On universal deformations in analysis of Signorini’s nonlinear theory of hyperelastic media

Abstract

It is shown that uniform compression/tension and simple shear as universal deformations are quite useful in studying Signorini's nonlinear theory of hyperelastic materials. They make it possible to formulate restrictions for the elastic constants of the theory and to explain the Poynting effect Keywords: Signorini's hyperelastic model, universal deformations, simple shear, uniform compression/ tension, restrictions for elastic constants, Poynting effect 1. Introduction. During the development of materials mechanics, different theoretical approaches were proposed to describe the nonlinear deformation of elastic materials [2,3,9,10,12,17,20,21,[26][27][28][29][30][31]. Most of the published studies use an approach based on the postulation of explicit nonlinear dependence of internal energy (potential energy or elastic potential) on the finite-strain tensor or its invariants. Such an approach applies to hyperelastic media where compressible and incompressible media (materials) are distinguished. Most popular are the Murnaghan, Rivlin-Sounders, Mooney, and John potentials. They stem from speculative reasoning (as termed by Lurie [10]), are phenomenological, and require (ultimately [10]) determining the physical constants present in their expressions by nonphenomenological (mainly experimental) methods.The Murnaghan potential (sometimes called five-constant potential) is most popular in studies of elastic waves or at least acoustic plane waves [4,5] and waves in prestressed materials [2,3]. One reason is that the dependence of the Murnaghan potential on the third algebraic invariant allows describing a number of significant wave effects [2,3]. With time, it became clear that Murnaghan, as well as his followers considered that the experimental determination of the physical constants (two Lamé constants and three Murnaghan constants) in the analytic expression of the potential is an important stage of research. By now, these constants have been determined for dozens of engineering materials.However, not all researchers who introduced potentials are known so well. The reason may be that the research on potentials commonly involves speculative analysis, and experiments are conducted only at the final stage of the research. As to when experimental observations should be done, Signorini said [10]: "The later, the better." Not all developers of potentials realized the importance of determining the physical constants in the expressions of potentials. For example, Signorini left no noticeable traces of his attempts to analyze the new constant appearing in the expression of the potential, which was later named after him. Despite a certain advantage of the Signorini potential (it includes only one constant of the third order, whereas the Murnaghan potential includes three such constants), the sympathies of researchers have been with the Murnaghan potential. The reason why Signorini lost this competition lies in the fact, well known from theoretical physics, that among approaches with equal capabilities of describing some physica...

Stability of Surface Rayleigh Waves in an Elastic Half‐Space

“…Assume that U is a family of solutions to (10a) (13a) as in Proposition 1, depending smoothly on the parameter X , and that (10b) (13b) admits a solution V , square integrable in Y and jointly smooth in (X 2 , X ). (18) where J is the (real) constant given by…”

“…It results to be a nonlocal generalization of the inviscid Burgers' equation of the same type studied by Hunter [15,16] and Benzoni-Gavage [17]. As a reference for the second-order elasticity theory, we cite [5,18].…”

Evolution of the theory of nonlinear waves in Murnaghan and Signorini materials

“…We will focus on the evolution of the waveprofile. Solutions (65) and (94) Thus, the first difference is that plane and cylindrical waves in a Murnaghan material are analyzed using different approaches.…”

“…Then the form of the linear part suggests that the cylindrical wave looks like a longitudinal plane wave with three more specific differences (than those mentioned above): the amplitude quickly decreasing with distance traveled and the phase variable k r t L --w p ( / ) 3 4 with nonzero shift ( / ) 3 4 p and high uncertainty at small radius. We revert to solutions (65) and (94). Noteworthy is the fact that formula (65) describes two main nonlinear effects for a longitudinal plane wave in a quadratic nonlinear elastic medium: self-generation and second-harmonic generation.…”

“…Therefore, results produced by the Murnaghan model are included here only for the sake of coherence and consistency. The present paper is mainly based on the author's publications [15,[25][26][27][28][29][30][31][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76] and consists of seven sections. The structure of the paper is described, features of the class of nonlinear elastic waves in materials described by elastic potentials are listed, and the main concepts are interpreted in Introduction.…”