March  2012, 17(2): 637-659. doi: 10.3934/dcdsb.2012.17.637

Acceleration waves in complex materials

1. 

School of Engineering and Applied Sciences, Harvard University, 29 Oxford st, Cambridge, MA 02138, United States

Received  September 2010 Revised  April 2011 Published  December 2011

A framework for modeling acceleration waves propagation in complex materials is presented. Coupled propagation of elasto-acoustic, microstructural and thermal waves is investigated in the full three dimensional case. The presence of microstructure inside each material element is taken into account without introducing additional hypotheses on the physical nature of the microstructure itself, thus obtaining a general theory that is suitable for the whole class of complex bodies. In particular, jump conditions across the discontinuity interface that identifies the acceleration wave are obtained and the amplitude evolution equation is derived.
Citation: Paolo Paoletti. Acceleration waves in complex materials. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 637-659. doi: 10.3934/dcdsb.2012.17.637
References:
[1]

H. Altenbach, V. A. Eremeyev, L. P. Lebedev and L. A. Rendón, Acceleration waves and ellipticity in thermoelastic micropolar media,, Archive of Applied Mechanics, 80 (2010), 217.  doi: 10.1007/s00419-009-0314-1.  Google Scholar

[2]

R. C. Batra, Thermodynamics of non-simple elastic materials,, J. Elasticity, 6 (1976), 451.  doi: 10.1007/BF00040904.  Google Scholar

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[4]

G. Capriz and E. G. Virga, On singular surfaces in the dynamics of continua with microstructure,, Quart. Appl. Math., 52 (1994), 509.   Google Scholar

[5]

G. Capriz and P. M. Mariano, Symmetries and Hamiltonian formalism for complex materials,, J. Elasticity, 72 (2003), 57.  doi: 10.1023/B:ELAS.0000018775.44668.07.  Google Scholar

[6]

P. J. Chen, Growth and decay of waves in solids,, in, (1973), 303.   Google Scholar

[7]

P. J. Chen, M. F. McCarthy and T. R. O'Leary, One-dimensional shock and acceleration waves in deformable dielectric materials with memory,, Arch. Ration. Mech. Anal., 62 (1976), 189.  doi: 10.1007/BF00248471.  Google Scholar

[8]

M. Ciarletta, G. Iovane and M. A. Sumbatyan, On stress analysis for crack in elastic materials with voids,, Internat. J. Engrg. Sci., 41 (2003), 2447.  doi: 10.1016/S0020-7225(03)00236-2.  Google Scholar

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M. Ciarletta and B. Straughan, Poroacoustic acceleration waves,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 3493.  doi: 10.1098/rspa.2006.1730.  Google Scholar

[10]

M. Ciarletta and B. Straughan, Thermo-poroacoustic acceleration waves in elastic materials with voids,, J. Math. Anal. Appl., 333 (2007), 142.  doi: 10.1016/j.jmaa.2006.09.014.  Google Scholar

[11]

M. Ciarletta and B. Straughan, Poroacoustic acceleration waves with second sound,, J. Sound Vibration, 306 (2007), 725.  doi: 10.1016/j.jsv.2007.06.015.  Google Scholar

[12]

M. Ciarletta, B. Straughan and V. Zampoli, Thermo-poroacoustic acceleration waves in elastic materials with voids without energy dissipation,, Internat. J. Engrg. Sci., 45 (2007), 736.  doi: 10.1016/j.ijengsci.2007.05.001.  Google Scholar

[13]

F. Davi and P. M. Mariano, Evolution of domain walls in ferroelectric solids,, J. Mech. Phys. Solids, 49 (2001), 1701.  doi: 10.1016/S0022-5096(01)00014-X.  Google Scholar

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A. C. Diebold, Subsurface imaging with scanning ultrasound holography,, Science, 310 (2005), 61.  doi: 10.1126/science.1119259.  Google Scholar

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S. Dost, Acceleration waves in elastic dielectrics with polarization gradient effects,, Internat. J. Engrg. Sci., 21 (1983), 1305.  doi: 10.1016/0020-7225(83)90127-1.  Google Scholar

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J. L. Ericksen, Liquid crystals with variable degree of orientation,, Arch. Ration. Mech. Anal., 113 (1991), 97.  doi: 10.1007/BF00380413.  Google Scholar

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C. de Fabritiis and P. M. Mariano, Geometry of interactions in complex bodies,, J. Geom. Phys., 54 (2005), 301.  doi: 10.1016/j.geomphys.2004.10.002.  Google Scholar

