Biomedical Engineering Reference
In-Depth Information
Young modulus and the Poisson ratio. In this chapter the rigid solid motion and
deformation are described, with emphasis on rotation, which plays an important
role in nonlinear continuum mechanics. Also the concepts of strain and stress in
nonlinear mechanics are introduced. The equilibrium and the constitutive equa-
tions are presented afterwards.
2.1.1 Kinematics
The general motion of a deformable body is represented in Fig.
2.1
. The body, in
the initial position t ¼ 0, is considered to be an assemblage of material particles,
labelled by the coordinates X, with respect to the Cartesian basis e. The current
position of a particle is defined at time t by the coordinates x.
The motion can be mathematically described by a mapping function / between
initial and current particle positions,
x ¼ /
ð
X
;
t
Þ
ð
2
:
1
Þ
It is considered the material description, the Lagragian description, since the
variation of the solid deformation is described with respect to the initial coordi-
nates X, at time t.
2.1.1.1 Deformation Gradient
The deformation gradient F, is a key quantity in finite deformation analysis, since
it is involved in all equations relating quantities before deformation (initial con-
figuration) with the correspondent quantities after the finite deformation (current
configuration). The deformation gradient tensor F can be defined as,
F ¼
o
/
oX
¼
r
/
ð
2
:
2
Þ
Alternatively to Eq. (
2.1
) the motion can be expressed by,
x ¼ x
ð
X
;
t
Þ
ð
2
:
3
Þ
which permits the deformation gradient to be written as,
F ¼
o
x
oX
ð
2
:
4
Þ
