On Thermomechanics of Elastic Stability [antikvár]

XfA. 21, 1970 On Thermomechaiiics of Elastic Stabilítv 539 small íf they are initially suífícíentlv smalL This críteríon is static in characíer. Hence one may consider 'vírtual' rather than the actual deviations of the solid from its equilibrium state ín decídíng on the posítíve-defínite nature of the potential energy, resultíng ín the ostiai static stability analysis, see Pearson [4] and Hill "51. In the study of the mechanics of deformable bodies it is common to represent the mechanical action of all the external effects upon a body by means of body forces and surface tractions. The símplest forces of this kind are dead loads which retain their magnitude as well as their initial direction as the body deforms. These forces are conservative in the sense that the work done by them in taking the body from a given configuratíon to a new one is independent of the path in physícal space. In generál however, the mechanical action of the environment upon a given deformable continuum may not be representable by dead loads. The continuum may interact with the environment and hence may be subjected to forces which depend not only upon the partícle posítíons ín physícal space, but alsó upon the configuratíon, velocity field and other kinevnatical quantities which describe the motion. The action of the environment upon the continuum is therefore dependent on the motion and hence the forces which are used to represent this action may be called motion-dependent loads. If the forces that are prescribed in this manner depend only upon the configuratíon, we shall refer to them as configuration-dependent forces, as proposed by Sewell [6]. In a similar way we define velocity-dependent and acceleration-dependent forces. It is clear that the work done by forces of this kind is in generál dependent upon the manner in which the body is carried from its initial configuratíon to a final one, as well as upon these two confígurations. Thus when the stability of an equilibrium configuratíon of, say, a finitely-deformed elastic body under such loads is being sought, one cannot expect to employ the energy method in the manner defined above for dead loads. Here, as has been first demonstrated by Ziegler in two memoirs [7, 8], the static approach becomes inadequate. We are therefore compelled to adopt a dynamic view of the stability phenomenon with all the added complications which arise from the energy dissipation that accompanies all dynamical processes. In this manner we are naturallv lead into the realm of the thermodynamics of deformation. In this paper, in order to explore the thermodynamical conditions of the stability of equilibrium states of elastic solids that are subjected to generál motion-dependent forces, we shall combine the results of the studies in [1-3] with those reported by Nemat-Nasser, who in [9] considered the local stability of a finitely deformed solid with internál energy dissipation but ignored all thermal effects. The present paper therefore represents further extension and generalization of the results recently reported by the author in [10]. As a consequence of our study we shall demonstrate that not only the usual viscous damping forces, but all sources which contribute to the entropy production during the deformation of the solid, may directly affect the stability of its given configuratíon. Hence the assessment of stability in the presence of generál motion-dependent forces requires a complete knowledge of the mechanisms which contribute to the generation of entropy within the body as it goes through nonequilibrium states. This is in contrast to the case of conservative loading. To measure the deviations from a given equilibrium configuratíon of the solid, various metrics may be introduced. Clearly enough the verdict of stability is therefore dependent on the nature of these metrics. A state which may be stable with