Vekua proposed one of the methods of constructing two-dimensional models of prismatic shells in [1]

Georgian Mathematical Journal Volume 10 (2003), Number 1, 17–36

INVESTIGATION OF TWO-DIMENSIONAL MODELS OF ELASTIC PRISMATIC SHELL

M. AVALISHVILI AND D. GORDEZIANI

Abstract. Statical and dynamical two-dimensional models of a prismatic elastic shell are constructed. The existence and uniqueness of solutions of the corresponding boundary and initial boundary value problems are proved, the rate of approximation of the solution of a three-dimensional problem by the vector-function restored from the solution of a two-dimensional one is estimated.

2000 Mathematics Subject Classification: 42C10, 65M60, 74K20.

Key words and phrases: Boundary and initial boundary value problems for elastic prismatic shell, Fourier–Legendre series, a priori error estimation.

1. Introduction

In mathematical physics and theory of elasticity one of the urgent issues is constructing and investigating lower-dimensional models. I. Vekua proposed one of the methods of constructing two-dimensional models of prismatic shells in [1].

It must be pointed out that in [1] boundary value problems are considered inC^k spaces and the convergence of sequences of approximations to exact solutions of the corresponding three-dimensional problems is not investigated. In the statical case the existence and uniqueness of a solution of the reduced two- dimensional problem in Sobolev spaces were investigated in [2] and the rate of approximation of an exact solution of a three-dimensional problem by the vector-function restored from the solution of the reduced problem in C^k spaces was estimated in [3]. Later, various types of lower-dimensional models were constructed and investigated in [4–18].

In the present paper we consider static equilibrium of a prismatic elastic shell and a dynamical problem of vibration of a shell. Due to I. Vekua’s reduction method we construct a statical two-dimensional model of the plate and inves- tigate the obtained boundary value problem. Moreover, if the solution of the original problem satisfies additional regularity properties, we estimate the accu- racy of its approximation by the vector-function restored from the solution of a two-dimensional problem. We reduce the dynamical three-dimensional problem for a prismatic shell to the two-dimensional one, prove the existence and unique- ness of the solution of the corresponding initial boundary value problem and show that the vector-function restored from the latter problem approximates the solution of the original problem. Also, under the regularity conditions on the solution of the original problem we obtain the rate of its approximation.

ISSN 1072-947X / $8.00 / c°Heldermann Verlag www.heldermann.de

Let us consider a prismatic elastic shell of variable thickness and initial con- figuration Ω ⊂ R³; x = (x1, x2, x3) ∈ Ω, Ω denotes the closure of the do- main Ω ⊂ R³ (the domain is a bounded open connected set with a Lipschitz- continuous boundary, the set being locally on one side of the boundary [25]).

Assume that the ruling of the lateral surface Γ of the plate is parallel to theOx3- axis (Ox1x2x3 is a system of Cartesian coordinates in R³) and the upper Γ⁺, lower Γ⁻surfaces of the plate are defined by the equationsx₃ =h⁺(x₁, x₂), x₃ = h⁻(x1, x2), h⁺(x1, x2) > h⁻(x1, x2), x1, x2 ∈ω, h¯ ⁻, h⁺ ∈C¹(¯ω), where ω ⊂ R² is a domain with boundaryγ. Let γ₀ denote the Lipschitz-continuous part ofγ with positive length.

In order to simplify the notation, we assume that the indices i, j, p, q take their values in the set{1,2,3}, while the indices α, β vary in the set {1,2}and the summation convention with respect to repeated indices is used. Also, the partial derivative with respect to the p-th argument ∂

∂x_p is denoted by ∂_p. For any domain D ⊂ R^s, L²(D) denote the space of square-integrable functions in D in the Lebesgue sense, H^m(D) = W^m,2(D) denotes the Sobolev space of order m and the space of vector-functions denote with H^m(D) = [H^m(D)]³, L²(D) = [L²(D)]³, s, m ∈N.

