Stability of elastic bodies under uniform compression (review)

Abstract

The main results on the three-dimensional theory of stability of compressible and incompressible hyperelastic simply connected bodies under uniform compression are analyzed. The problems are classified according to the type of loading (dead or follower loads, acting on the whole or a part of the surface) and the type of boundary conditions (the same conditions on the whole surface or different conditions on different parts of the surface). Approaches based on the three-dimensional linearized theory of stability (theory of finite subcritical deformations, first and second theories of small subcritical deformations, incremental-deformation theory, theory of small average rotations) and an approximate approach (linear equations; the loading parameter is approximately included in the boundary conditions) to solving these problems are discussed. Related problems (rock pressure manifestations, folding in the Earth's crust, wave-like formations on the surface of structural members, stability of laminated composite materials) are briefly commented. Regarding isotropic compressible and incompressible hyperelastic materials, the three-dimensional theory of elastic stability, theory of finite subcritical deformations, and first and second theories of small subcritical deformations are considered. The sufficient conditions for the applicability of the static (Euler's) method, the sufficient conditions for the stability of equilibrium state, and the general solutions to anti-plane, plane, and spatial problems for homogeneous subcritical states are formulated. The exact solutions, obtained with the above-mentioned general approaches, for compressible and incompressible isotropic bodies (strip, rectangular and circular plates, circular cylinder, sphere, and body of arbitrary geometry) under dead or follower loading are presented. These solutions were obtained for isotropic hyperelastic materials with an arbitrary elastic potential under uniform (hydrostatic, biaxial, or triaxial) pressure. The reviewed results were originally reported in the author's monograph (A. N. Guz, Stability of Elastic Bodies under Uniform Compression [in Russian], Naukova Dumka, Kyiv (1979)) and articles listed in the ReferencesKeywords: stability of simply connected isotropic hyperelastic bodies, three-dimensional linearized theory, uniform loading, dead follower loads Introduction.Here we provide an introductory discussion of the problem of interest and related problems, including a brief historical sketch, classification of problems, brief description of approaches, relation to phenomena observed in related fundamental and scientific-and-technological fields, and exemplary analysis of recent publications.1.1. Brief Historical Sketch. The present paper reviews results related to hyperelastic isotropic compressible or incompressible simply connected bodies with an arbitrary elastic potential. In this case (simply connected isotropic elastic bodies), uniform hydrostatic compression (of arbitrarily shaped bodies) and uniform uniaxial, biaxial, or triaxia...

Ultrasonic Nondestructive Method for Stress Analysis of Structural Members and Near-Surface Layers of Materials: Focus on Ukrainian Research (Review)

“…Item 6 and formulas (23), (25), (27) are not used for biaxial stresses. Item 6 and formulas (23), (24), (27) are not used for uniaxial stresses.…”

“…The basic equations are used to determine triaxial stresses (23), including biaxial stresses (24) and uniaxial stresses (25). These equations were derived using the theory outlined in Sec.…”

“…In contrast, there are two expressions in(30) for the UNDM, which makes it possible to separate the principal stresses, without the need to use additional experimental or numerical methods. The following expressions for uniaxial stresses( 25) can be derived from(28) or(30):…”