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Theoretical study of the geometrical non-linearity of the elastic properties of helical springs

We give a detailed derivation of the equation describing the elastic properties of the helical springs in the case of their large deformations. It is shown that the Taylor expansion for the restoring force contains, in addition to a linear term, a term quadratic in the strain. We discuss the physical consequences of this term. A surprising example of this is that for large values of the initial pitch angle during compression of spring, there may be an interval of strain values, for which the differential stiffness becomes negative. The issues outlined in this article will be useful for instructors and advanced undergraduates who deal with modern elasticity theory.