How is stress related to force and area in a solid body under equilibrium?

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Discussion Overview

The discussion revolves around the relationship between stress, force, and area in a solid body under equilibrium. Participants explore concepts related to internal stresses, equilibrium conditions, and the mathematical representation of these relationships through integrals and surface integrals.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions why the integral of delta P must equal P1 and P2, suggesting an alternative interpretation that delta P equals the sum of P4 and P3.
  • Another participant asserts that the diagram indicates internal stresses in a solid 3D body must sum to zero to maintain equilibrium, implying that any imbalance would lead to failure.
  • A different participant clarifies that when bisecting a body in equilibrium, the resultant surface integral must equal the forces P1 and P2 that were removed, emphasizing the importance of this equality for maintaining equilibrium.
  • One participant proposes that if stress is defined as force divided by area, integrating stress over a differential area (Delta A) could yield a new force equal to the sum of P1 and P2.

Areas of Agreement / Disagreement

Participants express differing interpretations of the relationships between stress, force, and area, indicating that multiple competing views remain without a consensus on the correct interpretation of the integral relationships.

Contextual Notes

Some assumptions about the definitions of stress and equilibrium conditions may be implicit in the discussion. The mathematical steps involving integrals and their implications for force and stress are not fully resolved.

v_pino
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Why is it the integral of delta P that must equal P1 and P2 in the second diagram (half of the original body)? I thought it is simply that Delta P = (P4 + P3).
 

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I don't really get what your asking? I think the diagram is basically just stating that internal stresses in a solid 3D body must sum to zero, if they do not then the body is exploding.
 
It says that if you bisect the body, that is already in equilibrium, the resultant surface integral has to equal P1+P2, which were removed, for the body to stay in equilibrium. It would also be correct to say what you are saying, but I think the main idea was that the surface integral has to be equal to what was removed.
 
So stress=force/area. Does it mean that if I find the stress over the area (Delta A) and integrate it with respect to Delta A, then I get the new force which is equal to the sum of P1 and P2?
 

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