Continuum Mech. Thermodyn. https://doi.org/10.1007/s00161-018-0633-y ORIGINAL ARTICLE Péter B. Béda Dynamic stability and bifurcation analysis in fractional thermodynamics Received: 13 September 2017 / Accepted: 9 February 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract In mechanics, viscoelasticity was the first field of applications in studying geomaterials. Further pos- sibilities arise in spatial non-locality. Non-local materials were already studied in the 1960s by several authors as a part of continuum mechanics and are still in focus of interest because of the rising importance of materials with internal micro- and nano-structure. When material instability gained more interest, non-local behavior appeared in a different aspect. The problem was concerned to numerical analysis, because then instability zones exhibited singular properties for local constitutive equations. In dynamic stability analysis, mathemati- cal aspects of non-locality were studied by using the theory of dynamic systems. There the basic set of equations describing the behavior of continua was transformed to an abstract dynamic system consisting of differential operators acting on the perturbation field variables. Such functions should satisfy homogeneous boundary con- ditions and act as indicators of stability of a selected state of the body under consideration. Dynamic systems approach results in conditions for cases, when the differential operators have critical eigenvalues of zero real parts (dynamic stability or instability conditions). When the critical eigenvalues have non-trivial eigenspace, the way of loss of stability is classified as a typical (or generic) bifurcation. Our experiences show that material non-locality and the generic nature of bifurcation at instability are connected, and the basic functions of the non-trivial eigenspace can be used to determine internal length quantities of non-local mechanics. Fractional calculus is already successfully used in thermo-elasticity. In the paper, non-locality is introduced via fractional strain into the constitutive relations of various conventional types. Then, by defining dynamic systems, sta- bility and bifurcation are studied for states of thermo-mechanical solids. Stability conditions and genericity conditions are presented for constitutive relations under consideration. Keywords Fractional strain · Thermal stress · Dynamic system 1 Introduction The roots of fractional calculus go back to Leibniz (1695) and Euler (1730) as a natural extension of calculus [1], and most definitions were already given in the golden age of analysis by Liouville, Riemann and others. In recent years, mechanics has brought it back into the center of interest, because it is a useful tool to model non-locality. Such non-locality is an old problem of solid mechanics [2] and is usually treated by using integrals and second gradients [3, 4]. The first application was in viscoelasticity [5], as a kind of non-local time effect. In material instability problems, spatial non-locality plays an important role in nonlinear bifurcation (post- bifurcation) investigations, while in several cases, local formulation of the basic equations of solid bodies may result in an indeterminate behavior [6]. In numerical studies, such behavior appears as mesh sensitivity, when Communicated by Attila R. Imre. P. B. Béda (B) Budapest University of Technology and Economics, M˝ uegyetem rkp. 1-3, Budapest 1111, Hungary E-mail: bedap@kme.bme.hu