12
2 Investigation of Rods in the Elastic Range
Fig. 2.1 Schematic
representation of a
continuum rod
Table 2.1 Different formulations of the basic equations for a rod (X -axis along the principal rod
axis), with L 1 (. . . ) =
d(... )
dX
Specific formulation
General formulation [1]
Kinematics
ε X (X ) =
du X (X )
dX
ε X (X ) = L 1 (u X (X ))
Constitution
σ X (X ) = Eε X (X )
σ X (X ) = Cε X (X )
Equilibrium
dσ X (X )
dX
+
p X (X )
A
= 0
or
dN X (X )
dX
+ p X (X ) = 0
L T
1 (σ X (X )) + b = 0
PDE
d
dX
E(X )A(X )
du X
dX
+ p X (X ) = 0
L T
1 (E AL 1 (u X (X ))) + p X = 0
E(X )A(X )
du X
dX − N X (X ) = 0
E AL 1 (u X (X )) − N X = 0
Table 2.2 Different formulations of the partial differential equation for a rod (X -axis: right facing)
Configuration
Partial differential equation
EA
E A
d 2 u X
dX 2 = 0
E
)
) (
(
X
X A
d
dX
E(X )A(X )
du X
dX
= 0
p
)
(X
X
E A
d 2 u X
dX 2 = −p X (X )
k )
(X
E A
d 2 u X
dX 2 = k(X )u X