prediction of crack path which is subjected to stress gradient and biaxial load.

Dear all.

I'm currently doing some research concerning crack path prediction in brittle thin film.

But I'm struggled to find efficient way to do it.

In reality my case: stress is higly confined within semi infinite narrow rectangular strip patten and out of this rectangular, stress is very low or near 0 compared to stress within the pattern. Crack starts from left side end and propagates within pattern toward right end side of pattern. Crack show wavy propagation behavior with regular period and amplitude. 

Simply express my situation, think of 1 dim crack (so only mode I and II) because it propagates through thin brittle film.

stress ditribution have y dependency and stress distribution is confined semi infinite narrow rectangular of width 2b. 

sigma yy =(sigma 0){1-cosh(y)/cosh(b)} or any other y dependent funcion which will be vanishes when y>b  and y<-b and smoothly increasing to maximum at y=0.  

sigma xx =(sigma 0){1-vcosh(y)/cosh(b)} or any other y dependent funcion which will be vanishes when y>b  and y<-b and smoothly increasing to maximum at y=0.  

v is constant. 

initiation of crack is arbitarily slanted with angle theta and length of d which is very smaller than 2b.

I want to get general expression of K or G in terms of x, y, theta so I can predict trajectary of crack which always favor the path which maximize G.

There is very same notion was reported with colloidal film drying.("Wavy cracks in drying colloidal films" Lucas Goehring, et al., Soft matter, 7, 7984 (2011))

But in that study they use serveral apporximatioin to show intrinsic nature of wavy pattern : that simply put, crack opening toward centerline(y=0, where stress is highest) will release more energy than other directions, so crack seems to reflect at certain outer boundary line(y=b, y=-b, where stress is nearly zero), anyway I love their approach but I want to expand this analogy to more general cases.

I know it is not easy(or nearly impossible) to directly get analytical solution for this. So at least If their is relatively general form of approximation for K or G that will be great help.

If I can get G then probably with that equaion I can compute crack path by numerical repeating approach. 

Is there any article or textbook that I can refer to this problem?

I welcome any comment or correction for error to the model in any form.

Thanks for reading through.

Best regards.

Myung Rae.