Nonlinear Acoustic Waves in Micro-inhomogeneous Solids

By Veniamin Nazarov and Andrey Radostin

Nonlinear Acoustic Waves in Micro-inhomogeneous Solids covers the broad and dynamic branch of nonlinear acoustics, presenting a wide variety of different phenomena from both experimental and theoretical perspectives.

The introductory chapters, written in the style of graduate-level textbook, present a review of the main achievements of classic nonlinear acoustics of homogeneous media. This enables readers to gain insight into nonlinear wave processes in homogeneous and micro-inhomogeneous solids and compare it within the framework of the book.

The subsequent eight chapters covering: Physical models and mechanisms of the structure nonlinearity of micro-inhomogeneous media with cracks and cavities; Elastic waves in media with strong acoustic nonlinearity; Wave processes in micro-inhomogeneous media with hysteretic nonlinearity; Wave processes in nonlinear micro-inhomogeneous media with relaxation; Wave processes in the polycrystalline solids with dissipative and elastic nonlinearity caused by dislocations; Experimental studies of the nonlinear acoustic phenomena in polycrystalline rocks and metals; Experimental studies of nonlinear acoustic phenomena in granular media; and Nonlinear phenomena in seismic waves are dedicated to the theoretical and experimental research of nonlinear processes, caused by longitudinal elastic waves propagation and interaction in the micro-inhomogeneous media with a strong acoustical nonlinearity of different types (elastic, hysteretic, bimodular, elastic quadratic and non-elastic).

This valuable monograph is intended for graduate students and researchers in applied physics, mechanical engineering, and applied mathematics, as well as those working in a wide spectrum of disciplines in materials science.

• Language: English
• Publisher: Wiley
• Release date: Nov 17, 2014
• ISBN: 9781118698327

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Introduction

Acoustics is a branch of continuum mechanics. Therefore, equations of hydrodynamics and theory of elasticity are employed to describe acoustic oscillations and waves in gases, liquids, and solids. The total system of equations consists of the equation of motion (Newton's second law), continuity equation, thermal-transport equation, and dynamic equation of state [1–6]. The first three equations of the system are universal and are, essentially, identical for every media. Acoustic properties of the particular medium, indeed, are engraved on its equation of state and, in general, different media are described by different equations of state.

All of the equations of continuum mechanics are nonlinear. Therefore, no exact solution to the system exists. In this connection an approximate approach is employed to describe wave processes in acoustics and the total system is simplified by small-parameter expansion to derive the wave equation. For liquids and gases this parameter is three-dimensional compression, c00-math-0001 , c00-math-0002 , where c00-math-0003 and c00-math-0004 are the perturbed and steady-state density of the medium; in the case of homogeneous solids this one is strain (longitudinal and shear). In a description of acoustic waves it can be assumed that no heat exchange occurs between the rarefaction and compression parts of medium during half of a wave period; the absorbed energy of the wave changes the equilibrium state of the medium weakly and its movement is close to adiabatic; in addition, dissipative processes due to viscosity and heat conductivity are linear.

There are two equivalent approaches when describing the movement of continuum media [1, 2]. The first one, Eurelian, is employed in hydrodynamics; it describes the movement of medium particles by fixed space coordinates, c00-math-0005 ( c00-math-0006 ), and time, c00-math-0007 . In the second one, Lagrangian, the independent variables are initial coordinates, c00-math-0008 , of a particle in a certain fixed instant of time, c00-math-0009 ; with time the particle moves in space and running coordinates are the functions of the initial coordinates (and time c00-math-0010 ): c00-math-0011 , where c00-math-0012 are vector components of a displacement of the particle in regard to its initial position. (It is notable that both of the approaches were proposed by Euler). Lagrangian coordinates are more applicable to describe wave processes in solids (particularly in the case of one-dimension problems). In linear approximation Eulerian and Lagrangian approaches are identical. However, if nonlinearity is taken into account, the corresponding equations in Eulerian and Lagrangian coordinates become different. Therefore, a derived solution in moving Lagrangian coordinates should be transformed into that in fixed Eulerian coordinates.

