Generic bifurcations in fractional thermo‐mechanics with peridyamic effects

Abstract

AbstractIn dynamic stability analysis, mathematical aspects of non‐locality are studied by using the theory of dynamical systems. The set of basic equations describing the behavior of continua is transformed to an abstract dynamical system. Such 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 nontrivial eigenspace, the way of loss of stability can be classified as a 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. In the paper, non‐local effects are introduced via fractional stress and strain into the equation of motion and into constitutive relations. Then, by defining dynamical systems, stability and bifurcation are studied for states of thermo‐mechanic solids. Conditions for stability and generic bifurcation are presented for constitutive relations under consideration.