Reverse Engineering Derivations - How?

[lanew]

Homework Statement

I have to reverse engineer a derivation in a paper and have no idea where to even start. The amount of terms are overwhelming and I can't imagine how the author even began.

Homework Equations

The following equations are known:
http://img7.imagebanana.com/img/jpfqw5qj/Selection_001.png

http://img7.imagebanana.com/img/v4cx2e07/Selection_009.png

http://img6.imagebanana.com/img/ape27n0j/Selection_002.png

Substituting them into the following equation gives an expression where $r$ needs to be isolated.

http://img6.imagebanana.com/img/gayq1jb5/Selection_008.png

The author makes many substitutions and results in the following simplified equation.

http://img6.imagebanana.com/img/5gmnl628/Selection_007.png

The substitutions are as follows:

http://img6.imagebanana.com/img/36qfovp8/Selection_004.png

http://img6.imagebanana.com/img/iinwallf/Selection_005.png

http://img7.imagebanana.com/img/japtx4e8/Selection_006.png

What I'm trying to do is figure out how he arrived at these coefficients [a], , [c], V, etc.

The Attempt at a Solution

I've substituted (1)--(5) into (6) but don't know where to go from here. The equation is huge and has many terms. I'm not sure how/where to begin.

Any help will be much appreciated. Thanks.

 

[NewtonianAlch]

Out of interest, for what course is this?

 

[lanew]

This is for a graduate level Fracture Mechanics course.

The title of the paper is: The effect of T -stress on crack-tip plastic zones under mixed-mode
loading conditions.