Modeling cylindrical waves in nonlinear elastic composites

A procedure of deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves in composite materials modeled by a mixture with two elastic constituents is outlined. Nonlinearity is introduced by metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential. It is the quadratic nonlinearity of all governing relations. For a configuration (state) dependent on the radial coordinate and independent of the angular and axial coordinates, quadratically nonlinear wave equations for stresses are derived and a relationship between the components of the stress tensor and partial strain gradient is established. Four combinations of physical and geometrical nonlinearities in systems of wave equations are examined. Nonlinear wave equations are explicitly written for three of the combinations __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 63–72, June 2007.     

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Plane-Strain State

A method is proposed for deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. The method is based on a rigorous approach of nonlinear continuum mechanics. Nonlinearity is introduced by means of metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. For a configuration (state) dependent on the radial and angle coordinates and independent of the axial coordinate, quadratically nonlinear wave equations for stresses are derived and stress-strain relationships are established. Four ways of introducing physical and geometrical nonlinearities to the wave equations are analyzed. For one of the ways, the nonlinear wave equations are written explicitly__________Translated from Prikladnaya Mekhanika,Vol. 41, No. 5, pp. 40–51, May 2005.     

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Axisymmetric and Other States

A rigorous approach of nonlinear continuum mechanics is used to derive nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. Nonlinearity is introduced by means of metric coefficients, the Cauchy—Green strain tensor, and the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. Quadratically nonlinear wave equations are derived for three states (configurations): (i) axisymmetric configuration dependent on the radial and axial coordinates and independent of the angular coordinate, (ii) configuration dependent on the angular coordinate, and (iii) axisymmetric configuration dependent on the radial coordinate. Four ways of introducing physical and geometrical nonlinearities to the wave equations are analyzed. Six different systems of wave equations are written __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 72–84, June 2005.     

The dynamic response of an incompressible non-linearly elastic membrane tube subjected to a dynamic extension

The dynamic response of an isotropic hyperelastic membrane tube, subjected to a dynamic extension at its one end, is studied. In the first part of the paper, an asymptotic expansion technique is used to derive a non-linear membrane theory for finite axially symmetric dynamic deformations of incompressible non-linearly elastic circular cylindrical tubes by starting from the three-dimensional elasticity theory. The equations governing dynamic axially symmetric deformations of the membrane tube are obtained for an arbitrary form of the strain-energy function. In the second part of the paper, finite amplitude wave propagation in an incompressible hyperelastic membrane tube is considered when one end is fixed and the other is subjected to a suddenly applied dynamic extension. A Godunov-type finite volume method is used to solve numerically the corresponding problem. Numerical results are given for the Mooney-Rivlin incompressible material. The question how the present numerical results are related to those obtained in the literature is discussed.     

Analysis of a quadratic nonlinear hyperelastic longitudinal plane wave

The perturbation (small-parameter) method is used to analyze the propagation of a harmonic longitudinal plane wave in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. The three first approximations are obtained, and the contribution of each of them into the wave pattern is analyzed. It is shown that the third approximation somewhat improves the prediction of the evolution of the initial waveprofile: the tendency to generate the second harmonic goes over into the tendency to generate the fourth harmonic Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 46–58, February 2009.     

Analysis of plane and cylindrical nonlinear hyperelastic waves in materials with internal structure

A comparative analysis of two types of hyperelastic waves—plane waves (with plane front) and cylindrical waves (with curved front)—is offered. The propagation of the waves is studied theoretically for quadratically nonlinear hyperelastic media and numerically for a class of unidirectional fibrous composite materials. Hyperelasticity is described using the classical Murnaghan potential and a structural model of the first order—the model of effective constants. The internal structure of materials is described by this model and is at the micro-or nanolevels in numerical analysis. Particular attention is given to the evolution of the wave profile. It is studied in three stages: (i) derivation of nonlinear wave equations, (ii) construction of solutions in the form of plane and cylindrical waves, and (iii) numerical analysis of the evolution of these waves in composites with microlevel (Thornel) or nanolevel (Z-CNT) fibers. The main similarities and differences between plane longitudinal and cylindrical waves are shown. The most unexpected result is the striking difference between the evolution patterns numerically observed for plane and cylindrical wave profiles __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 21–46, October 2006.     

