Aspects of the morphological character and stability of two-phase states in non-elliptic solids

Citation

Fried, Eliot (1991) Aspects of the morphological character and stability of two-phase states in non-elliptic solids. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/WJKM-GB39. https://resolver.caltech.edu/CaltechETD:etd-01302007-160351

Abstract

Part I. This work focuses on the construction of equilibrated two-phase antiplane shear deformations of a non-elliptic isotropic and incompressible hyperelastic material. It is shown that this material can sustain metastable two-phase equilibria which are neither piecewise homogeneous nor axisymmetric, but, rather, involve non-planar interfaces which completely segregate inhomogeneously deformed material in distinct elliptic phases. These results are obtained by studying a constrained boundary value problem involving an interface across which the deformation gradient jumps. The boundary value problem is recast as an integral equation and conditions on the interface sufficient to guarantee the existence of a solution to this equation are obtained. The contraints, which enforce the segregation of material in the two elliptic phases, are then studied. Sufficient conditions for their satisfaction are also secured. These involve additional restrictions on the interface across which the deformation gradient jumps-which, with all restrictions satisfied, constitutes a phase boundary. An uncountably infinite number of such phase boundaries are shown to exist. It is demonstrated that, for each of these, there exists a solution - unique up to an additive constant - for the constrained boundary value problem. As an illustration, approximate solutions which correspond to a particular class of phase boundaries are then constructed. Finally, the kinetics and stability of an arbitrary element within this class of phase boundaries are analyzed in the context of a quasistatic motion.

Part II. This work investigates the linear stability of an antiplane shear motion which involves a planar phase boundary in an arbitrary element of a wide class of non-elliptic generalized neo-Hookean materials which have two distinct elliptic phases. It is shown, via a normal mode analysis, that, in the absence of inertial effects, such a process is linearly unstable with respect to a large class of disturbances if and only if the kinetic response function - a constitutively supplied entity which gives the normal velocity of a phase boundary in terms of the driving traction which acts on it or vice versa - is locally decreasing as a function of the appropriate argument. An alternate analysis, in which the linear stability problem is recast as a functional equation for the interface position, allows the interface to be tracked subsequent to perturbation. A particular choice of the initial disturbance is used to show that, in the case of an unstable response, the morphological character of the phase boundary evolves to qualitatively resemble the plate-like structures which are found in displacive solid-solid phase transformations. In the presence of inertial effects a combination of normal mode and energy analyses are used to show that the condition which is necessary and sufficient for instability with respect to the relevant class of perturbations in the absence of inertia remains necessary for the entire class of perturbations and sufficient for all but a very special, and physically unrealistic, subclass of these perturbations. The linear stability of the relevant process depends, therefore, entirely upon the transformation kinetics intrinsic to the kinetic response function.

Part III. This investigation is directed toward understanding the role of coupled mechanical and thermal effects in the linear stability of an isothermal antiplane shear motion which involves a single planar phase boundary in a non-elliptic thermoelastic material which has multiple elliptic phases. When the relevant process is static - so that the phase boundary does not move prior to the imposition of the disturbance - it is shown to be linearly stable. However, when the process involves a steadily propagating phase boundary it may be linearly unstable. Various conditions sufficient to guarantee the linear instability of the process are obtained. These conditions depend on the monotonicity of the kinetic response function - a constitutively supplied entity which relates the driving traction acting on a phase boundary to the local absolute temperature and the normal velocity of the phase boundary - and, in certain cases, on the spectrum of wave-numbers associated with the perturbation to which the process is subjected. Inertia is found to play an insignificant role in the qualitative features of the aforementioned sufficient conditions. It is shown, in particular, that instability can arise even when the normal velocity of the phase boundary is an increasing function of the driving traction if the temperature dependence in the kinetic response function is of a suitable nature. Significantly, the instability which is present in this setting occurs only in the long waves of the Fourier decomposition of the moving phase boundary, implying that the interface prefers to be highly wrinkled.

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