Materials and Acoustics Handbook

By Michel Bruneau

Written by a group of acoustics and vibration specialists, this book studies the acoustic and vibrating phenomena that occur in diverse materials used for all kinds of purposes. The first part studies the fundamental aspects of propagation: analytical, numerical and experimental. The second part outlines industrial and medical applications. Covering a wide range of topics that associate materials science with acoustics, this will be of invaluable use to researchers, engineers, or practitioners in this field, as well as students in acoustics, physics, and mechanics.

• Language: English
• Publisher: Wiley
• Release date: May 10, 2013
• ISBN: 9781118622865

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Preface

This collective work tries to fill a gap in scientific publishing in the fields grouped under the expression Materials and Acoustics. At least, that was the starting objective of the editorial committee who gave us the mission of coordinating everything that needed to be done.

From the preliminary outlines of the book, to the creation of the index, this project has enjoyed varying fortunes. However, the result is such that the basics of the fields covering the themes of acoustics for materials and materials for acoustics, from current results of basic research (Parts 1 to 5 for the essentials) to applications (especially Parts 6 to 8), are presented in this work.

The construction of this book has been governed by a certain logic, following which the essential, imperative elements generally precede presentations of applications. However, the contribution of each author remains understandable on its own.

Ignoring the fact that no-one can ever be completely happy, we would sincerely like to thank every author for the confidence they have put in us and for their support in this collective work.

Michel BRUNEAU

Catherine POTEL

Part 1

Homogenous and Homogenous Stratified Media: Linear Model of Propagation

Chapter 1

Equations of Propagation ¹

1.1. Introduction

Acoustic (or elastic) waves are disturbances of a deformable medium (fluid or solid) propagating step by step into it by the actions of elementary particles on their neighbors. They do not exist in a vacuum.

We will consider the medium as continuous (continuum mechanics), that is to say we will take an interest in elementary volumes which are much larger than atoms or molecules; it will be assumed that this volume contains a great number of these elements. We will sometimes name this volume fluid or solid particles. The acoustic equations will therefore result from those of continuum mechanics.

We will first consider the medium as homogenous. In the absence of acoustic waves, it is in equilibrium. The acoustic waves will disturb this equilibrium but this disturbance will be small and will allow the equations to be linearized. The movement of the particles related to the wave will be described by a specific number of variables which will depend on the nature of the medium studied.

1.1.1. Fluid medium

In the case of fluid media, the classic variables we are interested in are the vector of the particle speed , the p ressure ρ and the density ; consequently, there are three variables among which one is vectorial, and we need the same number of equations in order to solve the problem. We will assume a certain number of simplifying hypotheses; we will specifically neglect the viscosity and will suppose that, in the absence of acoustic waves, the fluid is at rest (no flow) and that there is no external force. Then the variables considered satisfy three equations. One is vectorial, the mass conservation law, one is the fundamental equation of dynamics, general equations for all fluids, and the other is the medium state equation which characterizes its properties.

1.1.1.1. Mass conservation law

This relationship shows that the mass rise in a particular volume is opposed to the mass outflow through the surface surrounding the volume. Locally it is given by the following equation:

[1.1]

1.1.1.2. Fundamental equation of dynamics

In the absence of external forces, the only force applied on the elementary volume is the pressure gradient. This equation is therefore:

[1.2]

1.1.1.3. State equation

The mechanical properties of the fluid medium are characterized by its state equation. We are generally interested in sufficiently fast pressure variation (and consequently in density variation), which allows the thermal exchanges between the elementary volumes to be neglected, the transformation of each element being adiabatic. Then the state equation can be expressed as a relationship between pressure and density:

[1.3]

We will be interested only in fluids that have the following properties (barotropic ideal fluids): the function g(ρ) is positive as well as its first and second derivatives. We now introduce the following notation:

[1.4]

The quantity V0 has the dimension of a speed; that is the definition of the velocity of sound in a fluid. The compressibility factor of the fluid is defined by:

[1.5]

The velocity is then:

[1.6]

1.1.1.4. Linearization of equations

Linear acoustics studies motions with small amplitudes of the particles around a balance state characterized by a constant pressure p0 and density ρ0 , and a null flow speed. That is why these quantities are written as unchanging values to which are added small variations, that is to say:

[1.7]

with

[1.8]

Considering these equations and conserving only first-order terms, equations [1.1] to [1.3] become:

[1.9]

[1.10]

[1.11]

Considering p0 = g(ρ0) and = dg/dρ, these equations become:

[1.12]

[1.13]

[1.14]

There are five scalar equations for five unknown quantities p1 ,ρ1 and v1.

1.1.1.5. Velocity potential and Wave equation

It can be shown, from equation [1.13], that the particle speed vector is irrotational, that is to say it derives from a scalar potential . Equations [1.12] and [1.14] also allow us to calculate the variables p1 and ρ1 from this potential:

[1.15]

[1.16]

[1.17]

Transferring equations [1.15] and [1.17] in the linearized mass conservation law [1.12], and considering the relationship div φ = Δ φ where Δ φ refers to the Laplacian of φ, we obtain successively:

then:

[1.18]

Equation [1.18] is called the wave equation; it is applied here to the scalar variable φ. The three variables p1, ρ1 and v1² are deduced from φ by equations [1.15] to [1.17]. Furthermore, each of these variables satisfies the wave equation.

1.1.1.6. Energy conservation law

Combining the previously obtained equations, it is easy to deduce the energy conservation law. It is written:

[1.19]

This is a first order partial differential equation that contains only small terms of the second order. In this equation

is the bulk density of kinetic energy,

is the bulk density of potential energy.

This energy conservation law expresses that the energy variation in a closed volume is equal to the energy outflow of the volume considered.