Generation of the second,fourth, eighth,and subsequent harmonics by a quadratic nonlinear hyperelastic longitudinal plane wave

The perturbation (small-parameter) method is used to obtain the first three approximations for the problem of a harmonic longitudinal plane wave propagating in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. The subsequent approximations are discussed. The contribution of each approximation to the overall wave pattern is analyzed. It is shown that the third approximation corrects the prediction of the evolution of the initial wave profile. Which of the harmonics dominates depends on the distance traveled by the wave: the second harmonic is generated first, then it transforms into the fourth harmonic, and finally, as the distance increases, the eighth harmonic shows     

Thermal Fatigue Damage Assessment in an Isotropic Pipe Using Nonlinear Ultrasonic Guided Waves

The use of nonlinear ultrasonic waves has been accepted as a potential technique to characterize the state of material micro-structure in solids. The typical nonlinear phenomenon is generation of second harmonics. Second harmonic generation of ultrasonic waves propagation has been vigorously studied for tracking material micro-damages in unbounded media and plate-like waveguides. However, there are few studies of launching second harmonic guided wave propagation in tube-like structures. Considering that second harmonics could provide useful information sensitive for material degradation condition, this research aims at developing a procedure for detecting second harmonics of ultrasonic guided wave in an isotropic pipe. The second harmonics generation of guided wave propagation in an isotropic and stress-free elastic pipe is investigated. Flexible polyvinylidene fluoride (PVDF) comb transducers are used to measure fundamental wave and second harmonic one. Experimental results show that nonlinear parameters increase monotonically with propagation distance. This work experimentally verifies that the second harmonics of guided waves in pipe have the cumulative effect with propagation distance. The proposed procedure is applied to assessing thermal fatigue damage indicated by nonlinearity in an aluminum pipe. The experimental observation verifies that nonlinear guided waves can be used to assess damage levels in early thermal fatigue state by correlating them with the acoustic nonlinearity.     

Analyzing the propagation of a quadratic nonlinear hyperelastic cylindrical wave

The perturbation method (small-parameter method) is used to analyze the propagation of a cylindrical wave in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. The exact representation of the first-order solution in terms of squared Hankel functions of zero and first orders and their product is derived. A simplification of the new solution is considered     

Finite amplitude elastic waves due to non-uniform traction at the surface of a cylindrical cavity

This paper is concerned with two spatial dimension, finite amplitude wave propagation emanating from the surface of an initially circular cylindrical cavity in an unbounded isotropic compressible isotropic hyperelastic solid. The solid is initially in the natural reference configuration and the wave propagation is due to an azimuthally non-uniform, sudden application of compressive nominal traction at the surface of the cavity. Governing equations for the problem are obtained in Lagrangian form in terms of cylindrical polar coordinates and two different classes of strain energy functions are considered. Numerical solutions, for a particular application of traction, are obtained from a fully explicit finite difference scheme. It was found that the responses to the particular application of traction differed negligibly for the various strain energy functions considered.     

Comparative Analysis of the Profile Evolutions of an Elastic Harmonic Wave Caused by the Second and Third Harmonics

The influence of the second and third harmonics on the evolution of a harmonic longitudinal wave propagating through a nonlinearly elastic material has been simulated for real composite materials (the most typical plots are presented for a granulated composite with copper granules and molybdenum matrix). The frequency and initial amplitude are varied beginning from conditionally small values (at which visible distortions appear after a great number of oscillations) to extremely large values (at which the profile becomes distorted already after the second or third oscillation). Four and three different stages of profile evolution due to the influence of the second and third harmonics, respectively, are observed. It is found out that the effect of the initial amplitude on the evolution process is weaker for the second harmonic, and the effect of the frequency on the evolution process is weaker for the third harmonics. It is also revealed that the ranges of frequencies and initial amplitudes within which the evolution caused by different harmonics is very intensive are different—the effect of the third harmonic is stronger at larger values of both parameters. The effects of both harmonics are tantamount within the boundary ranges where the second harmonic is already predominant and the third harmonic is at the early stage of development     

Dynamic extension and twist of a non-linear elastic tube

The propagation of waves in a non-linear cylindrical elastic membrane is considered when one end is fixed and the other is subjected to a dynamic extension and twist. The governing equations are derived for a hyperelastic material with a general strain energy function. In order to obtain specific results the equations are specialised to deal with neo-Hookian materials and in this case we show that there are three real wave speeds in each direction along the cylinder. Numerical results are given and a limiting case considered which provides a check on these results.     

