Stress-strain; strain-displacement in 2-D; uniqueness

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SUMMARY

This discussion focuses on solving a 2-D linear elastic deformation problem using the finite difference method on a square flat plate with dimensions 1x1x0.1 meters. The problem involves determining displacement gradients from a stress matrix, which is symmetrical, leading to a non-unique solution due to having three known stresses and four unknown displacements. The participants discuss boundary conditions, including fixed displacements and stress boundary conditions, to ensure a unique solution. A compatibility equation is suggested as a potential additional condition to resolve the uniqueness issue.

PREREQUISITES
  • Understanding of linear elasticity principles
  • Familiarity with finite difference methods for numerical solutions
  • Knowledge of stress-strain relationships in 2-D systems
  • Experience with boundary condition applications in structural analysis
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  • Research the compatibility equations in elasticity theory
  • Study the finite difference method for solving partial differential equations
  • Explore methods for imposing boundary conditions in numerical simulations
  • Learn about the implications of shear stress boundary conditions on displacement calculations
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Mechanical engineers, structural analysts, and researchers involved in computational mechanics and numerical simulations of elastic materials.

  • #61
Chestermiller said:
I have nothing to add, except to say simplify as much as possible. But please check that heat balance equation. It seems to be missing a derivative on the conduction term.
I took a hiatus from looking at the structural problem for awhile and recently picked up again the past couple of weeks. I just realized what I think is a huge issue when solving the transient equation. The issue is these structural constants are HUGE (when I was previously working on this, I had largely neglected the constants). For example, steel's elastic modulus is about O(10^11), and its lame's constants are around ~O(10^8). In solving the transient equation, I have to consider stability issues. The max dt that I can use is essentially INVERSELY proportional to lame's constants, thus severely limiting my timestep size to the point that the computational time is impractical for the purpose of my code.

I could potentially approach this problem as steady state. However, temperature and slight pressure (also low pressure) charges really would suggest that this problem is transient.
 

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