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

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Discussion Overview

The discussion revolves around a 2-D planar problem involving stresses, strains, and displacements in the context of linear elasticity. Participants explore the relationship between displacement gradients and stress, particularly in finite difference numerical solutions for simulating linear elastic deformation of a square flat plate under compressive loads.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant notes that the stress matrix is symmetrical, leading to three knowns and four unknowns in the displacement gradients, raising the question of how to achieve uniqueness in the solution.
  • Another participant emphasizes that in structural problems, the focus is typically on solving for displacements rather than displacement gradients.
  • There is a discussion about the need for an additional condition to ensure a unique solution, with a participant suggesting that a compatibility equation might serve this purpose.
  • Participants describe the specific setup of a finite difference simulation involving a square flat plate with fixed boundary conditions and applied compressive loads.
  • Concerns are raised about the implications of boundary conditions requiring displacement functions in the z-direction, which could complicate the problem and shift it from 2-D to 3-D.

Areas of Agreement / Disagreement

Participants express differing views on whether the displacement gradients or displacements should be the primary focus in solving the problem. There is no consensus on the necessity or form of an additional condition to ensure uniqueness in the solution.

Contextual Notes

Participants acknowledge the limitations of their assumptions, particularly regarding the dimensionality of the problem and the implications of boundary conditions on the uniqueness of the solution.

  • #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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