0}^{L} (d θ_{mx} z_{μx} + d θ_{mx} z_{μx} + e {\overset{\cdot}{φ}}_{m} z_{μx}) dx \\ = \int_{0}^{L} q z_{μ} dx \forall m \geq μ, t \in I . \end{matrix}

Functions ${y_{μ}}, {z_{μ}}$ and ${ψ_{μ}}$ are dense in $L_{2} (I; V)$ and $L_{2} (I; H^{1} (0, L))$ respectively. Then the relations given by equations (15) to (17) follow. The initial conditions given by equations (8) tp (10) follow due to equation (21) and so Theorem 3.1 is proved.

4. Solving the original problem

The estimates given by equations (22) and (24) imply the $η$ independent estimates

\begin{matrix} \begin{matrix} ∥ u_{η} ∥_{L_{\infty} (I; V)}^{2} + ∥ {\overset{\cdot}{u}}_{η} ∥_{L_{\infty} (I; H)}^{2} + ∥ φ_{η} ∥_{L_{\infty} (I; H^{1} (0, L))}^{2} \\ + ∥ {\overset{\cdot}{φ}}_{η} ∥_{L_{\infty} (I; H)}^{2} + ∥ {\overset{\cdot\cdot}{φ}}_{η} ∥_{L_{2} (I; (H^{1} {(0, L)}^{*})}^{2} \\ + ∥ θ_{η} ∥_{L_{\infty} (I; H)}^{2} + ∥ θ_{η} ∥_{L_{2} (I; V)}^{2} + ∥ {\overset{\cdot}{θ}}_{η} ∥_{L_{2} (I; V^{*})}^{2} \\ \leq C_{1} \equiv C_{1} (f, p, q, u_{0}, v_{0}, φ_{0}, ω_{0}, θ_{0}) \end{matrix} \end{matrix}

(27)

for a solution ${u_{η}, φ_{η}, θ_{η}}, η > 0$ of the penalized problem. In order to gain the estimates of a penalty and acceleration terms for deflections ${u_{η}}$ we apply the assumption

w (x) \geq w_{0} > 0 \forall x \in (0, L)

and obtain

\begin{matrix} 0 \leq w_{0} \int_{Q} η^{- 1} u_{η}^{-} dx dt \leq \int_{Q} η^{- 1} u_{η}^{-} w dx dt \\ \leq \int_{Q} η^{- 1} u_{η}^{-} (w - u_{η}) dx dt . \end{matrix}

After inserting $y = w - u_{η}$ into the penalty equation we achieve using the estimate given by equation (27) the crucial estimates

\begin{matrix} \begin{matrix} ∥ η^{- 1} u_{η}^{-} ∥_{L_{1} (Q)} + ∥ ü_{η} ∥_{L_{1} (I; V^{*})} \\ \leq C_{5} \equiv C_{5} (f, p, q, u_{0}, v_{0}, φ_{0}, ω_{0}, θ_{0}) . \end{matrix} \end{matrix}

(28)

In order to get the subsequences in L₁ spaces due to the estimates given by equation (28) we apply the fact that they are the subsets of the dual spaces over C₀ spaces expressed in a form of measures. We set $M (Q)$ the set of all finite signed Borel measures on Q. For a Banach space X we set $M (I; X)$ the Banach space of all $X$ values measures on $I$ with a finite variation

| μ |_{X} (I) = ∥ μ ∥_{M (I; X)} .

The reader can consult the work by Bock et al.[4] (Appendix B) for details on vector valued measures (see also the work by Diestel and Uhl [7] or Dinculeanu [8]).

Let

η_{k} ↘ 0, k \to \infty .

We have for a subsequence of ${η_{k}}$ denoted again by ${η_{k}}$ the convergences

\begin{matrix} \begin{matrix} u_{η_{k}} ⇀^{*} u in L_{\infty} (I; V), \\ u_{η_{k}} \to u in C (Ī; H), \\ {\overset{\cdot}{u}}_{η_{k}} ⇀^{*} \overset{\cdot}{u} in L_{\infty} (I; H), \\ ü_{η_{k}} ⇀^{*} ü in M (I; V^{*}), \\ η_{k}^{- 1} u_{η_{k}}^{-} ⇀^{*} g in M (Q), \end{matrix} \end{matrix}

(29)

where g is a unique nonnegative measure expressing the contact force.

