A micromorphic constitutive framework for (non)linear softening modeling
Abstract
Nonlinear shear softening in soils and rocks is commonly described using empirical modulus-reduction laws. In this work, we investigate nonlinear softening within the Eringen--Mindlin micromorphic framework by interpreting its additional constitutive parameters in terms of coupling, internal relaxation, and micro-inertia. The micromorphic continuum is viewed as a dynamically enriched effective medium in which the macroscopic deformation interacts with an internal microstructural deformation field. This interaction naturally produces frequency-dependent effective elastic properties and provides a physical connection between macroscopic wave propagation and unresolved microstructural dynamics. We show that nonlinear shear softening can be described by a progressive reduction of the coupling between macroscopic and internal deformation. As this coupling weakens with increasing shear strain, the effective shear modulus decreases. In the low-frequency limit, the resulting constitutive relation takes a hyperbolic form closely related to the classical Hardin--Drnevich law. The model is validated against laboratory measurements of shear-modulus reduction in sand and reproduces the observed nonlinear behavior with only a small number of physically interpretable parameters. The proposed framework provides a new description of nonlinear softening and frequency-dependent behavior without requiring explicit knowledge of the underlying microstructural geometry. Classical empirical modulus-reduction curves emerge as a macroscopic consequence of the interaction between macroscopic and internal deformation.
Related
- Modeling soil volumetric water content based on climatic factors in the Tripoli region (Libya) using multiple linear regression
- Using GIS, remote-sensing and linear modeling to prioritize spatially explicit action against erosion and sediment transport within Lake Victoria watershed in Kenya
- An accurate and efficient framework for modeling multimetal competitive adsorption on clay minerals
- Tailored Graphene Oxide Incorporated PES/CA Mixed‐Matrix Loose Nanofiltration Membranes for Water Softening and Divalent Salt Removal
- A global internal tide modeling framework for improving satellite observations of fine-scale ocean circulation
BibTeX
@misc{abreu2026micromorphic,
title = {{A micromorphic constitutive framework for (non)linear softening modeling}},
author = {Rafael Abreu},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29503},
eprint = {2609.29503},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05802211v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A micromorphic constitutive framework for (non)linear softening modeling
AU - Rafael Abreu
PY - 2026
DA - 2026/09/01
PB - HAL
DO - 10.48550/arxiv.2609.29503
UR - https://hal.science/hal-05802211v1
AB - Nonlinear shear softening in soils and rocks is commonly described using empirical modulus-reduction laws. In this work, we investigate nonlinear softening within the Eringen--Mindlin micromorphic framework by interpreting its additional constitutive parameters in terms of coupling, internal relaxation, and micro-inertia. The micromorphic continuum is viewed as a dynamically enriched effective medium in which the macroscopic deformation interacts with an internal microstructural deformation field. This interaction naturally produces frequency-dependent effective elastic properties and provides a physical connection between macroscopic wave propagation and unresolved microstructural dynamics. We show that nonlinear shear softening can be described by a progressive reduction of the coupling between macroscopic and internal deformation. As this coupling weakens with increasing shear strain, the effective shear modulus decreases. In the low-frequency limit, the resulting constitutive relation takes a hyperbolic form closely related to the classical Hardin--Drnevich law. The model is validated against laboratory measurements of shear-modulus reduction in sand and reproduces the observed nonlinear behavior with only a small number of physically interpretable parameters. The proposed framework provides a new description of nonlinear softening and frequency-dependent behavior without requiring explicit knowledge of the underlying microstructural geometry. Classical empirical modulus-reduction curves emerge as a macroscopic consequence of the interaction between macroscopic and internal deformation.
ER -