Con¬ning attention to a single elemental material vector dX, the cor-

responding vector dx in the spatial con¬guration is conveniently written

as,

dx = F dX (3.10)

The inverse of F is,

‚X

F ’1 = φ’1

= (3.11a)

‚x

7

3.4 DEFORMATION GRADIENT

which in indicial notation is,

3

‚XI

’1

=

F E I — ei

‚xi

I,i=1

Remark 3.4.2. Much research literature expresses the relationship be-

tween quantities in the material and spatial con¬gurations in terms of the

general concepts of push forward and pull back. For example, the elemental

spatial vector dx can be considered as the push forward equivalent of the

material vector dX. This can be expressed in terms of the operation,

dx = φ— [dX] = F dX (3.12)

Inversely, the material vector dX is the pull back equivalent of the spatial

vector dx, which is expressed as* ,

dX = φ’1 [dx] = F ’1 dx (3.13)

—

Observe that in (3.12) the nomenclature φ— [ ] implies an operation that will

be evaluated in di¬erent ways for di¬erent operands [ ].

EXAMPLE 3.2: Uniform deformation

This example illustrates the role of the deformation gradient tensor F .

Consider the uniform deformation given by the mapping,

1

x1 = (18 + 4X1 + 6X2 )

4

1

x2 = (14 + 6X2 )

4

* In the literature φ— [ ] and φ’1 [ ] are often written, as φ— and φ— respectively.

—

8 KINEMATICS

EXAMPLE 3.2 (cont.)

which, for a square of side two units initially centred at X = (0, 0),

produces the deformation show below.

(5,5) `— (E2 ) (7,5)

e2

X2 x 2

e1 = `— (E1)

`

(2,2) (4,2)

(’1,1) E2 (1,1)

’1

` (e2 ) X1 x 1

— E1

’1

` (ei )

(’1,1) (1,1)

‚x1 ‚x1

23 3 ’3

‚X1 ‚X2 1 1

F ’1 =

F= = ;

2 3

‚x2 ‚x2 03 0 2

‚X1 ‚X2

Unit vectors E 1 and E 2 in the initial con¬guration deform to,

1 1 0 1.5

φ— [E 1 ] = F = ; φ— [E 2 ] = F =

0 0 1 1.5

and unit vectors in the current (deformed) con¬guration deform from,

’1

1 1 0

φ’1 [e1 ] = F ’1 φ’1 [e2 ] = F ’1

= ; =

— — 2

0 0 1 3

3.5 STRAIN

As a general measure of deformation, consider the change in the scalar prod-

uct of the two elemental vectors dX 1 and dX 2 shown in Figure 3.2 as they

deform to dx1 and dx2 . This change will involve both the stretching (that

is, change in length) and changes in the enclosed angle between the two

vectors. Recalling (3.7), the spatial scalar product dx1 · dx2 can be found

in terms of the material vectors dX 1 and dX 2 as,

dx1 · dx2 = dX 1 · C dX 2 (3.14)

9

3.5 STRAIN

where C is the right Cauchy“Green deformation tensor, which is given in

terms of the deformation gradient as F as,

C = FTF (3.15)

Note that in (3.15) the tensor C operates on the material vectors dX 1 and

dX 2 and consequently C is called a material tensor quantity.

Alternatively the initial material scalar product dX 1 · dX 2 can be ob-

tained in terms of the spatial vectors dx1 and dx2 via the left Cauchy“Green

or Finger tensor b as,*

dX 1 · dX 2 = dx1 · b’1 dx2 (3.16)

where b is,

b = FFT (3.17)

Observe that in (3.16) b’1 operates on the spatial vectors dx1 and dx2 , and

consequently b’1 , or indeed b itself, is a spatial tensor quantity.

The change in scalar product can now be found in terms of the material

vectors dX 1 and dX 2 and the Lagrangian or Green strain tensor E as,

1

(dx1 · dx2 ’ dX 1 · dX 2 ) = dX 1 · E dX 2 (3.18a)

2

where the material tensor E is,

1

E = (C ’ I)

2

Alternatively, the same change in scalar product can be expressed with

reference to the spatial elemental vectors dx1 and dx2 and the Eulerian or

Almansi strain tensor e as,

1

(dx1 · dx2 ’ dX 1 · dX 2 ) = dx1 · e dx2 (3.19a)

2

where the spatial tensor e is,

1

e = (I ’ b’1 )

2

* In C = F T F , F is on the right and in b = F F T , F is on the left.

10 KINEMATICS

X 3, x 3

n

`

d x = dsn

N

p

d X = dS N

time = t

X1 , x1

X 2 ,x2

P

time = 0

FIGURE 3.3 Change in length.

EXAMPLE 3.3: Green and Almansi strain tensors

For the deformation given in Example 3.2 the right and left Cauchy“

Green deformation tensors are respectively,

123 1 13 9

C = FTF = b = FFT =

;

239 4 99

from which the Green™s strain tensor is simply,

103

E=

437

and the Almansi strain tensor is,

10 9

e=

18 9 ’4

The physical interpretation of these strain measures will be demon-

strated in the next example.

Remark 3.5.1. The general nature of the scalar product as a measure

of deformation can be clari¬ed by taking dX 2 and dX 1 equal to dX and

11

3.5 STRAIN

consequently dx1 = dx2 = dx. This enables initial (material) and current

(spatial) elemental lengths squared to be determined as (see Figure 3.3),

dS 2 = dX · dX; ds2 = dx · dx (3.20a,b)

The change in the squared lengths that occurs as the body deforms from

the initial to the current con¬guration can now be written in terms of the

elemental material vector dX as,

12

(ds ’ dS 2 ) = dX · E dX (3.21)

2

which, upon division by dS 2 , gives the scalar Green™s strain as,

ds2 ’ dS 2 dX dX

= (3.22)

·E

2 dS 2 dS dS

where dX/dS is a unit material vector N in the direction of dX, hence,

¬nally,

1 ds2 ’ dS 2

= N · EN (3.23)

dS 2

2