Linking elastic moduli to shape fluctuations

I am simulating a polymer network shaped as a sphere. I would like to compute its elastic moduli (bulk and Young's, for example) not by their shear-strain definition but by looking at the fluctuations in volume and shape, which are estimated by using a convex hull construction.

The bulk modulus, being the inverse of the compressibility, can be readily computed by looking at the fluctuations of the volume, viz.:

$$\beta K = \frac{\langle V \rangle}{\langle V^2 \rangle - \langle V \rangle^2}$$

As for the Young's modulus, I haven't found any definitive answer on how to compute it in a similar way. The only papers I have found provide relations for 2D systems (for example here). I have tried to naively extend these relations to 3D, obtaining

$$\beta Y = \frac{1}{\langle V \rangle} \frac{\langle l \rangle^2}{\langle l^2 \rangle - \langle l \rangle^2}$$

I have no solid basis to support the above relation, and I would like to know whether it is correct and, if it is, how it can be derived.

• Young's modulus isn't very relevant for a sphere since it's defined to couple the uniaxial stress to the uniaxial strain for a long, thin object only. An alternative is to switch to the shear modulus (which gives a purer "shape vs. size" dichotomy with the bulk modulus); unfortunately, I can't immediately relate this parameter to shape fluctuations. Please take a look at Mietke's fine work here and the cited references, which may be useful. Commented Mar 16, 2018 at 16:39
• About the Young's modulus: it is in fact relevant also for spheres, as the Hertzian force felt by two objects of any shape (in the small deformation regime) depends on their Young's moduli and Poisson ratios, $\nu$. It is of course possible to convert the dependence on $Y$ and $\nu$ to any other pair of elastic moduli, but as far as I know the most used relation involves these two. About your advice: estimating the shear modulus might be well worth pursuing. Thanks a lot for the reference. I'll be sure to thoroughly read it. Commented Mar 16, 2018 at 16:59

3. Construct a Green-Lagran tensor $$\mathbf{C}$$ that measures the instantaneous deformation of an object. In order to do this, one has to first choose a stress-free reference configuration
4. Compute the tensor invariants that are related to the type of fluctuations you think are relevant. In my case I am interested in the volume and shape, to which we can associate the following invariants: \begin{align} J & = \sqrt{\mathrm{det}(\mathbf{C})}\\ I_1 & = \mathrm{tr}(\mathbf{C})J^{-2/3}\\ I_2 & = \frac{1}{2}\left[ \mathrm{tr}^2(\mathbf{C}) - \mathrm{tr}(\mathbf{C}^2)\right]J^{-4/3} \end{align}
5. Write down an expression for the energy of the system due to the thermal excitations of these quantities. Since I am dealing with a polymer network I use the phenomenological Mooney-Rivlin theory for rubber elasticity which can be written as: $$U(J, I_1, I_2) = U_0 + V\left(\frac{K}{2}(J - 1)^2 + C_{10}(I_1 - 3) + C_{01}(I_2 - 3)\right)$$ where $$K$$ is the bulk modulus and $$G = 2(C_{01} + C_{10})$$ is the shear modulus.
6. If we assume that the deformations are statistically independent, we can fit $$K$$, $$C_{10}$$ and $$C_{01}$$ by looking at the probability distributions of $$J$$, $$I_1$$ and $$I_2$$, as computed over all the equilibrium conformations.