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\vspace*{0.2cm}
EXTRACT  FROM   LECTURES

\vspace*{0.4cm}
{VARIATIONAL FORMULATION FOR REFINED THEORIES, GENERALIZED
  HELLINGER-REISSNER VARIATIONAL PRINCIPLE}

\vspace*{0.3cm}
{\it T. Vashakmadze}

\vspace*{0.3cm}
{\it I. Javakhishvili Tbilisi State University}
\end{center}

\vspace*{0.3cm}
\par
The variational formulation for construction of refined theories
has (see Chien Wei-zang [2], Ciarlet [3], 
Hellinger [4], Lukasievicz [5],
Reissner [8], Washizu [11]) very important meaning. There are also given
historical notes, commentaries, comparisions of different theories with
richest bibliography. As follow  from these works, variational formulation
has also other convenient properties (see e.g. [11], part A), but
I call readers attention to following well-known destinguishing f\'{e}atures
such as the possibility of investigation and approximate solution of initial
boundary value problems in most general  Sobolev spaces.
\par
For simplicity and definiteness we consider linear theory of elasticity,
when $\Omega_h$ is isotropic homogeneous elastic media with constant \index{constant!- elastic}
thickness. But the consideration of more general cases (such as, anisotropy
and non-linearity) doesn't represent any difficulties.
\par
Let the boundary conditions \index{boundary!- conditions} \index{conditions!- boundary} be such:
\par
\begin{equation}
u\big|_{S_1}=\sigma_n\big|_{S_2}=0, \ \ \ \ S_1\cup S_2=S, \ \ \ 
  \sigma_3\big|_{S^\pm}=g^\pm .
\end{equation}
\par Here and below are used notations of Vashakmadze [9].
\par
Following Mikhlin [6], let us consider the following functional:
\par
\begin{equation} 
\Phi[u]=\int\limits_{\Omega_h}(\frac{1}{2}\sigma e+uf)dv-\int\limits_{ S^+}
u\sigma_n ds-\int\limits_{ S^-}u\sigma_n ds,
\end{equation}
where $n$ is outward normal.
\par
If we use the third equality of Hooke's law, we have
$$
\begin{array}{l} \displaystyle
\Phi[u]=\frac{1}{2}\int\limits_{\Omega_h}\left[((\lambda^*+2\mu)u_{\alpha,\alpha}+
\lambda^* u_{3-\alpha},{3-\alpha})u_{\alpha,\alpha}+\mu(u_{1,2}+u_{2,1})^2+
\right. \\
\\ [-0.2cm]
\\ \displaystyle
\left. +\sigma_{\alpha,3}(u_{\alpha,3}+u_{3,\alpha})+ 
\frac{\lambda}{\lambda+2\mu}\sigma_{33}u_{\alpha,\alpha}+\sigma_{33}u_{3,3}\right]dv+\int\limits_{\Omega_h}ufdv-
\int\limits_{S^\pm}u\sigma_nds.
\end{array}
$$
\par
Assuming (see e.g. Berdichevski [1]):
\begin{equation}
u_i(x_1,x_2,t)=v_i(x_1,x_2)+tw_i(x_1,x_2)+r_i[u],
\end{equation}
%\nopagebreak[3]
for $\Phi[u]$ immediately follows:
\begin{equation}
\begin{array}{l} \displaystyle
\Phi[u]=\frac{1}{2}\int\limits_D\left[(\lambda^*+2\mu)[2hv_{\alpha,\alpha}^2+
\frac{2h^3}{3}w_{\alpha,\alpha}]+\lambda^*\left[2hv_{\alpha,\alpha}
v_{3-\alpha,3-\alpha}+ \right.\right. \\
\\ [-0.2cm] 
\displaystyle
\left. +\frac{2h^3}{3}w_{\alpha,\alpha}w_{3-\alpha,3-\alpha}\right]+
\mu\left(2hv_{\alpha,3-\alpha}^2+\frac{2h^3}{3}w_{\alpha,3-\alpha}^2\right)+\\ 
\\ [-0.2cm]
\displaystyle
+(v_{3,\alpha}+w_\alpha)\int\limits_{-h}^h\sigma_{\alpha,3}dt+ 
w_{3,\alpha}\int\limits_{-h}^ht\sigma_{\alpha,3}dt+
\frac{\lambda}{\lambda+2\mu}\left(v_{\alpha,\alpha}\int\limits_{-h}^h\sigma_{3,3}dt+
\right.\\
\end{array}
\end{equation}
$$
\begin{array}{l} 
\\ [-0.2cm] \displaystyle
\left.\left.+w_{\alpha,\alpha}\int\limits_{-h}^ht\sigma_{3,3}dt\right)+
w_3\int\limits_{-h}^h\sigma_{3,3}dt\right]dw+
\int\limits_D\left[v_i\int\limits_{-h}^hf_idt+
\right.
\end{array}
$$
$$
\begin{array}{l} 
\\ [-0.2cm] \displaystyle
\left.+w_i\int\limits_{-h}^htf_idt\right]dw-
\int\limits_{S^\pm}(v_i\pm hw_i)\sigma_{3n} ds+R[u],
\end{array}
$$
\par
where $r_{ij}[u]$ is defined from (3) evidently and
$$
R[u]=\int\limits_{\Omega_h}(\frac{1}{2}\sigma_{ij}r_{ij}[u]+f_ir_i[u])dv-
\int\limits_{S^\pm} \sigma_{3n} r_i[u]ds.
$$
\par
Now let us consider the difference
\begin{equation}
\Phi[v,w]=\Phi[u]-R[u].
\end{equation}
\par
This expression will be called {\bf Hellinger-Reissner generalized
two-dimensional functional.} Euler-Lagrange conditions give
us the following two-dimensional 
\index{dimensional} 
system of partial differential equations:
\begin{equation}
\begin{array}{l} \displaystyle
2h[\mu \Delta v_\alpha+(\lambda^*+\mu)\partial_\alpha(v_{\alpha,\alpha}+
v_{3-\alpha,3-\alpha})]+\frac{\lambda}{2(\lambda+2\mu)}\int\limits_{-h}^{h}
\sigma_{33,\alpha}dt=  \\
\\ [-0.2cm] \displaystyle
=\int\limits_{-h}^{h}f_\alpha dt-(g_\alpha^+-g_\alpha^-),\;\;\;
\alpha=1,2,\\
\\ [-0.2cm]
\displaystyle
\frac{1}{2}\int\limits_{-h}^{h}\sigma_{\alpha 3,\alpha}dt=
\int\limits_{-h}^{h}f_3dt-(g_3^+-g_3^-),\\
\\ [-0.2cm] \displaystyle
\frac{2h^3}{3}[\mu \Delta w_\alpha+(\lambda^*+\mu)\partial_\alpha
(w_{\alpha,\alpha}+w_{3-\alpha,3-\alpha})]-
\frac{1}{2}\int\limits_{-h}^{h}\sigma_{\alpha 3}dt+ \\
\\ [-0.2cm] \displaystyle
+\frac{\lambda}{2(\lambda+2\mu)}\int\limits_{-h}^{h}t\sigma_{33,\alpha}dt=
\int\limits_{-h}^{h}tf_\alpha dt-h(g_\alpha^++g_\alpha^-),\;\;\;\alpha=1,2.
\\
\\ [-0.2cm] \displaystyle
\frac{2h^3}{3} \Delta w_3-\frac{1}{2}\int\limits_{-h}^{h}\sigma_{33}dt=
\int\limits_{-h}^{h}tf_3 dt-h(g_3^++g_3^-).
\end{array}
\end{equation}

