BOUNDARY VALUE PROBLEMS IN (1.0) APPROXIMATION OF A MATHEMATICAL MODEL OF BARS^{1}

G. Jaiani

I.Vekua Institute of Applied Mathematics I.Javakhishvili Tbilisi State University

Generalizing an idea of I. Vekua [1] of constructing a version of the theory of plates and shells, in the elastic bar model suggested in [2,3] fields of displacements, strains and stresses of the three-dimensional theory of linear elasticity have been expanded into orthogonal double Fourier-Legendre series with respect to the variables along thickness, and width of the bar with a variable rectangular cross-section.

Now let us consider symmetric (2h̄_{i} = 0, i = 2, 3) bar in (1.0) approximation of the above model. The corresponding full system of equations has the following form:

(λ + 2μ)(h_{2}h_{3} v^{0,0}_{1,1}),_{1} +3λ(h_{2}h_{3} v^{1,0}_{3}),_{1} +X_{1}^{0}^{0,0} = 0,  (1)
μ(h_{2}h_{3}^{3} v^{1,0}_{3,1})_{,1} − 3(λ + 2μ)h_{2}h_{3} v^{1,0}_{3} − λh_{2}h_{3} v^{0,0}_{1,1} + h_{3}X_{3}^{0}^{1,0} = 0,  (2)
μ(h_{2}h_{3} v^{0,0}_{2,1})_{,1} + X_{2}^{0}^{0,0} = 0,  (3)
μ(h_{2}h_{3}^{3} v^{1,0}_{2,1})_{,1} − 3μh_{2}h_{3} v^{1,0}_{2} + h_{3}X_{2}^{0}^{1,0} = 0,  (4)
μ(h_{2}h_{3} v^{0,0}_{3,1})_{,1} + 3μ(h_{2}h_{3} v^{1,0}_{1})_{,1} + X_{3}^{0}^{0,0} = 0,  (5)
(λ + 2μ)(h_{2}h_{3}^{3} v^{1,0}_{1,1})_{,1} − μh_{2}h_{3} v^{0,0}_{3,1} − 3μh_{2}h_{3} v^{1,0}_{1} + h_{3}X_{1}^{0}^{1,0} = 0,  (6)

where

v_{k}^{r,s}(x_{1}) := \frac{u_{k}^{r,s}(x_{1})}{h_{2}^{r+1}(x_{1})h_{3}^{s+1}(x_{1})}

λ and μ are Lamé’s constants, index 1 after comma means differentiation with respect to x_{1},

X_{j}^{0}^{n_{3},n_{2}} := ∫_{h_{2}^{(−)}}^{h_{2}^{(+)}}[\sqrt{1 + \binom{(+)}{h_{3,1}}^{2}}Qh_{3}^{(+)} ; j(x_{1}, x_{2}, h_{3}^{(+)}) + (−1)^{n_{3}} \sqrt{1 + \binom{(−)}{h_{3,1}}^{2}}×

×Qh_{3}^{(−)} ; j(x_{1}, x_{2}, h_{3}^{(−)})] P_{n_{2}}(a_{2}x_{2} − b_{2})dx_{2} + ∫_{h_{3}^{(−)}}^{h_{3}^{(+)}}[\sqrt{1 + \binom{(+)}{h_{2,1}}^{2}} Qh_{2}^{(+)} ; j(x_{1}, h_{2}^{(+)}, x_{3})+

1This research was supported by a grant of Georgian government for universities

+(−1)^{n_{3}} \sqrt{1 + \binom{(−)}{h_{2,1}}^{2}} Qh_{3}^{(−)} ; j(x_{1}, h_{2}^{(−)}, x_{3})] P_{n_{3}}(a_{3}x_{3} − b_{3})dx_{3} + X_{j}^{n_{3},n_{2}}, i = 1, 2, 3,