Let us suppose that the material of the plate is elastic, homogeneous, isotropic andλ, µare the Lam´e constants of the material. The applied body force density is denoted byf = (f_i) : Ω×(0, T)→R³ and the surface force densities on the surfaces Γ⁺,Γ⁻ byg⁺ and g⁻, respectively, g^±= (g_i^±) : Γ^±×[0, T]→R³. The plate is clamped on the part Γ⁰ ={(x1, x2, x3)∈R³; (x1, x2)∈γ0, h⁻(x1, x2)≤ x3 ≤ h⁺(x1, x2)} of its lateral surface Γ, while the surface Γ¹ = Γ\Γ⁰ is free.

For the stress-strain state of the plate we have the following initial boundary value problem:

∂²ui

∂t² − X3

j=1

∂

∂x_j

©λepp(u)δij + 2µeij(u)ª

=fi(x, t), (x, t)∈ΩT, (1.1) u(x,0) =ϕ(x), ∂u

∂t(x,0) =ψ(x), x∈Ω, (1.2) u= 0 on Γ⁰_T = Γ⁰×[0, T],

X3

j=1

¡λe_pp(u)δ_ij + 2µe_ij(u)¢ n_j =







g_i⁺ on Γ⁺_T = Γ⁺×[0, T], g_i⁻ on Γ⁻_T = Γ⁻×[0, T], 0 on Γ¹_T = Γ¹×[0, T],

(1.3)

where ΩT = Ω ×(0, T), u = (ui) : ΩT → R³ is the unknown displacement vector-function, ϕ,ψ : Ω → R³ are initial displacement and velocity of the plate, n= (nj) denotes the unit outer normal vector along the boundary ∂Ω.

δij is the Kronecker symbol ande(u) ={eij(u)} is the deformation tensor e_ij(u) = 1

2

³∂u_i

∂x_j +∂u_j

∂x_i

´

, i, j= 1,2,3.

In Section 2 we study the statical case of problem (1.1)–(1.3) and construct a statical two-dimensional model of the plate, investigate the convergence of the sequence of vector-functions restored from the solutions of the correspond- ing boundary value problems to the solution of the original three-dimensional problem. In Section 3 we consider problem (1.1)–(1.3) in the suitable functional spaces, construct and investigate a hierarchic dynamical two-dimensional model of a prismatic shell.

2. Statical Boundary Value Problem

As we have mentioned in the introduction, in this section we study the statical case of problem (1.1)–(1.3).In this case the latter problem admits the following variational formulation: find a vector-function u∈V(Ω) ={v = (v_i)∈H¹(Ω), v = 0 on Γ⁰}, which satisfies the equation

B^Ω(u,v) =L^Ω(v), ∀v ∈V(Ω), (2.1) where

B^Ω(u,v) = Z

Ω

¡λe_pp(u)e_qq(v) + 2µe_ij(u)e_ij(v)¢ dx, L^Ω(v) =

Z

Ω

f_iv_idx+ Z

Γ⁺

g_i⁺v_idΓ + Z

Γ⁻

g_i⁻v_idΓ.

The variational problem (2.1) has a unique solution if λ ≥ 0, µ > 0, f ∈ L²(Ω), g^± ∈ L²(Γ^±), which is a unique solution of the following minimization problem: find u∈V(Ω) such that

J^Ω(u) = inf

v∈V(ΩJ^Ω(v), J^Ω(v) = 1

2B^Ω(v,v)−L^Ω(v), ∀v ∈V(Ω).

In order to reduce the three-dimensional problem (2.1) to a two-dimensional one, let us consider equation (2.1) on the subspace of V(Ω), which consists of polynomials of degree N with respect to the variable x₃, i.e.,

v_N = XN

r=0

a

³ r+1

2

´ _r

vP_r(ax₃−b), v= (^r v^r_i), r = 0, N,

wherea = 2

h⁺−h⁻, b= h⁺+h⁻

h⁺−h⁻, andP_r is a Legendre polynomial of degree r. Thus we get the following problem:

B^Ω(w_N,v_N) =L^Ω(v_N), ∀v_N ∈V_N(Ω), (2.2) V_N(Ω) =

n v_N =

XN

r=0

a

³ r+1

2

´ _r

vP_r(ax₃−b);

v∈r H¹(ω),v= 0 on^r γ0, r= 0, N o

.