I.1 Nonlinearity of Gases and Liquids

In gases and liquids, longitudinal acoustic waves (compression and rarefaction) propagate. In these waves, particles of medium make oscillations along a direction of wave propagation. The description of nonlinear acoustic waves in ideal gases and liquids is founded on the Taylor expansion of the adiabatic equation of state, c00-math-0013 , in terms of small three-dimensional compression, c00-math-0014 , where c00-math-0015 and c00-math-0016 are pressure and density, c00-math-0017 is entropy. In the quadratic approximation this can be written as:

I.1

equation

where c00-math-0019 is pressure at c00-math-0020 and c00-math-0021 is adiabatic sound velocity.

The equation of state for gases has the Poisson form: c00-math-0022 , where c00-math-0023 is the adiabatic exponent, c00-math-0024 and c00-math-0025 are the capacities per unit mass of the gases at constant pressure and volume, respectively. The nonlinearity of ideal gas is related to its heating and cooling at adiabatically fast compression and expansion under the action of the acoustic wave. The sound velocity in the gas is determined as c00-math-0026 , where c00-math-0027 is absolute temperature, c00-math-0028 is the gas constant, c00-math-0029 is molecular weight, and c00-math-0030 . It is worth noting that since c00-math-0031 , the equation of state for gases is always nonlinear. For air ( c00-math-0032 ) at temperature 20 °C and atmospheric pressure c00-math-0033 , the adiabatic exponent and the sound velocity are equal to c00-math-0034 and c00-math-0035 .

For liquids the analogous equation of state is used, so-called Tate's empirical formula, c00-math-0036 , where c00-math-0037 and c00-math-0038 are intrinsic pressure and exponent; these constants are weakly dependent on the temperature and can be measured by experiment. (For many liquids the pressure, c00-math-0039 , is about c00-math-0040 Pa and the value of c00-math-0041 in the range from 4 (as for liquid nitrogen) to 12 (as for mercury). For water the values of the constants are c00-math-0042 Pa, c00-math-0043 , c00-math-0044 m/s.) The expressions for c00-math-0045 and c00-math-0046 in the case of liquids are the same as for gases with c00-math-0047 and c00-math-0048 substituted instead of c00-math-0049 and c00-math-0050 . Nonlinear properties of gases and liquids can be characterized by the nondimensional parameter c00-math-0051 ; the form of this parameter is chosen in such a way to make easy the passage to the limit case of linear media, when c00-math-0052 , that is, c00-math-0053 corresponds to c00-math-0054 . Since c00-math-0055 , then liquids are more nonlinear than gases, c00-math-0056 . It also should be noted that nonlinearity of liquids is stipulated by the interaction of molecules.

I.2 Nonlinearity of Homogeneous Solids

Unlike gases and liquids, in solids there can be not only longitudinal but also shear elastic stresses for which c00-math-0057 . Therefore, in solids, shear (or transverse) waves, as well as longitudinal acoustic waves of compression and rarefaction, are possible. In these waves, the medium particles make oscillations in directions perpendicular to that of the propagation of a wave.

It is customary to describe propagation and interaction of acoustic waves in solids within the framework of the classical five-constant theory of elasticity [1, 3, 5–7]. This theory, being essentially mathematical, determines the nonlinear (in the quadratic approximation) equation of state (i.e., the dependence of the elastic stress tensor, c00-math-0058 , on the derivative, c00-math-0059 , of the components of the displacement vector, c00-math-0060 , with respect to Lagrangian coordinates, c00-math-0061 ) for ideal elastic isotropic media under adiabatic deformation:

I.2 equation

where c00-math-0063 is the internal energy of solid, c00-math-0064 is a strain tensor, and c00-math-0065 , c00-math-0066 .

