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134 DEFORMABILITY OF DISCRETE ELEMENTS
of this vector changes with the deformation of the discrete element. The orientation of
the initial frame is defined by a triad of non-unit vectors that are in general not parallel
to the respective axes of the Cartesian coordinate system. In addition, these vectors
are in general not orthogonal to each other. Initially, this frame coincides with the
initial frame. As the body deforms, this frame begins to differ from the initial frame.
As this frame is obtained through deformation of the initial frame, it is in further text
referred to as a deformed initial frame. A deformed initial frame is defined by a triad
of non-orthogonal non-unit vectors:
˘ ˘ ˘
(i, j, k) (4.10)
The global, local and initial frames are inertial frames, while the deformed local frame
and deformed initial frame are non-inertial frames.
The relationship between a local and deformed local frame can be obtained using the
deformation gradient. In the global frame, the deformation gradient tensor can be written
using the following matrix:
∂u ∂u ∂u
1 +
∂x ∂y ∂z
x + u(x, y, z)
∂v ∂v ∂v
F =∇f =∇ y + v(x, y, z) = 1 + (4.11)
∂x ∂y ∂z
z + w(x, y, z)
∂w ∂w ∂w
1 +
∂x ∂y ∂z
The same deformation gradient tensor can be written in the local frame using the follow-
ing matrix:
∂u ∂u ∂u
1 +
∂x ∂y ∂z
x + u(x, y, z)
∂v ∂v ∂v
F =∇f =∇ y + v(x, y, z) = 1 + (4.12)
∂x ∂y ∂z
z + w(x, y, z)
∂w ∂w ∂w
1 +
∂x ∂y ∂z
Triad vectors of the local frame are parallel to the axes of the corresponding Cartesian
coordinate system, and can be expressed in a matrix form:
1
i = 1i + 0j + 0k = 0 (4.13)
0
0
1
j = 0i + 1j + 0k = (4.14)
0
0
0
k = 0i + 0j + 1k = (4.15)
1