In problem (2.2) the unknown functionw_N ∈V_N(Ω) is of the form wN =

XN

r=0

a

³ r+1

2

´ _r

wPr(ax3−b), so we have to find the vector-function w~_N = (w, . . . ,⁰ w),^N

~

w_N ∈V~_N(ω) = ©

~v_N = (v, . . . ,^{0 N}v); v∈^r H¹(ω), v= 0^r on γ₀, r= 0, Nª , such that the corresponding w_N is a solution of problem (2.2).

Taking into account properties of the Legendre polynomials [19] equation (2.2) can be written in the following form

XN

r=0

³ r+ 1

2

´ Z

ω

a¡

λ θ^r (w~_N)θ^r (~v_N) + 2µe^r_ij (w~_N)e^r_ij (~v_N)¢

dx₁dx₂

= XN

r=0

³ r+1

2

´ Z

ω

af^r_iv^r_i dx₁dx₂+ XN

r=0

³ r+1

2

´ Z

ω

a˜g_i⁺k₊ v^r_i dx₁dx₂

+ XN

r=0

³ r+1

2

´ Z

ω

a˜g_i⁻k₋(−1)^{r r}v_i dx₁dx₂, ∀~v_N ∈V~_N(ω), (2.3) where ˜g_i^±(x₁, x₂) = g_i^±(x₁, x₂, h^±(x₁, x₂)), (x₁, x₂)∈ω, i= 1,3,

θr (~v_N) =e^r_ii(~v_N), er_ij (~v_N) = 1

2

³∂ v^r_i

∂xj

+∂ v^r_j

∂xi

´ +1

2 XN

s=r

(b^ris

vs_j +b^rjs

vs_i), brαr=−(r+ 1)∂_αh⁺−∂_αh⁻

h⁺−h⁻ , ^rb3r= 0,

brjs=















0, s < r,

−(2s+ 1)∂αh⁺−(−1)^s+r∂αh⁻

h⁺−h⁻ , j =α, s > r, (2s+ 1)1−(−1)^s+r

h⁺−h⁻ , j = 3, s > r, k_± =

r 1 +

³∂h^±

∂x1

´₂ +

³∂h^±

∂x2

´₂

, f^r_i= Z _h⁺

h⁻

f_iP_r(ax₃−b)dx₃, r, s = 0, N.

Thus three-dimensional problem (2.1) we have reduced to two-dimensional one. For the last problem (2.3) we obtain the existence and uniqueness of its solution. First we prove the inequalities of Korn’s type in this case.

Theorem 2.1. Assume that ω ⊂ R² is a bounded domain with Lipschitz boundary γ =∂ω.

a) There exists a constant α >0, which depends only on ω, such that XN

r=0

³ r+ 1

2

´ Z

ω

ae^r_ij (~v_N)e^r_ij (~v_N)dx₁dx₂+ XN

r=0

Z

ω

vr_iv^r_i dx₁dx₂ ≥αk~v_Nk²_H1(ω), for any ~vN ∈H¹(ω), where H¹(ω) = [H¹(ω)]^N+1.

b) There exists a constantα >0, which depends only on ω and γ0, such that XN

r=0

³ r+1

2

´ Z

ω

a e^r_ij (~v_N)e^r_ij (~v_N)dx₁dx₂ ≥αk~v_Nk²_H1(ω), ∀~v_N ∈V~_N(ω).