In cubic approximation, with respect to c00-math-0067 the internal energy, c00-math-0068 , is determined as a Taylor expansion in terms of the strain tensor invariants c00-math-0069 , c00-math-0070 , and c00-math-0071 :

I.3

equation

In this expansion the solid is assumed to be in equilibrium state, hence c00-math-0073 , c00-math-0074 . Introducing in Equation I.3 the notations c00-math-0075 , c00-math-0076 , c00-math-0077 , c00-math-0078 , c00-math-0079 yields:

I.4

equation

where c00-math-0081 and c00-math-0082 are the uniform compression and shear moduli, c00-math-0083 , c00-math-0084 , and c00-math-0085 are the Landau moduli; all of these are determined experimentally and their quantity—five—gave the name to the five-constant theory. Clearly, all of the elasticity moduli – c00-math-0086 , c00-math-0087 , c00-math-0088 , c00-math-0089 , and c00-math-0090 —correspond to their adiabatic values. Additionally, owing to the infinitesimal thermal expansion coefficient of solids, the adiabatic and isothermal values of the moduli c00-math-0091 , c00-math-0092 , c00-math-0093 , and c00-math-0094 differ insignificantly, while these values of the shear modulus, c00-math-0095 , are the same [1, 5].

Essentially, total lack of sound velocity dispersion (up to hypersound) is an inherent feature of homogeneous media, hence their linear ( c00-math-0096 ) and nonlinear c00-math-0097 elasticity moduli are independent of the frequency of the acoustic wave. [It is worth mentioning that for a description of the elastic properties of anisotropic solids—monocrystals—many more independent constants are required; in the general case (in the quadratic approximation), the number is greater than two hundred. Nevertheless, accounting for symmetries reduces this value abruptly; for instance, in the case of cubic crystals it is necessary to introduce no more than three linear and eight nonlinear elasticity moduli [1, 3, 5]. Thus, in spite of differences in chemical composition and structure, all monocrystals are described by the same matrix equation of state. The number and values of independent coefficients in this equation are determined by symmetry of the crystal and by the potential interaction of neighboring atoms].

Often, other pairs of independent moduli are used for characterization of linear properties of isotropic solids: Lamé coefficients c00-math-0098 and c00-math-0099 , and Young's modulus, c00-math-0100 , and Poisson's ratio, c00-math-0101 . Young's modulus determines the relationship between longitudinal stress, c00-math-0102 , and strain, c00-math-0103 , in the rod ( c00-math-0104 ), whereas Poisson's ratio determines the relationship between strains of lateral contraction, c00-math-0105 , and axial tension, c00-math-0106 . From thermodynamic relationships it follows that c00-math-0107 , c00-math-0108 , and Poisson's ratio can vary from c00-math-0109 to c00-math-0110 ; for homogeneous media its value belongs to the range c00-math-0111 , therefore also c00-math-0112 . The extreme case c00-math-0113 corresponds to going from solid to liquid ( c00-math-0114 ) and, in turn, the materials with c00-math-0115 are called water-like materials. The Murnaghan moduli ( c00-math-0116 , c00-math-0117 , and c00-math-0118 ) are sometimes used instead the Landau moduli ( c00-math-0119 , c00-math-0120 , and c00-math-0121 ) [8, 9]; they are simply related by the expressions c00-math-0122 , c00-math-0123 , and c00-math-0124 [1, 3, 5].

Substitution of Equation I.2 into Equation I.4 yields the equation of state for homogeneous perfectly elastic solids:

I.5

equation

It can be seen from this equation that the dependence c00-math-0126 contains a geometric nonlinearity, which is related to the nonlinearity of strain tensor c00-math-0127 and a physical (or material) one (the terms with moduli c00-math-0128 , c00-math-0129 , and c00-math-0130 ), so that even in the case of c00-math-0131 , it remains nonlinear.

In spite of a certain heaviness, Equation I.5 is a rather simple algebraic expression, determining the single valued relationship between c00-math-0132 and c00-math-0133 . For longitudinal stress, c00-math-0134 , and strains, c00-math-0135 , this equation has completely simple form, which can be received from the Taylor expansion of the continuously differentiable, that is, the analytical, function c00-math-0136 with respect to small strain c00-math-0137 ; assuming c00-math-0138 it can be written as:

I.6 equation

where c00-math-0140 is the elasticity modulus and c00-math-0141 is a dimensionless parameter of nonlinearity, c00-math-0142 .

