Nonlinear Hyperbolic Waves in Hyperelastic Solids

1993 ◽  
Vol 46 (12) ◽  
pp. 527-539
Author(s):  
J. B. Haddow
Keyword(s):  
Wave Propagation ◽  
Thermal Effects ◽  
Constitutive Relations ◽  
Spatial Dimension ◽  
Similarity Solutions ◽  
Nonlinear Wave Propagation ◽  
Longitudinal Wave Propagation ◽  
Hyperbolic Waves ◽  
Wave Problems ◽  
Rubberlike Materials

This paper considers hyperbolic, one spatial dimension nonlinear wave propagation in a hyperelastic solid, and a discussion of the basic theory is presented. Constitutive relations for compressible rubberlike materials, whose internal energies can be expressed as the sum of a function of specific volume only and a function of temperature only, are discussed. These relations are assumed for the analysis of a class of plane wave problems and similarity solutions are obtained. Thermal effects, including the effect of the jump in entropy across a shock for a problem of uncoupled longitudinal wave propagation, are taken into account, however heat conduction is neglected. Solutions for a piezotropic model, which is a model for which mechanical and thermal effects are uncoupled, are obtained for comparison purposes. An axisymmetric problem is also discussed.

scholarly journals Generalized wave propagation problems and discrete exterior calculus

2018 ◽  
Vol 52 (3) ◽  
pp. 1195-1218
Author(s):  
Jukka Räbinä ◽  
Lauri Kettunen ◽  
Sanna Mönkölä ◽  
Tuomo Rossi
Keyword(s):  
Wave Propagation ◽  
Harmonic Wave ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Time Stepping ◽  
Discrete Exterior Calculus ◽  
Exterior Calculus ◽  
Finite Dimensional ◽  
Order Of Magnitude ◽  
Wave Problems

We introduce a general class of second-order boundary value problems unifying application areas such as acoustics, electromagnetism, elastodynamics, quantum mechanics, and so on, into a single framework. This also enables us to solve wave propagation problems very efficiently with a single software system. The solution method precisely follows the conservation laws in finite-dimensional systems, whereas the constitutive relations are imposed approximately. We employ discrete exterior calculus for the spatial discretization, use natural crystal structures for three-dimensional meshing, and derive a “discrete Hodge” adapted to harmonic wave. The numerical experiments indicate that the cumulative pollution error can be practically eliminated in the case of harmonic wave problems. The restrictions following from the CFL condition can be bypassed with a local time-stepping scheme. The computational savings are at least one order of magnitude.

A Two-Dimensional Study of Finite Amplitude Sound Waves in a Trumpet Using the Discontinuous Galerkin Method

2014 ◽  
Vol 22 (03) ◽  
pp. 1450007 ◽  
Author(s):  
Janelle Resch ◽  
Lilia Krivodonova ◽  
John Vanderkooy
Keyword(s):  
Wave Propagation ◽  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Equations Of Motion ◽  
Numerical Data ◽  
Spatial Dimension ◽  
Finite Amplitude ◽  
Sound Waves ◽  
Nonlinear Wave Propagation

A model for nonlinear sound wave propagation for the trumpet is proposed. Experiments have been carried out to measure the sound pressure waveforms of the [Formula: see text] and [Formula: see text] notes played forte. We use these pressure measurements at the mouthpiece as an input for the proposed model. The compressible Euler equations are used to incorporate nonlinear wave propagation and compressibility effects. The equations of motion are solved using the discontinuous Galerkin method (DGM) and the suitability of this method is assessed. The third spatial dimension is neglected and the consequences for such an assumption are examined. The numerical experiments demonstrate the validity of this approach. We obtain a good match between experimental and numerical data after the dimensionality of the problem is taken into account.

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