Proof. Let us introduce the space E(ω) = ©

~v_N = (v, . . . ,^{0 N}v)∈L²(ω) = [L²(ω)]^N+1, e^r_ij (~v_N)∈L²(ω), r = 0, Nª . Then, equipped with the norm k.k defined by

k~v_Nk=©

|~v_N|²_0,ω +|e(~v_N)|²_0,ωª_1/2 , where

|~v_N|²_0,ω = XN

r=0

Z

ω

vr_iv^r_i dx₁dx₂,

|e(~v_N)|²_0,ω = XN

r=0

³ r+1

2

´ Z

ω

ae^r_ij (~v_N)e^r_ij (~v_N)dx₁dx₂,

the space E(ω) is a Hilbert space. Indeed, let us consider the Cauchy sequence {~v_N^(k)}^∞_k=1 in the space E(ω). By the definition of the norm k.k there exists vr_i∈L²(ω) ande^r_ij∈L²(ω) such that

vr^(k)_i →v^r_i, e^r_ij (~v_N^(k))→e^r_ij in L²(ω), ask → ∞.

Moreover, for anyϕ∈D(ω) (D(ω) is a space of infinitely differentiable functions with compact support in ω) the following equality is valid:

Z

ω

er_ij (~v_N^(k))ϕdx1dx2 = 1 2

Z

ω

³

−v^r(k)_i ∂jϕ−v^r(k)_j ∂iϕ +

XN

s=r

(b^ris

vs^(k)_j +b^rjs

vs^(k)_i )ϕ

´

dx₁dx₂, ∀k∈N.

Hence, passing to the limit as k → ∞, we obtain e^r_ij=e^r_ij (~v_N).

Let us show that the spaces E(ω) and H¹(ω) are isomorphic. It is clear that H¹(ω)⊂E(ω).Moreover, if we take~v_N ∈E(ω),then for any 1≤i, j, p≤3 and r= 0, N we get

∂p

vr_i∈H⁻¹(ω),

∂_j(∂_p v^r_i) =∂_j e^r_ip (~v_N) +∂_p e^r_ij (~v_N)−∂_i e^r_jp (~v_N)

−1 2

XN

s=r

¡∂_j(^rbis

vs_p +b^rps

vs_i) +∂_p(b^ris

vs_j +b^rjs

vs_i)−∂_i(b^rps

vs_j +b^rjs

vs_p)¢

∈H⁻¹(ω) since ify∈L²(ω), then ∂py∈H⁻¹(ω). By virtue of the lemma of Lions [20–22]

we have ∂_p v^r_i∈L²(ω), and therefore the spaces E(ω) and H¹(ω) coincide.

To prove the item a) of the theorem note that the identity mapping from H¹(ω) into E(ω) is injective, continuous and, by virtue of the preceding result, is surjective. Since both spaces are complete, the closed graph theorem [23]

shows that the inverse mapping is also continuous, which proves the desired inequality.

Now we will prove the item b) of the theorem. Notice that the semi-norm |.|

defined by |~v_N|= |e(~v_N)|_0,ω is the norm in the space V~_N(ω) when the measure of γ0 is positive. Indeed, if |e(~vN)|²_0,ω = 0, then [4]

v0₁ (x₁, x₂) = 1 a

³

b₃x₂+ 1

2(h⁺+h⁻)b₁+c₁

´ , v0₂ (x₁, x₂) = 1

a

³

−b₃x₁+ 1

2(h⁺+h⁻)b₂+c₂

´ , v0₃ (x₁, x₂) = 1

a

¡−b₁x₁−b₂x₂+c₃¢ ,

(x₁, x₂)∈ω, v1₁ (x₁, x₂) = b₁

3a², v¹₂ (x₁, x₂) = b₂

3a², v¹₃= 0, v^r_i= 0, i= 1,3, r= 2, N, for any real constantsb₁, b₂, b₃, c₁, c₂, c₃. Since ~v_N = 0 on γ₀ and the measure of γ₀ is positive, we get~v_N = 0 on ω.

To prove the inequality of the item b) we argue by contradiction. Then there exists a sequence {~v_N^k}^∞_k=1, ~v^k_N ∈ V~_N(ω) such that k~v^k_Nk_H¹_(ω) = 1 for all k ∈N, lim

k→∞|e(~v_N^k)|_0,ω = 0. Since the sequence{~v^k_N}^∞_k=1 is bounded in the space H¹(ω), a subsequence {~v^k_N^l}^∞_l=1 converges in L²(ω) by the Rellich–Kondraˇsov theorem. Each sequence {e^r_ij (~v^k_N^l)}^∞_l=1, r = 0, N, also converges in L²(ω). The subsequence {~v^k_N^l}^∞_l=1 is thus a Cauchy sequence with respect to the norm k.k.

