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\begin{center}
ON A MODEL OF A BAR WITH VARIABLE THICKNESS

\vspace*{0.3cm}
{\it G. Jaiani}

\vspace*{0.3cm}
{\it I. Vekua Institute of Applied Mathematics\\
I. Javakhishvili Tbilisi State University}
\end{center}

\vspace*{0.3cm}
\par Like I. Vekua's [1] method, when in order to construct theory of plates 
and shells all mechanical quantities of three-dimensional theory of elasticity 
have been expanded in the Fourier-series of Legendre polynomials with respect to 
the variable along thickness, and then only first $ N+1, \; N=0,1,\cdots, $ 
terms have been left, in the bar model under consideration all mechanical 
quantities have been expanded in double Fourier-series of Legendre 
polynomials with respect to the variables along thickness, and width of the 
bar, and then first $ (N_3+1)(N_2+1), \; N_3, \, N_2 =0,1,\cdots, $ terms have 
been left. This case will be called $ (N_3, N_2) $ approximation. In this 
approximation we have the system of $ 3(N_3+1)(N_2+1) $ equations as follows
$$
\Lambda_j(h_2^{2n_2+1}h_3^{2n_3+1}
 \; \mathop {v_{j,1}} \limits ^{\hspace{-1.2ex}n_3,n_2})_{,1}(x_1,t)+ 
\mathop {E_j} \limits ^{\hspace{-0.6ex}n_3,n_2}(\cdots \mathop {v_{k,1}}
\limits^{\hspace{-1.8ex}r,s}\cdots)+
\mathop {M_j} \limits ^{\hspace{-0.6ex}n_3,n_2}(\cdots \mathop {v_k}
\limits^{\hspace{-0.7ex}r,s}\cdots)+
\mathop {X_j^0}\limits^{\hspace{-0.7ex}n_3,n_2}
=$$
$$
=\rho h_2^{2n_2+1}h_3^{2n_3+1}\frac{\partial^2\; \mathop {v_j} \limits ^{\hspace{-0.7ex}n_3,n_2}}
{\partial t^2},\;\;
j=1,2,3,\;\;
n_i=\overline{0,N_i},\;\;i=2,3, \;\;\rho\;\; {\rm is\; density},
$$
\begin{equation}
\begin{array}{c}
\Lambda_j=
\left\{
\begin{array}{l}
\lambda+2\mu,\;\;\;\; j=1,\\
\mu, {\hspace{7ex}} j=2,3,
\end{array}  \right.\;\;
\mathop {v_{k}} \limits ^{\hspace{-0.7ex}r,s}=\frac{\mathop {u_k}\limits ^{\hspace{-0.7ex}{r,s}}}
{h_2^{r+1}h_3^{s+1}}
\end{array}
\end{equation}
$\lambda$ and $\mu$ are Lam\'{e} constants,
$\mathop {E_j} \limits ^{\hspace{-0.7ex}n_3,n_2},\;
\mathop {M_j} \limits ^{\hspace{-0.7ex}n_3,n_2} $ 
are certain linear functions of their arguments where 
$ r=\overline{0,N_3},$ $ s=\overline{0,N_2}$, $k=1,2,3,$ in general, with unbounded 
