e
ε
¼ 1 þ e
ð
Þ
or e ¼ e
ε
À 1
From Taylor series, we can write the following relation:
e
x
¼
X 1
k¼0
x
k
k!
¼ 1 þ x þ
x
2
2!
þ
x
3
3!
þ . . . . . .
As a result, we can write the following relation between true strain and engineering strain:
ε þ
E
2
2!
þ
ε
3
3!
þ . . . :: ¼ e
In three dimensions for small strain, this relationship can be generalized as
follows:
dE ij ¼ D ij dt
where D ij is the rate-of-deformation tensor and dt is the time increment.
It is important to point out that increments of natural strain, dE ij , are components
of a Cartesian tensor; as a result, the transformation formulas and principal axis
theory all apply. The quantities E ij defined by integration are not components of a
Cartesian tensor, because during the deformation, principal axes continuously rotate
at each increment (Malvern 1969).
2.10 Finite Strain and Deformation
There are many definitions of finite (large) strain. They can be categorized into two
classes.
1. Defining strain with respect to undeformed original configuration and geometry.
This approach is called Lagrangian formulation.
2. Defining strain with respect to deformed configuration and geometry. This
approach is called Eulerian formulation.
Large (finite) strain formulation is the easiest to define in terms of a deformation
gradient tensor. However, deformation gradient tensor includes the strain tensor and
the rotation tensor. As a result, this can make it tricky to be used in material
modeling. On the other hand, strain tensor only includes strains, where x, y is the
global Cartesian coordinate (referential description) system and r is the local coordinate (material description) system (Figs. 2.30 and 2.31).
Deformation equation for point B can be defined in local (material) coordinate
system and can be mapped onto referential coordinate system. For example,
46
2 Stress and Strain in Continuum
ε
¼ 1 þ e
ð
Þ
or e ¼ e
ε
À 1
From Taylor series, we can write the following relation:
e
x
¼
X 1
k¼0
x
k
k!
¼ 1 þ x þ
x
2
2!
þ
x
3
3!
þ . . . . . .
As a result, we can write the following relation between true strain and engineering strain:
ε þ
E
2
2!
þ
ε
3
3!
þ . . . :: ¼ e
In three dimensions for small strain, this relationship can be generalized as
follows:
dE ij ¼ D ij dt
where D ij is the rate-of-deformation tensor and dt is the time increment.
It is important to point out that increments of natural strain, dE ij , are components
of a Cartesian tensor; as a result, the transformation formulas and principal axis
theory all apply. The quantities E ij defined by integration are not components of a
Cartesian tensor, because during the deformation, principal axes continuously rotate
at each increment (Malvern 1969).
2.10 Finite Strain and Deformation
There are many definitions of finite (large) strain. They can be categorized into two
classes.
1. Defining strain with respect to undeformed original configuration and geometry.
This approach is called Lagrangian formulation.
2. Defining strain with respect to deformed configuration and geometry. This
approach is called Eulerian formulation.
Large (finite) strain formulation is the easiest to define in terms of a deformation
gradient tensor. However, deformation gradient tensor includes the strain tensor and
the rotation tensor. As a result, this can make it tricky to be used in material
modeling. On the other hand, strain tensor only includes strains, where x, y is the
global Cartesian coordinate (referential description) system and r is the local coordinate (material description) system (Figs. 2.30 and 2.31).
Deformation equation for point B can be defined in local (material) coordinate
system and can be mapped onto referential coordinate system. For example,
46
2 Stress and Strain in Continuum