On the surface instability of a highly elastic half-space

Abstract

A B S T R A C TThe phenomenon of surface instability of an isotropic half-space under biaxial plane stress is studied for compressible elastic materials in finite strain. Euler's method is used to derive the general form of the stability criterion, and analytical details are exhibited by special application to the class of hyperelastic Hadamard materials in two complementary cases: (i) the full solution is derived for the compressible, neo-Hookean members, and (it) the plane deformation solution is provided for every isotropic, elastic material and specific results are presented for the full Hadamard class. Results appropriate to incompressible Mooney-Rivlin materials are herein obtained as special limit cases. Several theorems are established and some of the conclusions are illustrated graphically. Z U S A M M E N F A S S U N GDas PhS.nomen yon OberflS.cheninstabilit~it yon einem isotropischen Halbraum unter zweiachsiger Flfichenanspannung wird ffir kompressibele, elaslische Materiale in begrenzter Belastung studiert. Eulers Methode wird angewendet, um die allgemeine Form von dem Stabilitfitskriterium abzuleiten und analytische Einzelheiten werden hervorgehoben durch besondere Anwendung mit der Klasse yon hyperelastischen Hadamard Materialen in zwei komplementfiren Fgllen: (i) die volle L6sung wird ffir die kompressiblen, Neo-Hookean Mitgliedern abgeleitet und (it) die Fl~ichendeformationl6sung wird ffir jedes, isotropisch es, elastisches Material geliefert und spezifische R esultate sind ffir die ganze Hadamard Klasse dargelegt. Ergebnisse eigen ffir inkompressibele Mooney-Rivlin Materiale sind hiermit erlangt als besondere Grenzf/ille. Mehrere Lehrsfitze sind festgesetzt und einige der Folgerungen sind illustriert veranschaulicht.

Plate waves in Hadamard materials

“…Thus in their work on the incompressible half-space, this is equivalent in our notation to the statement that a primary state is to be called unstable if we can find a pair of real values for kl, k2 for which oJ vanishes. We have seen that the condition for this is p2q < 0.296... < pq2 in agreement with [3]. Usmani and Beatty classify all other primary states as stable, with the reservation that "failure of a particular solution to provide an adjacent equilibrium position in Euler's method is a necessary but not sufficient condition for stability, so use of the term stable in our diagrams is not rigorously justified but merely conventional", [3], p. 260.…”

“…We have seen that the condition for this is p2q < 0.296... < pq2 in agreement with [3]. Usmani and Beatty classify all other primary states as stable, with the reservation that "failure of a particular solution to provide an adjacent equilibrium position in Euler's method is a necessary but not sufficient condition for stability, so use of the term stable in our diagrams is not rigorously justified but merely conventional", [3], p. 260. In fact some of these states are unstable as may be seen by the present analysis.…”

“…It is important now to assess the implications of these arguments for the stability of the material, and for this purpose it is convenient to recall first the work of Usmani and Beatty [3]. Their work is based on Euler's criterion of stability taken in the form: if for the same loading specified in K there exists an adjacent equilibrium configuration r* then the primary configuration K is called unstable for the prescribed loading.…”

“…Yet the perturbation is clearly unstable. We are forced to conclude, therefore, that the dispersion-equation argument is more useful for predicting instability, at any rate in this case, than is Euler's criterion in the form used in [3].…”

“…Usmani and Beatty in a recent paper [3] have examined the stability of a half-space of this material on the basis of Euler's criterion. In the present paper, we extend the results of [1] to the case in which the primary stresses are either compressive or tensile and may be unequal in magnitude.…”