Qh_{3}^{(+)} ; j, Qh_{3}^{(−)} ; j, Qh_{2}^{(+)} ; j, Qh_{2}^{(−)} ; j are components of surface forces acting on surfaces h_{3}^{(+)}, h_{3}^{(−)}, h_{2}^{(+)}, h_{2}^{(−)} respectively; the bar occupies the following domain

B := {(x_{1}, x_{2}, x_{3}) : 0 < x_{1} < l, h_{i}^{(−)}(x_{1}) < x_{i} < h_{i}^{(+)}(x_{1}), i = 2, 3, l = const};

2h_{i}(x_{1}) := h_{i}^{(+)}(x_{1})−h_{i}^{(−)}(x_{1}) ≥ 0, 2h̄_{i}(x_{1}) := h_{i}^{(+)}(x_{1})+h_{i}^{(−)}(x_{1}) ≥ 0, i = 2, 3; h_{i} ∈ C^{1};

X_{j}^{n_{3},n_{2}}(x_{1}) := ∫_{h_{2}^{(−)}}^{h_{2}^{(+)}} ∫_{h_{3}^{(−)}}^{h_{3}^{(+)}}X_{j}(x_{1}, x_{2}, x_{3})P_{n_{2}}(a_{2}x_{2} − b_{2})P_{n_{3}}(a_{3}x_{3} − b_{3})dx_{2}dx_{3}, j = 1, 2, 3,

X_{j} are volume forces; P_{n}(·) are Legendre Polynomials;

a_{i}(x_{1}) := \frac{1}{h_{i}(x_{1})}, b_{i}(x_{1}) := \frac{h̄_{i}(x_{1})}{h_{i}(x_{1})}, i = 2, 3;

u_{j}^{n_{3},n_{2}}(x_{1}) := ∫_{h_{2}^{(−)}}^{h_{2}^{(+)}} ∫_{h_{3}^{(−)}}^{h_{3}^{(+)}}u_{j}(x_{1}, x_{2}, x_{3})P_{n_{2}}(a_{2}x_{2} − b_{2})P_{n_{3}}(a_{3}x_{3} − b_{3})dx_{2}dx_{3}, j = 1, 2, 3,

u_{j}(x_{1}, x_{2}, x_{3}) are components of the displacement vector.

Boundary value problems for equation (3) have been solved in [2]. Therefore for this equation e.g. the Dirichlet problem is correct if only I^{0,0}_{0} < ∞, I^{0,0}_{ℓ} < ∞. For (4) the Dirichlet problem is correct if only I^{1,0}_{0} < ∞, I^{1,0}_{ℓ} < ∞ (see also [4],[5]), where

I_{0}^{N_{3},N_{2}} := ∫_{o}^{ε} h_{2}^{−2N_{2}−1}(τ)h_{3}^{−2N_{3}−1}(τ)dτ, ε = const > 0,

I_{ℓ}^{N_{3},N_{2}} := ∫_{ℓ−ε}^{ℓ} h_{2}^{−2N_{2}−1}(τ)h_{3}^{−2N_{3}−1}(τ)dτ, ε = const > 0.

Hence the approximate value of

u_{2}(x_{1}, x_{2}, x_{3}) ≅ \frac{1}{4} v^{0,0}_{2}(x_{1}) + \frac{3x_{3}}{4} v^{1,0}_{2}(x_{1})

can be preassigned on the bar ends if only I^{1,0}_{0} < +∞, I^{1,0}_{ℓ} < +∞.