Finite amplitude spherically symmetric wave propagation in a prestressed hyperelastic shell

Spherically symmetric finite amplitude wave propagation in a prestressed compressible hyperelastic spherical shell is considered. The prestress results from quasi-static application of internal pressure and a numerical solution for this elastostatic problem is obtained first. Dynamic change of the internal pressure results in the propagation of a spherically symmetric wave. A Godunov type finite difference scheme is proposed for the solution of the wave propagation problem and numerical results, which are valid until the first reflection, are presented for a particular isotropic strain energy function and for the special cases of sudden removal and sudden increase of the internal pressure.     

Non-linear combined axial and torsional shear wave propagation in an incompressible hyperelastic solid

Finite amplitude combined axial and torsional shear wave propagation in an incompressible isotropic hyperelastic solid is considered. When the strain energy function of the solid is a non-linear function offI₁,− 3) and (I₂− 3), where I₁, and I₂are the first and second basic invariants of the left Cauchy-Green tensor, the two second order partial differential equations governing the propagation of the axial and torsional waves are non-linear and coupled. These two coupled equations are equivalent to a hyperbolic system of first order partial differential equations and a modification of the MacCormack finite difference scheme is used to obtain numerical solutions of this system. Numerical results, which show the effect of the coupling, are presented for boundary-initial value problems of propagation into initially unstressed and initially stressed regions at rest.     

Generation of the second, fourth, and eighth harmonics by a hyperelastic longitudinal planewave: numerical simulation

The generation of the first, second, fourth, and eighth harmonics of a harmonic longitudinal plane wave propagating in quadratic nonlinear materials described by the classical Murnaghan model is numerically modeled. This situation was also analyzed analytically. The materials are fibrous composites with micro- and nanoscale structure. Using such materials introduces additional real restrictions on the main parameters and allows simulating various real situations: from the generation of all harmonics for feasible distances and times to the generation of the second harmonics for feasible distances and times. The contribution of each approximation to the overall wave pattern is analyzed. It is shown that these approximations affect the prediction of the evolution of the initial wave profile: first there is a tendency to the generation of the second harmonic which then transforms into a tendency to the generation of the fourth and eight harmonics     

Interesting effects in harmonic generation by plane elastic waves

The harmonics of plane longitudinal and trans-verse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic. This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a pri-mary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.     

Some Remarks Concerning a Boundary-Value Problem in Nonlinear Elastostatics

The paper is concerned with solvability of a mixed boundary value problem of place and traction that describes the straightening of annular cylindrical sectors composed of unconstrained isotropic hyperelastic solids. The existence and uniqueness of solutions is discussed and several examples are exhibited. This revised version was published online in July 2006 with corrections to the Cover Date.     

Wave propagation in circular cylindrical shells with periodic axial curvature

The propagation of longitudinal and flexural waves in axisymmetric circular cylindrical shells with periodic circular axial curvature is studied using a finite element method previously developed by the authors. Of primary interest is the coupling of these wave modes due to the periodic axial curvature which results in the generation of two types of stop bands not present in straight circular cylinders. The first type is related to the periodic spacing and occurs independently for longitudinal and flexural wave modes without coupling. However, the second type is caused by longitudinal and flexural wave mode coupling due to the axial curvature. A parametric study is conducted where the effects of cylinder radius, degree of axial curvature, and periodic spacing on wave propagation characteristics are investigated. It is shown that even a small degree of periodic axial curvature results in significant stop bands associated with wave mode coupling. These stop bands are broad and conceivably could be tuned to a specific frequency range by judicious choice of the shell parameters. Forced harmonic analyses performed on finite periodic structures show that strong attenuation of longitudinal and flexural motion occurs in the frequency ranges associated with the stop bands of the infinite periodic structure.