coeficients,
$$
\mathop {X_j^0}\limits^{\hspace{-0.7ex}n_3,n_2}:=\displaystyle \sqrt{1+
\left(\;\mathop {h_{3,1}}\limits^{\hspace{-1.8ex}(+)}\right)^2}\;
\int\limits_{\mathop {h_2}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_2}
\limits^{\hspace{-1ex}(+)}} Q_j^{\mathop {h_3}\limits^{\hspace{-1ex}(+)}}
(x_1,x_2,\mathop {h_3}\limits^{\hspace{-0.7ex}(+)})P_{n_2}(a_2x_2-b_2)dx_2+
$$
$$
\displaystyle+(-1)^{n_3}
\sqrt{1+\left(\;\mathop {h_{3,1}}\limits^{\hspace{-1.8ex}(-)}
\right)^2}\;
\int\limits_{\mathop {h_2}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_2}
\limits^{\hspace{-1ex}(+)}} Q_j^{\mathop {h_3}\limits^{\hspace{-1ex}(-)}}
(x_1,x_2,\mathop {h_3}\limits^{\hspace{-0.7ex}(-)})P_{n_2}(a_2x_2-b_2)dx_2+
$$
$$
\displaystyle
+\sqrt{1+\left(\;\mathop {h_{2,1}}\limits^{\hspace{-1.8ex}(+)}\right)^2}\;
\int\limits_{\mathop {h_3}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_3}
\limits^{\hspace{-1ex}(+)}} Q_j^{\mathop {h_2}\limits^{\hspace{-1ex}(+)}}
(x_1,\mathop {h_2}\limits^{\hspace{-0.7ex}(+)},x_3)P_{n_3}(a_3x_3-b_3)dx_3+
$$
$$
\displaystyle+(-1)^{n_3}
\sqrt{1+\left(\;\mathop {h_{2,1}}\limits^{\hspace{-1.8ex}(-)}
\right)^2}\;
\int\limits_{\mathop {h_3}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_3}
\limits^{\hspace{-1ex}(+)}} Q_j^{\mathop {h_3}\limits^{\hspace{-1ex}(-)}}
(x_1,\mathop {h_2}\limits^{\hspace{-0.7ex}(-)},x_3)P_{n_3}(a_3x_3-b_3)dx_3+
\mathop {X_j}\limits^{\hspace{-0.7ex}n_3,n_2}, \;\;i=1,2,3,
$$
$Q_j^{\mathop {h_3}\limits^{\hspace{-1ex}(+)}},\;Q_j^{\mathop {h_3}
\limits^{\hspace{-1ex}(-)}},\; Q_j^{\mathop {h_2}\limits^{\hspace{-1ex}(+)}},
\;Q_j^{\mathop {h_2}\limits^{\hspace{-1ex}(-)}}$ 
are components of surface forces acting on surfaces 
$\mathop {h_3}\limits^{\hspace{-0.7ex}(+)},\; 
\mathop {h_3}\limits^{\hspace{-0.7ex}(-)},\;
\mathop {h_2}\limits^{\hspace{-0.7ex}(+)},\;
\mathop {h_2}\limits^{\hspace{-0.7ex}(-)}$ respectively; the bar occupies the 
following domain
$$
\{(x_1,x_2,x_3):\;\;0<x_1<l, \;\;\mathop {h_i}\limits^{\hspace{-0.7ex}(-)}
(x_1)<x_i<\mathop {h_i}\limits^{\hspace{-0.7ex}(+)}(x_1),\;\;
i=2,3,\;\;l={\rm const}\};
$$
$$
2h_i(x_1):=\mathop {h_i}\limits^{\hspace{-0.7ex}(+)}-
\mathop {h_i}\limits^{\hspace{-0.7ex}(-)} \ge 0, \;\; i=2,3;\;\; h_i\in C^1;
