du
dr
dv
dr
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
¼
du
dx
du
dy
dv
dx
dv
dy
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
dx
dr
dy
dr
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
or
dU
dr
¼ J Á n ¼ u— Á e n
where
dU
dr is the column vector containing relative displacement vector components
and n is the column vector of direction cosines. J is called the Jacobian matrix, which
is an operator that can transform any unit relative displacements for an infinitesimal
line AB from spatial coordinate system x, y to local (material) coordinate system r.
Assume that the length of AB ! 0; as a result, all derivatives are evaluated at
point B.
We can split the Jacobian matrix into two components. One component will
include deformations around point B and the other will include rigid body relative
displacements. It is assumed that rigid body relative displacements do not cause any
strain in the body. This splitting process can be accomplished in the following
manner:
∂u
∂x
∂u
∂y
∂v
∂x
∂v
∂y
2
6
6
4
3
7
7
5 ¼
∂u
∂x
1
2
∂u
∂y
þ
∂v
∂x
1
2
∂v
∂x
þ
∂u
∂y
∂v
∂y
2
6
6
6
4
3
7
7
7
5
þ
0
1
2
∂u
∂y
À
∂v
∂x
1
2
∂v
∂x
À
∂u
∂y
0
2
6
6
6
4
3
7
7
7
5
Earlier we defined small strain components as follows:
E xx ¼
∂u
∂x
, ε yy ¼
∂v
∂y
, γ xy ¼
∂u
∂y
þ
∂v
∂x
For convenience, we will define shear strain E xy as half the decrease γ xy in the right
angle to be able to define strain vectors as components of a second-order strain
tensor, ε:
34
2 Stress and Strain in Continuum