Systems (5), (6) and (1), (2) can be correspondigly reduced to the following systems

3μ v^{1,0}_{1} = −μ v^{0,0}_{3,1} − h_{2}^{−1}(x_{1})h_{3}^{−1}(x_{1}) (∫_{x_{1}^{0}}^{x_{1}}X_{3}^{0}^{0,0}dx_{1} − C_{1}) , C_{1} = const, x_{1}^{0} ∈]0, ℓ[,

(λ + 2μ)(h_{2}h_{3}^{3} v^{0,0}_{3,11})_{,11} = 3X_{3}^{0}^{0,0} + 3(h_{3}X_{1}^{0}^{1,0})_{,1} − \frac{λ + 2μ}{μ} (h_{2}h_{3}^{3} (h_{2}^{−1}h_{3}^{−1}∫_{x_{1}^{0}}^{x_{1}}X_{3}^{0}^{0,0}dx_{1}−

−h_{2}^{−1}h_{3}^{−1}C_{1})_{,1})_{,11} ;

and

3λ v^{1,0}_{3} = −(λ + 2μ) v^{0,0}_{1,1} − h_{2}^{−1}(x_{1})h_{3}^{−1}(x_{1}) (∫_{x_{1}^{0}}^{x_{1}}X_{1}^{0}^{0,0}(t)dt − C_{2}) , C_{2} = const,

(λ + 2μ)μ(h_{2}h_{3}^{3} v^{0,0}_{1,11})_{,11} − 12(λ + μ)μ(h_{2}h_{3} v^{0,0}_{1,1})_{,1} =

= −μ (h_{2}h_{3}^{3} [(h_{2}^{−1}h_{3}^{−1}∫_{x_{1}^{0}}^{x_{1}}X_{1}^{0}^{0,0}(t)dt)_{,1} − C_{2} (h_{2}^{−1}h_{3}^{−1})_{,1}])_{,11} + 12\frac{(λ + μ)μ}{λ + 2μ}X_{1}^{0}^{0,0}+

+\frac{3λ^{2}}{λ + 2μ}X_{1}^{0}^{0,0} + 3λ(h_{3}X_{3}^{0}^{1,0})_{,1}.

The advantages of these systems consist in the following: the second equations are equations with respect to a only one unknown function and after their solution (see [6]) from the first equations correspondinly v^{1,0}_{1} and v^{1,0}_{3} can be readily calculated.

Remark 1. Let us consider (N_{3}, N_{2}) approximation of a model of bars given in [2,3]. Further the notation of above works will be used.

Let the width and the thickness of the bar be given by 2h_{i}(x_{1}) = h_{0}^{i}x_{1}^{α_{i}}, i = 2, 3 (h_{0}^{i} = const > 0, α_{i} = const ≥ 0), respectively.

Problem 1. Find bounded weighted double moments v_{j}^{n_{3},n_{2}} ∈ C^{3}(]0, ℓ[) ∩ C(]0, ℓ]), j = 1, 2, 3, n_{i} = \overline{0, N_{i}}, i = 2, 3, of displacement vector components u_{j}(x_{1}, x_{2}, x_{3}) under boundary conditions:

\begin{matrix}v_{j}^{n_{3},n_{2}}(0) = φ_{j}^{0}^{n_{3},n_{2}}, j = 1, 2, 3, n_{i} = \overline{0, N_{i}}, i = 2, 3, \\ if (2N_{2} + 1)α_{2} + (2N_{3} + 1)α_{3} < 1; \end{matrix}  (7)

v_{j}^{n_{3},n_{2}}(ℓ) = φ_{j}^{ℓ}^{n_{3},n_{2}}, j = 1, 2, 3, n_{i} = \overline{0, N_{i}}, i = 2, 3,

provided that the lateral boundaries (x_{i} = h_{i}^{(±)}(x_{1}), i = 2, 3, 0 < x_{1} < ℓ) of the

bar are arbitrarily loaded. φ_{j}^{0}^{n_{3},n_{2}}, φ_{j}^{ℓ}^{n_{3},n_{2}}, and loadings f_{j}^{i}^{(±)} are given.

Statement. Problem 1 is correct in the sense of Hadamard.

If we consider the limit N_{i} → +∞, i = 2, 3, then from (7) we have

α_{i} < \frac{1}{2N_{i} + 1}, and lim_{N_{i}→+∞}α_{i} = 0, i = 2, 3.

