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I'm going to follow a paper Phys. Rev. Lett. 101, 055504 that seems to answer this question very concisely. Usual Voight notation: $C_{ijkl} \to C_{mn}$ here, we define Voight and Reuss estimators as defined in Proc. Phys. Soc. A 65 349. For example : $$K^V= \frac{1}{9}\left(C_{11} + C_{22} + C_{33} + 2 (C_{12} + C_{23} + C_{31})\right)$$ and so on for ...

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http://en.m.wikipedia.org/wiki/File:HookesLawForSpring-English.png. I think this is also because of the spring constant which is I think is the gap present between the spring when it is coiled where the energy or the potential energy is stored and I don't think the atoms get affected

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Firstly, you can deform material permanently..spring is no exception. On the atomic level, you are working against Coulomb forces that bind the material id est, that form the lattice. One primitive cell is well defined by the conditions of minimal energy. You can describe this potential as a quadratic, so you get harmonic forces, but it is not truly ...

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You are asking two questions really 1) How is PE actually stored in a steel spring at the atomic level? The explanation for this lies in quantum mechanics 2) Could you explain in detail how/where potential energy is actually stored in a steel spring and why the material never surrenders to the bending forces taking a new shape? Replying to 1) ...

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horizontal spring exerts a force $F = (−kx, 0, 0)$ that is proportional to its deflection in the $x$ direction. The work of this spring on a body moving along the space curve $s(t) = (x(t), y(t), z(t))$, is calculated using its velocity, $v = (vx, vy, vz)$, to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}\mathrm\,{d}t =-\int_0^t kx v_x \mathrm\,{d}t = ...

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For the small strain setting, the trace of the strain tensor is a measure for the volume change, which should be zero for the incompressible material. That would be two times the lateral straining plus the longitudinal straining. Replacing the lateral straining by minus eta times longitudinal strain, you see immediately that eta must be 0.5. Imagine a ...

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I got a response here, but thank you very much for all the comments! http://math.stackexchange.com/questions/1006597/cauchy-momentum-equation-stress-tensor/1008627#1008627

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