$$
$$
\displaystyle
\mathop {X_j}\limits^{\hspace{-0.7ex}n_3,n_2}:=
\int\limits_{\mathop {h_2}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_2}
\limits^{\hspace{-1ex}(+)}}\; 
\int\limits_{\mathop {h_3}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_3}
\limits^{\hspace{-1ex}(+)}}
P_{n_2}(a_2x_2-b_2)P_{n_3}(a_3x_3-b_3) X_j dx_2dx_3,\;\;j=1,2,3,
$$
$X_j$ are volume forces; $P_n(\cdot)$ are Legendre Polynomials;
$$
a_i:=\displaystyle \frac {2}{\mathop {h_i}\limits^{\hspace{-0.7ex}(+)}-
\mathop {h_i}\limits^{\hspace{-0.7ex}(-)}},\;\;\;\;
b_i:=\displaystyle \frac {\mathop {h_i}\limits^{\hspace{-0.7ex}(+)}+
\mathop {h_i}\limits^{\hspace{-0.7ex}(-)}}
{\mathop {h_i}\limits^{\hspace{-0.7ex}(+)}-
\mathop {h_i}\limits^{\hspace{-0.7ex}(-)}},\;\;\;\;i=2,3;
$$
$$
\displaystyle
\mathop {u_j}\limits^{\hspace{-0.7ex}n_3,n_2}:=
\int\limits_{\mathop {h_2}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_2}
\limits^{\hspace{-1ex}(+)}} \;
\int\limits_{\mathop {h_3}\limits^{\hspace{-1ex}(-)}} \limits^{\mathop {h_3}
\limits^{\hspace{-1ex}(+)}}  u_j(x_1,x_2,x_3)
P_{n_2}(a_2x_2-b_2)P_{n_3}(a_3x_3-b_3)dx_2dx_3,\;\;j=1,2,3,
$$
$u_j(x_1,x_2,x_3)$ are components of displacement vector.
\par In $(0,0)$ approximation the system (1) will have the form
\begin{equation}
(h_2 h_3 v_{j,1}(x_1,t))_{,1}+\mathop {Y_j}\limits^{\hspace{-0.3ex}{0,0}}
=\Lambda_j^{-1}\rho h_2 h_3\frac{\partial^2  v_1(x_1,t)}
{\partial t^2},\;j=1,2,3,
\end{equation}
$$
v_j(x_1,t):=\displaystyle \frac {\mathop {u_j}\limits^{\hspace{-0.5ex}0,0}(x_1)}
{h_2(x_1)h_3(x_1)},\;\;\;\;\;\mathop {Y_1}\limits^{\hspace{-0.3ex}0,0}=
\displaystyle \frac {\mathop {X_1^0}\limits^{\hspace{-0.5ex}0,0}}
{\lambda+2\mu,}\;\;\;\;\;
\mathop {Y_i}\limits^{\hspace{-0.3ex}0,0}=
\displaystyle \frac {\mathop {X_i^0}\limits^{\hspace{-0.5ex}0,0}}
{\mu,}\;\;\;\;i=2,3.
$$
\par Let us consider statical case of (2). Obviously
$$
v_j(x_1)=-\int\limits_{x_1^0}^{x_1}\frac{d\tau}{h_2(\tau)h_3(\tau)}
\int\limits_{x_1^0}^{\tau}{\mathop {Y_j}\limits^{\hspace{-0.5ex}0,0}}(t)dt+
c_1^j\int\limits_{x_1^0}^{x_1}\frac{d\tau}{h_2(\tau)h_3(\tau)}+c_2^j,\eqno(*)
$$
$$
x_1^0=const\in]0,l[,\;\;\;c_\alpha^j=const,\;\;\alpha=1,2,\;\;j=1,2,3.
