Basic Continuum Kinematics - Continuum Mechanics of Anisotropic Materials - page 26

If the motion of one particle X of an object can be represented, then the motion
of all the particles of the object, X O (0), can be represented. A second coordinate

system with axes x i , i 1, 2, 3, is introduced to

represent the present position of the object O ( t ); this also represents the present
positions of the particles. The triplet ( x 1 , x 2 , x 3 ), denoted in the shorthand direct
notation by x , represents the place at time t of the particle X . The motion of the
particle X is then given by

¼ 1, 2, 3, and base vectors e i , i ¼

x 1 ¼ w 1 ð X I ; X II ; X III ; t Þ; x 2 ¼ w 2 ð X I ; X II ; X III ; t Þ; x 3 ¼ w 3 ¼ð X I ; X II ; X III ; t (2.1)

which is a set of three scalar-valued functions whose arguments are the particle X
and time t and whose values are the components of the place x at time t of the
particle X . Since X can be any particle in the object, X

O (0), the motion ( 2.1 )

describes the motion of the entire object x O ( t ) and ( 2.1 ) is thus referred to as the

motion of the object O . In the direct shorthand or vector notation ( 2.1 ) is written

x ¼ wð X ; t Þ for all X O ð 0 Þ: (2.2)

This is called the material description of motion because the material particles X
are the independent variables. Generally the form of the motion, ( 2.1 )or( 2.2 ), is
unknown in a problem, and the prime kinematic assumption for all continuum
models is that such a description of the motion of an object is possible.
However, if the motion is known, then all the kinematic variables of interest
concerning the motion of the object can be calculated from it; this includes
velocities, accelerations, displacements, strains, rates of deformation, etc. The
present, past, and future configurations of the object are all known. The philosophi-
cal concept embedded in the representation ( 2.2 ) of a motion is that of determinism.
The determinism of the eighteenth century in physical theory was modified by
humbler notions of “uncertainty” in the nineteenth century and by the discovery of
extreme sensitivity to starting or initial conditions known by the misnomer “chaos”
in the twentieth century. The quote of the Marquis Pierre-Simon de Laplace
(1759-1827) at the beginning of the chapter captures the idea of determinism
underlying the representation ( 2.2 ).
A translational rigid object motion is a special case of ( 2.2 ) represented by,

x ¼ X þ h ð t Þ for all X O ð 0 Þ; (2.3)

where h ( t ) is a time-dependent vector. A rotational rigid object motion is a special
case of ( 2.2 ) represented by

T

x ¼ Q ð t Þ X ; Q ð t Þ Q ð t Þ ¼ 1 for all X O ð 0 Þ; (2.4)

where Q ( t ) is a time-dependent orthogonal transformation. It follows that a general
rigid object motion is a special case of ( 2.2 ) represented by