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,