Li Yong-chi, T. C. T. Ting. Lagrangian Description of Transport Equations for Shock Waves in Three-Dimensional Elastic Solids[J]. Applied Mathematics and Mechanics, 1982, 3(4): 449-462.
 Citation: Li Yong-chi, T. C. T. Ting. Lagrangian Description of Transport Equations for Shock Waves in Three-Dimensional Elastic Solids[J]. Applied Mathematics and Mechanics, 1982, 3(4): 449-462.

# Lagrangian Description of Transport Equations for Shock Waves in Three-Dimensional Elastic Solids

• Publish Date: 1982-08-15
• A set of transport equations for the growth or decay of the amplitudes of shock waves along an arbitrary propagation direction in three-dimensional nonlinear elastic solids is derived using the Lagrangian coordinates. The transport equations obtained show that the time derivative of the amplitude of a shock wave along any propagation ray depends on(i) an unknown quantity immediately behind the shock wave,(ii) the two principal curvatures of the shock surface,(iii) the gradient taken on the shock surface of the normal shock wave speed and(iv) the inhomogeneous term,which is related to the motion ahead of the shock surface, vanishes when the motion ahead of the shock surface is uniform. Several choices of the propagation vector are given for which the transport equations can be simplified. Some universal relations, which relate the time derivatives of various jump quantities to each other but which do not depend on the constitutive equations of the material, are also presented.
•  [1] Herrman,W.and Nunziato,J.W.,Nonlinear constitutive equations in Dynamic Response of Materials to Intense Impulsive Loading.ed.by P.C.chen and A.K.Hopkins,Air Force Materials Laboratory,(l972),123-281. [2] Chen,P.J.and Gurtin,M.E.,On the growth of one-dimensional shock waves in materials with memorg,Arch.Rat.Mech.Anal.,Vol 36,(1970),33-46. [3] Chen,P.J.and Gurtin,M.E.,The growth of one-dimensional shock waves in elastic nonconductors.Int.J.Solids Structures,Vol.7,(1971),5-l0. [4] Chen,P.J.,One-dimensional shock waves in elastic nonconductors,Arch.Rat.Mech.Anal.,Vol.43,(197l).350-360. [5] Ting,T.C.T.,Further study on one-dimensional shock waves in nonliear elastic media.Q.Appl.Math.,Vol.37,No.4,(1980),421-429. [6] Chen,P.J.and Wright,T.W.,Three-dimensional shock waves and their behaviour in elastic fluids,Mechanics,Vol.10,(1975),232-238. [7] Bowen,R.M.,Chen,P.J.and McCarthy,M.F.,Thermodynamic influences on the behavior of curved shock waves in elastic fluids and the vorticity jump,J.of Elasticity,Vol.6,No.4,(1976). [8] Wright,T.W.,An intrinsic description of unsteady shock waves,Q.J.Mech.Appl.Math.,Vol.24,(1976),311-324. [9] Ting,T.C.T.,Intrinsic description of the three-dimensional shock waves in nonlinear elastic fluids.Int.J.Eng.Sci.,Vol.19,(1981),629-638. [10] Ting,T.C.T.and Li,Y.C.,Eulerian formulation of transport equations for three-dimensional shock waves in simple elastic solids.(submitted for publication). [11] Truesdell,C.A.and Toupin,R.A.,The classical field theories,Handbuch der physik,Ⅲ/I,Springer,(1960) 522,711,610,645., [12] McConnell,A.J.,Application of Tensor Analysis.Dover Publication,New York,(1957),197. [13] Bland,D.R.,On shock waves in hyperelastic media.in Second Order Effects in Elasticity and Fluid Dynamics.International Symposium,Haifa,Israel,April 23-27,ed.Markus Reiner and David Abir,(1962),93-108. [14] Guggenheimer,H.W.,Differential Geometry.McGraw-Hill Book Company,Inc.,New York,(1963),210．

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