I want to obtain stress - strain relations for a finite sized spring network and obtain the bulk modulus B and shear modulus G. I need some references. Any help would be appreciated.
In ab initio electronic structures calculations, the method of choice is the super-cell calculation, using a sufficiently large unit cell (a super cell) that is a multiple of the primitive unit cell corresponding to the system under investigation in the ordered phase. The same applies to a disordered network of springs. For details, you may wish to consult the following book and the reference therein:
DA Drabold, and SK Estreicher (Eds.), Theory of Defects in Semiconductors (Springer, Berlin, 2007).
Dear Sandipan,
I'm not sure, if I understood the spring network correctly. But the elasticity would depend on the loading direction, which is no more describable with only bulk and shear modulus. It would be appropriate to use anisotropic elasticity. The procedure in FEM is for example called homogenization. Loading of Your system with deformations should lead to reaction forces, which divided by the volume can be arranged to the elasticity matrix.
The computational homogenization techniques will solve your problem. You can carry out an RVE (representative volume element) analysis to obtain the 6x6 stiffness matrix, then figure out your bulk modulus and shear modulus. If your network is random enough, you might be able to get an equivalent isotropic material. If you want to avoid the difficulty of applying right boundary conditions, you can try SwiftComp, a general-purpose multiscale constitutive modeling code based on the recently discovered mechanics of structure genome. The code can be directly launched in the cloud at https://cdmhub.org/resources/scstandard. Plug ins for various commercial finite element packages are available.
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