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The undamped, finite amplitude horizontal motion of a load supported symmetrically between identical incompressible, isotropic hyperelastic springs, each subjected to an initial finite uniaxial static stretch, is formulated in general terms. The small amplitude motion of the load about the deformed static state is discussed; and the periodicity of the arbitrary finite amplitude motion is established for all such elastic materials for which certain conditions on the engineering stress and the strain energy function hold. The exact solution for the finite vibration of the load is then derived for the classical neo-Hookean model. The vibrational period is obtained in terms of the complete Heuman lambda-function whose properties are well-known. Dependence of the period and hence the frequency on the physical parameters of the system is investigated and the results are displayed graphically.

The finite amplitude, free radial oscillations of a thick-walled circular cylindrical tube are studied for an arbitrary incompressible, isotropic and homogeneous rubber-like material having limiting molecular chain extensibility. First, based on classical results for hyperelastic tubes, some results for thick-walled Mooney-Rivlin tubes are described graphically in the phase plane. Then the periodicity of the finite amplitude, free oscillations of a general limited elastic, thick-walled tube is studied; and some analytical results for the Gent model are illustrated in several numerical examples. Results for thick-walled Gent tubes are compared with those for corresponding Mooney-Rivlin tubes; and the motion of thin-walled Gent tubes is illustrated in the phase plane. Physical conclusions are presented. The period of small amplitude oscillations of an arbitrary elastic or limited elastic tube is derived from relations obtained by a linearization of a general class of equations of which the tube problem is a special case. Classical results of the linear theory are thereby recovered and compared with results for Mooney-Rivlin and Gent tubes.

The wave speed criterion is applied to discuss restrictions on the material response functions of an isotropic, Bell constrained material in order that small amplitude, plane waves may propagate in the material with real-wave speeds. The case of a body subjected to a biaxial stress is related to the elastic stability of a Bell constrained slab. We show that two cases among five reported by Beatty and Pan in study of the elastic stability of a thick plate cannot support such waves. These cases, therefore, are not possible and must be excluded as potential solutions. The question of the stability of the equibiaxial deformations of a Bell constrained material under an all-around Cauchy stress, suggested in primary work of Beatty and Hayes on homogeneous deformations of Bell materials, is investigated. It is shown that for a general Bell material model, the state of equibiaxial deformation under an all-around Cauchy stress is materially unstable, unless all stretches are equal to one, their value in the initial undistorted state.

The elastic stability of an internally constrained thick plate is investigated on the basis of the well-known Euler dead load criterion of existence of adjacent equilibrium states. Two classes of internally constrained materials are examined—the Bell constrained material characterized by tr V=3, and the incompressible material for which det V=1, V being the left-stretch tensor. Although this study focuses mainly on instability of a Bell constrained slab, some new results are presented as well for an incompressible neo-Hookean slab. Graphical illustrations of the critical stretch as a function of the slenderness measure of the slab are provided for two classes of Bell materials—the simple hyperelastic Bell material model and an empirical model first described in experiments by Bell. The oscillatory nature of the stability results for Bell’s empirical model differ from the monotone solutions found for both the simple hyperelastic Bell material and the neo-Hookean material models. Estimates for buckling loads of thin plates deduced from the thick-plate analysis are compared with classical results. The critical load found for the simple hyperelastic Bell material coincides with the result deduced from classical thin plate theory, while the solution found for Bell’s law, which is parabolic in the engineering strain, does not. The half-space stability problem is examined as an extension of results deduced for a thick plate based on Bell’s law.

Equations for small deformations superimposed on a finite deformation of an arbitrary isotropic and compressible, non-linearly elastic solid constrained by inextensible fibers are reviewed. The instability of a thick slab of infinite extent reinforced by strong fibers oriented through its thickness normal to the loading axis is then examined by application of Euler's criterion for dead loading. Solutions are obtained which suggest that special results in the literature on the tension instability of an ideal reinforced Blatz-Ko foamed rubber slab are questionable. It is proved that if the non-linearly elastic matrix material of the ideal reinforced slab satisfies the empirical inequalities and the tension-extension principle, the only mode of instability consistent with plane strain conditions occurs for compression loading, never for tension loading. The class of generalized Blatz-Ko materials that respect these rules is described; the foamed rubber model is not in this class.

Stability of the oscillatory motion of a body supported by simple shear springs is examined relative to a vehicle that executes a steady circular turn about an arbitrary axis fixed in the vertical plane. The problem is formulated for a general isotropic elastic material whose shear response function is a positive, even function of the amount of shear. A general sufficient condition bounding the normalized vehicle angular speed Ω by a certain critical angular speed Ωc, which depends on the vehicle and load suspension orientation design parameters alone, is obtained for periodic, and hence stable, free relative oscillations of the load for arbitrary initial data. Stability in the forced vibration problem also is discussed. For the special class of materials whose shear response function is a constant, a familiar class that includes the Mooney-Rivlin, Blatz-Ko, and Hadamard material models, for example, the free oscillatory motion of the load is stable if and only if Ω

The problem of the finite amplitude, free vertical oscillatory motion of a massattached to a neo-Hookean rubberlike string is solved exactly in terms of elementary functions and the Heuman lambda function, which is related to the elliptic integral of the third kind. Hence, the period of the oscillations for the various possible motions may be computed from tables of values of the complete lambda function. It is shown that the results differ significantly from those obtained elsewhere for a string having linearly elastic behavior; and all of the results are described graphically.

An exact energy criterion for elastic stability of a hyperelastic body subject to conservative, configuration-dependent forces is developed and subsequently expressed in terms of surface-traction potentials by aid of a recent general theorem due to Sewell. The surface potential for hydrostatic loading is formulated, and the aforementioned general criterion is specialized for these. The effect of rigid body rotations on the stability of a body subject on the boundary to both dead loads and hydrostatic loads is examined, and it is found desirable to restrict the class of virtual deformations so as to preserve equilibrium of moments. Several theorems concerning this case are then proved. Linearized criteria appropriate for both compressible and incompressible, hyperelastic materials under hydrostatic loads are derived and stability of the undistorted states of isotropic materials is investigated. Inequalities long accepted as characterizing reasonable material behavior in the classical linearized theory of elasticity are here proved valid for general isotropic, hyperelastic materials in finite strain.

Incremental stability criteria appropriate for compressible and incompressible, general isotropic elastic solids are used to study the stability of all undistorted states of an arbitrary body subject to dead loads corresponding to a uniform hydrostatic stress. General conditions sufficient for stability of all such hydrostatic stress states are derived. The results indicate a geometrical dependence for the apparent shear and bulk moduli. The relation of the main results (4.10) and (4.21) to uniqueness theorems in the theory of small deformations superimposed on large, to broad plausibility restrictions imposed on the elasticities so as to assure physically reasonable material behavior, and to the propagation of elastic waves is also outlined; these relations constitute only specific application of certain general theorems already in the literature. Finally, it is shown that the principal results coincide with estimates that may be gotten by aid of Holden's inequality.