# Analytic Solution for 3D Elastic Deformation

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Hey folks! I'm currently working on a deformation project, and my current testing application allows me to input the magnitude, position and direction of a force applied to an elastic deformable cube, and then deforms it accordingly. I was just curious whether anyone knew where I could find the analytic solution for such a problem in the case of small strain deformation? I'm wanting to compare the results I'm obtaining with the analytic solution, to determine my methods accuracy under a variety of situations. Thankyou very much for the help! James.

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I think you could find that in most any introductory book on structural finite element analysis, assuming your material is isotropic.

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Ah, thankyou very much! I'll take a trip into the library to see what I can find.

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Just to clarify what you should look for. An elastic material has a linear strain in response to small stress/force. Therefore for a 1 degree-of-freedom material (e.g., a linear spring), Hooke's law, x = F/k (displacement = force / stiffness) is the analytic solution. Or, written another way, displacement is a linear function of applied stress. The finite element books will present linear relationships between stress and strain, and for elastic materials these relationships are the analytic solution, as they are for the linear spring. It is those linear stress/strain relationships you are looking for. And, in fact, the material need not be isotropic, though the stiffness matrix is extremely simple for isotropic materials, but not so simple for those that are not isotropic.

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