[Paper Review] Fifth-degree elastic potential for predictive stress-strain relations and elastic instabilities under large strain and complex loading in Si

Hao Chen, Nikolai A. Zarkevich+3 more|arXiv (Cornell University)|Feb 14, 2020

Microstructure and mechanical properties37 references4 citations

TL;DR

This paper proposes a fifth-degree elastic potential for silicon (Si I) derived from density functional theory (DFT) data to accurately model large-strain stress-strain behavior and elastic instabilities under complex loading. By minimizing error across large strain ranges including instability points, the model achieves excellent agreement with DFT for Cauchy stress–Lagrangian strain curves, phase transition conditions, and shear instabilities, outperforming lower-order potentials that fail to capture these phenomena reliably.

ABSTRACT

Materials under complex loading develop large strains and often transition via an elastic instability, as observed in both simple and complex systems. Here, we present Si I under large strain in terms of Lagrangian strain by an $5^{th}$-order elastic potential found by minimizing error relative to density functional theory (DFT) results. The Cauchy stress-Lagrangian strain curves for arbitrary complex loadings are in excellent correspondence with DFT results, including elastic instability driving Si I$ ightarrow$II phase transformation (PT) and the shear instabilities. PT conditions for Si I$ ightarrow$II under action of cubic axial stresses are linear in Cauchy stresses in agreement with DFT predictions. Such elastic potential permits study of elastic instabilities and orientational dependence leading to different PTs, slip, twinning, or fracture, providing a fundamental basis for continuum simulations of crystal behavior under extreme loading.

Motivation & Objective

• To develop a predictive elastic potential capable of describing large-strain nonlinear elasticity and elastic instabilities in silicon (Si I) under complex loading conditions.
• To overcome the limitations of third- and fourth-order elastic constants, which fail to describe lattice instabilities at large strains.
• To provide a fundamental continuum framework for simulating phase transitions, slip, twinning, and fracture under extreme static and dynamic loading.
• To enable accurate prediction of phase transition conditions, such as Si I → Si II, under arbitrary stress states.
• To establish a higher-order elastic potential that captures nontrivial coupling between normal and shear stresses, crucial for material behavior under multidimensional loading.

Proposed method

• The elastic potential is formulated as a fifth-degree polynomial in Lagrangian strain components (6 independent components), using Voigt notation for compact representation.
• The coefficients of the fifth-degree potential are determined by minimizing the error between predicted and DFT-calculated Cauchy stress–strain curves over a large strain range, including instability points.
• The model uses DFT data as a reference for training, ensuring accuracy across both elastic and unstable regimes, including the onset of the Si I → Si II phase transition.
• Stress-strain responses under various complex loadings (uniaxial, biaxial, hydrostatic, and combined normal-shear loading) are computed and compared directly with DFT results.
• Phase transition conditions are derived analytically and validated against DFT predictions, showing linear dependence on Cauchy stresses under cubic axial loading.
• The method enables systematic study of how different loading paths influence shear strength, critical strain, and stability, including the emergence of plateau-like regions indicative of atomic plasticity.

Experimental results

Research questions

• RQ1 Can a fifth-degree elastic potential accurately predict stress-strain responses in silicon under large strains and complex loading, including elastic instabilities?
• RQ2 How does the inclusion of fifth-order terms improve the prediction of phase transition conditions (e.g., Si I → Si II) compared to lower-order elastic constants?
• RQ3 What is the influence of multidimensional loading (e.g., superposition of normal and shear strains) on the theoretical shear strength and critical strain for instability?
• RQ4 To what extent does the model capture nontrivial coupling between normal and shear stresses that affects stability and phase transformation pathways?
• RQ5 Can this approach serve as a general, predictive framework for continuum simulations of crystal behavior under extreme loading, including fracture and twinning?

Key findings

• The fifth-degree elastic potential reproduces DFT-calculated Cauchy stress–Lagrangian strain curves for arbitrary complex loadings with excellent agreement, even at strains exceeding 0.35.
• The model accurately predicts the Si I → Si II phase transition under cubic axial stresses, with transition conditions linear in Cauchy stresses, matching DFT predictions exactly.
• Superposition of uniaxial or biaxial compression along the shear direction reduces the ultimate shear strength by up to 2 GPa and alters the critical shear strain, with isotropic compression reducing it to 4.4 GPa (4.31 GPa from DFT).
• The model reveals that certain loading paths, such as η₁ = η₂ = -0.5η₄, significantly enhance the plateau-like region in shear stress-strain curves, indicating increased atomic ductility.
• Lower-order potentials (third- and fourth-degree) fail to reproduce the correct instability behavior and stress-strain curves at large strains, confirming the necessity of fifth-order terms.
• The model enables predictive simulation of multiple instabilities—phase transitions, amorphization, slip, twinning, and fracture—under orientationally dependent extreme loading, offering a new foundation for continuum mechanics.