Propagation and interaction of hyperbolic plane waves in nonlinear elastic solids

• Autorzy: Domański Wł.
• Języki publikacji: EN

Abstrakty

EN

Our aim is to create a mathematical theory of nonlinear plane waves which could have applications in many branches of nonlinear material science. Using asymptotic expansions we derive simplified models to complicated, dynamical, nonlinear phenomena (including wave interactions), described originally by large systems of nonlinear partial differential equations. We are particularly interested in applications to elastic solids. The models which we derive are weakly nonlinear. Weak nonlinearity means that we are interested in small amplitude solutions. This is more refined than linearized models. Asymptotic methods are very useful when we deal with complicated systems of PDEs. Weakly nonlinear geometric optic (WNGO) is suitable for nonlinear hyberbolic equations modelling wave type phenomena. The method works for small amplitude (weakly nonlinear) and high frequency waves. WNGO is based on the introduction of the additional "fast" independet variable and the use of a multiple scale analysis. It provides transport equations as the simplified asymptotic models for the evolution of wave profiles. The classical WNGO expansion results in the decoupled (in the nonresonant case) inviscid Burgers equations as the canonical asymptotic evolution equations for strictly hyberbolic and genuinely nonlinear waves. However in real models of continuum mechanics or physics it is quite common that the wave speeds may coincide ank often they are not monotonic in certain directions fo the assumptions of strict hyberbolicity and genuine nonlinearity are violated. This happent e.g. in nonlinear elastodynamics, the model which we are studying in the second part of this work. In order to take into consideration the loss of strict hyperbolicity and loss of genuine nonlinearity we modify theclassical WNGO expansion and derive new asymptotic evolution equations. The new reults include: 1. Derivation of a general structure of nonlinear plane waves' elastodynamics in terms of the strain energy function for an arbitrary direction of propagation, regardless of the anisotropy type. 2. Presentation of explicit nonlinear plane waves' elasktodynamics equations for a cubic crystal in the case - an arbitrary direction of propagation and a geometrically nonlinear but physically linear model. - an arbitrary direction of propagation in a cube face and both geometrically and physically nonlinear model, - three selected derections of pure mode propagation: (1,0,0), (1,1,0) and (1,1,1) with the inclusion of hihger order terms. 3. Derivation of the simlified evolution equations for nonlinear elastic waves in the isotropic and the cubic crystal cases. 4. Establishing a new model - the complex Burgers equation as the evolution equation describing interaction of a pair of transverse elastic waves propagatin along the three-fold symmetry axis in a cubic crystal. 5. Derivation of general analytical formulas for all interaction coefficients of nonlinear elastic plane waves propagating in any direction in an arbitrary hyperelastic medium. 6. Calculations of all interaction coefficients analytically for all considered models. 7. New formulations and interpretation of a null condition with the use of self-interaction coefficients. We focused in particular on cases where a local loss of strict hyperbolicity and/or genuine nonlinearity occurs. We showed that this happens for transverse or quasi-transverse elastic waves and implies the presence of a cubic nonlinearity in the evolution equations for decoupled (quasi)-transverse waves. However, we found cases when (quasi)-transverse waves do interact at a quadratically nonlinear level for certain directions in anisotropic media. We showed that this happens e.g. for a cubic crystal for a (1,1,1) direction. This fact is in contradiction to the isotropic case where it is known that shear elastic waves cannot interact at a quadratically nonlinear level see (75). We proved that the coupled systems with quadratic nonlinearity (complex Burgers equations) are new asymptotic models for interacting pairs of (quasi)-transverse waves propagating in the (1,1,1) direction in a cubic crystal. This is for the first time where complex Burgers equations appear in the context of nonlinear elasticity.

