Page 149 - The Combined Finite-Discrete Element Method

132 DEFORMABILITY OF DISCRETE ELEMENTS

are called material points, and bounded subregions of the body are called parts. Body
deforms via mapping
x = f(p) (4.2)

where f is one to one smooth mapping which maps B onto a closed region E,and
which satisfies
det ∇f(p)> 0 (4.3)

for any material point p. This condition simply states that no part with nonzero volume
can map into zero volume space, i.e. parts of the body occupy space before and after
deformation. The volume of such space may defer, but is always greater than zero.

4.2 DEFORMATION GRADIENT

Deformation can also be written as

x = f(p) = p + u(p) (4.4)

where u(p) is called displacement. Mapping

F(p) =∇f(p) = I +∇u (4.5)

describes change in deformation in the vicinity of each material point, and is referred to
as the deformation gradient.

4.2.1 Frames of reference
To describe the deformation of a particular discrete element in the vicinity of the material
point p, four reference frames are used in this chapter (Figure 4.2):

• Global frame: this reference frame is the frame defined by a triad of orthogonal unit
vectors that coincide with the axes of Cartesian coordinate system

(i, j, k) (4.6)

• Local frame: this reference frame is the frame associated with the initial position of
a particular discrete element. It is therefore fixed in space and does not move with
the discrete element. Very often this frame is made to coincide with the major axes of
inertia of a particular discrete element. This frame is defined by a triad of orthogonal
unit vectors
(i, j, k) (4.7)
• Deformed local frame: this frame is fixed to the material point p of the discrete element
and moves with that point. The base vectors of this frame also follow the deformation
in the vicinity of the point p. The origin of this frame thus coincides at all times with
the point x = f(p), while the direction of triad vectors and magnitude of this vectors