Deformation of a continuum element

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Discussion Overview

The discussion revolves around the deformation of a continuum element, specifically how to express the positions after deformation in terms of initial coordinates. The focus is on understanding the transformation of coordinates during deformation, with references to specific figures illustrating the problem.

Discussion Character

  • Homework-related
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant seeks to find the x coordinates after deformation, expressing them in terms of initial coordinates E1, E2, and E3, but is uncertain about the angles involved due to the deformation flipping to the opposite quadrant.
  • Another participant questions the definitions of E1, E2, and E3, indicating a need for clarification on these parameters.
  • A different participant observes that the deformation depicted does not appear to be pure shear, suggesting that the parameters m and n may represent stretches instead.
  • A further contribution asks for the equation of the position vector between two material points in the deformed configuration, given the stretching of the original position vector.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the nature of the deformation and the definitions of the parameters involved. There is no consensus on the interpretation of the deformation as pure shear or the correct expressions for the coordinates after deformation.

Contextual Notes

Limitations include potential ambiguities in the definitions of E1, E2, and E3, as well as the unclear nature of the deformation (whether it is pure shear or involves stretching). The discussion also highlights the dependence on the specific figures referenced, which are not provided in the text.

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Homework Statement



350qxj7.jpg


The continuum element shown in Figure (a) deforms to figure (b)

I'm looking to find the positions after deformation (x coordinates) by expressing them in terms of E coordinates.

Homework Equations


The Attempt at a Solution



x1 = E1 + E2
x2 = E1 + E2
x3 = E3

The first two equations for x I need help with, I'm not sure about the angles. I've done a previous question where the element deforms similarly (pure shear) but it stays within the same quadrant. The fact that this is flipped to the opposite quadrant is confusing me.

When the element deforms but stays in the same quadrant the answer is:

x1 = E1cosa + E2sina
x2 = E1sina + E2cosa
x3 = E3

ftd5ed.jpg
 
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What are E1, E2, and E3 supposed to be?
 
In your first figure, it doesn't look like the deformation is pure shear. Those parameters m and n look like stretches.

Chet
 
If the position vector between material points A and B is expressed as E1i in the original configuration of the body (figure a), and this vector stretches by an amount m to obtain the deformed configuration (figure b), what is the equation for the position vector between material points A and B in the deformed configuration?

Chet
 

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