The elastic nonlinearity of homogeneous solids is stipulated by the dependence of intermolecular forces on the displacement of molecules, the absolute value of the parameter c00-math-0143 therefore is sufficiently small and, as a rule, does not exceed 10 for liquids and also for many solids, except pyrex, c00-math-0144 [1]. For shear stresses, c00-math-0145 , and strains, c00-math-0146 ( c00-math-0147 ), Equation I.5 is linear at all ( c00-math-0148 ), since the shear stress is an odd function of shear strain through the space symmetry of shearing of homogeneous isotropic solids. The elastic nonlinearity for shear stress and strains appears in the third order of the Taylor expansion of the function c00-math-0149 with respect to the small c00-math-0150 ; the corresponding theory, therefore, is called nine-constant theory [1, 3]. It is worth noting that theoretical calculations within the frame of these theory are very cumbersome and corrections of the third order are smaller than that of the second one; in this connection description and analysis of nonlinear phenomena in homogeneous media (for longitudinal waves) are carried out in the frame of the five-constant theory.

In the description of deformation of nonideal (viscous-elastic) homogeneous media the elastic stress tensor, c00-math-0151 , should be replaced by the sum c00-math-0152 , where

c00-math-0153

is a viscous stress tensor, c00-math-0154 is particle velocity, c00-math-0155 and c00-math-0156 are shear and volume coefficients of viscosity (their values are determined experimentally), c00-math-0157 , c00-math-0158 [4, 5]. In the case of longitudinal strain (in the line of axis c00-math-0159 ) c00-math-0160 . The presence of viscosity results in linear damping of an acoustic wave and absorption (dissipation) of its energy; the similar additive effect is due to thermal conductivity [4, 5]. Thus, in Lagrangian coordinates, propagation of acoustic waves is actually determined by the equation of motion and by the equation of state (and, of course, by the boundary or initial conditions); in this case the equation of motion has following form [1, 6]:

I.7 equation

(Notice that the continuity equation in Lagrangian description determines the relationship between the density of a medium and its volume strain.) In the case of one-dimensional (in the line of axis c00-math-0162 ) longitudinal waves in a perfectly elastic solid, from Equations I.6 and I.7 and the continuity equation the following nonlinear equations can be derived for displacements of medium particles c00-math-0163 and density c00-math-0164 :

I.8 equation

I.9 equation

where c00-math-0167 is longitudinal wave velocity, c00-math-0168 .

Introducing new variables c00-math-0169 , c00-math-0170 in Equation I.8 allows derivation of the simple wave equation [2] for waves traveling in the positive direction of c00-math-0171 -axis:

I.10 equation

where c00-math-0173 . Equations I.8 and I.10 are basic equations of nonlinear acoustics of homogeneous perfect media (in the quadratic approximation). To transform derived solution of Equations I.8 and I.10 in Lagrangian coordinates for any arbitrary function c00-math-0174 to the one c00-math-0175 in Eulerian coordinates the following approximation can be used (when c00-math-0176 ) [1, 2]

I.11

equation

Linear media (and equations) possess the superposition property, which states that the net response of two or more input impacts (or initial waves) is equal to the sum of individual responses of each input impacts. In compliance with this property waves does not interact and propagate independently of one another. In nonlinear media, however, this is not the case. As well as the sum of individual responses of each input impacts (being something other than that in the linear case due to mutual influence) additional multiplicative responses (or secondary waves) arise due to the interaction of all conceivable wave disturbances. In spite of the smallness of the nonlinear term in the equation of state (Equation I.6) the amplitudes of secondary waves (initially absent) might reach measureable levels. In the case of real media the superposition property holds better as the amplitude of the wave gets smaller and linear approximation corresponds to the propagation of the waves with infinitesimal amplitudes.

Generally speaking, Equations I.8 and I.10 describes a rather narrow range of nonlinear wave phenomena, related substantially to the generation of waves with multiple and combinational frequencies, and distortion of wave profile resulting in the formation of ambiguity in it [1, 2, 5, 6]. In addition, there is an acousto-elastic effect [10, 11], consisting of a change in velocity of a weak (or probe) wave under the action of static load. It is worth noting that the probe wave propagates almost linearly with constant velocity and without distortion. (In nonlinear optics there is an analog of this effect, Pockels electro-optic effect).

