Reverse Engineering Derivations - How?

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SUMMARY

The discussion revolves around the challenge of reverse engineering derivations from a paper titled "The effect of T-stress on crack-tip plastic zones under mixed-mode loading conditions." The user is tasked with isolating the variable 'r' from a complex equation that involves multiple substitutions and coefficients [a], [b], [c], and V. The user has already substituted known equations into the main equation but is struggling to simplify and understand the derivation process. This is a graduate-level problem in Fracture Mechanics, requiring a solid grasp of advanced mathematical techniques.

PREREQUISITES
  • Understanding of advanced calculus and algebraic manipulation
  • Familiarity with Fracture Mechanics concepts
  • Experience with substitution methods in complex equations
  • Knowledge of the specific equations and terms used in the paper
NEXT STEPS
  • Study the derivation techniques used in Fracture Mechanics papers
  • Learn about isolating variables in complex equations
  • Research substitution methods in mathematical modeling
  • Examine case studies involving T-stress and crack-tip plastic zones
USEFUL FOR

This discussion is beneficial for graduate students in engineering, particularly those specializing in Fracture Mechanics, as well as researchers and professionals dealing with complex mathematical modeling in material science.

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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.
 
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Out of interest, for what course is this?
 
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.
 

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