PL

Praca poświęcona jest analizie propagacji i oddziaływania fal w nieliniowych materiałach sprężystych przy pomocy metody słabo nieliniowej optyki geometrycnej. Rozważono zarówno materiały izotropowe jak i kryształy o symetrii kubicznej. Celem pracy jest wyprowadzenie równań ewolucyjnych na amplitudy propagujących się fal oraz znalezienie analitycznych wzrów na współczynniki oddziaływania między falami. Praca składa się z dwóch części. W części pierwszej wprowadzono aparat matematyczny użyty w centralnej części drugiej. Rozdział pierwszy to wstęp, w którym przedstawione są motywacje, cele pracy, użyte metody i główne rezultaty. Rozdział drugi to przegląd zagadnień dotyczących hiperbolicznych praw zachowania. Nowe rezultaty zawarte są głównie w paragrafie 2.1.1. W rozdziale trzecim zaprezentowana jest metoda nieliniowej optyki geometrycznej. Nowe wyniki dotyczą lokalnej struktury układów 2 x 2 dla krotnych pierwiastków oraz modyfikacji asymptotyki w przypadku lokalnej utraty własności istotnej nieliniowości. W ramach rozdziału czwartego dokonano przeglądu nieliniowych równań cząstkowych, które mogą służyć jako kanoniczne asymptotyczne modele dla skomplikowanych dynamicznych zagadnień nieliniowych. Oprócz modeli hiperbolicznych zaprezentowano również modele dyssypatywne i dyspersyjne, w tym z nieliniową dyspersją. Część druga stanowi główną część rozprawy i zawiera zastosowania metod zaprezentowanych w części pierwszej do materiałów sprężystych. W rozdziale piątym rozważone są równania nieliniowej elastodynamiki (przy założeniu zarówno geometrycznej, jak i fizycznej nieliniowości). W tym rozdziale godny uwagi jest zwłaszcza paragraf 5.5, gdzie przedstawiono ogólną ideę wyprowadzenia równań nieliniowej hipersprężystości fal płaskich. Pokazano także oryginalny temat sprowadzając zagadnienie na wartości i wektory własne do badania macierzy 3 x 3. Warto podkreślić, iż cała analiza przeprowadzona jest dla dowolnego ośrodka i dowolnego kierunku propagacji płaskiej fali. W rozdziale szóstym omówiono związki konstytutywne dla nieliniowego ośrodka izotropowego. Rozdział siódmy zawiera dyskusję różnych postaci tzw. "null condition" - warunku zerowania. Warunek ten zapewnia gładkość rozwiązań nieliniowych dynamicznych równań sprężystości w ośrodku izotropowym przy małych danych początkowych, a co za tym idzie zapobiega powstawaniu fal uderzeniowych oraz udaremnia eksplozję rozwiązań. Oryginalnym wkładem autora jest zaproponowanie kilku alternatywnych postaci tego warunku bez użycia niezmienników oraz interpretacja warunku zerowania poprzez współczynniki samooddziaływania fal. W rozdziale ósmym wyprowadzono ogólne wzory na współczynniki oddziaływania falowego, abstrahując od rodzaju anizotropii materiału i kierunku propagacji fali płaskiej. Podano w ten sposób przepis na wyznaczanie współczynników w zależności od pochodnych funkcji energii. Korzystając z tego przepisu wyznaczono wszystkie współczynniki dla materiałów izotropowych podkreślając różnice jakie wprowadza fizyczna nieliniowość. W rozdziale dziewiątym wyprowadzono asymptotyczne równania ewolucyjne dla materiału izotropowego uwypuklając różnice między otrzymanymi nieliniowymi równaniami dla fal podłużnych i fal poprzecznych. W rozdziale 10, 11 i 12 znalazły się zasktosowania dla ośrodka anizotropowego - rezultaty te stanowią oryginalny wkład autora w dziedzinie propagacji i oddziaływania fal płaskich w krysztale o symetrii kubicznej. Rozważono trzy wybrane kierunki propagacji i wyprowadzono dla nich równania fal płaskich uwzględniając efekty wyższego rzędu, uzyskano równania asymptotyczne dla tych fal i obliczono analitycznie wszystkie współczynniki oddziaływania międzyfalowego, wykazując, że zależą one od stałych materiałowych. Dodatkowo uzyskano także równania fal płaskich dla dowolnego kierunku propagacji przy założeniu fizycznej liniowości oraz w ogólnym przypadku dla dowolnego kierunku leżącego w płaszczyźnie ściany bocznej kryształu. Rezultatem zasługującym na szczególne podkreślenie jest udowodnienie, że fale poprzeczne mogą ze sobą oddziaływać na poziomie kwadratowej nieliniowości w przeciwieństwie do tego, co zostało pokazane dla ośrodka izotropowego (75). W rozprawie wykazano, iż na to miejsce dla fal poprzecznych propagujących się w krysztale o symetrii kubicznej wzdłuż osi trzykrotnej symetrii (1,1,1). Wyprowadzono nowe równania ewolucyjne opisujące oddziaływanie tych fal (zespolone równania Burgersa).

Słowa kluczowe

EN

optics; geometrical optics; nonlinear equations

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Prace Instytutu Podstawowych Problemów Techniki PAN
• Rocznik: 2006
• Tom: nr 4
• Strony: 3--169
• Opis fizyczny: Bibliogr. 180 poz., rys.

Twórcy

autor

Domański Wł.

