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echoes

Extended Calculator of HOmogEnization Schemes: a C++ library with a Python interface for the analytical micromechanics of random media. Effective elastic, conduction and viscoelastic properties, and their derivatives with respect to the lower-scale moduli.

C++
Python
Micromechanics
Homogenization

echoes (Extended Calculator of HOmogEnization Schemes) estimates the effective properties of a random heterogeneous medium from the properties, shapes and volume fractions of its phases. It solves the Eshelby problem at second order (conductivity) and fourth order (elasticity), for isotropic or anisotropic reference media, and feeds the result to a wide family of mean-field schemes. Beyond the effective moduli themselves, it returns their derivatives with respect to the lower-scale moduli — the ingredient a secant-based nonlinear homogenization needs. The numerical core is C++; the modeling is written in Python.

Key features

  • A wide family of schemes — bounds, dilute and its dual, Mori–Tanaka, self-consistent and asymmetric self-consistent, Maxwell, Ponte Castañeda–Willis, differential
  • Eshelby at both orders — second order for conductivity, fourth order for elasticity; isotropic and anisotropic media; ellipsoids, cracks, n-layer spheres, generic inclusions
  • Time and frequency — aging linear viscoelasticity and complex moduli, so a creep or a dynamic modulus follows from the same microstructure
  • Derivatives of the moduli — the sensitivity of the macroscopic stiffness to the lower-scale moduli, which is what makes secant-based nonlinear homogenization possible
  • C++ core, Python interface — the numerics are compiled; the modeling is scripted next to NumPy and Matplotlib, in a notebook or a plain script

Installation

Binary wheels for Windows and Linux, Python 3.7 to 3.12, are attached to the Zenodo record the DOI below resolves to — pick the file matching your interpreter, then pip install it.

Documentation

The documentation is a course — tensor calculus, Eshelby and Hill tensors, concentration, schemes, viscoelasticity, nonlinear homogenization — with every tutorial runnable, at jfbarthelemy.github.io/echoes.

License

The wheels are distributed under the ECHOES Academic and Research License: free use for non-commercial research, academic and teaching purposes; any commercial use requires a separate written agreement with the copyright holder. The manual is licensed CC BY-SA 4.0.