Physical constants of an isotropic mixture

Rushchitskii, Ya. Ya.; Érgashev, B. B.

doi:10.1007/bf00888996

Cited by 2 publications

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“…Thus, the original equations (75), (76) and (77), (78) are not identical in structure (Eq. (78) contains the additional nonlinear term 4 3 11 3 1 cu u , , ) and solutions (79), (80) and (81), (82) describe not the same wave phenomena.…”

Section: Preliminary Comparison Of the Wave Equations For The Murnaghmentioning

confidence: 97%

“…Five families of universal deformations can be implemented in such materials, according to [76,87]. Universal deformations are those induced in any material by surface loads only.…”

Section: [ ]mentioning

confidence: 99%

“…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.…”

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confidence: 99%

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

Rushchitsky

2009

Int Appl Mech

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The development of the classical Signorini nonlinear elastic model is continued. Recent year's published results on a recently developed area of the theory of nonlinear waves are outlined from a unified position and analyzed. The paper consists of six sections: explanation of the features of the class of waves under consideration; detailed description of the Signorini model; general formulation of wave-propagation problems based on the Signorini potential; analysis of results on plane and cylindrical waves obtained with the Signorini model; comparison of these results with those obtained with the Murnaghan model; discussion of the possible lines for the development of the research.Introduction. After many decades of neglect, the Signorini model is addressed here to clarify its capabilities in analyzing plane waves. Antonio Signorini (1888Signorini ( -1963, an outstanding Italian scientist, is by no means forgotten in modern mechanics. He is mentioned and quoted repeatedly in a number of classical books on continuum mechanics. For example, Chapter 9 of the book [87] discusses and elaborates upon Signorini's idea of iterative solution of nonlinear boundary-value problems in elasticity. Trusdell starts from Signorini's publication [80]. Of recent popularity is the Signorini problem with so-called Signorini boundary conditions, which is analyzed by mathematicians who represent many countries and schools of thought [20,33,36,37,39,42,43,45,48,77,78,90].The paper is structured as follows. Its goal is to unite recent results on waves in materials described by the Signorini elastic potential. A feature of the associated class of nonlinear problems is that they originally dealt with materials described by the Murnaghan potential. A general view of the achievements in solving wave problems with this model is presented in the reviews [17,18,60,[68][69][70] and, partially, in the recent monographs [9,19,30,50]. 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. Section 1 describes the Signorini potential and gives some new comments on the Signorini constant, the Kelvin and Poynting effects, and Volterra distortions. Section 2 provides a general problem formulation for waves in nonlinear elasticity based on the Signorini potential and the transformation from the Eulerian to Lagrangian description of deformation, resulting in nonlinear wave equations in Lagrange coordinates. Section 3 preliminarily compares the nonlinear wave equations in the Murnaghan and Signorini models, including the solutions for longitudinal and transverse plane waves obtained with these models. Section...

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Section: Preliminary Comparison Of the Wave Equations For The Murnaghmentioning

confidence: 97%

“…Five families of universal deformations can be implemented in such materials, according to [76,87]. Universal deformations are those induced in any material by surface loads only.…”

Section: [ ]mentioning

confidence: 99%

mentioning

confidence: 99%

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

Rushchitsky

2009

Int Appl Mech

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“…The theory of effective moduli regards materials of complex internal structure with internal bonds as homogeneous elastic media. The possibility of generating universal deformations in such media was used by different authors to calculate specific effective constants by different methods [7,8,11,[17][18][19]. For example, the energy equivalence method includes assumptions on universal states as a necessary component.…”

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confidence: 99%

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

Rushchitsky

2007

Int Appl Mech

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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...

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Physical constants of an isotropic mixture

Cited by 2 publications

References 1 publication

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

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

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

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