The saga of internal variables of state in continuum thermo-mechanics (1893–2013)

The thermomechanics of continua with internal variables of state is often presented as an alternative to rational continuum mechanics or an extension of the classical linear theory of irreversible processes. It has become ubiquitous in the theories of dissipative mechanical phenomena such as viscoelasticity, plasticity, damage, phase transformations in deformable solids, and the rheology of many fluids. The present contribution highlights the historical development of this fruitful approach since the early works of Duhem, Bridgman, and Eckart, and then the quite definite framework more recently delineated in studies by Kestin and Muschik. It peruses the vast field of successful applications in both solid and fluid mechanics as well as in the theory of coupled fields in the second half of the twentieth century. It also necessarily envisages further generalizations of – deviations from, and contradictions with – this well defined framework with the introduction of gradients, and the difficulties posed by these in the interpretation of the variables. Globally, this pragmatic approach has proved its efficiency in dealing with complex behaviours of continua over a period that started more than a hundred years ago with ideas put forward by Duhem.“So hidden in this hocus-pocusThere lies the gift of double focus”(W.H. Auden, New Year Letter)