From Atoms to Continuum: Stress Control of Thermodynamic Equilibrium in Deforming Systems

The coupling between deformation, phase transformations, and chemical reactions governs key Earth processes, including mountain building, earthquakes, magma transport, reservoir stability, and glacier flow. Deformation of solids such as rocks, minerals, and ice generates non-hydrostatic stresses, yet fundamental disagreements persist on how stress controls the thermodynamics of solid–fluid equilibria and reactions. This knowledge gap limits our ability to predict the long-term stability of subsurface reservoirs critical to the energy transition, including nuclear waste repositories, CO₂ and hydrogen storage, and geothermal systems.

Thermodynamic equilibrium requires mechanical equilibrium. However, non-hydrostatic thermodynamic frameworks typically rely on simplified conceptual models that assume stresses within deforming solid grains are always homogeneous and equal to the far-field stress [1]. When extended to multigrain assemblages, the assumption of homogeneous stress in the solid matrix violates mechanical equilibrium and creates apparent inconsistencies between conditions of thermodynamic and mechanical equilibrium [2]. Such inconsistencies are often addressed by invoking the presence of fluid phases in rocks capable of sustaining non-hydrostatic stresses, as in pressure-solution theories [3].

Here, we show that atomistic molecular dynamics (MD) simulations provide an independent framework that avoids such assumptions by self-consistently simulating dissolution, precipitation, and stress evolution from atomic interactions [4,5]. By embedding meshes in large-scale MD simulations, we directly compute the continuum Cauchy stress field from atomic forces and velocities. These simulations reveal that stresses within solid phases of deforming multiphase systems are inherently heterogeneous. Stress patterns obtained from atomistic simulations quantitatively match analytical solutions and numerical continuum models, such as finite-element simulations, demonstrating that continuum mechanics accurately captures the stress state in agreement with atomistic descriptions.

MD simulations naturally capture elastic anisotropy, defect nucleation, stress heterogeneity, and interfacial instabilities, allowing mechanical and thermodynamic equilibrium to emerge spontaneously. Our results show that equilibrium in deforming rocks cannot be explained by normal-stress-only models. Instead, they confirm thermodynamic formulations [6,7] in which local equilibrium is governed by the full local stress state, resolving the apparent conflict between thermodynamics and mechanics and suggesting a significant shift in our understanding of deformation–reaction coupling in Earth materials.

References

1. Wheeler J (2020) Contrib Mineral Petrol 175:116

2. Hobbs BE, Ord A (2016) Earth-Sci Rev 163:190–233

3. Gratier J-P, Dysthe DK, Renard F (2013) Adv Geophys 54:47–179

4. Mazzucchelli ML, Moulas E, Kaus BJP, Speck T (2024) Am J Sci 324

5. Mazzucchelli ML, Moulas E, Schmalholz SM, et al. (2025) ESS Open Archive

6. Gibbs JW (1876) Trans Conn Acad Arts Sci 3:108–248

7. Frolov T, Mishin Y (2010) Phys Rev B 82(17), 1–14.