1.1.1.7. Remarks

It is essential that the evolution of acoustic perturbations is described by a wave equation which can be applied to any of the variables ϕ p1, ρ1 and or to the particular displacement vector 1. A direct consequence of linear acoustics is the irrotational property of the speed (or displacement) field of the acoustic perturbation; that explains the equivalence between the vectorial fields ( or 1)and the scalar fields p1 and ρ1. This will not be true in the case of solids, for which the problem will be a little more complicated.

Later, we will focus on small variations of physical quantities (until now denoted with subscript 1); they will be written without subscript, in order to simplify the forward expressions, keeping the subscript 0 for average values of these quantities.

1.1.2. Elastic solid

The mechanical properties of solids are basically different from those of fluids; they are expressed by relationships between internal stresses (more complex than a simple pressure) and local deformation of the solid around a point. Thus, it is necessary to introduce the strain tensor in order to characterize these deformations and the stress tensor to characterize the internal efforts. The acoustic field can no longer be expressed by a simple scalar variable; the base variable generally used is the particular displacement .

1.1.2.1. Strain tensor

The local deformations of a solid around a given point will naturally be expressed from the total differential of the displacement vector, which can be written in a Cartesian coordinate system:

[1.20]

is a second order tensor, which can be broken up as a sum of a symmetric tensor and a antisymmetric tensor:

[1.21]

The antisymmetric part corresponds to a rotation of the considered element, while the symmetric part corresponds to its deformation, that is to say it represents its length variation. This tensor is called the strain tensor and will be noted . Considering its symmetry, it has six independent components which are written in the considered Cartesian system:

[1.22]

1.1.2.2. Stress tensor

In the case of solids, the internal strains are due to molecular interaction forces with very short range. They can be modeled by forces acting through the surface separating two small adjacent elements of the solid. That is why they will be characterized by a surface density of forces at each point, that depends on the orientation of the normal to the considered elementary surface dS.

Figure 1.1.

Within the solid, the elementary surface dS divides, in the neighborhood of the considered point M, the material into two domains (a) and (b). On each side of dS, particles of both domains are found; particles of (a) apply an elementary force dS on the particles of (b). The vector , which has the dimension of a surface force, is the stress vector at the point M relative to the direction . According to the action-reaction principle, particles of (b) apply on particles of (a) an equal and opposite force. Let us note that in the solid case, the vector F is generally not collinear to ; its normal component (projection on ) corresponds to a traction-compression strain and its tangential component (projection on the plane normal to ) to a shear strain.

We can show that the fundamental equation of dynamics implies that the relationship between and is linear. Consequently, this relationship defines a tensor , of second order, named stress tensor. It can be shown that this tensor is symmetric.

It follows from the definition of this tensor that, in a Cartesian system, the components of the ith column vector of the matrix of the components of represent the components of the stress vector corresponding to a surface element perpendicular to the axis i. In the fluid case, this tensor is reduced to a spherical tensor whose components are equal to . The minus sign comes from the difference of sign convention adopted between stress in the case of solids and pressure in the case of fluids.

The mechanical properties of solids will be characterized by relationships between stress and strain (behavior law). We will first take an interest only in linear elasticity (as for the fluids, the stresses and strains will be considered small enough), that is to say that the stress and strain tensors will be deduced from each other by a linear transform (Hooke’s law).

1.1.2.3. Isotropic solids

1.1.2.3.1. Stress–strain relationship

If the solid is isotropic, that is to say if its properties are the same in every direction, it can be shown that these relationships are expressed by two constants only, for example the Lamé constants, which enables us to write them in the following way:

[1.23]

In all the following, indicial expressions will use the Einstein convention (also known as the Einstein summation): when an index variable appears twice in a single term, it implies that we are summing over all of its possible values. Using this convention, we obtain: Sll = S11 + S22 + S33.

These relationships can also be expressed as a function of the Young’s modulus E and Poisson’s ratio? in the following ways (direct and inverse):

[1.24]

[1.25]

The comparison of equations [1.23] and [1.25] allows us to deduce the relationships between these constants:

[1.26]

1.1.2.3.2. Equation of motion

In the solid case, the fundamental equation of dynamics is written:

[1.27]

where is the resultant of the exterior bulk forces which will be supposed static. Taking account only in small variations around the equilibrium position, let us separate stresses into two parts, one corresponding to static stresses at equilibrium (subscript 0) and another (subscript 1) corresponding to elastic waves:

[1.28]

The equation at equilibrium is written in the following way:

[1.29]

which allows us to write equation of motion [1.27] as a function of the only part corresponding to the perturbation of equilibrium in the following way:

[1.30]

As we are only interested in waves, thus to the part of the stress tensor with subscript 1, we now omit this subscript in order to simplify the forward expressions, as has been done for fluids.

Considering the stress–strain relationships given by equation [1.25] and the expression of the strain tensor as a function of the displacement vector [1.22], the right-hand side of [1.30] can be expressed as a function of the single variable .

The equation then becomes:

[1.31]

1.1.2.3.3. Elastic wave equation

Let us introduce the following quantities which have the dimensions of a speed:

[1.32]

That leads us to write equation [1.31] in the following way:

[1.33]

A vectorial field can be broken into two fields, one with null rotational and the other with null divergence. We can then write:

[1.34]

with:

[1.35]

From these relationships, it is possible to show that each of the two strain fields and L and T satisfies a wave equation. These equations are written:

[1.36]

[1.37]

The perturbation field is broken into two wave fields, the one whose curl is zero corresponds to waves propagating with speed VL; these waves will be named compression waves (associated strains are of traction-compression type). The other one, whose divergence is zero, corresponds to waves propagating with speed VT and these waves will be named shear waves (associated strains are of shear type).

Let us observe that the propagation speeds can be expressed from Lamé’s constants; they become:

[1.38]

The displacement field corresponding to compression waves being irrotational, derives from a scalar potential and can therefore be written:

[1.39]

The field corresponding to shear waves, having a null divergence, can be written as the curl of a vector , called potential vector of these waves:

[1.40]

These two potentials satisfy the wave equation, with a speed VL for φ and a speed VT for

Remarks

The variables φ and correspond to four scalar variables, whereas the displacement field depends only on three. An additional relationship can therefore be set between these variables, for example the relationship:

[1.41]

φ and are displacement potentials whereas, in the case of fluids, the potential φ was a velocity potential.

