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Kinematics of Deformation and Motion

4.1

Particles, Conﬁgurations, Deformation, and Motion

In continuum mechanics we consider

in the form of solids,

liquids, and gases. Let us begin by describing the model we use to represent

such bodies. For this purpose we deﬁne a material body

material bodies

as the set of

which can be put into a one-

to-one correspondence with the points of a regular region of physical space.

Note that whereas a particle of classical mechanics has an assigned mass, a

continuum particle is essentially a material point for which a density is

deﬁned.

The speciﬁcation of the position of all of the particles of

elements

, called

particles

material points,

with respect to

a ﬁxed origin at some instant of time is said to deﬁne the

of the

body at that instant. Mathematically, this is expressed by the mapping

conﬁguration

(

)

(4.1-1)

in which the vector function

assigns the position

relative to some origin

of each particle

of the body. Assume that this mapping is uniquely invert-

ible and differentiable as many times as required; in general, two or three

times will sufﬁce. The inverse is written

–1

(

)

(4.1-2)

and identiﬁes the particle

located at position

A change in conﬁguration is the result of a

displacement

of the body. For

example, a

is one consisting of a simultaneous trans-

lation and rotation which produces a new conﬁguration but causes no

changes in the size or shape of the body, only changes in its position and/or

orientation. On the other hand, an arbitrary displacement will usually

include both a rigid-body displacement and a

rigid-body displacement

deformation

which results in a

change in size, or shape, or possibly both.

is a continuous time sequence of displacements that

carries the set of particles

motion

of body

into various conﬁgurations in a stationary space.

Such a motion may be expressed by the equation

(

)

(4.1-3)

which gives the position

for each particle

for all times

, where

ranges

from –

. As with conﬁguration mappings, we assume the motion

function in Eq 4.1-3 is uniquely invertible and differentiable, so that we may

write the inverse

∞

to +

∞

–1

(

)

(4.1-4)

which identiﬁes the particle

We give special meaning to certain conﬁgurations of the body. In particular,

we single out a

located at position

at time

from which all displacements are reck-

oned. For the purpose it serves, the reference conﬁguration need not be one

the body ever actually occupies. Often, however, the

reference conﬁguration

initial conﬁguration,

that

is, the one which the body occupies at time

= 0, is chosen as the reference

conﬁguration, and the ensuing deformations and motions related to it. The

current conﬁguration

In developing the concepts of strain, we conﬁne attention to two speciﬁc

conﬁgurations without any regard for the sequence by which the second

conﬁguration is reached from the ﬁrst. It is customary to call the ﬁrst (ref-

erence) state the

is that one which the body occupies at the current time

undeformed conﬁguration,

and the second state the

deformed

conﬁguration

. Additionally, time is not a factor in deriving the various strain

tensors, so that both conﬁgurations are considered independent of time.

In ﬂuid mechanics, the idea of speciﬁc conﬁgurations has very little mean-

ing since ﬂuids do not possess a natural geometry, and because of this it is

the

velocity ﬁeld

of a ﬂuid that assumes the fundamental kinematic role.

4.2

Material and Spatial Coordinates

Consider now the reference conﬁguration prescribed by some mapping func-

tion

such that the position vector

of particle

relative to the axes

of Figure 4.1 is given by

(

)

(4.2-1)

In this case we may express

in terms of the base vectors shown in the

ﬁgure by the equation

X A ˆ

(4.2-2)

FIGURE 4.1

Position of typical particle in reference conﬁguration

and current conﬁguration

and we call the components

the

material coordinates,

or sometimes the

referential coordinates,

. Upper-case letters which are used as

subscripts on material coordinates, or on any quantity expressed in terms of

material coordinates, obey all the rules of indicial notation. It is customary

to designate the material coordinates (that is, the position vector

of the particle

) of each

particle as the

of that particle, so that in all subsequent conﬁg-

urations every particle can be identiﬁed by the position

name

label

it occupied in the

reference conﬁguration. As usual, we assume an inverse mapping

(

)

(4.2-3)

–1

so that upon substitution of Eq 4.2-3 into Eq 4.1-3 we obtain

[

–1

(

] =

(

)

(4.2-4)

which deﬁnes the motion of the body in physical space relative to the refer-

ence conﬁguration prescribed by the mapping function

Notice that Eq 4.2-4 maps the particle at

in the reference conﬁguration

onto the point

in the current conﬁguration at time

as indicated in

Figure 4.1 . With respect to the usual Cartesian axes

the current posi-

tion vector is

x ii

(4.2-5)

of the particle.

