**Journal of Theoretical**

and Applied Mechanics

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.

*Keywords*: brittle fracture, failure wave, internal variable, dissipative systems, variational principle

#### 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-VÂ´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