The compression waves can be expressed as a function of a scalar variable (for example the potential φ ); these waves are of the same kind as those which propagate in a fluid. Shear waves are expressed as a function of a vectorial variable (displacement or vector potential ) whose components are not independent (they are related by a relationship, for example equation [1.41]) and then are expressed as a function of two scalar variables.

1.1.2.4. Anisotropic solid

1.1.2.4.1. Stress–strain relationship

An anisotropic solid has, by definition, properties which depend on the direction in which we are interested. For example if traction is applied on such a solid, its reaction to this force will depend on the direction in which the force is applied. The parameters characterizing the local mechanical state are the same as those defined for isotropic solids (strain tensor and stress tensor), however the stress–strain relationships will be expressed by a larger number of parameters [ROY 96]. If we always consider only linear elasticity (Hooke’s law), as it has already been specified, the strain and stress tensors are deduced from one another by a linear transform; considering that the two tensors are of second order, this transform corresponds to a fourth-order tensor (called the elastic stiffness tensor), which is expressed theoretically with the help of 3⁴ = 81 components.

Then the relationships can be written:

[1.42]

or in Cartesian coordinates:

[1.43]

Actually, the stress and strain tensors being both symmetric, each depends only on six independent components, which reduces the number of independent components for the stiffness tensor to 36. The components obtained by commuting the first two or last two indices are equal:

[1.44]

Hooke’s law is written as a function of the displacements:

[1.45]

As , the two sums are equal:

[1.46]

As it is difficult to write all these components of the stiffness tensor in this form, equation [1.46] is often written in a matrix form using a 6 x 6 matrix, where each set of indices ij and kl is replaced by a single one, noted α and β and numbered from 1 to 6 with the help of the following relationships:

[1.47]

By using this notation for the components of the stress and strain tensor, the relationships are written:

[1.48]

where

Tα = Tijkl ,

and

[1.49]

It is possible to express the strains as function of the stresses by inverting the formula [1.43]. Then the flexibility tensor is sobtained:

[1.50]

This inversion can also be done using equation [1.48]. The matrix sαβ being the inverse of the matrix cαβ, we obtain:

[1.51]

with

[1.52]

where p is the number of indices greater than 3 in the pair αβ.

Some thermodynamic considerations [ROY 96], demonstrating the symmetry of the matrix cαβ (and sαβ), reduce again, in general, the number of elastic independent constants to a maximum of 21. This is equivalent to saying that the constants cijkl (and sijkl) are unchanged when the pairs of indices ij and kl are exchanged.

The maximum number, 21, of elastic independent constants is reached when the structure of the material has no symmetry (the material is then called triclinic). Such symmetries reduce this number, and, in practice, it is frequently smaller. We will not consider all the cases of possible symmetries, which are listed in reference [ROY 96], but we will simply give some common examples for some materials.

Orthotropic material

The structure of such a material possesses three binary axes of symmetry orthogonal with each other. The number of elastic independent constants is then reduced to 9. If the axes of the coordinates system are chosen parallel to these axes of symmetry, the table of elastic constants is:

[1.53]

Isotropic transverse material

The structure of this type of material possesses a rotation symmetry axis of any angle (the properties of this type of material are the same in each direction perpendicular to this axis). Then the number of elastic constants is reduced to 5. If the axis Ox3 is chosen parallel to this symmetry axis, the table is:

[1.54]

Isotropic material

As we have already seen, an isotropic material has the same properties in all directions. The number of elastic constants is then reduced to 2. The table can be written:

[1.55]

which yields the same result as equation [1.23] with

[1.56]

1.1.2.4.2. Elastic wave equation in an anisotropic solid

The movement equation is written in the same way as for the isotropic solid, if we focus only on small variations around equilibrium, it is written according to formula [1.30]:

[1.57]

The tensor representing the variation of the stress tensor with respect to the position of equilibrium, the subscript 1 has been omitted in order to simplify the equation.

Considering the expression of the stress tensor as a function of the displacements in expression [1.46], it becomes, with indicial writing:

[1.58]

This equation can be considered as a generalization of the wave equation which is applied for anisotropic solid media.

1.2. Solutions of the propagative equation: monochromatic waves, plane waves

Among the solutions of the propagative waves equations (from a general point of view, that is to say also in anisotropic media), there are two types which are of particular interest and use. The first concerns the temporal dependence of the solution: monochromatic waves, and the other its spatial dependence: the case of plane waves.

A monochromatic wave is a wave whose temporal dependence is sinusoidal (harmonic); it will be expressed with real notation as a sine or cosine, multiplied by any function of the spatial coordinates, or with complex notation as a complex exponential, multiplied by the same spatial function. The latter notation will be used, knowing that the real physical variable corresponds to the real part of this complex function. This notation is usable thanks to the linearity of the equations, and only for physical quantities associated with variables of degree one. For example, if the used variable is the displacement vector, this type of solution will be written:

[1.59]

or in indicial form:

[1.60]

where ω is the wave pulsation, its frequency ν is equal to ω/2π.

We chose to use the + sign before iωt in the exponential formula [1.59], another convention uses the – sign. These two conventions are strictly equivalent but cause sign changes in the resulting formulas and should therefore not be mixed.

A plane wave is a wave whose physical quantities are identical in planes parallel with each other. We call these planes, wave planes, and direction of propagation, their perpendicular direction. If is the unit vector of the propagation direction, the equation of the wave planes can be written in the form:

[1.61]

This type of solution can therefore be written:

[1.62]

The – sign corresponds to waves propagating in the direction of and the + sign to waves propagating in the opposite direction. V is the phase velocity. If the environment is isotropic (liquid or solid), it is the speed appearing in the wave equation [1.18], [1.36] and [1.37].