\par
If now we use formulae of type (2.15) and (2.16) (see [9]), assume also $w_3\equiv 0$,
from (5) follows parametrical representation of refined theories full
identical to $(P_h)$ schemes (see [9]).
\par
Thus, the following conclusions are true.
\par
{\bf The functional $\Phi[u]-R[u]$ corresponds to the refined theories (in wide sense)   
and scheme $(P_h)$ for this functional represents Euler's
equations.}
\par
In particular, from (6$_{3-5}$) equations follow:
\begin{equation}
\begin{array}{l}\displaystyle
\frac{2h^3}{3}[\mu \Delta w_\alpha+(\lambda^*+\mu)grad div w_+]-
\frac{\mu h}{(1+2\gamma)}(w_\alpha+v_{3,\alpha})=
\\ [-0.2cm] \displaystyle
=\int\limits_{-h}^{h}tf_\alpha dt-h(g_\alpha^++g_\alpha^-)-
\frac{\lambda}{2(\lambda+2\mu)}\int\limits_{-h}^{h}t\sigma_{33,\alpha}dt=
F_\alpha, \\
\\ [-0.2cm] \displaystyle
\frac{\mu h}{(1+2\gamma}[\Delta v_3+w_{\alpha,\alpha}]=
\int\limits_{-h}^{h}f_3 dt-(g_3^++g_3^-)=F_3.
\end{array}
\end{equation}
\par
This system, of course, gives (see e.g. [8]) when $\gamma=
0,1$, Mindlin's theory [7] for $ \displaystyle
\gamma=\frac{12-\pi^2}{2\pi^2}$, for theory of [11] - Vekua [10]
$\gamma=0$ 
(compare with [11], ch.8, \S 8.8. or [5], ch.2,
section 2.1; here it's discussed physical aspects of such differences).
\par
The form of system of differential equations of refined theories for anisotropic case,
as follows from [9], with respect to averaged deflection \index{deflection} \index{averaged!- deflection}
$v_3$ and components
of normals rotation $w_\alpha$ is such:
$$
\begin{array}{l} \displaystyle
\frac{2h^3}{3}\left[(c_{1\alpha}\partial_1+c_{\alpha 6}\partial_2)w_{1,\alpha}+
(c_{\alpha 6}\partial_1+c_{26}\partial_2)w_{2,\alpha}+(2c_{16}\partial_1+
c_{66}\partial_2)w_{1,3-\alpha}+ \right.\\
\\ [-0.2cm] \displaystyle
\left. +(c_{66}\partial_1+2c_{26}\partial_2)w_{2,3-\alpha}\right]-
\frac{h}{(1+2\gamma)\delta}\left[a_{3+\alpha,3+\alpha}(w_\alpha+
v_{3,\alpha})- \right.\\
\\ [-0.2cm] \displaystyle
\left. -a_{3+\alpha,6-\alpha}(w_{3-\alpha}+v_{3,3-\alpha})\right]=F_\alpha,
\end{array}
$$
$$
\frac{h}{(1+2\gamma)\delta}\left[(a_{44}\partial_{11}-2a_{45}\partial_{12}+
a_{55}\partial_{22})v_3+(a_{3+\alpha,3+\alpha}\partial_\alpha- 
a_{3+\alpha,6-\alpha}\partial_{3-\alpha})w_\alpha \right]= F_3.
$$
where, $a,\;\;b$ are elasticity and rigidity numbers defined from Hooke's law 
(about coefficient $c$ see [9])
$ \delta =a_{44}a_{55}-a_{45}^2$. 

