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Chiriţă S, Ghiba ID. Strong ellipticity and progressive waves in elastic materials with voids. Proc Math Phys Eng Sci 2009. [DOI: 10.1098/rspa.2009.0360] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
In the present paper, we investigate a model for propagating progressive waves associated with the voids within the framework of a linear theory of porous media. Owing to the use of lighter materials in modern buildings and noise concerns in the environment, such models for progressive waves are of much interest to the building industry. The analysis of such waves is also of interest in acoustic microscopy where the identification of material defects is of paramount importance to the industry and medicine. Our analysis is based on the strong ellipticity of the poroelastic materials. We illustrate the model of progressive wave propagation for isotropic and transversely isotropic porous materials. We also study the propagation of harmonic plane waves in porous materials including the thermal effect.
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Affiliation(s)
- Stan Chiriţă
- Faculty of Mathematics, Al. I. Cuza University of Iaşi, Blvd. Carol I, No. 11, 700506 Iaşi, Romania
| | - Ionel-Dumitrel Ghiba
- ‘Octav Mayer’ Mathematics Institute, Romanian Academy of Science, Iaşi Branch, Blvd. Carol I, No. 8, 700506 Iaşi, Romania
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Abstract
In this paper, we use the Green–Naghdi theory of thermomechanics of continua to derive a linear theory of thermopiezoelectricity of a body with inner structure. This theory permits propagation of thermal waves at finite speed. We establish a uniqueness result and a continuous dependence of the solutions upon initial data and body supplies. Some applications (the problem of a concentrated heat source, the problem of an impulsive body force, the deformation of a thick-walled spherical shell) are presented.
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Affiliation(s)
- D Ieşan
- Department of Mathematics, ‘Al.I. Cuza’ University700506 Iaşi, Romania
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Quintanilla R. On the impossibility of localization in linear thermoelasticity. Proc Math Phys Eng Sci 2007. [DOI: 10.1098/rspa.2007.0076] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
In the middle of the 1990s, Green & Naghdi proposed three theories of thermoelasticity that they labelled as types I, II and II. The type II theory, which is also called thermoelasticity without energy dissipation, is conservative and the solutions cannot decay with respect to time. It is well known that, in general, in the linear theories of thermoelasticity of types I and III, the solutions decay with respect to time. In many situations this decay is at least exponential. In this paper we study whether this decay can be fast enough to guarantee the solutions to be zero in a finite time. We investigate the impossibility of the localization in time of the solutions of linear thermoelasticity for the theories of Green & Naghdi. This means that the only solution that vanishes after a finite time is the null solution. The main idea is to show the uniqueness of solutions for the backward in time problem. To be precise, for type III thermoelasticity we will prove the impossibility of localization of solutions in the case of bounded domains, and for the type I thermoelasticity in the case of exterior domains, even when the solutions can be unbounded, whether the spatial variable goes to infinity.
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Affiliation(s)
- R Quintanilla
- Matemática Aplicada 2, ETSEIAT-UPCTerrassa, Barcelona 08222, Spain
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Jordan P. Growth, decay and bifurcation of shock amplitudes under the type-II flux law. Proc Math Phys Eng Sci 2007. [DOI: 10.1098/rspa.2007.1895] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
By replacing Fick's diffusion law with Green and Nagdhi's type-II flux law, a hyperbolic counterpart to the classical Fisher–KPP equation is obtained. In this article, an analytical study of this partial differential equation is presented with an emphasis on shock and related kinematic wave phenomena. First, an exact travelling wave solution (TWS) is derived and examined. Then, using singular surface theory, exact amplitude expressions for both shock and acceleration waves are obtained. In addition, the issue of shock stability is addressed and the limitations of the model are noted.
It is shown that discontinuity (i.e. shock) formation in the TWS occurs only when the propagation speed, which must exceed the characteristic speed, tends to the latter. It is also shown that the shock amplitude equation undergoes a transcritical bifurcation. Lastly, numerical simulations of acceleration waves in a simple model problem are presented.
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Affiliation(s)
- P.M Jordan
- Code 7181, Naval Research LaboratoryStennis Space Center, MS 39529-5004, USA
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Jordan P, Straughan B. Acoustic acceleration waves in homentropic Green and Naghdi gases. Proc Math Phys Eng Sci 2006. [DOI: 10.1098/rspa.2006.1739] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Acceleration and temperature rate waves in lossless Green and Naghdi gases are investigated. The exact equations of motion are also derived and then simplified under the finite-amplitude approximation. Bounds are established for the theory-specific coupling parameter, as well as several other quantities, and results are compared/contrasted with those for classical perfect gases.
