1 Particles, Configurations, Deformation, and Motion

A motion of body B is a continuous time sequence of displacements that

carries the set of particles X into various configurations in a stationary space.

Such a motion may be expressed by the equation

x = κ(X,t)



(4.1-3)



which gives the position x for each particle X for all times t, where t ranges

from – ∞ to + ∞. 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

X = κ–1(x,t)



(4.1-4)



which identifies the particle X located at position x at time t.

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

we single out a reference configuration from which all displacements are reckoned. For the purpose it serves, the reference configuration need not be one

the body ever actually occupies. Often, however, the initial configuration, that

is, the one which the body occupies at time t = 0, is chosen as the reference

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

current configuration is that one which the body occupies at the current time t.

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 (reference) state the 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 meaning 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 function Φ such that the position vector X of particle X relative to the axes

OX1X2X3 of Figure 4.1 is given by

X = Φ(X)



(4.2-1)



In this case we may express X in terms of the base vectors shown in the

figure by the equation

ˆ

X = XA I A



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(4.2-2)



FIGURE 4.1

Position of typical particle in reference configuration XA and current configuration xi.



and we call the components XA the material coordinates, or sometimes the

referential coordinates, of the particle X. 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 X) of each

particle as the name or label of that particle, so that in all subsequent configurations every particle can be identified by the position X it occupied in the

reference configuration. As usual, we assume an inverse mapping

X = Φ–1(X)



(4.2-3)



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

x = κ [Φ–1(X),t] = χ(X,t)



(4.2-4)



which defines the motion of the body in physical space relative to the reference configuration prescribed by the mapping function Φ.

Notice that Eq 4.2-4 maps the particle at X in the reference configuration

onto the point x in the current configuration at time t as indicated in

Figure 4.1. With respect to the usual Cartesian axes Ox1x2x3 the current position vector is

ˆ

x = xiei



(4.2-5)



where the components xi are called the spatial coordinates 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

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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 coordinatefree notation by the equivalent equations

xi = χi(XA, t)



or x = χ(X, t)



(4.2-6)



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

the alternative forms

xi = xi (XA, t)



or x = x(X, t)



(4.2-7)



with the understanding that the symbol xi (or x) on the right-hand side of

the equation represents the function whose arguments are X and t, while the

same symbol on the left-hand side represents the value of the function, that

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

follows.

Notice that as X ranges over its assigned values corresponding to the

reference configuration, while t simultaneously varies over some designated

interval of time, the vector function χ gives the spatial position x occupied

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

at t = t1, the function χ defines the configuration

x1 = χ(X, t1)



(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 = χ(X, 0) = X



(4.2-9)



at time t = 0.

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

vector XP, Eq 4.2-6 takes the form

xP = χ(XP, t)



(4.2-10)



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

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

of position, or

vP =

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dx P

 ∂χ 

˙

= χP =  

 ∂t  X= XP

dt



(4.2-11)



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

constant in taking the partial derivative of χ. 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

˙

v=x=



d x ∂χ ( X , t) ∂ x( X , t)

=

=

∂t

∂t

dt



(4.2-12)



Similarly, the acceleration field is given by

˙ ˙˙

a=v=x=



d 2x ∂χ 2 ( X , t)

=

∂ t2

dt 2



(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 impenetrability), 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 χ in Eq 4.2-6

must be single-valued and continuous, and must possess continuous derivatives with respect to space and time to whatever order is required, usually

to the second or third. Moreover, we require the inverse function χ –1 in the

equation

X = χ –1(x, t)



(4.2-14)



to be endowed with the same properties as χ. Conceptually, Eq 4.2-14 allows

us to “reverse” the motion and trace backwards to discover where the particle, 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

J=



∂χ i

≠0

∂X A



(4.2-15)



to be valid. This determinant may also be written as

J=



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∂ xi

∂ XA



(4.2-16)



Example 4.2-1

Let the motion of a body be given by Eq 4.2-6 in component form as

x1 = X1 + t2X2

x2 = X2 + t2X1

x3 = X3

Determine

(a) the path of the particle originally at X = (1,2,1) and

(b) the velocity and acceleration components of the same particle when t =

2 s.



Solution

(a) For the particle X = (1,2,1) the motion equations are

x1 = 1 + 2t2;



x2 = 2 + t2;



x3 = 1



which upon elimination of the variable t gives x1 – 2x2 = –3 as well as x3 = 1

so that the particle under consideration moves on a straight line path in the

plane x3 = 1.

