A constitutive relation describing the shear-banding transition

Area of Science:

• Fluid dynamics
• Rheology
• Materials science

Background:

• Shear banding is a critical phenomenon in materials science, requiring extensions to standard constitutive relations for accurate description.
• Understanding the underlying physics of shear banding is crucial for predicting material failure and designing new materials.

Purpose of the Study:

• To propose and analyze a novel hydrodynamic extension to constitutive relations for describing shear banding.
• To investigate the physical origins and mathematical similarities of shear banding to spinodal decomposition.
• To explore the kinetics and conditions leading to shear banding under different experimental controls.

Main Methods:

• Analysis of the linearized Navier-Stokes equation in a two-plate geometry.
• Comparison with the Cahn-Hilliard equation for spinodal decomposition.
• Derivation of a modified Maxwell equal area construction.
• Numerical simulation of the shear-banding transition kinetics.

Main Results:

• The proposed model shows similarities to the Cahn-Hilliard equation, predicting a most rapidly growing wavelength for shear band formation.
• A modified Maxwell construction indicates true shear bands do not coexist under controlled stress conditions.
• Numerical simulations reveal multiple stationary states under controlled shear and the possibility of shear banding via "spinodal demixing" or "condensation."

Conclusions:

• The hydrodynamic model provides a framework for understanding shear banding, linking it to spinodal decomposition phenomena.
• The study highlights the distinct behaviors of shear banding under controlled shear versus controlled stress conditions.
• Further investigation into flow gradients in other directions is suggested for controlled stress scenarios.