Crack Paths 2012

r

1(8)

Figure 1: Point P in local coordinates near the crack front F.

the FRENET-SERRETformulas from differential geometry [4] lead to

dt

dn

db

5(8) I —'1(8)n(8),

5(8) I /1(8)t($)+ rtsibtsii

5(8) I —7(8)n(8),

where n(s) is the curvature and 7(5) is the torsion of the curve I‘ (the crack front)

at arc length 5. A covariant basis {g1,g2, g3} of (positive orientated) curvilinear

coordinates at a point @(y) I 06 G T is given by gj : GyjOQg), here

81(0) I 71(8),

82(0) I (1 + .1/16(8))t(8)+

T(8)(y1b(8) + 0211(8)),

83(0) I 41(8)

The components gij

z:

- gj of the RIEMANNIANmetric tensor are

1

1027(5)

0

can): two)(n+ononttortehor)—oao

0

—y17'(s)

1

The contravariant basis, defined by the relation gi - gj : 6,-9- with the K R O N E C K E R

symbol 6,], can also be explicitly calculated:

1

1ns_yeo>s ,

I____

_

g(”**() go)“L g(” 90)

go»

where g(y) : (1 + y1n(s)) is the JACOBIAN, see e g Because E is considered

to be smooth, local coordinates are uniquely defined at any arc length 5. Neverthe

less, the determinant

of the JACOBIANmatrix vanishes at g1 : — $and the

transformation (2) is valid only in a possibly small vicinity around the crack front.

547