Similarity representation of wave propagation in a nonlinear viscoelastic rod on a group theoretic basis

Frydrychowicz, W.; Singh, M. C.

doi:10.1016/0307-904x(86)90060-0

Cited by 8 publications

(2 citation statements)

References 5 publications

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“…This seems to be relevant in the mathematical modelling of the behaviour of some metals and polymers [3,10,12]. Several papers have been devoted to the investigation of system (4.1-4.3) either within the framework of Lie-group analysis [1,11,14,15] or within reduction procedures [19,26,27]. The eigenvalues (characteristic wave speeds) of the matrix 0 0-1 -10 0 -</ > 0 0. where p and q must be determined by satisfying conditions (2.8-2.10).…”

Section: Rate-type Materialsmentioning

confidence: 99%

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A method for finding exact solutions to hyperbolic systems of first-order PDEs

Fusco

1996

IMA Journal of Applied Mathematics

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Differential constraints are used as a means of developing a systematic method for finding exact solutions to quasilinear nonautonomous hyperbolic systems of first-order partial differential equations (PDEs) involving two independent variables. The leading assumption of the hyperbolicity of the basic system together with a strict compatibility argument permits characterization of the most general class of quasilinear first-order constraint equations, which can be appended to the governing mathematical model under consideration. Furthermore, the exact solutions can be found by integrating the resulting overdetermined and consistent system of PDEs. When the number of auxiliary constraint equations is equal to that of the field equations, the solutions provided by the 'nonclassical' group method for PDEs are recovered; for a very special form of the appended constraint equations simple wave solutions to hyperbolic systems of PDEs are also obtained. Within the present theoretical framework a model describing rate-type media is considered and several classes of initial-and/or boundary-value problems are solved.

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Section: Rate-type Materialsmentioning

confidence: 99%

“…where m and 8 are constants. Boundary conditions like (4.39) and (4.40) are usually assumed in the study of longitudinal impact in semi-infinite bars [11,15]. In particular, the case m < 0 is consistent with an applied velocity impact which is infinitely large at / = 0 and which then decays with time.…”

Section: Rate-type Materialsmentioning

confidence: 99%