The simplest nonlinear phenomenon caused, initially, by harmonic wave propagation is the second harmonic generation. In the ideal media with quadratic elastic nonlinearity (at small distances from the source, well before the shock front formation), the amplitude of the second harmonic is proportional to the amplitude of the initial wave squared; propagating with a constant velocity c00-math-0178 without attenuation, the amplitudes of the highest harmonics of order c00-math-0179 are proportional to the c00-math-0180 -th power of the amplitude of the initial wave therewith. Accounting for the linear loss (viscous or heat-conducting), that is, supplementing the nonelastic terms proportional to c00-math-0181 , c00-math-0182 , c00-math-0183 and c00-math-0184 [1–6], in the equation of state (Equation I.5) and in the wave equations (Equations I.8 and I.10) reduces the nonlinear phenomena intensity but does not change above mentioned trends for the higher harmonics amplitudes. It is significant that nonlinear behavior in solids manifests itself weaker than in liquids, since the velocities of longitudinal waves are c00-math-0185 times c00-math-0186 greater than that in liquids, and hence (in the case of equal parameters of nonlinearity and amplitudes of vibrational speed at fundamental frequency) the amplitude of vibrational speed at the frequency of the second harmonic in solids will be smaller than that in liquids by a factor of c00-math-0187 .

I.3 Micro-inhomogeneous Solids. General Considerations

In micro-inhomogeneous [12, 13] (or mesoscopic [14–17]) media, which include most types of rock and soils, some polycrystalline metals and composite materials, the range of nonlinear wave phenomena is wider and its intensity is several degrees higher. [The term micro-inhomogeneous media refers to those containing defects that are greater in size than the interatomic distances but less than the acoustic wavelength. In these media there are many defects per wavelength and their spatial distribution is statistically homogeneous. Thus, on average, it is possible to consider this medium acoustically homogeneous or macro-homogeneous on a scale larger than the size of the defects but smaller than the wavelength. In the long-wave approximation micro-inhomogeneous media can be considered as homogeneous ones, and the scattering processes of waves on defects can be neglected.] To obtain the nonlinear equation of state for a micro-inhomogeneous medium it is necessary to take into consideration its complex structure and the presence of the nonlinear microdefects, such as cracks, cavities, grains, dislocations, and so on. So the approach based on the equations of the five-constant theory is not applicable. The equations of state of these media correspond to the type and number (i.e., concentration) of defects present in their structure and, as a rule, are nonanalytic, that is, not smooth and not differentiable. For example, the presence of cracks with smooth surfaces in a solid may lead to a difference between its elastic moduli of compression and tension; a granular structure of a material may change the power of the nonlinear term in the equation of state, specifically, from an integer of 2, as in the five-constant elasticity theory, to a fractional power of c00-math-0188 (as follows from Hertz contact theory [5]). One-dimensional defects of the crystal lattice, that is, dislocations, lead to a hysteretic (ambiguous) and dissipative (nonelastic) nonlinearity for polycrystals, and so on [18, 19]. The effective nonlinear parameter of micro-inhomogeneous media thereforre exceeds the corresponding parameter of homogeneous media and materials by two or three orders of magnitude. In addition, nonlinear acoustic properties of micro-inhomogeneous media (as opposed to homogeneous ones) depend on acoustic wave frequency, that is, dispersion of nonlinearity occurs. (Generally speaking, not only nonlinear parameters but also linear ones (damping constant and phase velocity) are frequency dependent; the latter influences the nonlinear phenomena dynamics also, although to a lesser degree.) Therefore, the character of the nonlinear phenomena accompanying the propagation and interaction of elastic waves may be not only quantitatively but also qualitatively different for different micro-inhomogeneous media. All these facts can be used in the diagnostics and nondestructive testing of such media. From the viewpoint of the latter applications, a favorable factor is that the nonlinear acoustic properties of such media are more sensitive to the presence of defects, as compared to the linear ones.

In accordance with the aforementioned peculiaritie, the strong acoustic nonlinearity of micro-inhomogeneous media will be called the nonclassic, as opposed to the classic, weak elastic nonlinearity of homogeneous media. It is worth noting that nonclassic nonlinearity is also small as it has place in homogeneous media, in that the nonlinear term c00-math-0189 in the equation of state c00-math-0190 for micro-inhomogeneous medium is always less than linear one for the strains typical for acoustic waves, that is, c00-math-0191 , therefore c00-math-0192 , c00-math-0193 , c00-math-0194 .