[18]

M. Fabrizio, F. Franchi and B. Straughan, On a model for thermo-poroacoustic waves,, Internat. J. Engrg. Sci., 46 (2008), 790.  doi: 10.1016/j.ijengsci.2008.01.016.  Google Scholar

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L. Fiorino and P. Giovine, Nano-pores in thermoelastic materials,, Proceedings of XVII$^o$ AIMETA, (2005).   Google Scholar

[20]

L. Fiorino and P. Giovine, Some remarks on acceleration waves in porous solids,, in, (2008), 280.   Google Scholar

[21]

M. Garai and F. Pompoli, A simple empirical model of polyester fibre materials for acoustical applications,, Applied Acoustics, 66 (2005), 1383.   Google Scholar

[22]

A. E. Green and N. Laws, On the entropy production inequality,, Arch. Ration. Mech. Anal., 45 (1972), 47.  doi: 10.1007/BF00253395.  Google Scholar

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A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 432 (1991), 171.  doi: 10.1098/rspa.1991.0012.  Google Scholar

[24]

P. M. Jordan, Growth and decay of acoustic acceleration waves in Darcy-type porous media,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2749.  doi: 10.1098/rspa.2005.1477.  Google Scholar

[25]

R. A. Lemons and C. F. Quate, Acoustic microscope--scanning version,, Applied Physics Letters, 24 (1974), 163.  doi: 10.1063/1.1655136.  Google Scholar

[26]

T. Manacorda, "Introduzione alla Termomeccanica dei Continui,", Pitagora, (1979).   Google Scholar

[27]

P. M. Mariano, Multifield theories in mechanics of solids,, Adv. in Appl. Mech., 38 (2002), 1.  doi: 10.1016/S0065-2156(02)80102-8.  Google Scholar

[28]

P. M. Mariano and L. Sabatini, Homothermal acceleration waves in multifield theories of continua,, Internat. J. Non-Linear Mech., 35 (2000), 963.  doi: 10.1016/S0020-7462(99)00071-2.  Google Scholar

[29]

P. M. Mariano and F. L. Stazi, Computational aspects of the mechanics of complex materials,, Arch. Comput. Methods Eng., 12 (2005), 391.  doi: 10.1007/BF02736191.  Google Scholar

[30]

P. M. Mariano, Mechanics of complex bodies: Commentary on the unified modeling of material substructures,, Theor. Appl. Mech., 35 (2008), 235.  doi: 10.2298/TAM0803235M.  Google Scholar

[31]

P. M. Mariano, Cracks in complex bodies: Covariance of tip balances,, J. Nonlinear Sci., 18 (2008), 99.  doi: 10.1007/s00332-007-9008-4.  Google Scholar

[32]

W. Maysenhölder, A. Berg and P. Leistner, "Acoustic Properties of Aluminium Foams-Measurements and Modelling,", Proceedings of CFA/DAGA, (2005).   Google Scholar

[33]

R. J. Meyer, Ultrasonic drying of saturated porous solids via second sound, 2006., Available from: \url{http://www.freepatentsonline.com/6376145.html}., ().   Google Scholar

[34]

K. Mitra, S. Kumar, A. Vedavarz and M. K. Moallemi, Experimental evidence of hyperbolic heat conduction in processed meat,, Journal of Heat Transfer, 117 (1995), 568.  doi: 10.1115/1.2822615.  Google Scholar

[35]

M. Ostoja-Starzewski and J. Trebicki, Stochastic dynamics of acceleration waves in random media,, Mechanics of Materials, 38 (2006), 840.  doi: 10.1016/j.mechmat.2005.06.022.  Google Scholar

[36]

G. F. Raiser, J. L. Wise, R. J. Clifton, D. E. Grady and D. E. Cox, Plate impact response of ceramics and glasses,, Journal of Applied Physics, 75 (1994), 3862.  doi: 10.1063/1.356066.  Google Scholar

[37]

L. Sabatini and G. Augusti, Acceleration Waves in Thermoelastic Beams,, Meccanica, 35 (2000), 519.  doi: 10.1023/A:1010592409020.  Google Scholar

[38]

J. J. Vadasz, S. Govender and P. Vadasz, Heat transfer enhancement in nano-fluids suspensions: Possible mechanisms and explanations,, International Journal of Heat Mass Transfer, 48 (2005), 2673.  doi: 10.1016/j.ijheatmasstransfer.2005.01.023.  Google Scholar

[39]