A monochromatic plane wave is, in complex notation, of the form:

[1.63]

The vector the wave number vector, is parallel to the direction of propagation, and its module is the wave number k = ω V . The wavelength is then defined by the formula λ = 2π/k and represents the spatial periodicity of the wave. The vector is a constant vector characterizing the direction of motion of the particles; it can be written in the form of its module a multiplied by a unit vector , called the polarization vector . It can also be complex in which case the movement of particles is elliptical rather than linear.

Note that these types of solutions have no physical reality: monochromatic waves because they cover all the times from minus infinity to plus infinity, and planes waves for a similar reason but in the geometric space. Their interest, however, is double: on one hand, they approximately approach a number of practical cases (the waves will be almost monochromatic and/or almost plane) and on the other hand, because Fourier transformations can decompose any field into these types of waves.

1.2.1. Fluid medium or isotropic solid

If the environment is a fluid or an isotropic solid, the propagation equations are wave equations (one for fluids, two independent for isotropic solids). Searching for solutions in the form of monochromatic waves [1.59] leads to the equation for the displacement vector:

[1.64]

where k = ω/V is the wave number. Equation [1.64] is called the Helmholtz equation.

1.2.1.1. Fluid medium

What has just been described for the displacement vector also applies to the velocity vector (these two vectors, in the case of monochromatic waves, are formed from one another simply by multiplying by iω ) but also applies to the velocity potential defined in 1.1.1.5.

We have then, according to equations [1.15] and [1.16]:

[1.65]

[1.66]

[1.67]

The particle velocity and particle displacement vectors are parallel to the wave number vector, and hence to the direction of propagation. These waves (compression waves) will be called longitudinal waves. The term longitudinal wave is often used synonymously with compression wave whereas, strictly speaking, it applies only to waves whose displacement vector is at every point parallel to the direction of propagation, such as plane waves.

1.2.1.2. Isotropic solid

We have seen in paragraph 1.1.2.3.3. that, in the case of isotropic solid media, there are two types of waves, one which is derived from a scalar potential φ and the other derived from a vectorial potential . The first wave is the same kind as waves in fluids; the displacement vector is written for monochromatic plane waves:

[1.68]

These are the longitudinal waves.

The second type is written:

[1.69]

The divergence of T is zero, which leads to:

[1.70]

The second type is written:The displacement vector is then perpendicular to the direction of propagation. Such waves (shear waves) are called transverse waves. In the plane perpendicular to the direction of propagation, the displacement vector has two components corresponding to two polarizations. As for longitudinal waves, strictly speaking, the term transvers waves applies only for waves whose displacement vectors are at any point perpendicular to the direction of propagation, such as plane waves.

1.2.2. Anisotropic solid

The equation of propagation is no longer in this case the wave equation, it is equation [1.58]. Let us again seek solutions in the form of plane waves that we write in the general form (limited to waves propagating in the direction and using the indicial form):

[1.71]

1.2.2.1 Christoffel tensor

Inserting the solution given by [1.71] in the equation of propagation [1.58], gives:

[1.72]

Let us introduce the second order tensor defined by:

[1.73]

Then the former equation, called Christoffel equation, is written:

[1.74]

This equation shows that the direction of displacement given by the vector of components ai (polarization) is an eigenvector of the tensor of components Γ:il with the eigenvalue ρ V².

Consequently, speed and polarizations of plane waves propagating in a direction in an anisotropic solid (whose stiffness tensor components are cijkl ) can be obtained by searching the eigenvalues and eigenvectors of the tensor of components Γil . There are, in general, for a given direction, three propagation velocities, which are the roots of the secular equation:

[1.75]

formulating the compatibility condition of the three homogenous equations [1.74], where the notation corresponds to the determinant of the square matrix A.

δil represents the components of the Kronecker tensor, equal to one if i is equal to l and zero if i is not equal to l.

An eigenvector defining the direction of the particle displacement vector of the wave corresponds to each velocity.

In addition, the Christoffel tensor is symmetric: in fact, by swapping the first two indices, the last two or the pairs ij and kl, the coefficients cijkl are unchanged, then successively:

[1.76]

Its eigenvalues are therefore real and there is always an orthonormed basis of eigenvectors. Moreover, we can show [ROY 96] that its eigenvalues are always positive, which defines a real phase velocity V if the elastic stiffness tensor is real (in the case of non-absorbent materials seen so far).

In general, for each direction there are three waves propagating at three different speeds, whose displacement vectors are orthogonal with each other: they are eigenvectors of the Christoffel tensor. Thereafter the polarization of the waves will be represented by a normed vector (η) and the displacement vector will be equal to: (η)a (η) where (η)a is the amplitude of the wave η. Figure 1.2 represents these polarizations for a given direction of propagation of the wave.

Figure 1.2. Polarization of the waves propagating in a given direction

The wave whose direction of polarization is the closest to the direction of propagation will be called quasi-longitudinal and the other two quasi-transverse (slow and fast). These terms are commonly used but are not very appropriate.

If an eigenvalue is twofold, a plane of eigenvectors will then be associated with this value and any vector belonging to this plane will be a possible polarization of a wave whose speed corresponds to this value.

The isotropic solid is obviously a special case of an anisotropic solid; the components of the stiffness tensor are then given by the following formula:

[1.77]

and the Christoffel tensor is written:

[1.78]

Then there is a twofold eigenvalue associated with a plane of polarization of the waves perpendicular to the direction of propagation (it corresponds to the transverse waves), and one single value associated with a polarization parallel to the direction of propagation corresponding to longitudinal waves.

