Fundamentals - Preventive Biomechanics - page 35

Fig. 3.12 Movement of a
material body and trajectory
or path of a material point X

3.2.3.4 Displacement Field

During the transformation of a body from ICFG into CCFG every material point X
is displaced, which through the displacement vector u

u ¼ x X or u ð X ; t Þ¼ v ð X ; t Þ v ð X ; t 0 Þ ð 3 : 48 Þ

is represented in the form of (vectorial) differences of both position vectors x and
X. The entire displacement vectors of a body are termed the displacement field
(see Fig. 3.11 b).

3.2.3.5 Deformation and Displacement Gradient

Deformation Gradient. The spatial derivation of motion x ¼ v ð X ; t Þ plays a
central role in continuum mechanics. The deformation gradient F is generated
through formal differentiation of ( 3.47 ) with respect to X and constitutes the most
important ''building block'' of continuum mechanics. A more precise term may be
configuration gradient, since the change or gradient between configurations ICFG
and CCFG is described.

F ð X ; t Þ : ¼ ox

oX ¼ o v ð X ; t Þ

! x r: ð 3 : 49 Þ

oX

In ( 3.49 ), the formal partial differentiation o = oX is to be performed with the
help of the concrete gradient operation ðÞr with the (material) N ABLA -operator r
represented e.g. in an OBS. Its representation (note that in ( 3.49 ), the differenti-
ation is made with respect to the material position vector X and to the material
coordinates X i , respectively) is given by

r : ¼ o

oX i e i :

ð 3 : 50 Þ

Note: The N ABLA symbol arose from similar symbols used in the Quaterion-
analysis of William Rowan Hamilton. Peter Guthrie Tait introduced the symbol in