Asymptotically approximate model equations for nonlinear dispersive waves in incompressible elastic rods

Dai, Hui–Hui; Huo, Yi

doi:10.1007/bf01182157

Cited by 46 publications

(37 citation statements)

References 11 publications

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“…For such axisymmetric problems, S = R 2 will be a more natural radial variable than R. 16 Also, as in Ref. 16, we introduce the transformation U = RV (Z, S,T).…”

Section: B Longitudinal Waves With Small But Finite Amplitudementioning

confidence: 99%

“…(21)- (25) are complex two-dimensional (2D) nonlinear partial differential equations, which are still too difficult to get an analytical solution. For a slender rod, to further simplify the equations, we adopt the asymptotical method introduced by Dai and Huo 16 to tackle such a complicated system. First, the unknowns (w, v andp) can be expanded in the neighborhood of s = 0 as follows:…”

Section: B Longitudinal Waves With Small But Finite Amplitudementioning

confidence: 99%

“…16, we introduce the transformation U = RV (Z, S,T). These two changes of variables are now used to simplify the governing equations (12), (13) and (15)- (17).…”

Section: B Longitudinal Waves With Small But Finite Amplitudementioning

confidence: 99%

“…Among them, the investigation of solitary waves in rods gained a particular popularity due to the simple one-dimensional geometry. [11][12][13][14][15][16] Besides, Yong and LeVeque investigated the longitudinal elastic strain solitary waves in a one-dimensional periodically layered medium. 17 Maugin studied the possibilities of existence of solitary surface waves travelling over a substrate.…”

Section: Introductionmentioning

confidence: 99%

“…In the limit of finite-small amplitude and long wavelength, we simplify the three-dimensional (3D) nonlinear governing equations to one-dimensional (1D) ones by making use of the asymptotic expansions of variables as in Dai and Huo. 16 Then, using the reductive perturbation method gives rise to the far-field equation (the KdV-Burgers equation). Finally, two kinds of explicit wave solutions are presented, namely the kink and kink-like waves, which correspond to the saddle-node heteroclinic orbit and the saddle-focus heteroclinic orbit of the equation, respectively.…”

Section: Introductionmentioning

confidence: 99%

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Kink and kink-like waves in pre-stretched Mooney-Rivlin viscoelastic rods

2015

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The present paper theoretically investigates kink and kink-like waves propagating in pre-stretched Mooney-Rivlin viscoelastic rods. In the constitutive modeling, the Cauchy stress tensor is assumed to consist of an elastic part and a dissipative part. The asymptotic method is adopted to simplify the nonlinear dynamic equations in the limit of finite-small amplitude and long wavelength. Using the reductive perturbation method, we further derive the well-known far-field equation (i.e. the KdV-Burgers equation), to which two kinds of explicit traveling wave solutions are presented. Examples are given to show the influences of pre-stretch and viscosity on the wave shape and wave velocity. It is shown that pre-stretch could be an effective method for modulating the two types of waves. In addition, such waves may be utilized to measure the viscosity coefficient of the material. The competition between the effects of pre-stretch and viscosity on the kink and kink-like waves is also revealed.

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“…For such axisymmetric problems, S = R 2 will be a more natural radial variable than R. 16 Also, as in Ref. 16, we introduce the transformation U = RV (Z, S,T).…”

Section: B Longitudinal Waves With Small But Finite Amplitudementioning

confidence: 99%

Section: B Longitudinal Waves With Small But Finite Amplitudementioning

confidence: 99%

“…16, we introduce the transformation U = RV (Z, S,T). These two changes of variables are now used to simplify the governing equations (12), (13) and (15)- (17).…”

Section: B Longitudinal Waves With Small But Finite Amplitudementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Kink and kink-like waves in pre-stretched Mooney-Rivlin viscoelastic rods

2015

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Buckling and post-buckling of a compressible slab under combined axial compression and lateral load

Wang

2017

Z. Angew. Math. Mech.

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In this paper, we study the buckling and post-buckling of a compressible hyperelastic slab under both axial compressive force and distributed force on two lateral surfaces. The combined series-asymptotic expansions method is used to obtain the simplified model equations. We further assume that the lateral load is uniformly distributed and is a constant. We discuss three cases: (I) axial compression dominant, (II) lateral compression dominant and (III) uniform compression. In case (I), the lateral stress is given as a constant and no buckling bifurcation will occur when the axial stress is not imposed. In this case, the lateral stress may be compressive or tensile stress. In case (II), the axial compressive stress is given as a constant and no buckling bifurcation will occur when the lateral compressive stress is not imposed. In case (III), the slab is uniformly compressed at two lateral boundaries and two ends of the slab. For all three cases, with sliding-sliding end conditions at two ends, we obtain the approximate expressions of the critical stress values and mode numbers from linear bifurcation analysis and obtain the approximate analytical post-buckling solutions by using the method of multiple scales. We find both supercritical and subcritical bifurcations may occur depending on the material constants, slenderness ratio of the slab and bifurcation modes. The analytical post-buckling solutions are compared with the numerical solutions of the model equations and good agreements are found.

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Approximate modeling of wave processes in elastoplastic solids

Myagkov

2012

Acta Mech

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Asymptotically approximate model equations for nonlinear dispersive waves in incompressible elastic rods

Cited by 46 publications

References 11 publications

Kink and kink-like waves in pre-stretched Mooney-Rivlin viscoelastic rods

Kink and kink-like waves in pre-stretched Mooney-Rivlin viscoelastic rods

Buckling and post-buckling of a compressible slab under combined axial compression and lateral load

Approximate modeling of wave processes in elastoplastic solids

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