Solid Mechanics Fundamentals - Meshless Methods in Biomechanics - page 33

which can be developed as,
dL 1 ¼ Z
X

dB T rdX þ Z B T drdX ¼ K T du

ð 2 : 84 Þ

X

Using Eqs. ( 2.35 ) and ( 2.70 ) the following relation is obtained,

dr ¼ cBdu ð 2 : 85 Þ

In the deforma ti on matrix B only the nonlinear part dB NL is dependent of u,
Eq. ( 2.80 ), thus dB ¼ dB NL and therefore,
dL 1 ¼ Z
X

dB NL r dX þ Z
X

B T c B dX

ð 2 : 86 Þ

Where the stiffness matrix can be presented as,

K T ¼ K r þ K 0 þ K NL ð 2 : 87 Þ

Being,

K r ¼ Z
X

dB NL r dX

ð 2 : 88 Þ

K 0 ¼ Z
X

B 0 cB 0 dX ð 2 : 89 Þ

K NL ¼ Z
X

ffi dX

B 0 cB NL þ B NL cB NL þ B NL cB 0 ð 2 : 90 Þ

The initial stress matrix or geometric matrix K r is defined as,
K r du ¼ Z
X

G T dA T r dX ð 2 : 91 Þ

The variation of matrix A in order to u can be defined as,

2
4

3
5

dh x 0

3 1

0
3 1

dh y 0

3 1

dh z

½ ½ ½

0
3 1

dh y 0

3 1

dh x dh z 0

3 1

dA T ¼ ð 2 : 92 Þ

½ ½ ½

0
3 1

0
3 1

dh z 0

3 1

dh y dh x

½ ½ ½