Effective moduli of porous elastoplastic rocks: micromechanical modeling and numerical verification

The macroscopic properties of porous rocks are governed by their complex microstructure and the constitutive behavior of the solid matrix. Traditional effective media models, often based on analytical solutions for isolated cavities or inclusions, are typically limited to linear elastic or viscous rheologies. However, the elastic-plastic response characteristic of rocks is rarely captured by these approaches. Building on our previous work [1], which provided an analytical solution for elastoplastic compaction and revealed how the effective bulk modulus depends on both elastic properties and yield strength, we extend the framework to include shear effects. Our model demonstrates that plastic yielding couples shear and volumetric responses, leading to phenomena such as shear-enhanced compaction and significant deviations from traditional linear predictions.

Here, we extend this framework [1] by deriving the effective shear modulus and exploring how plastic yielding jointly controls both bulk and shear responses [2]. We systematically test the limits of the analytical solution against high-resolution numerical models of multiple interacting voids, in both 2D (cylindrical/elliptical voids) and 3D (spherical voids) geometries. Our results show that the elastoplastic analytical solution remains valid for rocks with higher porosities of up to 20%, extending beyond typical dilute-distribution assumptions. For cylindrical voids in the 2D case, the merging of plastic zones leads to a sharp decrease in the effective bulk modulus and the onset of full pore collapse. The analytical solution for the critical pressure at which full pore collapse occurs agrees well with the numerical results. For spherical voids arranged in 3D configurations, plastic zone merging leads to a more gradual reduction in the effective bulk modulus. Furthermore, under the influence of shear stresses, the development of aligned plastic zones leads to stress-induced anisotropy in initially isotropic materials.

For numerical homogenization, we employ two complementary approaches: a GPU‑accelerated finite‑difference solver (dynamic relaxation method) and a finite/spectral‑element solver (based on CAE Fidesys [3] computational kernel). To ensure the reliability of the calculated effective moduli, we implement periodic boundary conditions, which we find essential for minimizing spurious boundary effects in both 2D and 3D simulations. The developed numerical tools are capable of handling complex rock microstructures, broadening the potential applications of this modeling approach.

MY acknowledges the support by the Russian Science Foundation under grant 24-77-10022.

 

1. Yarushina, V.M., Podladchikov Y.Y., & Wang L.H. (2020). Model for (de)compaction and porosity waves in porous rocks under shear stresses. Journal of Geophysical Research: Solid Earth, 125(8), e2020JB019683.

2. Yakovlev, M.Ya., Yarushina, V.M., Bystrov, I.D., Nikitin, L.S., & Podladchikov, Yu.Yu. (2025). Benchmarking effective moduli in porous elastoplastic materials. International Journal of Mechanical Sciences, 306, 110854.

3. https://cae-fidesys.com/