In describing the wave processes in micro-inhomogeneous media with the strong acoustic nonlinearity analytically, the physical (i.e. material or structural) nonlinearity of the state equation is assumed to predominate over the geometric kinematic nonlinearity of the equations of motion and boundary conditions, so that the latter can be ignored. In this approximation, the strain tensor, c00-math-0195 , is a linear function of the components of the displacement vector, c00-math-0196 , with respect to Lagrangian coordinates c00-math-0197 : c00-math-0198 , and the equations of the elasticity theory in the Lagrangian and Eulerian forms thus coincide.

Some features of stationary (i.e., independent on spatial coordinate c00-math-0199 ) acoustic waves in these media can be pointed out [20]. In this case, for one-dimensional longitudinal waves the governing equations have forms:

I.12 equation

I.13 equation

I.14 equation

where c00-math-0203 is perturbation of medium density, c00-math-0204 is particle velocity, c00-math-0205 is longitudinal component of the viscous stress tensor. The form of the function c00-math-0206 is not specified, as long as classic model of viscosity cannot be applicable, and it is assumed that c00-math-0207 if c00-math-0208 . Note that in this case the continuity equation (Equation I.13) is linear. As in hydrodynamics [4], one of the solutions of Equations I.12 is a shock wave (the compression shock). The width of the shock is a finite quantity in real media; it is determined by the shock amplitude and the nonlinear and dissipative properties of a medium. Nevertheless, description of the shock wave can be simplified substantially, if the structure of the shock can be ignored and the shock is considered as discontinuity, in a mathematical sense. Seeking the solution to Equations I.12 in the form of stationary wave propagating with a constant velocity c00-math-0209 , that is, supposing that c00-math-0210 , c00-math-0211 , c00-math-0212 , c00-math-0213 are the functions of the traveling coordinate c00-math-0214 only, and after integrating over c00-math-0215 in the limits from c00-math-0216 to c00-math-0217 , the boundary conditions at the discontinuity are obtained:

I.15 equation

I.16 equation

I.17 equation

where indexes 1 and 2 are related to the quantities behind and before the shock, therewith c00-math-0221 , c00-math-0222 . These equations determine the expression for the velocity of the shock as:

I.18 equation

The latter has a pictorial interpretation. In the quasi-static approximation the slope ratio of the tangent to the curve c00-math-0224 is proportional to the local sound velocity squared, c00-math-0225 . Moreover, in accordance with Equation I.18 the shock velocity squared, c00-math-0226 , is proportional to the slope ratio of the secant joining points 1 and 2 of the curve c00-math-0227 (Figure I.1). For any evolutional shock the condition of stability c00-math-0228 [4] must be fulfilled. Therefore, c00-math-0229 and c00-math-0230 increase when c00-math-0231 (Figure I.1a) and decrease when c00-math-0232 (Figure I.1b). It follows from Equations I.15 and I.16 that:

I.19 equation

that is, increase of c00-math-0234 at shock corresponds to a rarefaction shock wave ( c00-math-0235 , c00-math-0236 ) (Figure I.1a), whereas decrease of c00-math-0237 corresponds to a compression shock wave ( c00-math-0238 , c00-math-0239 ) (Figure I.1b).

c00f001

Figure I.1 Examples of the dependencies c00-math-0240 (solid lines) and shock velocity secants (dashed lines). Indexes 1 and 2 are related to the quantities behind and before the stable shock, respectively

Thus, the shock waves of different types occur depending on the sign of the effective parameter of medium quadratic nonlinearity c00-math-0241 . Their propagation is attended with energy loss and entropy, c00-math-0242 , increase [ c00-math-0243 ] at the vicinity of the shock. The energy loss are determined by the area enclosed by the curve c00-math-0244 and the secant joining the points 1 and 2 (Figure I.1) [2, 4]. Generally speaking, in the description of shock waves with discontinuity the reflection of the wave overtaking the shock with local velocity c00-math-0245 as well as changes in the continuous part of wave should be taken into account [2, 4]. However, it can be shown that reflection coefficient c00-math-0246 in this case is determined by the formula:

I.20 equation

When the values of the velocities c00-math-0248 and c00-math-0249 are close