A Unified Formulation of Gibbs Free Energy for Solids and Fluids

The Gibbs free energy plays a central role in the modeling of materials within both solid and fluid mechanics, dating back to the seminal work of Gibbs (1873). While its formulation in fluid mechanics is well established and largely undisputed, the situation in solid mechanics is considerably less uniform. Over the past decades, a variety of formulations have been proposed, see, e.g., Truesdell and Toupin (1969), Landau and Lifschitz (1975), Lurie (2005), often based on different choices of kinematic variables, stress measures, and thermodynamic potentials. These approaches are frequently incompatible and may lead to conceptual inconsistencies and therefore have been criticized, see, e.g., Silhavy (1997), Dreyer and Duderstadt (2008), particularly when attempting to establish a unified description that consistently encompasses both solids and fluids. In this work, existing formulations of the Gibbs free energy for solids are critically analyzed and compared with their counterparts in fluid mechanics. Building on these considerations, a geometrically nonlinear formulation of the Gibbs free energy is proposed. The proposed formulation provides a unified thermodynamic structure that avoids the inconsistencies of earlier approaches and offers a transparent transition between solid-like and fluid-like behavior. Several applications are discussed to illustrate the versatility of the framework and its implications for constitutive modeling in nonlinear continuum mechanics.

References

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