Crack Paths 2006

M U S K H E L I S H V I LCIO’ SM P LFEUXN C T I AO NA L Y S I S

Since the IC method measures displacement fields, it is advantageous to derive the

relationship between displacements and SIF. This can be done using Muskhelishvili’s

complex function analysis. Let’s consider a plane body lying in a complex plane Z.

Displacements in Z can be expressed through two analytical functions of complex

variable, q) I q)(z) and \|lI\|/(z), where zIx+iy, as follows [1 1]:

2M”+ iv) = Z¢(Z) — Z¢'(Z) — V/(Z)

(2)

where

,u =

Z = 3 — 4v (plane strain); )5 I i i(plane stress)

(3)

2(1 + v) ’

+ V

u and v are displacements and 61.]. are stresses, i,jI1,2. E is the Young’s modulus and v is

the Poisson’s ratio. The overbar denotes the complex conjugate and the prime denotes

the first derivative.

The function

m Z=0)(C)=R U K

(4)

Where m is a shape parameter, 0§m§1, and R is a scale parameter which maps a unit

circle onto an elliptical contour, L, in plane Z. If mIl then L is an idealized crack of

zero thickness and length 4R.

The analytical functions in the mapping plane (MC) and MC)can be represented by an

infinite Fourier series:

+00 +00 ¢= 2b,?‘

(5)

Where ak and bk are complex coefficients.

It can be shownafter a lengthy derivation that (2), (4) and (5) and the satisfaction of

the traction-free crack boundary condition lead to the following system of real linear

equations written for p experimental sampling points:

N _

N .

.

2A2“; ZBIBFZW

<6)

kI-N kI-N

N _

N .

j

2 eta; zdkfifzw/

<7)

kI-N

kI-N