(b) By Eqs 4.2-12 and 4.2-13 the velocity and acceleration fields are given in

component form, respectively, by

v1 = 2tX2

v2 = 2tX1

v3 = 0



and



a1 = 2X2

a2 = 2X1

a3 = 0



so that for the particle X = (1,2,1) at t = 2

v1 = 8

v2 = 4

v3 = 0



and



a1 = 4

a2 = 2

a3 = 0



Example 4.2-2

Invert the motion equations of Example 4.2-1 to obtain X = χ –1(x, t) and

determine the velocity and acceleration components of the particle at x (1,0,1)

when t = 2 s.



Solution

By inverting the motion equations directly we obtain

X1 =

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x1 − t 2 x2

;

1− t4



X2 =



x2 − t 2 x1

; X 3 = x3

1− t4



which upon substitution into the velocity and acceleration expressions of

Example 4.2-1 yields



v1 =



v2 =



(



2t x2 − t 2 x1



(



1− t4



2t x1 − t 2 x2

1− t



4



)

)



a1 =



and



v3 = 0



a2 =



(



2 x2 − t 2 x1



(



1− t4



2 x1 − t 2 x2

1− t



4



)

)



a3 = 0



For the particle at x = (1,0,1) when t = 2 s

v1 =



16

15



v2 = −



a1 =



4

15



and



v3 = 0



4.3



8

15



a2 = −



2

15



a3 = 0



Lagrangian and Eulerian Descriptions



If a physical property of the body B such as its density ρ, or a kinematic

property of its motion such as the velocity v, is expressed in terms of the

material coordinates X, and the time t, we say that property is given by the

referential or material description. When the referential configuration is taken

as the actual configuration at time t = 0, this description is usually called the

Lagrangian description. Thus, the equations



ρ = ρ (XA,t) or ρ = ρ (X,t)



(4.3-1a)



vi = vi (XA,t)



(4.3-1b)



and

or v = v(X,t)



chronicle a time history of these properties for each particle of the body. In

contrast, if the properties ρ and v are given as functions of the spatial

coordinates x and time t, we say that those properties are expressed by a

spatial description, or as it is sometimes called, by the Eulerian description. In

view of Eq 4.2-14 it is clear that Eq 4.3-1 may be converted to express the

same properties in the spatial description. Accordingly, we write



ρ = ρ (X,t) = ρ [χ –1 (x,t),t] = ρ*(x,t)

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(4.3-2a)



and

v = v(X,t) = v[χ –1(x,t),t] = v*(x,t)



(4.3-2b)



where the asterisk is appended solely for the purpose of emphasizing that

different functional forms result from the switch in variables. We note that

in the material description, attention is focused on what is happening to the

individual particles during the motion, whereas in the spatial description

the emphasis is directed to the events taking place at specific points in space.



Example 4.3-1

Let the motion equations be given in component form by the Lagrangian

description

x1 = X1et + X3 (et – 1)

x2 = X2 + X3 (et – e –t)

x3 = X 3

Determine the Eulerian description of this motion.



Solution

Notice first that for the given motion x1 = X1, x2 = X2 and x3 = X3 at t = 0, so

that the initial configuration has been taken as the reference configuration.

Because of the simplicity of these Lagrangian equations of the motion, we

may substitute x3 for X3 into the first two equations and solve these directly

to obtain the inverse equations

X1 = x1e –t + x3 (e –t – 1)

X 2 = x 2 + x 3 (e –t – e t )

X3= x 3



Example 4.3-2

For the motion of Example 4.3-1 determine the velocity and acceleration

fields, and express these in both Lagrangian and Eulerian forms.



Solution

From the given motion equations and the velocity definition Eq 4.2-12 we

obtain the Lagrangian velocity components,



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v 1 = X 1e t + X 3e t

v2 = X3 (et + e –t)

v3 = 0

and from Eq 4.2-13 the acceleration components

a1 = (X1 + X3)et

a2 = X3 (et – e –t)

a3 = 0

Therefore, by introducing the inverse mapping equations determined in

Example 4.3-1 we obtain the velocity and acceleration equations in Eulerian

form,

v1 = x1 + x3



a1 = x1 + x3



v2 = x3(et + e –t)



and



v3 = 0



4.4



a2 = x3 (et – e –t)

a3 = 0



The Displacement Field



As may be seen from Figure 4.1, the typical particle of body B undergoes a

displacement

u=x–X



(4.4-1)



in the transition from the reference configuration to the current configuration. Because this relationship holds for all particles it is often useful to

analyze deformation or motion in terms of the displacement field of the body.

