$3\mathrm{.}7($Plane Elasticity$)$ The governing equations of plane $($i$.$e$.$ two$-$dimensional$)$ elasticity

problems are summarized below.

Equilibrium of forces

$\frac{\text{d}\text{e}\text{l}{}_{}}{\text{d}\text{e}\text{l}\text{x}}+\frac{\text{d}\text{e}\text{l}{}_{xy}}{\text{d}\text{e}\text{l}\text{y}}+f_x=0$

PROBLEMS

$\frac{\text{d}\text{e}\text{l}{}_{xy}}{\text{d}\text{e}\text{l}\text{x}}+\frac{\text{d}\text{e}\text{l}{}_{yy}}{\text{d}\text{e}\text{l}\text{y}}+f_y=0$

where $({}_{},{}_{yy},{}_{xy})$ are the stress components and $f_x$ and $f_y$ are the components

of the body force vector $($measured per unit volume$)$ along the $x-$ and $y-$directions,

respectively.

Stress$-$displacement $($or constitutive$)$ relations

$\{[{}_{}],[{}_{yy}],[{}_{xy}]\}=\left[\begin{array}{ccc}{c}_{11}& c_{12}& 0\\ c_{12}& c_{22}& 0\\ 0& 0& c_{66}\end{array}\right]\{[\frac{del{u}_x}{\text{d}\text{e}\text{l}\text{x}}],[\frac{del{u}_y}{\text{d}\text{e}\text{l}\text{y}}],[\frac{del{u}_x}{\text{d}\text{e}\text{l}\text{y}}+\frac{del{u}_y}{\text{d}\text{e}\text{l}\text{x}}]]$

where $c_{ij}(c_{ji}=c_{ij})$ are the elasticity $($material$)$ constants for an orthotropic medium

with the material principal directions $(x_1,x_2,x_3)$ coinciding with the coordinate axes

$(x,y,z)$ used to describe the problem and $(u_x,u_y)$ are the displacements. The $c_{ij}$ can

be expressed in terms of the engineering constants $(E_1,E_2,$

$u{?}_{(}()()12),G_{12})$ for an orthotropic

material. For plane stress problems the elastic constants are given by

$c_{11}=\frac{E_1}{(1-)}$

$u{?}_{(}()()12)$

$u{?}_{(}()()21),c_{22}=\frac{E_2}{(1-)}$

$u{?}_{(}()()12)$

$u{?}_{(}()()21),c_{12}=$

$u{?}_{(}()()12)c_{22}=$

$u{?}_{(}()()21)c_{11},c_{66}=G_{12}$

There are four independent material constants for plane stress case: $E_1,E_2,$

$u{?}_{(}()()12),$ and

$G_{12}.$ For isotropic case we have

$c_{11}=c_{22}=\frac{E}{(1-)}$

$u{?}^{(}()()2),c_{12}=\frac{?}{(1-)}$

$u{?}^{(}()()2)$

$uE,c_{66}=G$

For plane strain problems they are given by

$c_{11}=(\frac{1-}{(1-)}$

$u{?}_{(}()()23)$

$u{?}_{(}()()32)-$

$u{?}_{(}()()13)$

$u{?}_{(}()()31)-$

$u{?}_{(}()()12)$

$u{?}_{(}()()21)-2$

$u{?}_{(}()()12)$

$u{?}_{(}()()23)$

$u{?}_{(}()()31)$

$u{?}_{(}()()23)$

$u{?}_{(}()()32)-$

$u{?}_{(}()()13)$

$u{?}_{(}()()31)-$

$u{?}_{(}()()12)$

$u{?}_{(}()()21)-2$

$u{?}_{(}()()12)$

$u{?}_{(}()()23)$

$u{?}_{(}()()31))$

$u{?}_{(}()()23)$

$u{?}_{(}()()32)$

Thus, there are seven independent material constants for the plane strain case: $E_1,E_2,$

$E_3($to determine

$u{?}_{(}()()31)$ and

$u{?}_{(}()()32))$

$u{?}_{(}()()12),$

$u{?}_{(}()()13),$

$u{?}_{(}()()23),$ and $G_{12}.$ For isotropic case, the constitutive

equations reduce to

$c_{11}=c_{22}=(\frac{1-}{(1+)}$

$u(1-2$

$u)$

$u)(1-2$

$u)$

$u$

Boundary conditions

${}_{}{n}_x+{}_{xy}{n}_y=$hat$(t{)}_x,{}_{xy}{n}_x+{}_{yy}{n}_y=$hat$(t{)}_y$

where $(n_x,n_y)$ denote the components $($or direction cosines$)$ of the unit normal vector

on the boundary $$;hat$(t{)}_x$ and hat$(t{)}_y$ denote the components of the specified traction vector,

and hat$(u{)}_x$ and hat$(u{)}_y$ are the components of specified displacement vector. Only one element

of each pair, $(u_x,t_x)$ and $(u_y,t_y),$ may be specified at a boundary point. Eliminate

the stresses from Eqs. $(1)$ and $(2)$ by substituting the stress$-$displacement relations $(3).$

Develop weak$-$form Galerkin finite element model of the resulting equations.