1.2.2.2. Poynting vector – energy velocity

Bulk densities of kinetic energy and potential energy respectively are classically defined by:

[1.79]

Then, starting from the propagation equation, it is possible to establish the following relationship [ROY 96]:

[1.80]

This relationship may also be expressed locally using an integral form. By integrating on a volume V bounded by a closed surface ∑, and applying the theorem of the divergence, it becomes successively:

[1.81]

with

[1.82]

and by inverting the order of the integral and derivation, this equation can be written as:

[1.83]

This equation describes an energy conservation where , the Poynting vector, is a surface density of energy flows. It expresses the variation, during the time interval dt, of the total energy (sum of kinetic energy and potential energy) as equal to the outflow of energy through the surface ∑.

The speed of energy transportation (or speed energy) is then defined as the ratio of this vector to the total energy density:

[1.84]

In the case of a plane wave, all variables depending on the displacement vector can be rewritten, using the expression given in equation [1.62]; then we can demonstrate that, for this type of wave, the bulk densities of kinetic and potential energy are equal, and equation [1.84] becomes:

[1.85]

Remember that Pj and Pk are the components of the polarization vector (unitary vector of the direction of particular displacement).

The energy velocity indicates the direction of travel of the energy that is, in general, different from the propagation direction of the wave defined by the vector .

Similarly, it is easy to show that the projection of the energy velocity on the direction of propagation is equal to the phase velocity of the plane wave (Figure 1.3):

[1.86]

Figure 1.3. Projection of the energy velocity on the direction of propagation

This implies that the energy velocity is always greater than or equal to the phase velocity.

In conclusion, in an anisotropic medium, unlike in isotropic media, a plane wave propagating in a given direction generally has a speed of energy transportation (energy velocity) whose direction is different from the propagation direction.

1.2.2.3. Slowness surface

The slowness surface corresponds to the points M obtained by drawing, from a given origin O, a vector (slowness vector) whose direction is the direction of propagation, and of module the inverse of the phase velocity, for all directions of propagation , i.e.:

[1.87]

For each direction of propagation, generally, three waves propagate at three different phase velocities; consequently, this surface is composed of three distinct layers: one corresponds to quasi-longitudinal waves and the other two to quasitransverse waves. If, as it is often the case, the phase velocity of longitudinal waves is lower than these of transverse waves, the corresponding layer is then located inside the two others.

The calculation of the differential of the slowness vector yields the relationship:

[1.88]

This relationship applies to all vectors of the plane tangential to the slowness surface. It follows that the energy velocity is normal to the slowness surface.

The phase velocity remaining unchanged if we invert the direction of propagation, the slowness surfaces are symmetric with respect to the origin O. This comes from the fact that the stiffness tensor affecting the propagation properties is a tensor of even order (order 4). When the material considered has, in addition, a particular symmetry axis, slowness surfaces have the same symmetry.

In the case of fluids, a single wave propagates in a given direction. The slowness surface has only one layer, which is a sphere because the medium is isotropic. The normal to the slowness surface is parallel to the corresponding radius, so the energy velocity is parallel to the direction of propagation. For isotropic solids for which two types of waves propagate, the slowness surface is composed of two spheres: one corresponds to longitudinal waves and the other to transverse waves. The former is external because the transverse waves are slower than longitudinal waves. For the same reasons as above, in this case also, the energy velocity is parallel to the direction of propagation.

Figure 1.4. Slowness surface of an unidirectional carbon/epoxy composite: top right corresponds to the quasi-longitudinal waves, top left to the quasi-transverse fast waves and down to the quasi-transverse slow waves. These three layers have the same scale in s/km

Figure 1.4 represents the slowness surface of a unidirectional carbon/epoxy composite (carbon fibers embedded in an epoxy resin), whose fibers are oriented vertically. We observe the three layers (plotted separately for readability) corresponding to quasi-longitudinal waves (the smallest since it is the fastest wave), as well as quasi-transverse fast and slow waves. In this case, these layers are axisymmetric since the material is transverse isotropic.

Figure 1.5. Slowness surface of an unidirectional carbon/epoxy composite, cut following a plane containing the axis parallel to the fibers

Figure 1.5 shows a cut of this surface on which the three layers are superimposed.

The strong anisotropy of this material may be noted on these figures. The layer corresponding to the quasi-longitudinal waves (furthest inside in Figure 1.5) shows a phase velocity that is much higher in the direction of the fibers than in the perpendicular direction. The anisotropy of quasi-transverse waves is less marked.

The wave surface corresponds to the points M obtained by drawing a vector , equal to the energy velocity, from a given origin O. The vector joining O to any point on the wave surface represents the distance traveled by the energy of the wave during the unitary time.

Figure 1.6 presents, on the left hand-side, a partial cut of the wave surface of the same material as above (layer of quasi-transverse fast waves of unidirectional carbon/epoxy composite) and on the right hand-side the cut corresponding to the slowness surface (slow curve). We observe some singular points on the cut of the wave surface; they correspond to the inflection points of the slowness curve. We also find that for some directions, there are three values of the energy velocity (for example: Oa, Ob and Oc) for a single layer of the wave surface. These three values of the energy velocity correspond to three directions of propagation identified by the vectors on the corresponding slowness curve. We observe, in fact, that the normals to tangents of this curve at the three points A, B and C are parallel to the energy velocities we considered. So there are more than three waves that propagate with an energy velocity parallel to this direction.

To simplify the presentation, these remarks were made in a case where the energy velocity vector is in the same plane as the slowness curve considered (section of a slowness surface). In general, the vector is outside this plane and the representation should be three-dimensional.

Figure 1.6. On the left, a partial cut of the layer of the surface wave corresponding to the quasi-transverse fast waves, on the right, a cut of the slowness surface corresponding to the unidirectional carbon/epoxy

1.3. Bibliography

[ROY 96] Royer D., Dieulesaint E., Ondes élastiques dans les solides, Tome 1: Propagation libre et guidée, Masson, 1996.

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1Chapter written by Jean-FranÈois DE BELLEVAL, Catherine POTEL and Philippe GATIGNOL.