\vspace*{0.8cm}
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.3cm}
\end{center}
\par
[1] Berdichevski L., The Approach to the Dynamic Theory of 
            Thin Elastic Plates.  {\it MTT. N6, 1973}: 99-109. 
\par
[2] Chien Wei-zang., Variational Principles and Generalized
               Variational Principles for Nonlinear Elasticity with
              Finite Displacement. {\it Applied  Math. Mech.}, {\bf 9}(1), 1988. 
\par
[3] Ciarlet Ph., {\it Mathematical Elasticity:} V.II, Theory of Plates.
      Elsevier. Amst.-Lond.-N.-Y., 1997.
\par
[4], Hellinger E., Die Allgemeine Ansatze der Mechanik der Kontnua.
          Leipzig: {\it Encyclop\"{a}die der Mathematishen Wissenschaften,
          v.IV/4, 1914}: 602-694.
\par
[5] Lukasiewicz S.,{\it  Local Loads in Plates and Shells.} Warszawa,
    Leyden: Noordhoff Inter. Publishing,  1979.
\par
[6] Mikhlin S., {\it Variational Methods in Mathematical Physics.} M.: 
    Nauka, 1970.  
\par
[7] Mindlin R., Influence of Rotatory Inertia and Shear on Flexsual
              Motions of Izotropic Elastic Plate.{\it J.Appl.Mech.} {\bf 18}  1951: 31-38.
\par
[8] Reissner E., On a Variational Theorem of Elasticity. {\it J.
        Math. Phys.} {\bf 29},  1950: 90-95.
\par
[9] Vashakmadze T., {\it Some Problems of Mathematical Theory of Anisotropic 
elastic plates.} Tbilisi University Press, 1986.
\par
[10] Vekua I., {\it Shell Theory: General Methods of Construction.}
     Pitman Advance Publ. Prog. B.-L.-M, 1985.
\par
[11] Washizu K., {\it Variational Methods in Elasticity and
     Plasticity.} Pergamon Press. Oxford, 1970. 
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