The mechanical behaviour of fibres - Biomechanics: Concepts and Computation - page 63

2 )

This implies that (recall that a · a =| a || a |=| a |

= ( R + δ ) · ( R + δ )
= R · R + 2 R · δ + δ · δ . (4.27)
If the end point displacements are sufficiently small, the inner product δ · δ

2

will
be small compared to the other inner products in the above expression and may be
neglected. Therefore, to a good approximation we have

≈ R · R + 2 R · δ .

2

(4.28)

Consequently, the current length may be written as

R

· R 2 R · δ = + . (4.29)

This may be rewritten in a more convenient form, bearing in mind that each of the
inner products yields a scalar:

R

· R 1

2 R · δ
R · R

= +

R · R

1 + 2 R

· δ

=

· R .

R

is a small number, then √ 1 , hence if δ If α + 2 α ≈ 1 + α is sufficiently small the

current length may be approximated by

· R 1

.

R

R · δ
R

≈ + (4.30)

· R

Using

0 =| R

R

· R , |= (4.31)

this can be rewritten as:

R

· δ
0

= 0 + . (4.32)

The stretch λ may now be expressed as:

R · δ

0 = 1 +

λ = . (4.33)

0

Recall that the force-stretch relation for a spring is given by:

F B = c ( λ − 1) a ,

(4.34)

where a denotes the vector of unit length pointing from point A to point B and

where F B is the force acting on point B.