Variational Methods in Thermoelasticity and Dynamics

Variational methods provide a unifying framework for modelling the interaction of thermal and elastic phenomena in solids and fluids. Rooted in the principle of least action, these approaches cast governing equations as extrema of energy or action functionals, enabling systematic derivation of coupled mechanical and thermal evolution laws. Extensions of classical Hamiltonian and Lagrangian formalisms now incorporate irreversible processes through entropy production constraints, non-holonomic conditions and higher-order gradients in displacement and temperature. Such generalised variational principles underpin modern analyses of strain-gradient and microstructured media, thermopiezoelectric materials and Cosserat continua. They also guide the development of stable numerical schemes for complex boundary-value problems and foster optimisation of materials with tailored thermal-mechanical responses. The global significance of this research spans aerospace structures exposed to extreme heat loads, energy-conversion devices, microelectromechanical systems and novel metamaterials where accurate coupling of heat and stress at multiple scales is essential.

Research from Nature Portfolio

No recent Nature Portfolio content available.

Variational Methods in Thermoelasticity and Dynamics publication trend

The graph below shows the total number of articles in variational methods in thermoelasticity and dynamics across all publications each year (not limited to Nature Index journals).

Technical terms

Variational principle: A statement that the true evolution of a system extremises an action or energy functional, from which governing equations can be derived.

Lagrangian: A function of field variables, their rates and spatial gradients whose integral over time defines the action to be extremised.

Hamiltonian: The Legendre transform of the Lagrangian, representing the total energy of a system in terms of canonical variables and generating time evolution via Hamilton’s equations.

Strain gradient: Higher-order spatial derivatives of displacement fields that capture size-dependent and microstructural effects in continuum theories.

Entropy production inequality: A thermodynamic constraint enforcing the second law by bounding irreversible processes, used to extend variational formulations to non-equilibrium systems.

References

1. From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective. Entropy (2018).
2. Incremental least action principle in the framework of thermodynamics of irreversible processes. Physical Review Research (2020).
3. A strain gradient problem with a fourth-order thermal law. Journal of Computational and Applied Mathematics (2024).
4. Uniqueness of Solutions in Thermopiezoelectricity of Nonsimple Materials. Entropy (2022).
5. Continuous dependence in thermopiezoelectricity of nonsimple materials. Meccanica (2024).
6. On the thermal stresses in chiral porous elastic beams. Continuum Mechanics and Thermodynamics (2023).
7. Behaviour of solutions for a thermoelastic Cosserat medium with temperature gradients. Continuum Mechanics and Thermodynamics (2024).