Chapter 2

Interaction of a Plane Wave and a Plane Interface ¹

2.1. Introduction

Mechanical structures and the materials themselves (composites for example), are often composed of different media linked with each other through interfaces of various kinds (welds, bondings, etc.). In addition, these structures are limited by surface boundaries that separate them from the environment. It is essential to understand how acoustic waves behave when meeting such surfaces.

After having established the equations of propagation in an infinite medium and studied the particular case of plane waves and monochromatic waves, we are now studying the interaction of these waves with interfaces separating two media. We consider the special case where the surfaces are plane and where the incident wave is itself a plane wave. In addition, although this is not always necessary, the problem will be restricted to the case of monochromatic waves.

When two semi-infinite distinct media (liquid, isotropic or anisotropic solid) are separated by an interface, the interaction of a plane oblique and monochromatic wave with the interface generates reflected waves in medium and transmitted waves in medium (Figure 2.1). Knowing the characteristics of the incident wave (propagation direction, polarization, amplitude) and the properties of both materials constituting the semi-infinite media (density, elastic constants), the writing of the boundary conditions on the interface allows the determination of the characteristics of the transmitted and reflected waves. In terms of non-destructive testing, the understanding of these phenomena is fundamental for the interpretation of various echoes, reflected or transmitted from the control sample.

Figure 2.1. Interaction of a plane monochromatic oblique wave with a plane interface separating two distinct media. The plane (O x1 x2) is the plane of the interface

As a starting point, we review the various boundary conditions that may appear in the problems of acoustic in fluids or solids.

2.1.1. Boundary conditions in acoustics

2.1.1.1. General background of boundary conditions in linear physics

The boundary conditions in acoustics concern surfaces limiting the propagation medium, such as walls or free surfaces, or surfaces between two media with different physical characteristics. The latter are called interfaces.

The boundary conditions, which have to be written, deal with the physical quantities related to the acoustic field: displacements or particle velocity, fluctuations of pressure or stress. They can be kinematic, when expressing conservation of mass, non-penetration of bodies, or dynamic when they translate, in a simplified form, the fundamental principle of dynamics. More complex circumstances will also be explored when the boundary conditions represent the impact caused by the presence of a medium or structure whose details are to be ignored, while we wish to take into account its only influence on the neighboring propagation media. This will be the case of impeding walls or interfaces between two solids showing an effect of elastic bonding.

Linearization rule at the boundaries. Before going into the details of the equations representing various types of circumstances, it is necessary to formulate a general rule, which is valid within the scope of linear physics and which simplifies the mathematical representation of boundary conditions.

The surfaces involved participate most often in vibratory movement of the acoustic field around them. Naturally, it is the case for free surfaces and interfaces, whose particles oscillate with the acoustic phenomenon affecting them. On the other hand, walls can be fixed. But they can also vibrate, either actively if they are part of a vibrating source of the acoustic field, or passively if sufficiently light (a vibration is then transmitted to the walls by the acoustic medium they surround). The latter is known as vibro-acoustic coupling. If the surface occupies the position Σ0 in its resting state (without acoustic phenomenon), it occupies a position Σ(t) , which varies with time, in the presence of the acoustic wave.

The general rule (valid in the frame of the linearized theory under the assumption of low amplitudes of the vibratory phenomenon) comes from the fact that the boundary conditions affecting the surface can be simply written, at any time t, on Σ0 corresponding to its resting position, instead of the instantaneous distorted surface Σ(t).

To convince ourselves, let us look at the condition of equal pressure which should be written on both sides of an interface between two fluids (1) and (2) in vibratory motion (Figure 2.2), as a consequence of the action–reaction reciprocity principle.

Figure 2.2. The interface Σo and the corresponding distorted surface Σ(t) at time t

If Mo is any point of this interface in its resting position Σo , this point occupies (a position M(t) )on Σ(t) during the vibration. The equality of (total) pressure at point M, on both sides of the interface Σ(t):

[2.1]

breaks up, according to equation [1.7], showing the reference pressure and acoustic pressure (here we temporarily reintroduce the subscript 1 when referring to acoustic disturbances):

[2.2]

According to the assumptions made on the reference state (see 1.1.1.4), the pressures are constant (in space and time). This reference state being a state of equilibrium, the equality of pressure on both sides of the interface Σ0 should be ensured,

[2.3]

such that condition [2.1] on the total pressures reduces to the equality of acoustic pressures:

[2.4]

For the moment, this condition is also written on the distorted interface Σ(t). (However, the point M(t) is deduced from its reference position M0 by a displacement which is supposed to be a small perturbation, as well as the fluctuating quantities introduced in section 1.1.1.4.

Symbolically, we can write:

[2.5]

and specify boundary condition [2.5] in the form:

[2.6]

Then a first-order development of this equation yields

[2.7]

However, the second terms of each member above are infinitely small and of second order, as were those that have been neglected in the linearization process of the propagation equations in section 1.1.1.4. It would therefore be unrealistic to keep them at the interface condition.

Thus, in the frame of linear acoustics, the pressure condition at the interface between two media is expressed by the equality of acoustic pressures on the interface in its position at rest:

[2.8]

It is clear that the scope of the argument developed here is actually general, and that the conclusion extends to all possible boundary conditions we have to write, under the assumptions of linear acoustics.

We shall now write the boundary conditions for a few typical cases. The subscript 1 for acoustic perturbations will again be implied.

2.1.1.2. Various border surface conditions

In this context, borders are surfaces that limit the domain of acoustic propagation in space. Thus, fixed or vibrating free surfaces or walls are considered as borders.

2.1.1.2.1. Free surfaces

Free surfaces are borders on which the propagation medium applies no mechanical stress, during its vibratory motion, other than those resulting from equilibrium conditions in the reference state. By contrast, the vibratory motion of such a border is completely free: it depends only on the acoustic movement of the medium. Strictly speaking, a free surface is found only in the case of a medium in contact with a vacuum; in practice such conditions are written on an interface separating the considered medium from a much less dense medium.