• Instytut Podstawowowych Problemów Techniki Polskiej Akademii Nauk

Bibliografia

• [1] R. ABEYARATNE and J. K. KNOWLES, "Kinetic relations and the propagation of phase boundaries in solids", Arch. Rational Mech. AnaL, 114, 119-154, (1991).
• [2] R. AGEMI, "Global existence of nonlinear elastic waves", Inven. Math., 142, 225-250, (2000).
• [3] S. ALINHAC, Blowup for Nonlinear Hyperbolic Eąuations, Progress in Nonlinear Dif-ferential Eąuations and Their Applications, 17, Birkhauser, Boston, (1995).
• [4] S. S. ANTMAN, Nonlinear Problems of Elasticity, Appl. Math. Sci., 107, Springer-Verlag, New York, (1995).
• [5] M. ARTOLA and A. MAJDA, "Nonlinear development of instabilities in supersonic vortex sheets, I: the basie kink modes", Phys.D., 28, 253-281, (1987).
• [6] B. A. AULD, Acoustic Fields and Waves in Solids, 1 k 2, John Wiley &; Sons, New York, (1973).
• [7] M. BAKER and J. E. ERICKSEN, "Ineąualities restricting the form of the stress-deformation relations for isotropic elastic solids and Reiner-Rivlin fluids" J. Wash. Acad. Sci., 44, 33-35, (1954).
• [8] J. M. BALL, "Convexity conditions and existence theorems in nonlinear elasticity" Arch. Rat Mech. AnaL, 63, 337-403, (1977).
• [9] J. M. BALL, "Remarks on blow-up and nonexistence theorems for nonlinear evolution eąuations", Quart. J. Math., 28, 473-486, (1977).
• [10] J. M. BALL, "Nonlinear elasticity and materials science; a survey of some recent devel-opments", 93-119, in Nonlinear Mathematics and Its Applications, edited by P. J. Aston, Cambridge University Press, (1996).
• [11] W. BEN YOUSSEF and T. COLIN, "Rigorous derivation of Korteweg-de Vries-type systems from a generał class of nonlinear hyperbolic systems", preprint, Universite de Bordeaux I, (2004).
• [12] H. A. BETHE, The Theory of Shock Waves for an Arbitrary Eąuation of State, Technical Report PB-32189, Clearinghouse for Federal Scientific and Technical Information, U. S. Dept. of Commerce, Washington D. C, (1942).
• [13] S. BIANCHINI and A. BRESSAN, MVanishing viscosity solutions to nonlinear hyperbolic systems", Annal. Math., 161, 223—342, (2005).
• [14] F. BIRCH, "Finite elastic strain of cubic crystals", Phys. Rev., 1, 809-824, (1947).
• [15] A. BLOKHIN, Y. TRAKHININ, Well-posedness of linear hyperbolic problems: theory and applications, Sobolev Inst. of Math., Russian Acad. Sci., Novosibirsk, (2005).
• [16] D. R. BLAND, Nonlinear Dynamie Elasticity, Blaisdel Publishing Company, Woltham, (1969).
• [17] G. BOILLAT, "Chocs caracteristiąues", C. R. Acad. Sci. Paris Ser. A-B, 274, A1018-A1021, (1972).
• [18] J. BOUSSINESQ, "Theorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiąuant au liąuide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond", J. Math. Pures Appl, 17, 55-108, (1872).
• [19] J. BOUSSINESQ, "Essai sur la theorie des eaux courants", J. Mem. Acad. Sci. Inst. Nat. France, 123, 1-608, (1877).
• [20] M. A. BREAZALE and J. PHILIP, "Determination of third-order elastic constants from ultrasonic harmonie generation measurements", 1-60, in Physical Acoustics, XVII, edited by W. P. Mason and R. N. Thurston, McGraw-Hill, New York, (1984).
• [21] A. BRESSAN, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy problem, Oxford University Press, Oxford, (2000).
• [22] M. BRIO and J. K. HUNTER, "Asymptotic eąuations for conservation laws of mixed type", Wave Motion, 6, 57-64, (1992).
• [23] K. BRUGGER, "Thermodynamic definition of higher order elastic coefficients", Phys. Rev., 33, 1611-1622, (1964).
• [24] R. CAMASSA, J. M. HYMAN and B. P. LUCE, "Nonlinear waves and solitions in physical systems", Physica D, 123, 1-20, (1998).
• [25] R. CAMASSA and D. D. HOLM, " An integrable shallow water eąuation with peaked solitons", Phys. Rev. Lett., 71, 1661-1664, (1993).
• [26] S. CATHELINE, J -L. GENISSON, M. TANTER and M. FINK, "Observation of shock transverse waves in elastic media", Phys. Rev. Lett, 91, 164301-1-164301-4, (2003).
• [27] S. CATHELINE, J -L. GENISSON and M. FINK, "Measurement of elastic nonlinearity of soft solid with transient elastography", J. Acoust. Soc. Am., 114, 3087-3091, (2003).
• [28] J. CHALLIS, "On the velocity of sound", Phil. Mag., 3, 494-499, (1848).
• [29] G.-Q. CHEN, "Euler eąuations and related hyperbolic conservation laws", vrevrint (2005).