According to the inequality of the item a) we have that the subsequence {~v_N^kl}^∞_l=1 is a Cauchy sequence with respect to the normk.k_H¹_(ω) too. The space V~_N(ω) is complete as a closed subspace of H¹(ω) and hence there exists ~v_N ∈ V~N(ω) such that

~v_N^kl →~vN in H¹(ω), as l→ ∞, and |e(~v_N)|_0,ω = lim

l→∞|(~v_N^kl)|_0,ω = 0. Therefore ~v_N = 0, which contradicts the relations k~v^k_N^lk_H¹_(ω)= 1 for all l≥1. ¤ On the basis of Theorem 2.1 we prove the theorem on the existence and uniqueness of a solution of problem (2.3).

Theorem 2.2. Assume that the Lam´e constants λ ≥ 0, µ > 0,f ∈ L²(Ω), g^± ∈L²(Γ^±), then the symmetric bilinear form B_N^Ω :V~_N(ω)×V~_N(ω)→R,

B_N^Ω(~u_N, ~v_N) = XN

r=0

³ r+ 1

2

´ Z

ω

a¡

λθ^r (~u_N)θ^r (~v_N) + 2µe^r_ij (~u_N)e^r_ij (~v_N)¢

dx₁dx₂ is continuous and coercive, the linear form L^Ω_N :V~_N(ω)→R,

L^Ω_N(~v_N) = XN

r=0

³ r+1

2

´ Z

ω

af^r_iv^r_i dx₁dx₂+ XN

r=0

³ r+1

2

´ Z

ω

a˜g_i⁺k₊v^r_i dx₁dx₂

+ XN

r=0

³ r+1

2

´ Z

ω

a˜g_i⁻k₋(−1)^{r r}v_i dx₁dx₂

is continuous. The two-dimensional problem (2.3) has a unique solution w~_N ∈ V~N(ω), which is also a unique solution to the following minimization problem:

~

wN ∈V~N(ω), JN(w~N) = inf

~vN∈V~N(ω)JN(~vN), J_N(~v_N) = 1

2B_N^Ω(~v_N, ~v_N)−L^Ω_N(~v_N), ∀~v_N ∈V~_N(ω).

Proof. By the inequality of the item b) of Theorem 2.1, the bilinear formB^Ω_N is coercive

B_N^Ω(~v_N, ~v_N)≥2µ XN

r=0

³ r+ 1

2

´ Z

ω

ae^r_ij (~v_N)e^r_ij (~v_N)dx₁dx₂ ≥2µαk~v_Nk²_H1(ω). Therefore, applying the Lax–Milgram theorem we obtain that problem (2.3) has a unique solution w~_N, which can be equivalently characterized as a solution of the minimization problem of the energy functional J_N(~v_N). ¤ Thus we have reduced the three-dimensional problem (2.1) to the two-dimen- sional one and for the latter problem proved the existence and uniqueness of its solution. For the reduced two-dimensional problem (2.3) the following theorem is true.

Theorem 2.3. If all the conditions of Theorem 2.2 hold, then the vector- function w_N =

XN

r=0

a

³ r+1

2

´ _r

w P_r(ax₃−b)corresponding to the solution w~_N = (w, . . . ,⁰ w)^N of the reduced problem (2.3) tends to the solution u of the three- dimensional problem (2.1) w_N →u in the space H¹(Ω) as N → ∞. Moreover, if u∈H^s(Ω), s≥2, then the following estimate is valid:

ku−wNk²_H¹_(Ω)≤ 1

N^2s−3q1(h⁺, h⁻, N), q₁(h⁺, h⁻, N)→0 as N → ∞.

