Clarification on Definition of Green Strain

AI Thread Summary
Green strain is defined in continuum mechanics by the equation ds^2 - dS^2 = 2dX·E·dX, where the factor of 2 is crucial. This factor ensures that the Green strain tensor reduces to the conventional linear strain tensor when displacements are small. The necessity of the factor stems from maintaining consistency in the mathematical formulation of strain in both small and large deformation scenarios. Understanding this relationship is essential for accurate modeling in continuum mechanics. The discussion emphasizes the importance of the factor in bridging different strain definitions.
EvanZ
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Hey, everyone. I'm new here. This is a continuum mechanics question. If it's the wrong forum, please let me know, of course.

The question is pretty simple, but since I haven't been able to find an answer, I'm looking for "crowd help". Green strain \mathbf{E} is usually defined implicitly by the following equation:

ds^2-dS^2=2\mathbf{dX}\cdot \mathbf{E}\cdot\mathbf{dX}

For the life of me, I can't explain why the 2 is necessary or where it comes from. I assume there's a good reason, though. Any thoughts are appreciated.
 
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The two is there so that it reduces to the usual linear strain tensor in the limit of small displacements.
 
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