Homework help: anisotropic elasticity equations

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SUMMARY

The discussion centers on the application of anisotropic elasticity equations as presented in Lekhnitskii's textbook. The user is struggling to derive the correct equilibrium equations for an orthotropic body, specifically encountering issues with the combination of strain-displacement equations and Hook's Law. The equation presented, A11.Uxx + A12.Vyx + A13.Wzx + A66.(Uyy + Vyx) + A55.(Wzx + Uzz) = 0, highlights the user's confusion regarding cross derivatives and their elimination through continuity equations. The community is called upon to assist in identifying the user's error in this derivation process.

PREREQUISITES
  • Understanding of anisotropic elasticity principles
  • Familiarity with strain-displacement equations
  • Knowledge of Hook's Law in elasticity
  • Basic grasp of equilibrium equations in mechanics
NEXT STEPS
  • Review the derivation of strain-displacement equations in anisotropic materials
  • Study the application of Hook's Law in orthotropic elasticity contexts
  • Examine methods for eliminating cross derivatives in elasticity equations
  • Explore continuity equations and their role in mechanical equilibrium
USEFUL FOR

Students and researchers in mechanical engineering, particularly those focused on elasticity theory, as well as professionals dealing with material behavior in orthotropic contexts.

santamartina
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Homework Statement


Hello, I'm studying anisotropic elasticity. One of the books I'm using is Lekhnitskii's. The book presents the general equations of the theory of elasticity for an orthotropic body as follows:

Homework Equations


upload_2018-7-13_9-56-33.png


The Attempt at a Solution


However, when I combine the strain-displacement equations and the Hook's Law and then replace them into the equilibrium equations I cannot get the same expression. My reasoning looks like this (for example, the first equation of the system):

A11.Uxx + A12.Vyx + A13.Wzx + A66.(Uyy + Vyx) + A55.(Wzx + Uzz) = 0

Obviously, I'm doing something wrong. What is my mistake?

Many thanks for your kind help.
 

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Maybe they are able to eliminate the cross derivatives by combining the equations with the continuity equation.
 

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