• [30] G.-Q. CHEN and H. FRID, "Divergence-measure fields and hyperbolic conservation laws", Arch. Rational Mech. Anal., 147, 89—118, (1999).
• [31] Y. CHOQUET-BRUHAT, "Ondes asymptotiąues et approchees pour systemes non-lineaires d'equations aux derivees partielles nonlineaires", J. Math. Pure et Appl., 48, 17-158, (1969).
• [32] P. G. CIARLET, Mathematical Elasticiły, 1: Three Dimensional Elasticity, North Holland, Amsterdam, (1988).
• [33] W. D. COLLINS, "One-dimensional non-linear wave propagation in incompressible elastic materials", Quart. J. Mech. Appl. Math., 19, 25-328, (1966).
• [34] R. COURANT and K. O. FRIEDRICHS, Supersonic Flow and Shock Waves, Springer-Verlag, New York, (1948).
• [35] C. DAFERMOS, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, Heidelberg, (2000).
• [36] S. DEMOULINI, D. M. A. STUART, A. E. TZAVARAS, "A variational approximation scheme for three-dimensional elastodynamics with polyconvex energy", Arch. Rational Mech. Anal., 157, 325-344, (2001).
• [37] M. DESTTRADE, G SACCOMANDI, "Solitary and compact-like shear waves in the bulk of solids", preprint, (2006).
• [38] R. DIPERNA, "Measure-valued solutions to conservation laws", Arch. Ration. Mech. Anal, 88, 223—270, (1985).
• [39] R. DIPERNA and A. MAJDA, "The validity of geometrie optics for weak solutions of conservation laws", Comm. Math. Phys., 98, 313-347, (1985).
• [40] W. DOMAŃSKI, Admissible Nonhomogeneous Boundary Conditions for Linear Sym-metric Hyperbolic Systems with Applications in Continuum Mechanics, and Electrody-namics, Ph.D. Thesis, Warsaw, (1988), in Polish.
• [41] W. DOMAŃSKI, "Stability of shock fronts for multidimensional conservation laws", Bonn University, SFB, 256, 1-41, (1989).
• [42] W. DOMAŃSKI, "Surface and boundary waves for linear hyperbolic systems. Applications to basics eąuations of electrodynamics and mechanics of continuum", J. Tech. Phys., 30, 283-300, (1989).
• [43] W. DOMAŃSKI, "Asymptotic eąuations for nonlinear magnetoelasticity", 68-72, in Proc. of Third Polish-Japanese Joint Seminar on Modeling and Control of Electromag-netic Phenomena, Kazimierz, (1993).
• [44] W. DOMAŃSKI, "Simplified asymptotic eąuations for nonlinear magnetoelasticity", J. Tech. Phys., 35, 39-47, (1994).
• [45] W. DOMAŃSKI, "Weakly nonlinear magnetoelastic waves", in Nonlinear Waves in Solids, edited by J. L. Wegner and F. R. Norwood, ASME Book No AMR 137, 241-246, (1995).
• [46] W. DOMAŃSKI and W. KOSIŃSKI, "Asymptotic eąuations of geometrie optics for two models of hyperbolic heat waves", Arch. Mech., 47, 745-753, (1995)
• [47] W. DOMAŃSKI and T. F. JABŁOŃSKI, "Resonant interaction coefficients for non- linear electro-magnetic waves; part I: in dielectrics", 188-190, in The Digesł of Fourth Polish-Japanese Joint Seminar on Electromagnetic Phenomena Applied to Technology, Oita, (1995).
• [48] W. DOMAŃSKI and T. F. JABŁOŃSKI, "Resonant interaction coefficients for nonlinear electro-magnetic waves; part II: in magnetics", 191-193, in The Digest of Fourth Polish-Japanese Joint Seminar on Modeling and Control of Electromagnetic Phenomena, Oita, (1995).
• [49] W. DOMAŃSKI and T. F. JABŁOŃSKI, "Resonant interaction coefficients for nonlinear electromagnetic waves in dielectrics and magnetics", JSAEM Stud. in Appl. Elec-tromagn., 4, 68-72, (1996).
• [50] W. DOMAŃSKI, W. KOSIŃSKI and T. F. JABŁOŃSKI, "Symmetrization of a heat conduction model for a rigid medium", Arch. Mech., 48, 541-550, (1996).
• [51] W. DOMAŃSKI and T. F. JABŁOŃSKI, "Formation of singularities for nonlinear electromagnetic waves", 96-98, in Proc. of 5th Polish-Japanese Seminar on Electromagnetics in Science and Technology, Gdańsk, (1997).
• [52] W. DOMAŃSKI and T. F. JABŁOŃSKI, "Resonances for nonlinear electromagnetic I69 waves in a reciprocal chiral medium", 99-101, in Proc. of 5th Polish-Japanese Seminar on Electromagnetics in Science and Technology, Gdańsk, (1997).
• [53] W. DOMAŃSKI, "Propagation of weakly nonlinear elastic waves in a cubic crystal", 96-97, in Book of Abstracts of Seventh International Conference on Hyperbolic Problems, Zurich, (1998).
• [54] W. DOMAŃSKI, "Asymptotic eąuations for weakly nonlinear elastic waves in a cubic crystal", Int. Ser. Num. Math., 129, 233-241, (1999).
