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KINEMATICS




3.1 INTRODUCTION

It is almost a tautology to say that a proper description of motion is fun-
damental to ¬nite deformation analysis, but such an emphasis is necessary
because in¬nitesimal deformation analysis implies a host of assumptions that
we take for granted and seldom articulate. For example, we have seen in
Chapter 1, in the simple truss example, that care needs to be exercised when
large deformations are anticipated and that a linear de¬nition of strain is
totally inadequate in the context of a ¬nite rotation. A study of ¬nite de-
formation will require that cherished assumptions be abandoned and a fresh
start made with an open (but not empty!) mind.
Kinematics is the study of motion and deformation without reference to
the cause. We shall see immediately that consideration of ¬nite deformation
enables alternative coordinate systems to be employed, namely, material
and spatial descriptions associated with the names of Lagrange and Euler
respectively.
Although we are not directly concerned with inertial e¬ects, neverthe-
less time derivatives of various kinematic quantities enrich our understanding
and also provide the basis for the formulation of the virtual work expres-
sion of equilibrium, which uses the notion of virtual velocity and associated
kinematic quantities.
Wherever appropriate, nonlinear kinematic quantities are linearized in
preparation for inclusion in the linearized equilibrium equations that form
the basis of the Newton“Raphson solution to the ¬nite element equilibrium
equations.




1
2 KINEMATICS




Time Length Area Volume Density

0 S A V 0

q
t s a v




Q
X3
p
x3
time = t
E3
x2
e3 e2
E2 X 2 P
x1
e1
E1
time = 0
X1

FIGURE 3.1 General motion of a deformable body.



3.2 THE MOTION

Figure 3.1 shows the general motion of a deformable body. The body is
imagined as being an assemblage of material particles that are labeled by the
coordinates X, with respect to Cartesian basis E I , at their initial positions
at time t = 0. Generally the current positions of these particles are located,
at time = t, by the coordinates x with respect to an alternative Cartesian
basis ei . In the remainder of this text the bases E I and ei will be taken to
be coincident. However the notational distinction between E I and ei will
be retained in order to identify the association of quantities with initial or
current con¬gurations. The motion can be mathematically described by a
mapping φ between initial and current particle positions as,

x = φ(X, t) (3.1)

For a ¬xed value of t the above equations represent a mapping between
the undeformed and deformed bodies. Additionally, for a ¬xed particle X,
Equation (3.1) describes the motion or trajectory of this particle as a func-
tion of time. In ¬nite deformation analysis no assumptions are made re-
garding the magnitude of the displacement x ’ X, indeed the displace-
ment may well be of the order or even exceed the initial dimensions of
3
3.3 MATERIAL AND SPATIAL DESCRIPTIONS



the body as is the case, for example, in metal forming. In in¬nitesimal
deformation analysis the displacement x ’ X is assumed to be small in
comparison with the dimensions of the body, and geometrical changes are
ignored.


3.3 MATERIAL AND SPATIAL DESCRIPTIONS

In ¬nite deformation analysis a careful distinction has to be made between
the coordinate systems that can be chosen to describe the behavior of the
body whose motion is under consideration. Roughly speaking, relevant
quantities, such as density, can be described in terms of where the body
was before deformation or where it is during deformation; the former is
called a material description, and the latter is called a spatial description.
Alternatively these are often referred to as Lagrangian and Eulerian descrip-
tions respectively. A material description refers to the behavior of a material
particle, whereas a spatial description refers to the behaviour at a spatial
position. Nevertheless irrespective of the description eventually employed,
the governing equations must obviously refer to where the body is and hence
must primarily be formulated using a spatial description.
Fluid mechanicians almost exclusively work in terms of a spatial de-
scription because it is not appropriate to describe the behavior of a material
particle in, for example, a steady-state ¬‚ow situation. Solid mechanicians,
on the other hand, will generally at some stage of a formulation have to con-
sider the constitutive behavior of the material particle, which will involve
a material description. In many instances “ for example, polymer ¬‚ow “
where the behavior of the ¬‚owing material may be time-dependent, these
distinctions are less obvious.
In order to understand the di¬erence between a material and spatial
description, consider a simple scalar quantity such as the material density ρ:
(a) Material description: the variation of ρ over the body is described with
respect to the original (or initial) coordinate X used to label a material
particle in the continuum at time t = 0 as,
ρ = ρ(X, t) (3.2a)
(b) Spatial description: ρ is described with respect to the position in space,
x, currently occupied by a material particle in the continuum at time t
as,
ρ = ρ(x, t)
In Equation (3.2a) a change in time t implies that the same material
4 KINEMATICS



particle X has a di¬erent density ρ. Consequently interest is focused on
the material particle X. In Equation (3.2b), however, a change in the
time t implies that a di¬erent density is observed at the same spatial
position x, now probably occupied by a di¬erent particle. Consequently
interest is focused on a spatial position x.