$$
Let further
$$
{\mathop {I_0}\limits^{N_3,N_2}}:=\int\limits_{o}^{\varepsilon}
h_2^{-2N_2-1}(\tau)h_3^{-2N_3-1}(\tau)d\tau,\;\;\varepsilon=const>0,
$$
$$
{\mathop {I_\ell}\limits^{N_3,N_2}}:=\int\limits_{\ell-\varepsilon}^{\ell}
h_2^{-2N_2-1}(\tau)h_3^{-2N_3-1}(\tau)d\tau,\;\;\varepsilon=const>0,
$$
\begin{equation}
v_j(0)=\varphi_j^0\;\;\;\;\;{\rm if}\;\;\;\;
{\mathop {I_0}\limits^{0,0}}<+\infty,\;\;\;\;\;j=1,2,3,
\end{equation}
\begin{equation}
v_j(l)=\varphi_j^l\;\;\;\;\;{\rm if}\;\;\;\;\;
{\mathop {I_\ell}\limits^{0,0}}<+\infty,\;\;j=1,2,3,
\end{equation}
\begin{equation}
v_j(x_1)=O(1),\;\;\;x_1\to 0_+\;\;\;\;\;{\rm if}\;\;{\mathop {I_0}\limits^{0,0}}=+\infty,\;\;j=1,2,3,
\end{equation}
\begin{equation}
v_j(x_1)=O(1),\;\;\;x_1\to l_-\;\;\;\;\;{\rm if}\;\;
{\mathop {I_\ell}\limits^{0,0}}=+\infty,\;\;j=1,2,3,
\end{equation}
\begin{equation}
\mathop {X_{1j}}\limits^{\hspace{-0.5ex}0,0}(0)=
\Lambda_j h_2 h_3 v_{j,1}\vert_{x_1=0}=\psi_j^0,\;\;\;
{\rm if}\;\;
{\mathop {I_0}\limits^{0,0}}\le+\infty,\;\;\;
j=1,2,3,
\end{equation}
\begin{equation}
\mathop {X_{1j}}\limits^{\hspace{-0.5ex}0,0}(l)=
\Lambda_j h_2 h_3 v_{j,1}\vert_{x_1=l}=\psi_j^l,
\;\;\;{\rm if}\;\;
{\mathop {I_\ell}\limits^{0,0}}\le+\infty,\;\;\;
j=1,2,3,
\end{equation}
$\varphi_j^0,\; \psi_j^0, \; \varphi_j^l, \;\psi_j^l,\; j=1,2,3,$ 
are given constants.
\par Let $\mathop {Y_{j}}\limits^{\hspace{-0.5ex}0,0}\in L([0,l]),$ 
and  if ${\mathop {I_0}\limits^{0,0}}=+\infty$ $({\mathop {I_\ell}\limits^{0,0}}=+\infty)$, it is such that iterated integral in ($*$) is bounded by
$x_1\rightarrow 0_+$ $(l_-)$.
\par If $\displaystyle \frac {1}{h_2\cdot h_3}$ is locally summable in
$]0,l[$ then
for regular solutions $(v_j\in$ $\in C^2(]0,l[))$
 only the following problems are correct:
(2), (3), (4) $(v_j\in$ $\in C([0,l]))$; (2), (5), (4) $(v_j\in C(]0,l]))$;
 (2), (3), (6) $(v_j\in C([0,l[))$; (2), (5), (6) ($v_j$ is bounded); (2), (3), (8)
$(v_j\in C([0,l[),$ $h_2h_3v_{j,1}\in C([0,l])$;
(2), (7), (4) $(v_j\in C(]0,l]),$ $h_2h_3v_{j,1}\in C([0,l])$; the mixed problem when on the one end of the bar for some 
displacement components the Dirichlet and for others either the Neumann or
(5), (6) type conditions 
are given but on the other end for first components either the Neumann or the Dirichlet
 or (5), (6) type conditions and for the second components the Dirichlet conditions 
(the Neumann and (5), (6) type conditions are not admissible) are
given, are correct.
\par In the case (2), (5), (6) we have the solution up to the rigid transfer.
\par If $\displaystyle \frac {1}{h_2\cdot h_3}$ has nonsummable singularities 
in $]0,l[$
then only arbitrary three 
(but one for any fixed $j$) out of six conditions of (3), (4) can be given (in 
particular, it contains cases when either only (3) or only (4) can be given).
