Potential Energy Stored by Elastic

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Discussion Overview

The discussion centers around calculating the potential energy stored in a piece of elastic when it is stretched and twisted. Participants explore the theoretical frameworks and formulas applicable to this scenario, including aspects of elasticity theory and potential energy calculations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks a formula to calculate the potential energy stored in elastic when stretched and twisted, noting a lack of resources on this specific problem.
  • Another participant suggests that elastic behaves like a spring, proposing that the potential energy stored can be calculated using the formula (1/2)kx², where k is a constant specific to the elastic.
  • A different participant recommends "Continuum Mechanics" in the Schaum's Outline series for foundational knowledge on elasticity theory.
  • One participant discusses the use of structural moduli to relate strains to forces, suggesting that integrating these from a relaxed state to the specified deformation can yield the work done, interpreted as potential energy.
  • Another participant expresses skepticism about the influence of the order of deformations on the work calculated, particularly regarding twisting, and notes that spring potential energy may only provide a first-order approximation for significant strains.
  • Additionally, a participant raises concerns that the "small deformations theory" may not be applicable for the discussed deformations, advocating for the use of a fully nonlinear tensor for stress-strain relationships.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of various theories and formulas to the problem at hand. There is no consensus on a single approach or formula for calculating the potential energy stored in the elastic under the specified conditions.

Contextual Notes

Limitations include potential dependencies on the definitions of elasticity and the assumptions regarding deformation types. The discussion acknowledges that significant strains may necessitate higher-order corrections beyond simple spring models.

Mikoden
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Hi.

I'm trying to calculate the amount of P.E. stored by a piece of elastic. I've been looking for formulas but all i can find is how to calculate gravitational potential energy and spring potential energy. The piece of elastic is to be streched by about 1-2cm (from an original 30cm length) along the horizontal, and then twisted a lot.

Any links to sites talking about this sort of problem would be handy, as well as any help that can be posted here. I'm after a formula of some sort to theoretically calculate the energy that is going to be stored, so that i have theory to back up my assumptions.

Thanks,
 
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Basically, an elastic is just a spring. The force necessary to stretch a spring (or elastic) a distance x is kx (k is a constant depending on the particular spring (or elastic)) and the work done (therefore potential energy stored) is (1/2)kx2.
 
If you are interested in learning some of the basics of general elasticity theory,
"Continuum Mechanics" in the Schaum's Outline's series might be a good start.
 
The structural moduli will allow you to translate the strains (Young's for stretch and shear? for twist) into required forces. If you integrate these from some relaxed state up to the deformation you specify, then they should give you the work required to cause the deformation. If the order of the deformations has no influence on the work calculated (for which I am suspicious of the twisting), then you can interpret this as an increase in the potential energy.

I think the spring potential energy will only give you a first order approximation. If your strains are a significant fraction of original length, then I suspect that you will incur significant (at least) second order corrections.
 
Just adding to turin's comment:
For the type of deformations you're talking about, I suspect that the
"small deformations theory" used widely would be invalid, and that stress-strain relationships should utilize the fully nonlinear tensor of relative displacements
 

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