Although it is not necessary to superpose the material and spatial coordinate

axes as we have done in Figure 4.1 , it is convenient to do so, and there are

no serious restrictions for this practice in the derivations which follow. We

where the components

are called the

spatial coordinates

emphasize, however, that the material coordinates are used in conjunction

with the reference conﬁguration only, and the spatial coordinates serve for

all other conﬁgurations. As already remarked, the material coordinates are

therefore time independent.

We may express Eq 4.2-4 in either a Cartesian component or a coordinate-

free notation by the equivalent equations

(

)or

(

)

(4.2-6)

It is common practice in continuum mechanics to write these equations in

the alternative forms

(

)or

(

)

(4.2-7)

with the understanding that the symbol

(or

) on the right-hand side of

the equation represents the

function

whose arguments are

and

, while the

same symbol on the left-hand side represents the

of the function, that

is, a point in space. We shall use this notation frequently in the text that

follows.

Notice that as

value

ranges over its assigned values corresponding to the

reference conﬁguration, while

simultaneously varies over some designated

interval of time, the vector function

occupied

at any instant of time for every particle of the body. At a speciﬁc time, say

gives the spatial position

, the function

deﬁnes the conﬁguration

(

t 1 )

(4.2-8)

In particular, at t = 0, Eq 4.2-6 deﬁnes the initial conﬁguration which is often

adopted as the reference conﬁguration, and this results in the initial spatial

coordinates being identical in value with the material coordinates, so that in

this case

x =

( X , 0) = X

(4.2-9)

at time t = 0.

If we focus attention on a speciﬁc particle X P having the material position

vector X P , Eq 4.2-6 takes the form

x P =

( X P , t )

(4.2-10)

and describes the path or trajectory of that particle as a function of time. The

velocity v P of the particle along its path is deﬁned as the time rate of change

of position, or

∂

(4.2-11)

where the notation in the last form indicates that the variable X is held

constant in taking the partial derivative of

. Also, as is standard practice,

the super-positioned dot has been introduced to denote differentiation with

respect to time. In an obvious generalization, we may deﬁne the velocity ﬁeld

of the total body as the derivative

vx x

∂

(,)

∂

(,)

== =

(4.2-12)

∂

Similarly, the acceleration ﬁeld is given by

avx x

∂

(,)

˙˙

=== =

(4.2-13)

∂

and the acceleration of any particular particle determined by substituting its

material coordinates into Eq 4.2-13.

Of course, the individual particles of a body cannot execute arbitrary

motions independent of one another. In particular, no two particles can

occupy the same location in space at a given time (the axiom of impenetra-

bility), and furthermore, in the smooth motions we consider here, any two

particles arbitrarily close in the reference conﬁguration remain arbitrarily

close in all other conﬁgurations. For these reasons, the function

in Eq 4.2-6

must be single-valued and continuous, and must possess continuous deriv-

atives with respect to space and time to whatever order is required, usually

to the second or third. Moreover, we require the inverse function

–1 in the

equation

X =

–1 ( x , t )

(4.2-14)

to be endowed with the same properties as

. Conceptually, Eq 4.2-14 allows

us to “reverse” the motion and trace backwards to discover where the par-

ticle, now at x , was located in the reference conﬁguration. The mathematical

condition that guarantees the existence of such an inverse function is the

non-vanishing of the Jacobian determinant J . That is, for the equation

∂χ

∂

≠

(4.2-15)

to be valid. This determinant may also be written as

= ∂

∂

(4.2-16)

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