\par The following boundary conditions can be similary set in the $(N_3,N_2)$
approximation:
\begin{equation}
{\mathop {v_j}\limits^{n_3,n_2}}(0)={\mathop {\varphi_j^0}\limits^{n_3,n_2}},\;\;
j=1,2,3,\;\;n_i=\overline{0,N_i},\;\;i=2,3,\;\;{\rm if}\;\;
{\mathop {I_0}\limits^{N_3,N_2}}\hspace{-0.2cm}<+\infty;
\end{equation}
\begin{equation}
{\mathop {v_j}\limits^{n_3,n_2}}(\ell)={\mathop {\varphi_j^\ell}\limits^{n_3,n_2}},\;\;
j=1,2,3,\;\;n_i=\overline{0,N_i},\;\;i=2,3,\;\;{\rm if}\;\;
{\mathop {I_\ell}\limits^{N_3,N_2}}\hspace{-0.2cm}<+\infty;
\end{equation}
$$
{\mathop {v_j}\limits^{n_3,n_2}}(x_1)=0(1),\;x_1\rightarrow 0_+,\;
j=1,2,3,\;n_i=\overline{0,N_i},\;i=2,3,\;{\rm if}\;
{\mathop {I_0}\limits^{N_3,N_2}}\hspace{-0.3cm}=+\infty;
\;\;(11)_0
$$
$$
{\mathop {v_j}\limits^{n_3,n_2}}(x_1)=0(1),\;x_1\rightarrow \ell_-,\;
j=1,2,3,\;n_i=\overline{0,N_i},\;\;i=2,3,\;{\rm if}\;
{\mathop {I_\ell}\limits^{N_3,N_2}}\hspace{-0.3cm}=+\infty;
\;\;(11)_\ell
$$
$$
{\mathop{lim}\limits_{x_1\rightarrow 0_+}}h_2^{n_2}h_3^{n_3} {\mathop 
{X_{1j}}\limits^{n_3,n_2}}(x_1)=
{\mathop{lim}\limits_{x_1\rightarrow 0_+}}\left\{\Lambda_jh_2^{2n_2+1}h_3^{2n_3+1}
{\mathop {v_{j,1}}\limits^{\hspace{-0.7ex}{n_3,n_2}}}-
\Lambda_j{\mathop\sum\limits^3_{k=2}}\;{\mathop\sum\limits^{N_k}_{s=n_k+1}}
h_2^{\delta_{k2}s+(\delta_{k3}+1)n_2+1}\cdot\right.
$$
$$
\cdot h_3^{\delta_{k3}s+(\delta_{k2}+1)n_3+1}h_k^{-1}\;{\mathop {b^k_s}\limits^{n_k}}\;\;
{\mathop{v_j}\limits^{\delta_{k2}n_3+\delta_{k3}s,\delta_{k2}s+\delta_{k3}n_2}}-
\eqno{(12)}
$$
$$
-\left[\delta_{j1}\lambda{\mathop\sum\limits^3_{i=2}}+\delta_{ij}(\delta_{j2}+
\delta_{j3})\mu\right]{\mathop\sum\limits^{N_i}_{s=n_i+1}}
h_2^{\delta_{i2}s+(\delta_{i3}+1)n_2+1}
h_3^{(\delta_{i2}+1)n_3+\delta_{i3}s+1}h_i^{-1}{\mathop b\limits^{n_i}}_{is}\cdot
$$
$
\left.