(2.4)

If, additionally,kuk_H^s_(Ω)≤c,wherecis independent ofh= max

(x1,x2)∈¯ω(h⁺(x₁, x₂)−

h⁻(x1, x2)), then the following estimate holds:

ku−wNk²_E(Ω) ≤ h^2(s−1)

N^2s−3q2(N), q2(N)→0 as N → ∞, where kvk_E(Ω) =p

B^Ω(v,v), ∀v ∈V(Ω).

Proof. By virtue of Theorem 2.2 w~_N is a solution of the minimization problem of the energy functional J_N,i.e.

J_N(w~_N) = 1

2B_N^Ω(w~_N, ~w_N)−L^Ω_N(w~_N)

≤J_N(~v_N) = 1

2B_N^Ω(~v_N, ~v_N)−L^Ω_N(~v_N), ∀~v_N ∈V~_N(ω). (2.5) Taking into account that

B_N^Ω(~v_N, ~v_N) =B^Ω(v_N,v_N), L^Ω_N(~v_N) =L^Ω(v_N), ∀~v_N ∈V~_N(ω), where v_N =

XN

r=0

a

³ r+ 1

2

´ _r

v P_r(ax₃−b), and applying (2.5), we obtain B^Ω(u−wN,u−wN)≤B^Ω(u,u)−2L^Ω(vN) +B^Ω(vN,vN).

From the latter inequality we have

B^Ω(u−w_N,u−w_N)≤B^Ω(u−v_N,u−v_N), ∀v_N ∈V_N(Ω). (2.6) Since γ₀ is Lipschitz-continuous, by the trace theorems for Sobolev spaces [24], for any v ∈ H¹(Ω), v = 0 on Γ⁰, there exists a continuation v˜ ∈H¹₀(Ω₁) of the function v, where Ω₁ ⊃ Ω, ∂Ω₁ ⊃ Γ⁰. From the density of D(Ω₁) in H¹₀(Ω₁) we obtain that the space of infinitely differentiable functions in Ω, which are equal to zero on Γ⁰, is dense in V(Ω), and since the set of polynomials is dense inL²(−1,1),we conclude that [

N≥1

V_N(Ω) is dense in V(Ω), and therefore w_N →u in the space H¹(Ω) as N → ∞.

Let us prove estimate (2.4).Suppose that u ∈H^s(Ω), s≥2, then ε_N =u−u_N =u−

XN

r=0

a

³ r+1

2

´ r

u P_r(ax₃−b), u=^r Z _h⁺

h⁻

uP_r(ax₃−b)dx₃. Applying the properties of Legendre polynomials [19], we obtain

kε_Nk²_L2(Ω) = X∞

r=N+1

Z

ω

a³ r+1

2

´

(u^r_i)²dx₁dx₂,

°°

°∂ε_N

∂x₃

°°

°²

L²(Ω) = X∞

r=N

Z

ω

a

³ r+1

2

´

(∂3^ru_i)²dx1dx2+ Z

ω

aN(N−1)

4 (∂^N3u_i)²dx1dx2

+ Z

ω

aN(N + 1)

4 (^N+1∂₃u_i)²dx₁dx₂,

°°

°∂ε_N

∂x_α

°°

°²

L²(Ω ≤2

µ X_∞

r=N+1

Z

ω

a

³ r+ 1

2

´

(∂_α^ru_i)²dx₁dx₂ +

Z

ω

aN + 1 4

³ N

³ ∂˜h

∂x_α

´₂

+ (N + 2)

³ ∂¯h

∂x_α

´₂´

(∂^N₃u_i)²dx₁dx₂ +

Z

ω

aN + 1 4

³

(N + 2)³ ∂˜h

∂xα

´₂

+N³ ∂¯h

∂xα

´₂´

(^N+1∂₃u_i)²dx₁dx₂

¶ , where α= 1,2, ¯h= ¹₂(h⁺+h⁻), ˜h= ¹₂(h⁺−h⁻).

