<i>Introduction to Wave Propagation in Nonlinear Fluids and Solids </i>

Drumheller, Douglas S.; Harris, John G.

doi:10.1121/1.1448515

Cited by 24 publications

(41 citation statements)

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“…The Riesz fractional integral and derivative [46] has been used to circumvent this problem, which is more appropriate for defining the physics of a problem in spatial coordinates. Thus in a linear elastic material, a point in the material gets information carried by both left and right running characteristics [17]. Therefore the Riesz-Feller fractional integral and derivative [46] are used based on Jumarie's modified Riemann-Liouville derivative.…”

Section: Fractional Calculusmentioning

confidence: 99%

“…It has been known that a material point in a linear elastic body gets information from either side of the point with left and right running characteristics [17] which makes the symmetry properties of the operators used in modeling the problem important [33]. Local fractional derivatives, which are used in modeling continuum problems [39,50], are based on one sided fractional derivatives and do not have the symmetry property.…”

Section: Introductionmentioning

confidence: 99%

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Wave Propagation in Heterogeneous Media with Local and Nonlocal Material Behavior

Aksoy¹

2015

J Elast

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Wave propagation in heterogeneous solids has been an interest of researchers due to industrial applications. Some of the heterogeneous materials can exhibit power law scaling in material behavior which can be characterized by the fractal dimension of the microstructure. In this study, wave propagation in heterogeneous media with self-similar structure is investigated via fractional calculus along with space-time discontinuous Galerkin method. One and two dimensional problems are studied to demonstrate the capability of the proposed model in modeling heterogeneous media. The results show that the proposed model is a good candidate for modeling the mechanical behavior of disordered materials.

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Section: Fractional Calculusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Wave Propagation in Heterogeneous Media with Local and Nonlocal Material Behavior

Aksoy¹

2015

J Elast

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“…The compression modulus of such a fluid increases with increasing pore pressure and contributes to the pressure dependence of the total stiffness of the soil. As a consequence of the increasing stiffness in compression, a continuous compression wave propagating in the soil steepens and turns into a shock wave after a finite time of propagation [7][8][9][10], similar to a compression wave propagating in gas [11]. The distance travelled by a continuous wave before it turns into a shock wave (so-called critical distance) depends on the rate of boundary loading and decreases with increasing rate.…”

Section: Introductionmentioning

confidence: 99%

Blast-induced waves in soil around a tunnel

Osinov

2010

Arch Appl Mech

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The paper is concerned with the numerical modelling of waves in saturated granular soil around a cylindrical tunnel with a tubbing-type lining. The waves are induced by a short-term large-amplitude load on the tunnel wall. The soil is assumed to be fully saturated or contain a small amount (a few volume percent) of free gas in the pore water. The behaviour of the solid skeleton is described by a constitutive relation of the hypoplasticity theory. The presence of gas makes the soil stiffness stronger pressure dependent and leads to the formation of a shock front in the soil in the immediate vicinity of the tunnel. Free gas in the pore water is shown to be an adverse factor from the viewpoint of both the residual state of the soil and the deformation of the lining. A permanent reduction of the effective stresses in the soil caused by the wave is stronger for the soil with gas. For a sufficiently large loading amplitude, the wave may lead to the vanishing of the effective stresses and liquefaction of the soil. It is also shown that the presence of gas increases the displacement and the velocity of the tubbings.

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“…The nonlinear theory of waves in materials is a developing division of mechanics. The recent advances in this theory are described in [2,6,[8][9][10]16]. Waves with curvilinear fronts have been studied much less than waves with plane fronts.…”

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confidence: 98%

“…For convenience, we will write the solution for a quadratically nonlinear elastic longitudinal plane wave [10]:…”

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confidence: 99%

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Primary Analysis of Evolution

Rushchitsky

2005

Int Appl Mech

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The propagation and interaction of hyperelastic cylindrical waves are studied. Nonlinearity is introduced by means of the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. To analyze wave propagation, an asymptotic representation of the Hankel function of the first order and first kind is used. The second-order analytical solution of the nonlinear wave equation is similar to that for a plane longitudinal wave and is the sum of the first and second harmonics, with the difference that the amplitudes of cylindrical harmonics decrease with the distance traveled by the wave. A primary computer analysis of the distortion of the initial wave profile is carried out for six classical hyperelastic materials. The transformation of the first harmonic of a cylindrical wave into the second one is demonstrated numerically. Three ways of allowing for nonlinearities are compared Keywords: nonlinear continuum mechanics, rigorous approach, nonlinear hyperelastic cylindrical waves, distortion, initial wave profile, second harmonic generation Introduction. The nonlinear theory of waves in materials is a developing division of mechanics. The recent advances in this theory are described in [2,6,[8][9][10]16]. Waves with curvilinear fronts have been studied much less than waves with plane fronts. Therefore, analysis of nonlinear cylindrical waves appears to be appropriate as expanding our knowledge of waves in materials.In the previous publications [17,18], an attempt was made to write, based on the general principles of the nonlinear mechanics of hyperelastic continua, the nonlinear equations of motion in cylindrical coordinates, with nonlinearity described by the Murnaghan potential and only quadratic nonlinearity retained in all the analytical representations of mechanical (displacement, strain, and stress) fields. Finally, all the equations of motion written in terms of displacements turned out to be quadratically nonlinear. Note that the assumption of quadratic nonlinearity is acceptable enough, and most problems on nonlinear waves in materials are solved under this assumption [3,4,6,13,14].Let us use the last three equations from [18], which represent an axisymmetric (depending only on the coordinate r and having Oz as a symmetry axis) state of the continuum. As indicated in [18], this state is typical of the classical cylindrical wave or Volterra distortions in a hollow cylinder. Therefore, we will further use these equations to analyze the evolution of a hyperelastic cylindrical wave.1. Wave Equations. Recall that we use the cylindrical coordinate system θ 1 = r, θ 2 = ϑ, θ 3 = z.As indicated earlier, we are considering an axisymmetric configuration (state) with Oz as a symmetry axis.

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Introduction to Wave Propagation in Nonlinear Fluids and Solids

Cited by 24 publications

References 0 publications

Wave Propagation in Heterogeneous Media with Local and Nonlocal Material Behavior

Wave Propagation in Heterogeneous Media with Local and Nonlocal Material Behavior

Blast-induced waves in soil around a tunnel

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Primary Analysis of Evolution

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