Thus, Problem 1 tends to the following three-dimensional model Problem 2. Find

u_{i}(x_{1}, x_{2}, x_{3}) ∈ C^{3}(V) ∩ \{ \begin{matrix}C(\overline{V}) if α_{i} = 0, i = 2, 3, \\ C(\overline{V} \\{0, x_{2}, x_{3}\}) if α_{2} + α_{3} > 0, \end{matrix}

bounded in the latter case, under boundary conditions

u_{j}(0, x_{2}, x_{3}) = φ_{j}^{0}(x_{2}, x_{3}), j = 1, 2, 3, if α_{i} = 0, i = 2, 3;

u_{j}(ℓ, x_{2}, x_{3}) = φ_{j}^{ℓ}(x_{2}, x_{3}), j = 1, 2, 3,

X_{ij}(x_{1}, x_{2}, x_{3}) |_{x_{i}={(±) \atop h_{i}}(x_{1})} = f_{j}^{i}^{(±)}(x_{1}, x_{5−i}), j = 1, 2, 3, i = 2, 3.

Remark 2. In the forthcoming paper the similar mathematical model of bars with the variable cross-section of general form will be constructed.

Remark 3. In [7] to deduce the system of one-dimensional equations of ceramic bar of uniform cross-section from the three-dimensional equations of thermopiezoelasticity, the method of power series expansion is used for the variables along the bar thickness and width. In contrast to this, by using the method of the present paper, one-dimensional equations of a ceramic bar of a variable cross-section can be deduced.

Remark 4. If we waive the assumption that ℓ is much greater than width and thickness which is necessary in case of bars then B should be understood as an essentially three-dimensional elastic body. Generalizing the idea of this paper we could expand the displacement vector components into triple Fourier-Legendre series. This approach leads in the general case to the infinite system of algebrical equations. In case of a prismatic body (a_{j} = const, j = 1, 2, 3), a_{1} = ℓ^{−1} (N_{3}, N_{2}, N_{1}) approximation is equivalent to the polynomial approximation of solution of three-dimensional boundary value problems of the linear theory of elasticity. In dynamical case (0, 0, 0) approximation characterizes rigid transfer of the body under action of volume forces; (1.1.1) approximation coincides with the homogeneous deformation of the body. This approach is convinient especially in point of view of numerical solution of the threedimensional problems.

Volume 3, 1999 11

R e f e r e n c e s

[1] Vekua I.N., Shell Theory: General Methods of Construction. Pitman Advanced Publishing Program, Boston-London-Melburne, 1985, 1-287.

[2] Jaiani G.V., On a Model of a Bar with Variable Thickness, BULLETIN of TICMI, 2, 1998, 36-40. (electronic version: http://www.viam.hepi. edu.ge/others/TICMI).

[3] Jaiani G.V., On a Mathematical Model of a Bar with Variable Rectangular Crosssection, Universitaet Potsdam, Institut fuer Mathematik, Preprint 98/21, 1998, 1-24.

[4] Kiguradze I.T., Shekhter B.L., Singular Boundary Value Problems for Ordinary Differential Equations of Second Order. Itogi Nauki i Tekhniki, Seria, Sovremennje problemi Matematiki, Noveishje Dostizhenia, Moskva, 30, 1987, 105-202.

[5] Jaiani G.V., The First Boundary Value Problem of Cusped Prismatic Shell Theory in Zero Approximation of Vekua Theory, Proceedings of I.Vekua Institute of Applied Mathematics, 29, 1988, 5-38, (Russian, Georgian and English summaries).

[6] Jaiani G.V., Bending of an Orthotropic Cusped Plate, Universitaet Potsdam, Institut fuer Mathematik, Preprint 98/23, 1998, 1-30.

[7] Askar Altay G., Dökmeci M.C., Numerical Algorithms for Dynamics of Thermopiezoceramic Bars. Journal of Applied Mathematics and Mechanics (ZAMM) 77, 6, 1997, 429-445.