It should be mentioned that u=r

Z _h⁺

h⁻

uPr(ax3−b)dx3 = 1 a(2r+ 1)

¡ ^r−1

∂3u−∂^r+13u¢

, r= 1,2, . . . , N, and thus we have

ku^r k²_L2(ω)≤ c r^2s

Xr+s

k=r−s

°°

°1 a^s

∂₃. . . ∂k ₃

| {z }

s

u

°°

°²

L²(ω), (2.7) where c=const is independent of r, h⁺, h⁻. Therefore from (2.7) we get

kεNk²_L²_(Ω) ≤ 1

N^2sq(h⁺, h⁻, N),

°°

°∂ε_N

∂x_i

°°

°²

L²(Ω) ≤ 1

N^2s−3q(h⁺, h⁻, N),

q(h⁺, h⁻, N)→0 as N → ∞,

where i = 1,2,3. Taking into account (2.6) and the coerciveness of B^Ω, we obtain

ku−w_Nk²_H1(Ω) ≤ 1

N^2s−3q₁(h⁺, h⁻, N), q₁(h⁺, h⁻, N)→0 as N → ∞. (2.8) From (2.7) we also have

kε_Nk²_L2(Ω) ≤ h^2s N^2sq(N¯ ),

°°

°∂ε_N

∂x_i

°°

°²

L²(Ω) ≤ h^2(s−1) N^2s−3q(N),¯

¯

q(N)→0 as N → ∞, i= 1,2,3, (2.9)

where h= max

(x1,x2)∈¯ω2˜h(x1, x2).

From inequalities (2.9) we obtain the second inequality of the theorem ku−w_Nk²_E(Ω)≤ h^2(s−1)

N^2s−3q₂(N), q₂(N)→0 as N → ∞.

3. Dynamical Initial Boundary Value Problem

Now we proceed to studying the dynamical problem (1.1)–(1.3), constructing and investigating a dynamical two-dimensional model of a prismatic shell. This

problem admits the following variational formulation: findu∈C⁰([0, T];V(Ω)), u⁰ ∈C⁰([0, T];L²(Ω)), which satisfies the equation

d

dt(u⁰,v)_L²_(Ω)+B^Ω(u,v) = L^Ω(v), ∀v∈V(Ω) (3.1) in the sense of distributions in (0, T) and the initial conditions

u(0) =ϕ, u⁰(0) =ψ, (3.2) where ϕ∈V(Ω), ψ ∈L²(Ω).

Note that the formulated three-dimensional dynamical problem (3.1), (3.2) has a unique solution u for λ ≥ 0, µ > 0, f ∈ L²(Ω×(0, T)), g^±, ∂g^±

∂t ∈ L²(Γ^±×(0, T)), which satisfies the following energetical identity: ∀t∈[0, T],

(u⁰(t),u⁰(t))_L²_(Ω)+B^Ω(u(t),u(t)) = (ψ,ψ)_L²_(Ω)+B^Ω(ϕ,ϕ) + 2 ˜L^Ω(u)(t), where

L˜^Ω(u)(t) = Z _t

0

¡f(τ),u⁰(τ)¢

L²(Ω)dτ+¡

g⁺(t),u(t)¢

L²(Γ⁺)

+¡

g⁻(t),u(t)¢

L²(Γ⁻)−¡

g⁺(0),u(0)¢

L²(Γ⁺)−¡

g⁻(0),u(0)¢

L²(Γ⁻)

− Z _t

0

³∂g⁺

∂t (τ),u(τ)

´

L²(Γ⁺)dτ− Z _t

0

³∂g⁻

∂t (τ),u(τ)

´

L²(Γ⁻)dτ, ∀t∈[0, T].