EXAMPLE 3.1: Uniaxial motion
This example illustrates the di¬erence between a material and a spa-
tial description of motion. Consider the mapping x = (1 + t)X de¬n-
ing the motion of a rod of initial length two units. The rod expe-
riences a temperature distribution given by the material description
T = Xt2 or by the spatial description T = xt2 /(1 + t), see diagram
below.
t (X = 1,T = 9) (X = 2,T = 18)
3
(X = 1,T = 4) (X = 2,T = 18)
2
(X = 1,T = 1) (X = 2,T = 2)
1
X, x

7
2 3 4 5 6 8
0 1
The diagram makes it clear that the particle material coordinates (la-
bel) X remains associated with the particle while its spatial position x
changes. The temperature at a given time can be found in two ways, for
example, at time t = 3 the temperature of the particle labeled X = 2
is T = 2 — 32 = 18. Alternatively the temperature of the same particle
which at t = 3 is at the spatial position x = 8 is T = 8—32 /(1+3) = 18.
Note that whatever the time it makes no sense to enquire about par-
ticles for which X > 2, nor, for example, at time t = 3 does it make
sense to enquire about the temperature at x > 8.


Often it is necessary to transform between the material and spatial de-
scriptions for relevant quantities. For instance, given a scalar quantity, such
as the density, a material description can be easily obtained from a spatial
description by using motion Equation (3.1) as,
ρ(X, t) = ρ(φ(X, t), t)
Certain magnitudes, irrespective of whether they are materially or spa-
tially des- cribed, are naturally associated with the current or initial con-
5
3.4 DEFORMATION GRADIENT



X 3, x 3



`
q1
d x1
p

d x2
x
q2
Q1
X d X1
P
time = t
dX 2
X1 , x1 Q2
X 2 ,x2


time = 0

FIGURE 3.2 General motion in the neighborhood of a particle.



¬gurations of the body. For instance the initial density of the body is a
material magnitude, whereas the current density is intrinsically a spatial
quantity. Nevertheless, Equations (3.2a“c) clearly show that spatial quan-
tities can, if desired, be expressed in terms of the initial coordinates.



3.4 DEFORMATION GRADIENT

A key quantity in ¬nite deformation analysis is the deformation gradient
F , which is involved in all equations relating quantities before deformation
to corresponding quantities after (or during) deformation. The deformation
gradient tensor enables the relative spatial position of two neighboring par-
ticles after deformation to be described in terms of their relative material
position before deformation; consequently, it is central to the description of
deformation and hence strain.
Consider two material particles Q1 and Q2 in the neighborhood of a
material particle P ; see Figure 3.2. The positions of Q1 and Q2 relative to
P are given by the elemental vectors dX 1 and dX 2 as,

dX 1 = X Q1 ’ X P ; dX 2 = X Q2 ’ X P (3.3a,b)
6 KINEMATICS



After deformation the material particles P , Q1 , and Q2 have deformed to
current spatial positions given by the mapping (3.1) as,
xp = φ(X P , t); xq1 = φ(X Q1 , t); xq2 = φ(X Q2 , t) (3.4a,b,c)
and the corresponding elemental vectors become,
dx1 = xq1 ’ xp = φ(X P + dX 1 , t) ’ φ(X P , t) (3.5a)
dx2 = xq2 ’ xp = φ(X P + dX 2 , t) ’ φ(X P , t) (3.5b)
De¬ning the deformation gradient tensor F as,
‚φ
F==φ (3.6)
‚X
then the elemental vectors dx1 and dx2 can be obtained in terms of dX 1
and dX 2 as,
dx1 = F dX 1 ; dx2 = F dX 2 (3.7a,b)
Note that F transforms vectors in the initial or reference con¬guration into
vectors in the current con¬guration and is therefore said to be a two-point
tensor.

Remark 3.4.1. In many textbooks the motion is expressed as,
x = x(X, t) (3.8)
which allows the deformation gradient tensor to be written, perhaps, in a
clearer manner as,
‚x
F= (3.9a)
‚X
In indicial notation the deformation gradient tensor is expressed as,
3
‚xi
F= FiI ei — E I ; FiI = ; i, I = 1, 2, 3
‚XI
i,I=1

where lowercase indices refer to current (spatial) Cartesian coordinates,

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