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I have a question that has been bothering me for a long time. I know very well how to take the continuum limit for a chain of masses connected by springs. But in my recent project I would like to extend this to a bulk of masses (3D) connected by springs and then apply the continuum limit. Is that even possible? can someone recommend a book, paper, notes, etc. to me where this is done?

thanks in advance

LLanadas
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  • This is a standard problem in solid-state physics, solved when the phonon spectrum is calculated. Please have a look at textbooks like Ashcroft-Mermin. – flaudemus Feb 18 '19 at 12:49
  • Thanks a lot for this recommendation, it was exactly what I was looking for. – LLanadas Feb 28 '19 at 09:55

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If by applying the continuum limit you mean going from the standard mass-spring model to the usual displacement/stress-strain formalism of elasticity, then the short answer is 'No, that cannot be done.'.

In fact, that very question was already bothering Augustus Love some 100 years ago. He demonstrated in his "A treatise on the mathematical theory of elasticity" that such a spring model can approximate isotropic materials with any Young's modulus, but their Poisson's ratio is fixed to $\nu = 1/4$ (if you drop the isotropy requirement, the story is similar: it can only reproduce 15 of the expected 21 degrees of freedom of the constitutive tensor). To paraphrase him (italics are mine):

The theory of Elasticity has sometimes been based on that hypothesis concerning the constitution of matter, according to which bodies are regarded as made up of material points, and these points are supposed to act on each other at a distance, the law of force between a pair of points being that the force is a function of the distance between the points, and acts in the line joining the points. It is a consequence of this hypothesis that the coefficients in the function W (the strain energy density, the coefficients of which are, for the linear case, the dofs of the constitutive tensor) are connected by six additional relations, whereby their number is reduced to 15 (instead of the usual 21 - isotropy is not assumed).

That said, there have been some recent attempts to extend the usual spring models beyond that constraint, typically by considering additional volumetric terms (see, e.g., here - this paper in particular focuses more on the 2D case, but the approach in 3D is similar).

The Theoretical Astronaut
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  • Yes, it was exactly my hope to model in this way the usual displacement/stress-strain formalism of (linear) elasticity. My objective was, with such a model, to find out how displacement and rotations interact with each other. Or in other words, find the Lagranian of a (linear) elastic body including rotations. The main goal will be to establish a quantum theory which describes the coupling between phonons and angular momentum. Unfortunately it seems a bit hopeless. – LLanadas Feb 18 '19 at 13:36