• [55] W. DOMAŃSKI, "Resonant interactions of nonlinear transverse waves in incompressible magnetoelasticity", J. Tech. Phys., 40, 297-307, (1999).
• [56] W. DOMAŃSKI, "Weakly nonlinear elastic piane waves in a cubic crystal", Contemp. Math., 255, 45-61, (2000).
• [57] W. DOMAŃSKI, "Propagation and interaction of finite amplitudę elastic waves in a cubic crystal", 249-252, in Nonlinear Acoustics at the Tum of the Millenium, edited by W. Lauterborn and T. Kurz, American Institute of Physics, 524, Melville, New York, (2000).
• [58] W. DOMAŃSKI and T. F. JABŁOŃSKI, "On resonances of nonlinear elastic waves in a cubic crystal", Arch. Mech., 53, 91-104, (2001). [771
• [59] W. DOMAŃSKI, "Waves in fluid and solid media", J. Acoust. Soc. Am., 109, 2500-2500, (2001).
• [60] W. DOMAŃSKI, "Some remarks on magnetoelastic shocks", 31-32 in Applications of Electromagnetism in Technics and Informatics, PTZE Warszawa, (2004).
• [61] W. DOMAŃSKI, "On stability of shock waves for a perfect magnetoelasticity", 99-110, in Proc. of the Srd International Symposium on Trends in Continuum Physics, TRECOP'04, edited by B Maruszewski, W Muschik, A Radowicz, Poznań (2004)
• [62] W. DOMAŃSKI and R. W. OGDEN, "On the nuli condition for nonlinearly elastic solids", Arch. Mech., 58, 339-361, (2006).
• [63] W. DOMAŃSKI, "Propagation and interaction of weakly nonlinear elastic piane waves in a cubic crystal", submitted.
• [64] W. DOMAŃSKI and R. YOUNG, "Interactions of piane waves in nonlinear elasticity", submitted.
• [65] W. DOMAŃSKI and A. N. NORRIS, "Degenerate weakly nonlinear elastic piane waves", submitted.
• [66] J.Y. DUQUESNE and B. PERRIN, "Interaction of collinear transverse acoustic waves in cubic crystals", Phys. Rev. B, 63, (6): art. no. 064303 FEB 1, (2001).
• [67] S. P. DYAKOV, "On the stability of shock waves", J. Exp. Theor. Phys., 27, 288-296, (1954), in Russian.
• [68] A. G. ERINGEN and E. S. SUHUBI, Elastodynamics, Academic Press, New York, (1974).
• [69] F. I. FEDOROV, Theory of Elastic Waves in Crystals, Plenum Press, New York, (1968).
• [70] S. FRIEDLAND, J. W. ROBBIN and J. SYLWESTER, "On the crossing rule", Comm. Pure Appl Math., 37, 19—38, (1984).
• [71] K. O. FRIEDRICHS and P. LAX, "Systems of conservation eąuations with a convex extension", Proc. Nat. Acad. Sci. U.S.A., 68, 1636-1688, (1971).
• [72] J. GLIMM, "The interactions of nonlinear hyperbolic waves", Comm. Pure Appl. Math., 41,569-590,(1988).
• [73] J. GLIMM and D. H. SHARP, Multiscale Science, preprint, State University of New York, Stony Brook, (1997).
• [74] S. K. GODUNOV, "The problem of generalized solution in the theory of ąuasi-linear eąuations in gas dynamics", Russ. Math. Surveys, 17, 145-156, (1962).
• [75] Z. A. GOLDBERG, "On selflnteraction of piane longitudinal and transverse waves", Soviet. Phys. Acoust, 6, 307-310, (1960), in Russian.
• [76] A. E. GREEN and J. E. ADKINS, Large Elastic Deformations, Clarendon Press, Oxford, (1970).
• [77] R. A. GUYER and P. A. JOHNSON, "Nonlinear mesoscopic elasticity: evidence for a new class of materials", Phys. Today, 30-36, (April 1999).
• [78] M. F. HAMILTON, Y. A. ILINSKI and E. A ZABOLOTSKAYA, "Separation of com-pressibility and shear deformation in the elastic energy density", J. Acoust. Soc. Am., 116, 41-44, (2004).
• [79] A. HANYGA, Mathematical Theory of Non-Linear Elasticity, PWN Polish Scientific Publishers, Warsaw, Ellis Horwood Limited Publishers, Chichester, (1985).
• [80] R. B. HETNARSKI and J. IGNACZAK, Mathematical Theory of Elasticity, Taylor and Francis, New York, (2004).
• [81] M. H. HOLMES, Introduction to Perturbation Methods, Springer-Verlag, Berlin Heidelberg, (1995).
• [82] M. H. HOLZAPFEL, Nonlinear Solid Mechanics, John Wiley & Sons Ltd, Chichester, (2000).
• [83] T. J. R. HUGHES, T KATO, J. E. MARSDEN, "Well-posed quasi linear second -order hyperbolic systems with applications to non - linear elastodynamics and generał relativity", Arch. Rational Mech. Anal, 63, 273-294, (1976).
• [84] J. K. HUNTER, "Interaction of elastic waves", Stud. Appi Math., 86, 281-314, (1992).
• [85] J. K. HUNTER and J. KELLER, "Weakly nonlinear high freąuency waves", Comm. Pure Appl. Math., 36, 547-569, (1983).
• [86] J. K. HUNTER, A. MAJDA and R. R. ROSALES, "Resonantly interacting weakly nonlinear hyperbolic waves, II: several space variable", Stud. Appl. Math., 75, 187-226, (1986).
