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It is known that the Hooke's law relating the restoring force of a spring to the distance of retraction from the equilibrium position, is only an approximation.

That is, the equation $F=-kx$ is only the linear term that approximates the relationship, but gets less accurate the more the spring is retracted from the equilibrium position. There is also an elastic limit for the spring, which makes the relationship vertically asymptotic at distances large enough and the law completely breaks down there.

The relationship looks to me more like a tangent function than a line when considering the increasing deviation and asymptotic behavior.

Since $F=-kx$ is only an approximation, then what is the full story? What is the actual relationship? I couldn't find the answer anywere on the internet. Is it a transcendental function, or perhaps some non-elementary function? Or does the function defining the relationship depend on the material used and local physical quantities?

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You can think of doing a series expansion like $$ F=-k_1x + k_2x^2+k_3x^3+\ldots $$ but, given no additional information, would need to determine experimentally the various coefficients as a function of the “spring” you have using a force meter.

In general, if $V(x)$ is the potential energy, and $x_0$ is such that $V’(x_0)=0$ then $$ V(x-x_0)= V(x_0)+\frac{1}{2}V^{\prime\prime }(x_0)(x-x_0)^2+\frac{1}{3!} V^{\prime\prime\prime}(x_0)(x-x_0)^3+\ldots $$ so that any locally concave potential will result is an approximately linear for near the minimum $x_0$. Hence, beyond the ideal spring, the Lennard-Jones molecular potential $$ V(x)=4\epsilon\left(\left(\frac{\sigma}{x}\right)^{12}-\left(\frac{\sigma}{x}\right)^6\right) $$ will produce an approximately linear restoring force near $x_0=2^{1/6}\sigma$. The Morse potential and a whole host of others, also have a quadratic approximation (leading to linear force) but the coefficients depend on the potential. Clearly then this depends critically on “what the spring is”, i.e. on the specific molecular potential, although they all have a regime where the force will be linear. There’s a good discussion of this on this page.

Where I work we actually couple penduli not with springs - they would sag because of gravity and buckle rather than compress - but using copper strips that have been bent in a V shape. For sufficiently small bending near the “equilibrium” shape, we measured the restoring force using an accelerometer as being nearly linear over a reasonable range of deformation, including compressions.

Getting two copper strips to have the same “effective $k$ value” is a bit tricky as it seems depend on a number of factors - thickness of the actual strip, width of the strip, bend angle in the V - but it’s good enough to investigate the coupling of penduli as if they we coupled by a spring.

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The answer isn't well defined, since the full behaviour is different for different materials. It is not that the function is some mysterious transcendental function, but rather some function typically described by a series of powers of x with coefficients dependent on the material.

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    $\begingroup$ And more importantly, when you do include those nonlinear terms, you stop using the term Hooke's Law - it only applies to the linear regime. $\endgroup$ – Emilio Pisanty Feb 9 at 10:53
  • $\begingroup$ That makes sense. Is there however a function that seems to approximate this concept better for a large class of materials? $\endgroup$ – KKZiomek Feb 9 at 14:19
  • $\begingroup$ @KKZiomek - See ZeroTheHero's more extensive answer. $\endgroup$ – Anders Sandberg Feb 9 at 18:17
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We can all agree that for sufficiently small deformations, any stable solid follows Hooke's Law with a well-defined stiffness (as derived here). It's remarkable that this condition holds for any material and any geometry.

When it comes to larger deformations, there's a variety of ways that Hooke's Law can fall apart:

  • Sufficiently large changes in the macroscale geometry disrupt the loading conditions: A classic example here is the stiffening of a membrane under pressure (as the lateral loading transitions to stretching of the membrane material):

    The reverse example is the buckling of a column as the compressive loading transitions to bending:

    • The material permanently deforms: In a ductile crystal, the stress might be high enough that dislocations start to move, resulting in shear slip and plastic deformation. Here, the stress-strain curve changes direction at this yield point:

      In a brittle ceramic, in contrast, the strain energy stored in the material might grow so large that it becomes energetically preferable to make a new surface, which corresponds to fracture. In this case, the linear stress-strain region simply ends:

      Finally, in viscoelastic materials (and all solids are viscoelastic over sufficiently long time periods), the material will creep under a constant stress, which is clear violation of Hooke's Law. Consider the deformation of these lead pipes after decades of gravity exposure:

    • We leave the nearly symmetric region of the strain vs. energy curve: For sufficiently large deformations, we can no longer assume that an energy minimum is parabolic, which is equivalent to saying that the stiffness becomes strain dependent and that Hooke's Law breaks down. For crystals, the energy minimum is that of the pair potential:

      For polymers, the kinked chains may have straightened (this is the microscale version of Reason 1 above) so that we're now pulling on the molecular backbone rather than just uncoiling the polymer. The general result is stiffening, here shown for an elastomer:

So Hooke's Law can break down in various ways that depend on the material, the time scale, and the loading geometry, at a minimum.

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