\cdot\hspace{1.3cm}{\mathop {v_{\delta_{j1}i+\delta_{j2}+\delta_{j3}}}
\limits^{ {\hspace{-1.5cm}
\delta_{i2}n_3+\delta_{i3}s,\delta_{i2}s+\delta_{i3}n_2} } }\right\}
={\mathop{\psi_j^0}\limits^{n_3,n_2}}\hspace{-0.2cm},\;
j=1,2,3,\;n_i=\overline{0,N_i}\;(i=2,3),\;{\rm if}\;
{\mathop{I_0}\limits^{N_3,N_2}}\hspace{-0.3cm}\le+\infty;
$
$$
{\mathop{lim}\limits_{x_1\rightarrow l_-}}h_2^{n_2}h_3^{n_3} {\mathop {X_{1j}}\limits^{n_3,n_2}}(x_1)=
{\mathop{\psi_j^\ell}\limits^{n_3,n_2}}\hspace{-0.2cm},\;
j=1,2,3,\;n_i=\overline{0,N_i}\;(i=2,3),\;{\rm if}\;
{\mathop{I_\ell}\limits^{N_3,N_2}}\hspace{-0.3cm}\le+\infty;\;(13)
$$
where ${\mathop{\varphi_j^0}\limits^{n_3,n_2}}$, ${\mathop{\varphi_j^\ell}\limits^{n_3,n_2}}$, 
${\mathop{\psi_j^0}\limits^{n_3,n_2}}$, ${\mathop{\psi_j^\ell}\limits^{n_3,n_2}}$
are given constants,
$$
{\mathop {b_s^k}\limits^{n_k}}:=\left(s+\frac{1}{2}\right)\left[
{\mathop {h_{k,1}}\limits^{\hspace{-2ex}(+)}}-
(-1)^{s+n_k}\;\;{\mathop {h_{k,1}}\limits^{\hspace{-2ex}(-)}}\right],\;\;
{\mathop {b_{is}}\limits^{n_i}}:=
\left(s+\frac{1}{2}\right)\left[1-
(-1)^{n_i+s}\right].
$$
\par
\par The boundary value problems (1),(9),(10); (1),(9),(13); (1),(10),(12);
(1),(9), $\rm(11)_\ell$; (1),(10),$\rm(11)_o$ are uniquely solvable. The problem 
(1),$\rm(11)_o$,$\rm(11)_\ell$ is solvable up to the rigid motion.
\par
If the bar has cusped end (i.e. at this end at least one of $h_i$, $i=2,3$, 
vanishes), and $N_2,N_3\rightarrow +\infty$, in limit case (which obviously
coincides with three-dimensional case) the boundary conditions (9), or (10), or
both disappear, and will be replaced by boundedness of $\mathop {v_j}\limits^{n_3,n_2}$,
$j=1,2,3$, $n_2,n_3=0,1,...,$ i.e. by boundedness of displacement vector in a 
neighbourhood of the corresponding end of the bar.
\par
If in a neighbourhood of a cusped end stresses are bounded then  at the above
end all moments of stress vector will be equal to zero. Non-zero stress vector moments
given at a cusped bar end mean that this end is loaded by concentrated surface force,
and concentrated moments of coresponding order.
\par
{\bf Remark.} The $(N_3,N_2)$ approximation is equavalent to the assumption
$$
u_j(x_1,x_2,x_3)\cong {\mathop\sum\limits^{N_2}_{n_2=0}}\hspace{0.3cm}
{\mathop\sum\limits^{N_3}_{n_3=0}}
\left(n_2+\frac{1}{2}\right)\left(n_3+\frac{1}{2}\right)
\frac{1}{2^{n_2+n_3}n_2!n_3!}\cdot
$$
$$
\cdot\frac{d^{n_2}[(x_2-{\overline h}_2)^2-h_2^2]^{n_2}}{dx_2^{n_2}}
\frac{d^{n_3}[(x_3-{\overline h}_3)^2-h_3^2]^{n_3}}{dx_3^{n_3}}
{\mathop{v_j}\limits^{n_3,n_2}}(x_1),\;\;j=1,2,3,
$$
where 
$$2{\overline h}_i(x_1):={\mathop h\limits^{(+)}}{\hspace{-0.15cm}_i}(x_1)+
{\mathop h\limits^{(-)}}{\hspace{-0.14cm}_i}(x_1),
\;\; i=2,3.$$

\vspace*{0.4cm}
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.3cm}
\end{center}
\par
[1] {Vekua, I.N. Shell Theory: General Methods of Construction. Pitman
Advanced Publishing Program, Boston-London-Melburne, 1985, pp. 1-287.}

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