• [87] A. JEFFREY and T. TANIUTI, Non-linear Wave Propagation, Academic Press, New York, (1964).
• [88] H. K. JENSSEN, "Blowup for systems of conservation laws", SIAM J. Math. Anal., 31, 894-908, (2000).
• [89] X. JIANG and R. W. OGDEN, "On azimuthal shear of a circular cylindrical tubę of compressible elastic materiał", Quart. J. Mech. Appl. Math., 51, 143-158, (1998).
• [90] F. JOHN, "Formation of singularities in one-dimensional nonlinear wave propagation", Comm. Pure Appl. Math., 27, 377-405, (1974).
• [91] F. JOHN, Nonlinear Wave Eąuations. Formation of Singularities, Lehigh University, University Lecture Series, A.M.S. Providence, 1990.
• [92] J. N. JOHNSON and R. CHERET, Classic Papers in Shock Compression Science, Springer-Verlag, New York, (1998).
• [93] J. N. JOHNSON and R. CHERET, "Shock waves in solids: an evolutionary perspective", Shock Waves, 9, 193-200, (1999).
• [94] P. A. JOHNSON and R. A. GUYER, "Nonlinear mesoscopic elasticity: evidence for a new class of materials", Phys. Today, 30-36, (April 1999).
• [95] J.-L. JOLY, G. MfiTIVIER and J. RAUCH, "Resonant one dimensional nonlinear geometrie optics", J. Func. Anal, 114, 106-231, (1993).
• [96] J.-L. JOLY, G. MfiTIVIER and J. RAUCH, "A nonlinear instability for 3 x 3 systems of conservation laws ", Comm. Math. Phys., 162, 47-59, (1994).
• [97] G. L. JONES and D. R. KOBETT, "Interaction of elastic waves in an isotropic solid", J. Acoust. Soc. Am., 35, 5-10, (1963).
• [98] T. KATO, "The Cauchy problem for ąuasi-linear symmetric hyperbolic systems", Arch. Rat Mech. Anal, 58, 181-205, (1975).
• [99] D. J. KAUP, A. REIMAN, A. BERS, "Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium. Rev. Modern Phys., 51, 275-309, (1979).
• [100] B. KEYFITZ and H. KRANZER, "A system of non-strictly hyperbolic conservation laws arising in elasticity theory", Arch. Rational Mech. Anal., 72, 219-241, (1980).
• [101] KLAINERMAN "The null condition and global existence to nonlinear wave equa-tions", Lect. in Appl. Math., 23, 293-326, (1986).
• [102] V. M. KONTOROVICH, "On the ąuestion of stability of shock waves", J. Exp. Theor. Phys., 33, 1525-1526, (1957), in Russian.
• [103] D. J. KORTEWEG and G. DE VRIES, "On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves", Phil. Mag., 39, 422-443, (1895).
• [104] W. KOSIŃSKI, Field Singularities and Wave Analysis in Continuous Mechanics, PWN Polish Scientific Publishers, Warsaw, Ellis Horwood Limited Publishers, Chichester, (1986).
• [105] H. O. KREISS, "Initial boundary value problem for hyperbolic systems", Comm. Pure Appl. Math., 23, 277-298, (1970).
• [106] A. G. KULIKOVSKII and E. I. SVESHNIKOVA, Nonlinear Waves in Elastic Media, CRC Press, Boca Raton, (1995).
• [107] L. D. LAND AU, "On shock waves at large distances from their place of origin", J. Phys. USSR, 9, 495-500, (1945).
• [108] L. D. LAND AU and E. M. LIFSCHITZ, Theory of Elasticity, Pergamon Press, Oxford, (1956).
• [109] P. D. LAX, "Hyperbolic systems of conservation laws II", Comm. Pure Appl. Math., 10, 537-566, (1957).
• [110] P. D. LAX, Hyperbolic Systems of Conseruation Laws and the Mathematical Theory of Shock Waves, CBMS-NSF Regional Conference Series in Applied Mathematics, 11, SIAM, Philadelphia, (1973).
• [111] P. D. LAX, "The multiplicity of eigenvalues", Buli. AMS., 6, 213-214, (1982).
• [112] P. G. LEFLOCH, Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves, Birkauser, Heidelberg, (2002).
• [113] M. J. LIGHTILL, "A method of rendering approximate solutions to physical problems uniformly valid", Phil. Mag., 40, 1179-1201, (1949).
• [114] T. P. LIU amd K. ZUMBRUN, "Nonlinear stability of an undercompressive shock for complex Burgers eąuation", Commun. Math. Phys., 168, 163-186, (1995).
• [115] V. E. LJAMOVT Polarisation Effects and Anisotropy of Interacłions of Acoustic Waves in Crystals, Moscow Univ., Moscow, (1983), in Russian.
• [116] A. MAJDA, The Stability of Multidimensional Shock Fronts, Mem. Amer. Math. Soc. 275, AMS, Providence, (1983).
• [1171 A. MAJDA, The Ezistence of Multidimensional Shock Fronts, Mem. Amer. Math. Soc. 281, AMS, Providence, (1983).
• [118] A. MAJDA, Compressible Fluid Flow and Conseruation Laws in Several Space Dimen-sions, Appl. Math.Sci., 53, Springer-Verlag, New York, (1984).