As in the statical case, to reduce the three-dimensional problem (3.1), (3.2) to a two-dimensional one, let us consider equation (3.1) on the subspaceV_N(Ω) (V_N(Ω) is defined in Section 2), and choose ϕ,ψ as elements of V_N(Ω) and H_N(Ω), respectively, where

H_N(Ω) =

½ v_N =

XN

k=0

a³ k+1

2

´ _k

v P_k(ax₃−b); v∈^k L²(ω), k= 0, . . . , N

¾ . Consequently, we consider the following variational problem: find w_N ∈ C⁰([0, T];V_N(Ω)), w⁰_N ∈C⁰([0, T];H_N(Ω)), which satisfies the equation

d

dt(w⁰_N,vN)_L²_(Ω)+B^Ω(wN,vN) = L^Ω(vN), ∀vN ∈VN(Ω), (3.3) in the sense of distributions in (0, T) and the initial conditions

w_N(0) =ϕ_N, w⁰_N(0) =ψ_N, (3.4) where ϕ_N ∈V_N(Ω), ψ_N ∈H_N(Ω).

It must be pointed out that in problem (3.3),(3.4) the unknown is the vector- function w_N(t) =

XN

k=0

a

³ k + 1

2

´ k

w (t)P_k(ax₃ − b) and so this problem is

equivalent to the following one: find w~_N = (w, . . . ,⁰ w)^N ∈ C⁰([0, T];V~_N(ω)),

~

w⁰_N ∈C⁰([0, T]; [L²(ω)]^N+1),which satisfies the equation d

dt(P ~w_N⁰ , ~vN)_[L²_(ω)]^N+1+B_N^Ω(w~N, ~vN) =L^Ω_N(~vN), ∀~vN ∈V~N(ω), (3.5) in the sense of distributions in (0, T) and the initial condition

~

wN(0) = ϕ~N, w~_N⁰ (0) =ψ~N, (3.6) where ϕ~N = (ϕ, . . . ,⁰ ϕ)^N ∈V~N(ω), ~ψN = (ψ⁰, . . . ,ψ^N)∈[L²(ω)]^N+1,

ϕ_N = XN

k=0

a

³ k+ 1

2

´ _k

ϕPk(ax3−b),ψ_N = XN

k=0

a

³ k+1

2

´ _k

ψPk(ax3−b), P = (P_rl), P_rl =a

³ r+1

2

´

δ_rl, r, l = 0, . . . , N, and B_N^Ω, L^Ω_N are defined in Section 2.

Thus we get a two-dimensional model of the prismatic shell. To investigate problem (3.5), (3.6) let us consider a more general variational problem and formulate the theorem on the existence and uniqueness of its solution, from which we obtain the corresponding result for (3.5), (3.6).

Let us suppose that V and H are Hilbert spaces, V is dense in H and con- tinuously imbedded into it. The dual space of V is denoted by V⁰ and H is identified with its dual with respect to the scalar product, then

V ,→H ,→V⁰

with continuous and dense imbeddings. The scalar product and the norm in the space V is denoted by ((., .)), k.k, while in the space H by (., .) and|.|.Denote the norm in the space V⁰ by k.k_∗,and the dual relation between the spaces V⁰ and V by h., .i.

Assume that A, B, L are linear continuous operators such that

B =B₁+B₂, B₁ ∈L(V, V⁰), B₂ ∈L(V, H)∩L(H, V⁰), A, L∈L(H, H), B₁ is self-adjoint andB₁+λI is coercive for some real numberλ,Ais self-adjoint and coercive, i.e.,

b₁(u, v) =b₁(v, u), |b₁(u, v)| ≤c_b1kukkvk,

b₁(u, u)≥βkuk²−λ|u|², β >0, ∀u, v ∈V,

|b₂(˜u,v)| ≤˜

(c_b2k˜uk|˜v|, ∀u˜∈V,˜v ∈H, c_b2|˜u|k˜vk, ∀u˜∈H,v˜∈V, a(u1, v1) = a(v1, u1), a(u1, u1)≥α|u1|², α >0,

|a(u₁, v₁)| ≤c_a|u₁||v₁|,|l(u₁, v₁)| ≤c_l|u₁||v₁|, ∀u1, v1 ∈H,

(3.7)

where b₁(u, v) = hB₁u, vi, b₂(u, v) = hB₂u, vi, l(u₁, v₁) = (Lu₁, v₁), a(u₁, v₁) = (Au₁, v₁), b(u, v) = b₁(u, v) +b₂(u, v), ∀u, v ∈V, u₁, v₁ ∈H.