# Multi-Scale Theory of Elasticity for Geomaterials

## TL;DR

This paper develops a cross-scale elasticity theory for geomaterials that aligns with thermodynamics, incorporates volume ratios, and extends to soft condensed matter, addressing limitations of classical models.

## Contribution

It introduces a thermodynamically compatible elasticity theory for geomaterials that includes volume ratios and distinguishes deformation components, broadening applicability.

## Key findings

- The theory is compatible with empirical models and thermodynamics.
- It incorporates volume ratios as internal variables.
- It extends elasticity theory to soft condensed matter.

## Abstract

The modern theory of elasticity and the first law of thermodynamics are cornerstones of engineering science that share the concept of reversibility. Engineering researchers have known for four decades that the modern theory violates the first law of thermodynamics when applied to the more commonly accepted empirical models of geomaterial stiffness. This paper develops a cross-scale theory of elasticity that is compatible with the empirical models and the first law of thermodynamics. This theory includes a material sample's total-volume to solid-volume ratio as an independent internal variable, distinguishes deformation into uniform and contraction-swelling components, introduces a uniformity surface that partitions stress space into contraction and swelling sub-domains, couples the macroscopic properties to the volume ratio and extrapolates the accepted empirical models to states that include shear stress. This paper broadens the scope of the theory of elasticity to include soft condensed matter.