and Applied Mechanics
56, 3, pp. 657-674, Warsaw 2018
DOI: 10.15632/jtam-pl.56.3.657
A continuum description of failure waves
failure wave, which however, can not be assigned to any of the classical wave. In this paper,
we build a thermodynamically consistent theory based on the idea that a failure wave is
analogous to a deflagration wave. Our theory admits, as special cases, the classical models
of Feng and Clifton. Two fundamental thermodynamic functions form the basis of our theory.
This approach allows for the construction of a new variational principle and a Lagrangian
formulation. Finally, the theory is linearized to interpret specific experimental observations.
References
Beardsley C.L., Anderson D.D., Marsh J.L., Brown T.D., 2005, Interfragmentary surface
area as an index of comminution severity in cortical bone impact, Journal of Orthopaedic Research, 23, 3, 686-690
Bebernes J., Eberly D., 2013, Mathematical Problems from Combustion Theory, Applied Mathematical
Sciences, Vol. 83, Springer Science & Business Media
Biot M.A., 1955, Variational principles in irreversible thermodynamics with application to viscoelasticity,
Physical Review, 97, 6, 1463
Biot M.A., 1956, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics, 27, 3, 240-253
Biot M.A., 1970, Variational Principles in Heat Transfer: a Unified Lagrangian Analysis of Dis-
sipative Phenomena, DTIC Document
Bless S.J., Brar N.S., 2007, Failure waves and their effects on penetration mechanics in glass
and ceramics, [In:] Shockwave Science and Technology Reference Library, 105-141, Springer
Bourne N.K., Rosenberg Z., 1996, The dynamic response of soda-lime glass, Proceedings of the
Conference of the American Physical Society Topical Group on Shock Compression of Condensed
Matter, 370, 1, 567-572
Callister W.D., Rethwisch D.G., 2011,Materials Science and Engineering, Vol. 5, JohnWiley
& Sons, NY
Chen Z., Feng R., Xin X., Shen L., 2003, A computational model for impact failure with shear-
-induced dilatancy, International Journal for Numerical Methods in Engineering, 56, 14, 1979-1997
Chorin A.J., Marsden J.E., 1990, A Mathematical Introduction to Fluid Mechanics, 3rd edit.,
Springer
Clavin P., Searby G., 2016, Combustion Waves and Fronts in Flows: Flames, Shocks, Detona-
tions, Ablation Fronts and Explosion of Stars, Cambridge University Press
Clifton R.J., 1993, Analysis of failure waves in glasses, Applied Mechanics Reviews, 46, 12, 540-546
Courant R., Friedrichs K.O., 1991, Supersonic Flow and Shock Waves, Applied Mathematical
Sciences, Vol. 21, Springer Science & Business Media
Dafermos C., 2005, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin
Feng R., 2000, Formation and propagation of failure in shocked glasses, Journal of Applied Physics, 87, 4, 1693-1700
Feng X.W., Liu Z.F., Chen G., Yao G.W., 2012, Experimental investigation on delayed failure
of alumina under shock compression, Advances in Applied Ceramics, 111, 4, 237-242
Fung Y.-C., Tong P., 2001, Classical and Computational Solid Mechanics, Vol. 1,World Scientific
Goldstein H., 1965, Classical Mechanics, Pearson Education India
Gurtin M.E., Fried E., Anand L., 2010, The Mechanics and Thermodynamics of Continua,
Cambridge University Press
Kanel G.I., Rasorenov S.V., Fortov V.E., 1991, The failure waves and spallations in homogeneous
brittle materials, Shock Compression of Condensed Matter, 199, 1, 451-454
Kanel G.I., Razorenov S.V., Savinykh A.S., Rajendran A., Chen Z., 2005, A study of
the failure wave phenomenon in glasses compressed at different levels, Journal of Applied Physics, 98, 11, 113523
Lebon G., Jou D., Casas-V´azquez J., 2008, Understanding Non-Equilibrium Thermodynamics,
Springer
Lee J.J.-W., Constantino P.J., Lucas P.W., Lawn B.R., 2011, Fracture in teeth a diagnostic
for inferring bite force and tooth function, Biological Reviews, 86, 4, 959-974
Liebe T., Steinmann P., Benallal A., 2001, Theoretical and computational aspects of a thermodynamically
consistent framework for geometrically linear gradient damage, Computer Methods
in Applied Mechanics and Engineering, 190, 49, 6555-6576
Marsden J.E., Hughes T. Jr., 1994, Mathematical Foundations of Elasticity, Courier Corporation
Maugin G.A., 1990, Internal variables and dissipative structures, Journal of Non-Equilibrium
Thermodynamics, 15, 2, 173-192
Maugin G.A., 1992, The Thermomechanics of Plasticity and Fracture, Vol. 7, Cambridge University
Press
Murakami S., 2012, Continuum Damage Mechanics: a Continuum Mechanics Approach to the
Analysis of Damage and Fracture, Springer
Nedjar B., 2002, A theoretical and computational setting for a geometrically nonlinear gradient
damage modelling framework, Computational Mechanics, 30, 1, 65-80
Neuberg J.W., Tuffen H., Collier L., Green D., Powell T., Dingwell D., 2006, The
trigger mechanism of low-frequency earthquakes on Montserrat, Journal of Volcanology and Geo-
thermal Research, 153, 1, 37-50
Partom Y., 1998, Modeling failure waves in glass, International Journal of Impact Engineering, 21, 9, 791-799
Rayleigh L., 1945, Theory of Sound, Vol. 1, Macmillan, London, reprinted 1945 by Dover, New
York
Sapozhnikov O.A., Maxwell A.D., MacConaghy B., Bailey M.R., 2007, A mechanistic
analysis of stone fracture in lithotripsy, The Journal of the Acoustical Society of America, 121, 2, 1190-1202
Walley S.M., 2013, An introduction to the properties of silica glass in ballistic applications,
Strain, 50, 470-500
Wei H., Samulyak R., 2014, Mass-conservative network model for brittle fracture, Journal of
Coupled Systems and Multiscale Dynamics, 2, 2, 79-90
Zhang Z.-X., 2016, Rock Fracture and Blasting: Theory and Applications, Butterworth-Heinemann
Ziegler H., 2012, An Introduction to Thermomechanics, Vol. 21, Elsevier