Before stating a variational formulation of the original problem given by equations (1) to (10) we introduce some sets and spaces of vector valued functions

$C_{w} (Ī; V)$ the set of weakly continuous functions.

$K : = {y \in C_{w} (Ī; V) + w; y \geq 0}$ - a shifted cone.

$R_{w} (Ī, H)$ the set of all weakly right continuous mappings.

$M_{0} (Q)$ the set of signed measures M on Q fulfilling

|\int_{Q} φ dM| \leq c ∥ φ ∥_{C_{0} (I; H)} \forall φ \in C_{0} (Q) .

We are looking for a solution u in the shifted cone

$Y = {u \in w + W; u \geq 0}$

W = {v \in C_{w} (Ī; V), \overset{\cdot}{v} \in R_{w} (Ī, H), \overset{\cdot\cdot}{v} \in M_{0} (Q)} .

(30)

We state the following variational formulation of the dynamic contact problem given by equations (1) to (10).

Problem $P$ . Look for

{u, φ, θ} \in Y \times L_{\infty} (I; H^{1} (0, L)) \times L_{2} (I; V)

such that ${\overset{\cdot}{φ}, θ} \in C {(\bar{I}; H)}^{2}$ ,

{\overset{\cdot\cdot}{φ}, \overset{\cdot}{θ}} \in L_{2} (I; H^{1} {(0, L)}^{*}) \times L_{2} (I; V^{*}),

the inequality and equations

\begin{matrix} \begin{matrix} \int_{Q} (y - u) dü \\ \geq \int_{Q} (f (y - u) - κ (u_{x} + φ) (y_{x} - u_{x})) dx dt, \end{matrix} \end{matrix}

(31)

\begin{matrix} \begin{matrix} \int_{I} {⟨ a \overset{\cdot\cdot}{φ}, ψ ⟩}_{- 1} dt \\ + \int_{Q} (b φ_{x} ψ_{x} + κ (u_{x} + φ) ψ - c θ_{x} ψ) dx dt \\ = \int_{Q} pψ dx dt, \end{matrix} \end{matrix}

(32)

\begin{matrix} \begin{matrix} \int_{I} ⟨ \overset{\cdot}{θ}, z ⟩ dt + \int_{Q} (d θ_{x} z_{x} + e \overset{\cdot}{φ} z_{x}) dx dt \\ = \int_{Q} qz dx dt \end{matrix} \end{matrix}

(33)

hold for any ${y, ψ, z} \in K \times L_{2} (I; H^{1} (0, L)) \times L_{2} (I; V)$ and the initial conditions given by equations (8) to (10) remain.

Using the penalized Problem $P_{η}$ with the convergences verified above we obtain in a similar way as in the work by Bock et al. [4] the existence of a solution to the original problem given by equations (1) to (10).

Theorem 4.1. There exists a solution ${u, φ, θ}$ of the Problem $P$ .

Remark 4.2. The set $W$ in equation (30) can be expressed also in the form

W = {v \in L_{\infty} (I; V) : \overset{\cdot}{v} \in L_{\infty} (I, H^{1} (0, L)), \overset{\cdot\cdot}{v} \in M_{0} (Q)} .

The previous representation given by equation (30) enables us to express directly the initial conditions.

Remark 4.3. Boundary conditions given by equations (5) and (6) for a simply supported beam and zero boundary thermal strain resultant enable us to get the a priori estimates in the previous and this chapter in an easy way. It is possible to consider also other types of boundary conditions with a bit more complicated deriving of a priori estimates inevitable for the convergence process.

5. Conclusion

We have formulated in a classical and a weak (variational) form the dynamic problem for a thermoelastic Mindlin-Timoshenko beam in a contact with a rigid obstacle. With respect to noncontinuity of velocities is the second-time derivative of the deflection function for the acceleration expressed as a vector measure in variational form of the problem. We verified the existence of a weak solution. It is a challenge to solve the same problem for a Mindlin–Timoshenko plate due to the lack of suitable imbedding formulas.

Footnotes

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Ministry of Education of Slovak Republic under Grant of Science and Technology Assistance Agency (grant number APVV-0246-12 and VEGA 1/0819/17).

References

Eck

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