Constitutive Modeling with Single and Dual Internal Variables.

Area of Science:

• Continuum Mechanics
• Thermodynamics of Materials

Background:

• Phenomenological constitutive models utilize internal variables for material behavior.
• Existing models often employ a single internal variable formalism based on Coleman and Gurtin's thermodynamic approach.

Purpose of the Study:

• To explore the extension of internal variable theory to dual internal variables.
• To differentiate constitutive modeling using single versus dual internal variables.
• To present a thermodynamically consistent framework for internal variable treatment.

Main Methods:

• Application of the Clausius-Duhem inequality for a thermodynamically consistent framework.
• Utilizing the Onsagerian procedure and extra entropy flux for evolution equations of observable but uncontrollable internal variables.
• Illustrating distinctions with examples from heat conduction, linear thermoelasticity, and viscous fluids.

Main Results:

• Single internal variable models lead to parabolic evolution equations.
• Dual internal variable models result in hyperbolic evolution equations.
• A framework is presented requiring minimal a priori knowledge for internal variable treatment.

Conclusions:

• Dual internal variables provide a novel approach to constitutive modeling.
• The derived framework offers a consistent method for analyzing internal variables.
• The parabolic versus hyperbolic nature of evolution equations is a key distinction.