Nonequilibrium Thermodynamics in Solid Mechanics

Solids that are driven beyond their elastic limit exhibit strongly disspative and irreversible dynamical behaviors. Such behaviors call for the development of nonequilibrium approaches that go beyond standard equilibrium thermodynamics. In a recent work we have developed an internal-variable, effective-temperature non-equilibrium thermodynamics for glass-forming and polycrystalline materials driven away from thermodynamic equilibrium by external forces [1, 2]. The basic idea is that the slow configurational (structural) degrees of freedom of such materials are weakly coupled to the fast kinetic-vibrational degrees of freedom and therefore these two subsystems can be described by different temperatures during deformation. The configurational subsystem is defined by the mechanically stable positions of the constituent atoms, i.e. the "inherent structures", and is characterized by an effective temperature. The kinetic-vibrational subsystem is defined by the momenta and the displacements of the atoms at small distances away from their stable positions, and is characterized by the bath temperature.

In glass-forming materials, the configurational degrees of freedom include structural objects such as Shear-Transformation-Zones (STZ) and vacancy-like defects. In polycrystalline materials, the configurational degrees of freedom include structural defects such as interstitials, dislocations, disclinations, stacking faults and grain boundaries. A continuum level description of the configurational subsystem contains coarse-grained internal variables that are state variables that account for the effect of the evolving structure on various macroscopic mechanical properties. We highlighted the need for understanding how both energy and entropy are shared by the different components of the system and used the first and second laws of thermodynamics to constrain the equations of motion for the internal variables and the effective temperature.

The proposed framework should be supplemented with physics-based models for describing specific phenomena and systems. It was recently applied to three different problems:

A. Plastic deformation of amorphous materials [3].

The theory was based on the Shear-Transformation-Zones (STZ) model, which was already shown to be in agreement with a wide range of amorphous plasticity phenomena, including shear banding instabilities.

B. Dislocation-mediated plasticity and strain-hardening of polycrystalline materials [4].

The theory was shown to be in agreement with experimental strain-hardening data for Cu over a wide range of temperatures and strain rates. Furthermore, the transition between stage II and stage III hardening, including the observation that this transition occurs at smaller strains for higher temperatures, was predicted. Finally, power-law rate hardening in the strong-shock regime was explained.

C. A thermo-mechanical memory effect in glassy polymers [5].

The memory effect, the so-called "Kovacs effect", reveals some of the most subtle and important nonequilibrium features of glassy materials in which they deform irreversibly and remember their thermo-mechanical histories. The developed theory was shown to be in good quantitative agreement with extensive molecular dynamics simulations of ortho-terphenyl (OTP). 

[1] E. Bouchbinder and J.S. Langer, Nonequilibrium Thermodynamics of Driven Amorphous Materials I : Internal Degrees of Freedom and Volume Deformation, Phys. Rev. E 80, 031131 (2009).

[2] E. Bouchbinder and J.S. Langer, Nonequilibrium Thermodynamics of Driven Amorphous Materials II : Effective-Temperature Theory, Phys. Rev. E 80, 031132 (2009).

[3] E. Bouchbinder and J.S. Langer, Nonequilibrium Thermodynamics of Driven Amorphous Materials III : Shear-Transformation-Zone Plasticity, Phys. Rev. E 80, 031133 (2009).

[4] J.S. Langer, E. Bouchbinder and T. Lookman, Thermodynamic Theory of  Dislocation-mediated Plasticity, arXiv:0908.3913 (2009).

[5] E. Bouchbinder and J.S. Langer, Nonequilibrium Thermodynamics of the Kovacs Effect, arXiv:1001.3701 (2010).