D. K. Wilson, Simple, relaxational models for the acoustical properties of porous media,, Applied Acoustics, 50 (1997), 171.  doi: 10.1016/S0003-682X(96)00048-5.  Google Scholar

[40]

T. W. Wright, Acceleration waves in simple elastic materials,, Arch. Ration. Mech. Anal., 50 (1973), 237.  doi: 10.1007/BF00281508.  Google Scholar

show all references

References:
[1]

H. Altenbach, V. A. Eremeyev, L. P. Lebedev and L. A. Rendón, Acceleration waves and ellipticity in thermoelastic micropolar media,, Archive of Applied Mechanics, 80 (2010), 217.  doi: 10.1007/s00419-009-0314-1.  Google Scholar

[2]

R. C. Batra, Thermodynamics of non-simple elastic materials,, J. Elasticity, 6 (1976), 451.  doi: 10.1007/BF00040904.  Google Scholar

[3]

G. Capriz, "Continua with Microstructure,", Springer Tracts in Natural Philosophy, 35 (1989).   Google Scholar

[4]

G. Capriz and E. G. Virga, On singular surfaces in the dynamics of continua with microstructure,, Quart. Appl. Math., 52 (1994), 509.   Google Scholar

[5]

G. Capriz and P. M. Mariano, Symmetries and Hamiltonian formalism for complex materials,, J. Elasticity, 72 (2003), 57.  doi: 10.1023/B:ELAS.0000018775.44668.07.  Google Scholar

[6]

P. J. Chen, Growth and decay of waves in solids,, in, (1973), 303.   Google Scholar

[7]

P. J. Chen, M. F. McCarthy and T. R. O'Leary, One-dimensional shock and acceleration waves in deformable dielectric materials with memory,, Arch. Ration. Mech. Anal., 62 (1976), 189.  doi: 10.1007/BF00248471.  Google Scholar

[8]

M. Ciarletta, G. Iovane and M. A. Sumbatyan, On stress analysis for crack in elastic materials with voids,, Internat. J. Engrg. Sci., 41 (2003), 2447.  doi: 10.1016/S0020-7225(03)00236-2.  Google Scholar

[9]

M. Ciarletta and B. Straughan, Poroacoustic acceleration waves,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 3493.  doi: 10.1098/rspa.2006.1730.  Google Scholar

[10]

M. Ciarletta and B. Straughan, Thermo-poroacoustic acceleration waves in elastic materials with voids,, J. Math. Anal. Appl., 333 (2007), 142.  doi: 10.1016/j.jmaa.2006.09.014.  Google Scholar

[11]

M. Ciarletta and B. Straughan, Poroacoustic acceleration waves with second sound,, J. Sound Vibration, 306 (2007), 725.  doi: 10.1016/j.jsv.2007.06.015.  Google Scholar

[12]

M. Ciarletta, B. Straughan and V. Zampoli, Thermo-poroacoustic acceleration waves in elastic materials with voids without energy dissipation,, Internat. J. Engrg. Sci., 45 (2007), 736.  doi: 10.1016/j.ijengsci.2007.05.001.  Google Scholar

[13]

F. Davi and P. M. Mariano, Evolution of domain walls in ferroelectric solids,, J. Mech. Phys. Solids, 49 (2001), 1701.  doi: 10.1016/S0022-5096(01)00014-X.  Google Scholar

[14]

A. C. Diebold, Subsurface imaging with scanning ultrasound holography,, Science, 310 (2005), 61.  doi: 10.1126/science.1119259.  Google Scholar

[15]

S. Dost, Acceleration waves in elastic dielectrics with polarization gradient effects,, Internat. J. Engrg. Sci., 21 (1983), 1305.  doi: 10.1016/0020-7225(83)90127-1.  Google Scholar

[16]

J. L. Ericksen, Liquid crystals with variable degree of orientation,, Arch. Ration. Mech. Anal., 113 (1991), 97.  doi: 10.1007/BF00380413.  Google Scholar

[17]

C. de Fabritiis and P. M. Mariano, Geometry of interactions in complex bodies,, J. Geom. Phys., 54 (2005), 301.  doi: 10.1016/j.geomphys.2004.10.002.  Google Scholar

[18]

M. Fabrizio, F. Franchi and B. Straughan, On a model for thermo-poroacoustic waves,, Internat. J. Engrg. Sci., 46 (2008), 790.  doi: 10.1016/j.ijengsci.2008.01.016.  Google Scholar

[19]

L. Fiorino and P. Giovine, Nano-pores in thermoelastic materials,, Proceedings of XVII$^o$ AIMETA, (2005).   Google Scholar

