plane_cst-lst - Plane Problems Constitutive Equations...

1
Plane Problems: Constitutive Equations
■
Constitutive equations for a linearly elastic and isotropic
material in plane stress (i.e., σ
z
= τ
xz
= τ
yz
=0):
where the last column has the initial (thermal) strains which are
0 ,
xy0 0 0
= = = γ α ε ε T
y x
∆
• Rewriting in a compact form and solving for the stress vector,
where

2
Plane Problems: Approximate Strain-Displacement Relations
• From the above, by definition
, ,
x
v
y
u
y
v
x
u
xy y x
∆
∆
∆
∆
∆
∆
∆
∆
+ ≈ ≈ ≈ γ ε ε

3
Plane Problems: Strain-Displacement Relations
■
As the size of the rectangle goes to zero, in the limit,

4
Plane Problems: Displacement Field Interpolated
■
Interpolating the displacement field, u(x,y) and v(x,y), in the
plane finite element from nodal displacements,
where entries of matrix
N
are the shape (interpolation) functions N
i
.
■
From the previous two equations,
where
B
is the strain-displacement matrix .

5
Stiffness Matrix and strain energy
■
Strain energy density of an elastic material (energy/volume)
ε ε