We may write the displacement vector u in component form by either of the

equivalent expressions

ˆ

ˆ

u = ui ei = uAI A



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(4.4-2)



Additionally, with regard to the material and spatial descriptions we may

interpret Eq 4.4-1 in either the material form

u(X,t) = x(X,t) – X



(4.4-3a)



u(x,t) = x – X(x,t)



(4.4-3b)



or the spatial form



In the first of this pair of equations we are describing the displacement that

will occur to the particle that starts at X, and in the second equation we

present the displacement that the particle now at x has undergone. Recalling

that since the material coordinates relate to positions in the reference configuration only, and hence are independent of time, we may take the time

rate of change of displacement as an alternative definition for velocity. Thus,

du d (x − X ) dx

=

=

=v

dt

dt

dt



(4.4-4)



Example 4.4-1

Obtain the displacement field for the motion of Example 4.3-1 in both material and spatial descriptions.



Solution

From the motion equations of Example 4.3-1, namely,

x1 = X1et + X3 (et – 1)

x2 = X2 + X3 (et – e –t )

x3 = X3

we may compute the displacement field in material form directly as

u1 = x1 – X1 = (X1 + X3)(et – 1)

u 2 = x 2 – X 2 = X3(e t – e –t )

u3 = x3 – X3 = 0

and by using the inverse equations from Example 4.3-1, namely,

X1 = x1e –t + x3(e –t – 1)

X 2 = x 2 + x 3 (e –t – e t)

X3 = x 3

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we obtain the spatial description of the displacement field in component

form

u1 = (x1 + x3) (1 – e –t)

u 2 = x 3 (e t – e –t )

u3 = 0



4.5



The Material Derivative



In this section let us consider any physical or kinematic property of a continuum body. It may be a scalar, vector, or tensor property, and so we

represent it by the general symbol Pij… with the understanding that it may

be expressed in either the material description

Pij… = Pij… (X,t)



(4.5-1a)



Pij… = Pij… (x,t)



(4.5-1b)



or in the spatial description



The material derivative of any such property is the time rate of change of that

property for a specific collection of particles (one or more) of the continuum

body. This derivative can be thought of as the rate at which Pij… changes

when measured by an observer attached to, and traveling with, the particle

or group of particles. We use the differential operator d/dt, or the superpositioned dot to denote a material derivative, and note that velocity and

acceleration as we have previously defined them are material derivatives.

When Pij… is given in the material description of Eq 4.5-1a, the material

derivative is simply the partial derivative with respect to time,

d

∂

[Pij ... (X , t )] =

[P ...(X , t )]

dt

∂ t ij



(4.5-2)



since, as explained earlier, the material coordinates X are essentially labels

and do not change with time. If, however, Pij… is given in the spatial form

of Eq 4.5-1b we recognize that the specific collection of particles of interest

will be changing position in space and we must use the chain rule of differentiation of the calculus to obtain

d xk

d

∂

∂

[ Pij ... (x , t )] = [ Pij ... (x , t )] +

[ Pij ... (x , t )]

dt

∂t

∂ xk

dt

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(4.5-3)



In this equation, the first term on the right-hand side gives the change

occurring in the property at position x, known as the local rate of change; the

second term results from the particles changing position in space and is

referred to as the convective rate of change. Since by Eq 4.2-12 the velocity is

defined as v = dx/dt (or vk = dxk/dt), Eq 4.5-3 may be written as

d

∂

∂

[ P ... (x , t )] = [ Pij ... (x , t )] +

[ P ... (x , t )] vk

dt ij

∂t

∂ xk ij



(4.5-4)



from which we deduce the material derivative operator for properties expressed

in the spatial description

∂

d

∂

= + vk

dt ∂t

∂ xk



or



∂

d

= + v⋅ ١

dt ∂t



(4.5-5)



The first form of Eq 4.5-5 is for rectangular Cartesian coordinates, while the

second form is coordinate-free. The del operator ( ١ ) will always indicate

partial derivatives with respect to the spatial variables unless specifically

stated.



Example 4.5-1

Let a certain motion of a continuum be given by the component equations,

x1 = X1e –t,



x2 = X2et,



x3 = X3 + X2(e –t – 1)



and let the temperature field of the body be given by the spatial description,



θ = e –t (x1 – 2x2 + 3x3)

Determine the velocity field in spatial form, and using that, compute the

material derivative dθ/dt of the temperature field.



Solution

Note again here that the initial configuration serves as the reference configuration so that Eq 4.2-9 is satisfied. When Eq 4.5-2 is used, the velocity

components in material form are readily determined to be

v1 = –X1e –t,



v2 = X2et,



v3 = –X2e –t



Also, the motion equations may be inverted directly to give

X1 = x1et,



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X = x2e –t,



X3 = x3 – x2(e –2t – e –t)