3.3 Analysis of Deformation
73
Inserting these relations into Eqs. (3.35) and (3.37) 1 yields
λ x =
∂x
∂X
2
+
∂y
∂X
2
1/2
λ y =
∂x
∂Y
2
+
∂y
∂Y
2
1/2
.
3.3.3 Deformation in 3D
Besides being limited to two dimensions, the equations derived above are restricted
to Cartesian coordinates. Next, we define deformation measures in 3D without
regard to any specific coordinate system. This is where tensor analysis enters the
picture.
If we try to extend our prior 2D strain analysis to 3D without using tensors, the
complex geometry soon becomes unwieldy. Tensors ease our pain considerably, but
this comes at a cost. While direct tensor notation maintains some of the physical
insight that may be lost with indicial notation, deriving the fundamental relations
from geometry like that depicted in Fig. 3.6 can be instructive. For this reason, we
first analyzed the 2D case to show how the various deformation measures are related
to the geometry. While the following 3D analysis may seem quite different, the
geometric interpretations are essentially the same.
Deformation Gradient Tensor
In 2D, the line elements dX and dY at a point in an undeformed body transform into
dx and dy in the deformed body. From the geometry of Fig. 3.6c, these elements can
be expressed in the form of the vectors
dX = dX e x ,
dY = dY e y
dx = dx e
x ,
dy = dy e
y .
During deformation, the vectors dX and dY become the vectors dx and dy,
respectively. Since a second-order tensor transforms a vector into another vector,
this mapping can be described by the relations dx = F ·dX and dy = F ·dY, where
F is the deformation gradient tensor.
There is nothing special about the elements dX and dY. Line elements of any
orientation located at a given point in a body undergo transformations defined by
73
Inserting these relations into Eqs. (3.35) and (3.37) 1 yields
λ x =
∂x
∂X
2
+
∂y
∂X
2
1/2
λ y =
∂x
∂Y
2
+
∂y
∂Y
2
1/2
.
3.3.3 Deformation in 3D
Besides being limited to two dimensions, the equations derived above are restricted
to Cartesian coordinates. Next, we define deformation measures in 3D without
regard to any specific coordinate system. This is where tensor analysis enters the
picture.
If we try to extend our prior 2D strain analysis to 3D without using tensors, the
complex geometry soon becomes unwieldy. Tensors ease our pain considerably, but
this comes at a cost. While direct tensor notation maintains some of the physical
insight that may be lost with indicial notation, deriving the fundamental relations
from geometry like that depicted in Fig. 3.6 can be instructive. For this reason, we
first analyzed the 2D case to show how the various deformation measures are related
to the geometry. While the following 3D analysis may seem quite different, the
geometric interpretations are essentially the same.
Deformation Gradient Tensor
In 2D, the line elements dX and dY at a point in an undeformed body transform into
dx and dy in the deformed body. From the geometry of Fig. 3.6c, these elements can
be expressed in the form of the vectors
dX = dX e x ,
dY = dY e y
dx = dx e
x ,
dy = dy e
y .
During deformation, the vectors dX and dY become the vectors dx and dy,
respectively. Since a second-order tensor transforms a vector into another vector,
this mapping can be described by the relations dx = F ·dX and dy = F ·dY, where
F is the deformation gradient tensor.
There is nothing special about the elements dX and dY. Line elements of any
orientation located at a given point in a body undergo transformations defined by