• [119] A. MAJDA and R.R. ROSALES, "Resonantly interacting weakly nonlinear hyperbolic waves, I: a single space variable", Stud. Appl. Math., 71, 149-179, (1984).
• [120] A. MAJDA, R. R. ROSALES and M. SCHONBECK, "A canonical system of integrod-ifferential eąuations arising in resonant nonlinear acoustics", Stud. Appl. Math., 79, 205-262, (1988).
• [121] G. A. MAUGIN, Nonlinear Waves in Elastic Solids, Oxford University Press, Oxford, (1999).
• [122] J. MAŁEK, J. NEĆAS, M. ROKYTA, M. RUZICKA, Weak and Measure-valued Solu-tions to Evolutionary PDE, Chapman and Hall, London, (1996).
• [123] J. E. MARSDEN and T. J. R. HUGHES, Mathematical Foundations of Elasticity, Pren-tice Hall, Englewood Cliffs, (1983).
• [124] G. METWIER, "Stability of multi-dimensional weak shocks", Comm. Partial Diff. Eqs., 15, 983-1028, (1990).
• [125] G. METWIER, "The błock structure condition for symmetric hyperbolic systems" , preprint, (1999).
• [126] S. MULLER, "Variational models for microstructure and phase transitions", 85-210, in Calculus of Variations and Geometrie Evolution Problems, Lecture Notes in Maths 1713, Springer-Verlag, Berlin, (1999).
• [127] F. D. MURNAGHAN, Finite Deformation of an Elastic Solid, John Wiley k Sons, New York, (1951).
• [128] K. A. NAUGOLNYKH and L. A. OSTOVSKY, NonlinearrWave Processes in Acoustics, Cambridge University Press, Cambridge, (1998).
• [129] S. NOELLE, Cauchy Problems for the Complez Burgers Eeąuation in One and Two Space Dimensions, Ph.D. thesis, Courant Institute, New York University, (1990).
• [130] S. NOELLE, "Development of singularities for the complex Burgers eąuation", Nonlinear Analysis: Theory, Methods, Applications, 26, 1313-1321, (1996).
• [131] S. NOELLE, "Radially symmetric solutions for a class of hyperbolic systems of conser-vation laws", ZAMP, 48, 676-679, (1997).
• [132] A. N. NORRIS, "Finite amplitudę waves in solids", 263—277, in Nonlinear Acoustics, edited by M. F. Hamilton and D. T. Blackstock, Academic Press, San Diego, (1999).
• [133] A. N. NORRIS, "Acoustic axes in elasticity", 315-328, Wave Motion, 40, (2004).
• [134] R. W. OGDEN, Non-linear Elastic Deformations, Dover paperback edition, (1997).
• [135] R. W. OGDEN, Nonlinear Elasticity with Application to Materiał Modelling, Lecture Notes 6, Center of Excellence for Advanced Materials and Structures, AMAS, IFTR PAS, Warsaw, (2003).
• [136] L. A. OSTROYSKY and A. I. POTAPOV, Modulated waves. Theory and Applications, The John Hopkins University Press, Baltimore, London, (1999).
• [137] R. RACKE, Lectures on Nonlinear Evolution Eąuations. Initial Value Problems, Vieweg Verlag, Braunschweig, Wiesbaden, (1992).
• [138] J. RAUCH, "BV estimates fail for most ąuasilinear hyperbolic systems in dimensions greater than one", Comm. Math. Phys., 106, 481-484, (1986).
• [139] J. RAUCH, "Finite speed for symmetrizable hyperbolic systems", preprint, Department of Mathematics, University of Michigan, (2005).
• [140] P. ROSNEAU, "Nonlinear dispersion and compact structures", Phys. Rev. Lett., 73, 1737-1741, (1994).
• [141] P. ROSNEAU, "On solitons, compactons, and Lagrange maps", Phys. Lett. A, 211, 265-275, (1996).
• [142] P. ROSNEAU, "On nonanalytic solitary waves formed by a nonlinear dispersion", Phys. Lett. A, 230, 305-318, (1997).
• [143] P. ROSNEAU and J. M. HYMAN, "Compactons: solitons with finite wavelength", Phys. Rev. Lett, 70, 564-567, (1993).
• [144] P. ROSNEAU, J. M. HYMAN and M. STALEY, "On multi-dimensional compactons", preprint, (2006).
• [145] D. ROYER and E. DIEULESAINT, Elastic Waves in Solids I. Free and Guided Propa-gation, Springer Verlag, Berlin, Heidelberg, (2000).
• [146] A. M. SAMSONOV, "Nonlinear strain waves in elastic waveguides" in Nonlinear Waves in Solids, edited by A. Jeffrey and J.Engelbrecht, Springer, Heidelberg, (1994).
• [147] J. P. SCHAUDER, Oeuvres, PWN, Warszawa, (1978).
• [148] S. SCHOCHET, "Resonant nonlinear geometrie optics for weak solutions of conservation laws", J. Diff. Eq., 113, 473-504, (1994).
• [149] D. SCHAEFFER, M. SHEARER, "The classification of 2 x 2 systems of nonstrictly hyperbolic conservation laws, with application to oil recovery", Comm. Pure Appl. Math., 40, 141-178, (1987).
