Exponential stability in Mindlin's Form II gradient thermoelasticity with microtemperatures of type III

Area of Science:

• Continuum Mechanics
• Materials Science
• Thermodynamics

Background:

• Increasing use of materials with microstructural features at mechanical and thermal levels.
• Need for advanced theories to model complex material behaviors.
• Existing theories may not fully capture micro-inertia and microtemperature effects.

Purpose of the Study:

• Derive a nonlinear strain gradient theory for thermoelastic materials incorporating microtemperatures and micro-inertia.
• Investigate the well-posedness of the derived model.
• Establish conditions for exponential stability.

Main Methods:

• Application of Mindlin's Form II gradient elasticity theory for elastic behavior.
• Utilizing Green and Naghdi's type III entropy balance for thermal behavior.
• Employing semigroup theory to analyze the well-posedness of the governing equations.

Main Results:

• Derivation of a nonlinear strain gradient theory for thermoelastic materials with microtemperatures and micro-inertia.
• Demonstration that the coupled equations lack exponential decay due to high-order derivatives and microtemperatures.
• Identification of a specific frictional damping mechanism that achieves exponential stability.

Conclusions:

• The developed theory provides a comprehensive framework for modeling advanced thermoelastic materials.
• The study highlights the challenges in achieving exponential decay in such complex systems.
• A practical damping strategy is proposed to ensure stability in microstructured thermoelastic materials.