[20]

L. Fiorino and P. Giovine, Some remarks on acceleration waves in porous solids,, in, (2008), 280.   Google Scholar

[21]

M. Garai and F. Pompoli, A simple empirical model of polyester fibre materials for acoustical applications,, Applied Acoustics, 66 (2005), 1383.   Google Scholar

[22]

A. E. Green and N. Laws, On the entropy production inequality,, Arch. Ration. Mech. Anal., 45 (1972), 47.  doi: 10.1007/BF00253395.  Google Scholar

[23]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 432 (1991), 171.  doi: 10.1098/rspa.1991.0012.  Google Scholar

[24]

P. M. Jordan, Growth and decay of acoustic acceleration waves in Darcy-type porous media,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2749.  doi: 10.1098/rspa.2005.1477.  Google Scholar

[25]

R. A. Lemons and C. F. Quate, Acoustic microscope--scanning version,, Applied Physics Letters, 24 (1974), 163.  doi: 10.1063/1.1655136.  Google Scholar

[26]

T. Manacorda, "Introduzione alla Termomeccanica dei Continui,", Pitagora, (1979).   Google Scholar

[27]

P. M. Mariano, Multifield theories in mechanics of solids,, Adv. in Appl. Mech., 38 (2002), 1.  doi: 10.1016/S0065-2156(02)80102-8.  Google Scholar

[28]

P. M. Mariano and L. Sabatini, Homothermal acceleration waves in multifield theories of continua,, Internat. J. Non-Linear Mech., 35 (2000), 963.  doi: 10.1016/S0020-7462(99)00071-2.  Google Scholar

[29]

P. M. Mariano and F. L. Stazi, Computational aspects of the mechanics of complex materials,, Arch. Comput. Methods Eng., 12 (2005), 391.  doi: 10.1007/BF02736191.  Google Scholar

[30]

P. M. Mariano, Mechanics of complex bodies: Commentary on the unified modeling of material substructures,, Theor. Appl. Mech., 35 (2008), 235.  doi: 10.2298/TAM0803235M.  Google Scholar

[31]

P. M. Mariano, Cracks in complex bodies: Covariance of tip balances,, J. Nonlinear Sci., 18 (2008), 99.  doi: 10.1007/s00332-007-9008-4.  Google Scholar

[32]

W. Maysenhölder, A. Berg and P. Leistner, "Acoustic Properties of Aluminium Foams-Measurements and Modelling,", Proceedings of CFA/DAGA, (2005).   Google Scholar

[33]

R. J. Meyer, Ultrasonic drying of saturated porous solids via second sound, 2006., Available from: \url{http://www.freepatentsonline.com/6376145.html}., ().   Google Scholar

[34]

K. Mitra, S. Kumar, A. Vedavarz and M. K. Moallemi, Experimental evidence of hyperbolic heat conduction in processed meat,, Journal of Heat Transfer, 117 (1995), 568.  doi: 10.1115/1.2822615.  Google Scholar

[35]

M. Ostoja-Starzewski and J. Trebicki, Stochastic dynamics of acceleration waves in random media,, Mechanics of Materials, 38 (2006), 840.  doi: 10.1016/j.mechmat.2005.06.022.  Google Scholar

[36]

G. F. Raiser, J. L. Wise, R. J. Clifton, D. E. Grady and D. E. Cox, Plate impact response of ceramics and glasses,, Journal of Applied Physics, 75 (1994), 3862.  doi: 10.1063/1.356066.  Google Scholar

[37]

L. Sabatini and G. Augusti, Acceleration Waves in Thermoelastic Beams,, Meccanica, 35 (2000), 519.  doi: 10.1023/A:1010592409020.  Google Scholar

[38]

J. J. Vadasz, S. Govender and P. Vadasz, Heat transfer enhancement in nano-fluids suspensions: Possible mechanisms and explanations,, International Journal of Heat Mass Transfer, 48 (2005), 2673.  doi: 10.1016/j.ijheatmasstransfer.2005.01.023.  Google Scholar

[39]

D. K. Wilson, Simple, relaxational models for the acoustical properties of porous media,, Applied Acoustics, 50 (1997), 171.  doi: 10.1016/S0003-682X(96)00048-5.  Google Scholar

[40]

T. W. Wright, Acceleration waves in simple elastic materials,, Arch. Ration. Mech. Anal., 50 (1973), 237.  doi: 10.1007/BF00281508.  Google Scholar

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