• [150] J. SCHRÓDER and P. NEFF, "Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions", Int. J. Solids Structures, 40, 401-445, (2003).
• [151] D. SERRE, Systems of Conservation Laws, 1 & 2, Cambridge Univeristy Press, Cambridge, (1999).
• [152] A. SHADI TAHVILDAR-ZADEH, "Relativistic and non-relativistic elastodynamics with smali shear strains", Ann. Insi H. Poincare Phys. Theor., 69, 275-307, (1998).
• [153] T. C. SIDERIS, "The nuli condition and global existence of nonlinear elastic waves", Inven. Math., 123, 323-342, (1996).
• [154] T. C. SIDERIS, "Nonresonance and global existence of prestressed nonlinear elastic waves", Ann. Math., 151, 849-874, (2000).
• [155] M. ŚILHAVY, The Mechanics and Thermodynamics of Continuous Media, Springer-Verlag, Berlin, (1997).
• [156] G. F. SMITH and R. S. RIVLIN, "The strain-energy function for anisotropic elastic materials", Trans, of the American Math Soc, 88, 175—193, (1958).
• [157] J. SMOLLER, Shock Waves and Reaction-Diffusion Eąuations, Springer-Verlag, New York, (1983).
• [158] E. TADMOR, M. RASCLE, P. BAGHERINI, "Compensated compactness for 2D con-servation laws", J. Hyp. Diff. Eq., 2, 697-712, (2005).
• [159] T. TANIUTI and K. NISHIHARA, Nonlinear Waves, Pitman Advanced Publishing Program, Boston, (1983).
• [160] L. TARTAR, "The compensated compactness method applied to systems of conservation laws", 263—285, in Systems of Nonlinear PDEs, NATO Advanced Science Institutes Series, 3, Oxford, (1983).
• [161] M. E. TAYLOR, Partial Differential Eąuations III, Nonlinear Eąuations, Springer-Verlag, New York, (1984).
• [162] J. J. TELEGA and S. JEMIOŁO, "Polyconvexity for anisotropic elastic materials un-dergoing finite deformations. Part I. General setting and existence theorem", 432-438, in Theoretical Foundations of Civil Engineering, edited by W. Szczęśniak, OW PW, Warszawa, (2002).
• [163] P. A. THOMPSON, "A fundamental derivative in gas dynamics", Physics of Fluids, 14, 1843-1849, (1971).
• [164] R. N. THURSTON, Waves in Solids, in Mechanics of Solids, 4, 109-308, edited by C. Truesdell, Springer Verlag, Berlin, (1984).
• [165] K. VAN DEN ABEELE. P. A. JOHNSON, A. SUTIN, "Nonlinear Elastic Wave Spec-troscopy (NEWS) Techniąues to Discern Materiał Damage, Part I: Nonlinear Wave Modulation Spectroscopy (NWMS)", Res. Nondestr. Eval., 12, 17—30, (2000).
• [166] M. WADATI, Y. H. ICHIKAWA, and T. SCHIMIZU, "Cusp soliton of a new integrable nonlinear evolution eąuation", Próg. Theor. Phys., 64, 1959-1967, (1980).
• [167] Z. WESOŁOWSKI, Zagadnienia dynamiczne nieliniowej teorii sprężystości, PWN, Warszawa, (1974), in Polish
• [168] Z. WESOŁOWSKI, "Nonlinear interaction between two shear waves in elastic materials" Eng. Trans., 39, 97-110, (1991).
• [169] G. B. WHITHAM, Linear and Nonlinear Waves, John Wiley and Sons, New York (1974).
• [170] S. WOLFRAM, Mathematica A System for Doing Mathematics by Computer, Addison Wesley, Redwood City, California, (1991).
• [171] J. XIN, "Some remarks on the nuli condition for nonlinear elastodynamic system", Chin. Ann. Math. Ser. £,23, 311-316, (2002).
• [172] R. YOUNG, "Blowup of solutions and boundary instability in nonlinear hyperbolic eąuations", Comm. Math. Sci., 1, 269-292, (2003).
• [173] R. YOUNG and W. SZELIGA, "Blowup with smali BV data in hyperbolic conservation laws" Arch. Rational Mech. Anal, 179, 31-54, (2005).
• [174] E. A ZABOLOTSKAYA, M. F. HAMILTON, Y. A. ILINSKI, and G. D. MEGAN, "Modelling of nonlinear shear waves in solids", J. Acoust. Soc. Am., 116 2809-2813' (2004).
• [175] N. J. ZABUSKY and M. D. KRUSKAL, "Interaction of solitons in a collisionless plasma and the recurrence of initial states", Phys. Rev. Lett., 15, 240-243, (1965).
• [176] L. K. ZAREMBO and V. A. KRASILNIKOV, Introduction to Nonlinear Acoustics, Nauka, Moscow (1966), in Russian.
• [177] L. K. ZAREMBO and V.A. KRASILNIKOV, " Nonlinear phenomena in the propa-gation of elastic waves in solids" Sov. Phys. Usp., 13, 718-797, (1971).
• [178] Y. B. ZELDOVICH, "On the possibility of rarefaction shock waves," J. Exp Theor Phys., 4, 363-364, (1946), in Russian.
• [179] K. ZUMBRUN, "N-waves in elasticity", Comm. Pure Appl. Math., 46, 75-95, (1993).
• [180] K. ZUMBRUN, Asymptotic Behavior for Systems of Nonconvex Conservation Laws, Ph.D. Thesis, Courant Institute, New York, (1990).