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Affiliation(s)
- P.M Jordan
- Code 7181, Naval Research Laboratory, Stennis Space CenterMS 39529, USA
| | - B Straughan
- Department of Mathematical Sciences, Durham UniversityDurham DH1 3LE, UK
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Abstract
A model for acoustic waves in a porous medium is investigated. Due to the use of lighter materials in modern buildings and noise concerns in the environment, such models for poroacoustic waves are of much interest to the building industry. The model has been investigated in some detail by P. M. Jordan. Here we present a rational continuum thermodynamic derivation of the Jordan model. We then present results for the amplitude of an acceleration wave making no approximations whatsoever.
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Affiliation(s)
- M Ciarletta
- Dipartimento di Ingegneria dell'Informazione e Matematica Applicata, Universitá di Salerno84100 Salerno, Italy
| | - B Straughan
- Department of Mathematical Sciences, University of DurhamDurham DH1 3LE, UK
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Quintanilla R, Straughan B. Bounds for some non-standard problems in porous flow and viscous Green–Naghdi fluids. Proc Math Phys Eng Sci 2005. [DOI: 10.1098/rspa.2005.1508] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
A. E. Green, F. R. S. and P. M. Naghdi developed a new theory of continuum mechanics based on an entropy identity rather than an entropy inequality. In particular, within the framework of this theory they developed a new set of equations to describe viscous flow. The new theory additionally involves vorticity and spin of vorticity. We here derive energy bounds for a class of problem in which the ‘initial data’ are given as a combination of data at time
t
=0 and at a later time
t
=
T
. Such problems are in vogue in the mathematical literature and may be used, for example, to give estimates of solution behaviour in an improperly posed problem where one wishes to continue a solution backward in time. In addition, we derive similar energy bounds for a solution to the Brinkman–Forchheimer equations of viscous flow in porous media.
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Affiliation(s)
- R Quintanilla
- Departamento Matematica Aplicada 2, E.T.S. d'Enginyers Industrials de Terrassa, Universidad Politecnica de CatalunyaColón 11, Terrassa, 08222 Barcelona, Spain
| | - B Straughan
- Department of Mathematical Sciences, Durham UniversityDurham DH1 3LE, UK
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Jordan P. Growth and decay of acoustic acceleration waves in Darcy-type porous media. Proc Math Phys Eng Sci 2005. [DOI: 10.1098/rspa.2005.1477] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
The propagation of acoustic waves in a fluid that saturates a Darcy-type porous medium is considered under finite-amplitude theory. The equation of motion is derived, an acceleration wave analysis is carried out, and a travelling wave solution (TWS) is obtained. In addition, analytical findings are supported with numerical work generated by a simple, but effective, finite-difference scheme and results obtained are compared with those of the nonporous and linear cases. Most notably, this analysis reveals the following: (i) that the equation of motion is a new, hyperbolic form of Kuznetsov's equation; (ii) that finite-time blow-up of the wave amplitude is possible even if dissipation is present; (iii) the presence of a porous medium increases, with respect to the nonporous case, the rate at which amplitude growth/decay occurs; (iv) in the case of porous media propagation, not all compressive acceleration waves suffer blow up; and (v) that there exists a connection between acceleration waves and TWSs.
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Affiliation(s)
- P.M Jordan
- Code 7181, Naval Research Laboratory, Stennis Space CenterMS 39529, USA
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Quintanilla R, Straughan B. Energy bounds for some non-standard problems in thermoelasticity. Proc Math Phys Eng Sci 2005. [DOI: 10.1098/rspa.2004.1381] [Citation(s) in RCA: 35] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
A.E. Green F.R.S. and P.M. Naghdi developed two theories of thermoelasticity, called type II and type III, which are likely to be more natural candidates for the identification of a thermoelastic body than the usual theory. We here derive energy bounds for a class of problem in which the ‘initial data’ are given as a combination of data at time
t
=0 and at a later time
t
=
T
. Such problems are in vogue in the mathematical literature and may be used, for example, to give estimates of solution behaviour in an improperly posed problem where one wishes to continue a solution backward in time.
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Affiliation(s)
- R Quintanilla
- Departamento Matematica Aplicada 2, E.T.S. d'Enginyers Industrials de, Universidad Politecnica de Catalunya, Colón 11, Terrassa, 08222 Barcelona, Spain and
- Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK
| | - B Straughan
- Departamento Matematica Aplicada 2, E.T.S. d'Enginyers Industrials de, Universidad Politecnica de Catalunya, Colón 11, Terrassa, 08222 Barcelona, Spain and
- Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK
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Puri P, Jordan PM. On the propagation of plane waves in type–III thermoelastic media. Proc Math Phys Eng Sci 2004. [DOI: 10.1098/rspa.2004.1341] [Citation(s) in RCA: 51] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Affiliation(s)
- P. Puri
- Department of Mathematics, University of New Orleans, New Orleans, LA 70148, USA
| | - P. M. Jordan
- Code 7181, Naval Research Laboratory, Stennis Space Center, MS 39529, USA
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