The free surface conditions on the mechanical stress are expressed as follows:

for a fluid medium: cancelation of the acoustic pressure

[2.9]

for a solid medium: cancelation of the (acoustic) normal stress vector

[2.10]

where is the fluctuation of the stress tensor introduced in [1.28]. The linearization rule at the borders applies here: the current point M0 is located on the free surface at rest Σ0 and is the unitary vector normal to Σ0 at M 0.

2.1.1.2.2. Border walls of a fluid medium

Various types of boundary conditions can be considered on a wall limiting fluid propagation media.

The perfectly reflective wall represents an ideal situation in which the fluid particles cannot penetrate into the wall. From an acoustical point of view, it may also be said that such a wall has a rigid property, although it remains deformable from a mechanical point of view. The term perfectly rigid wall is then also often used. The fluid being assumed as not viscous (see section 1.1.1), it is appropriate to write the classic sliding condition of the fluid. If the wall is fixed, this condition is expressed by:

[2.11]

In practice, such a circumstance occurs when the fluid is in contact with an extremely dense and non-porous material, which does not participate in the vibratory motion. It is the extreme opposite to the free surface condition.

When the perfectly reflective wall has a vibratory motion, the sliding condition of the fluid must be expressed in relative velocity. If M 0 is the speed of the point M 0 considered as belonging to the wall, then, with the linearization rule at the borders, the sliding condition is written:

[2.12]

It should be noted that, as a consequence of the linearization rule at the borders, these sliding conditions may be indifferently expressed in terms of velocity, as

above, or in terms of displacement. In fact, if (M0) represents the fluid displacement at M0 on the wall, we can write, showing the temporal dependence explicitly:

[2.13]

A simple integration over t allows us to deduce from [2.11] or [2.12] similar conditions expressed in terms of the displacement at M 0.

Another point about the conditions of free surfaces or perfectly reflective walls is worth raising. As they have been written, the boundary conditions for these somewhat ideal situations do not involve any characteristic lengths. If the border, due to its geometry, does not introduce any length scale (which will be the case for an infinite plane boundary), the problem of the interaction of an acoustic wave with this border will be independent of the wavelength, and consequently of the frequency.

However, this observation is no more valid in the more realistic situation of impeding walls that we now consider. Such boundaries are representative of walls with a partial penetration of fluid particles (for example due to some porosity, possibly including a dissipation mechanism of acoustic energy, by viscosity or thermal effect). The penetrable nature of the wall imposes a proportionality relationship between the variations of the pressure in its neighborhood and the penetration velocity of particles. It is clear that the penetration or extraction direction of particles plays an essential role in this mechanism and that, in those

circumstances, the choice of the unitary normal vector , of no consequence in equations [2.10], [2.11] and [2.12], must here be precisely defined. Usually, the unitary vector is shown penetrating into the wall (thus away from the fluid medium). The condition of impeding wall, written under the assumption of a vibrating wall, is then expressed in the following form:

[2.14]

The coefficient of proportionality Zw , which reflects the physical nature of the wall, is called wall impedance. This impedance has the physical dimension of an acoustic impedance, such as the characteristic impedance of the fluid medium Z0=ρ 0V0. It is measured in Rayleigh units (1 Rayl = 1 kg / m²s ).

Generally, because the physical properties of an impeding wall reveal length scales (thickness, porosity), boundary condition [2.14] depends on frequency, and it can be expressed in the simple multiplicative form above only in the frequency domain, using the complex representation of physical quantities.

It must therefore be assumed that, in practice, wall impedance is a complex-valued function of the frequency, which can be written:

[2.15]

The real part Xw (ω) , called wall resistance, is due to the effects of acoustic energy dissipation. This quantity is always positive (or zero), with the choice that was made for the unitary normal vector , penetrating into the wall.

The imaginary part Yw (ω) , called wall reactance, does not have an imposed sign. It represents advancing or delaying mechanisms that can be observed between pressure effects and input–output movements of fluid particles.

2.1.1.2.3. Border walls of a solid medium

The case of a wall surrounding a solid medium is less usual, and we will refer to two specific situations, corresponding to fixed walls.

The fixed wall with embedding, or perfect adhesion, requires the solid medium having zero displacement at the contact with the border:

[2.16]

However, no conditions are imposed on the normal stress vector. That is, again, the extreme opposite to the free surface, which led to condition [2.10].

Another ideal situation, but intermediate between these two extreme cases, is the fixed wall with sliding. Displacements of solid particles are allowed along the wall. However, the solid can only exert a normal action on the wall. Such a situation is reflected in the next two conditions:

[2.17-a]

[2.17-b]

The first equation expresses the geometric condition of sliding, written in terms of displacement. The second requires the two components of the normal stress vector in the plane that is tangent to the wall at the current point M0 to be zero.

2.1.1.3. Usual interface conditions

Regarding solids involved in the formation and implementation of materials, the problems related to the interfaces between two solids or between a solid and a fluid are of a more practical interest. We will begin by quickly mentioning the simple case of the interface between two fluids.

Two types of conditions usually occur along an interface. The first one is a kinematic condition that expresses the non-penetration of media. It may be written in terms of velocity or in terms of displacement, due to the linearization rule at the borders, and referring to the comment that was made earlier (relationship [2.12]). The second condition stems from the dynamic laws applied to the interface itself, considered as a mechanical system of mass zero. It follows from the fundamental principle that, at any moment and any point of the interface, stress forces carried by each of the media should counterbalance each other.

2.1.1.3.1. Interface separating two fluid media

The fluids being supposed to be not viscous, the kinematic condition of non-penetration of the media leads us to write that the fluids slip one over the other. This condition is usually expressed in terms of particle velocities. The dynamic balance condition of stress forces amounts to the equality of acoustic pressure, as it has been shown earlier to illustrate the linearization rule at the borders.

In summary, using the same notations as introduced earlier, the conditions at an interface between two fluids (1) and (2) will be written:

[2.18-a]

[2.18-b]

2.1.1.3.2. Interface separating a fluid and a solid

Solid structures which are studied are normally immersed in a fluid (liquid or gas) and such interfaces play the role of entry or exit areas for the acoustic phenomena involving the structure.

Although one of the media is a fluid, it is preferable to write the kinematic condition in terms of displacement. This condition, which expresses the sliding of the fluid along the solid, its motion being relative to the solid, results in the equality of normal displacements of the two fluid and solid particles located at M0 on both sides of the interface.

Moreover, as the pressure force exerted by the fluid is purely normal, the dynamic condition requires that the two components of the normal stress vector located in the plane tangent to the interface are zero. The normal component of this stress vector must balance the pressure exerted by the fluid. It should be noted, in this balanced relationship of forces, that the difference of sign convention between the definition of the pressure in a fluid and the stress vector for a solid yields a minus sign here.

The conditions for an interface between a fluid (1) and a solid (2) will then be written:

[2.19-a]

[2.19-b]

[2.19-c]

2.1.1.3.3. Interfaces separating two elastic solids

Such interfaces, which are fundamental for the mechanical behavior of materials composed of several solid components (sandwiches, multilayered media, composites), can be of quite different physical natures. We consider for the moment the simple cases of i) an interface with perfect adhesion and ii) a sliding interface. A special section (2.1.1.4) will be dedicated to the more complex case of bonding interfaces.

The two specific cases considered here are the generalization, in the case of the junction between two solids, of the two types of wall conditions introduced in section 2.1.1.2.3.

The interface of perfect adherence, which reflects the existence of an embedding or a weld between the two solids, is expressed first by the equality of displacements of particles in contact , belonging respectively to solids (1) and (2), at any point M0 on the interface Σ0 and at any time during the vibratory motion. This kinematic requirement is supplemented by the dynamic equation expressing the equality of stress vectors normal to Σ0 at M0:

[2.20-a]

[2.20-b]

The sliding interface allows relative motions of the particles of each solid relative to the other one in the plane tangent at M0 to the interface Σ0. Only the normal component of the displacements has to satisfy the continuity condition. However, the solids do not exert any shear action on one another: the components of the normal stress vectors must be zero in the tangent plane. Equality of mutually applied efforts is expressed only by the continuity of the normal components of these vectors:

[2.21-a]

[2.21-b]

[2.21-c]

It should be noted, in a similar way to what was observed on the usual border conditions, that the various interface situations analyzed in section 2.1.1.3 lead to relationships in which no reference length is involved. Therefore, the interface equations so-written do not, in themselves, induce a dependence on the wavelength, nor consequently on the frequency.

It is different for bonding interfaces studied in the next section.

2.1.1.4. Conditions on solid/solid interface of bonding type

Hypotheses of perfect adhesion or sliding contact, between two solids, describe ideal situations which appear to be specific, extreme cases of reality, in which a slight normal relative shift between the two solids is possible. We shall now study the more general circumstance under the term: contacts of bonding type.

In practice, the two solids are linked together with a thin layer which is the result of the process of contact, either by surface modification of these materials, or by the addition of a third material, the glue. The latter has a certain thickness d and has an elastic or more generally visco-elastic behavior (we shall return to this later, taking into account the viscous effects inside the collage). In the context of the vibro-acoustic assumptions, the behavior of the glue will be considered as linear.

The thickness of the glue is generally sufficiently small, compared to the wavelengths involved in the vibro-acoustic signal, as to be negligible, and we can consequently neglect the inertial effects of the glue. This will be explained below in a formal statement, see equations [2.26] and [2.27].

Under these simplifying hypotheses, the behavior of the thin layer of glue, treated like a single interface, can be described by a rheological model without mass. Using a linear law of behavior, this model connects the displacement discontinuity on both sides of the interface to the normal stress vector, which is continuous at the crossing of the glue layer (whose mass has been neglected):

[2.22]

where is a three-dimensional second order tensor with real components (see later in this section), which represents the (by surface unit) stiffness of the glue, and where the normal unitary vector is oriented from (1) to (2) so that the components of the tensor (or rather their real parts) are positive.

It is usual to assume that the effects of compression and shear on the glue layer are uncoupled, and that the effects of shear are isotropic in the plane tangent to M0 at the interface (see for example [ROK 81] or [PIL 82]). In a local orthonormed coordinate system with as third vector, the stiffness tensor takes the following diagonal form:

[2.23]

reducing the characterization of the collage to the knowledge of two parameters: the normal stiffness KN and the tangential stiffness KT . The conditions through the interface described by [2.22] are then uncoupled and may be written:

[2.24-a]

[2.24-b]

Three limit cases can be underlined in the general context of these relationships:

– When KN and KT tend towards infinity, relationships [2.24] imply the

continuity of the displacement vector. We find again the perfect adherence condition [2.20-a].

– If KN tends towards infinity and KT towards zero, relationship [2.24-a] reduces to the continuity of the normal component of displacement [2.21-a], whereas relationship [2.24-b] leads to the nullity of the tangential components of the normal stress vector, thus to relationships [2.21-c]. We again find the situation of the sliding interface.

– Finally, when KN and KT both tend towards zero, the normal stress vectors cancel both. This is the case of total decohesion between the two solids.

Note 1. The thickness d of the adhesive layer is not involved in writing the boundary conditions [2.24]. However, if we want to determine the stiffness parameters KN and KT of the rheological model from the Lamé coefficients λc , μc of the bonding material, we will write:

[2.25]

where ρc is the density of the glue, and VLc and VTc are the velocities of longitudinal and transverse waves in the bonding material. Following [BAL 03], we can introduce dimensionless stiffness, depending on the angular frequency ω , in the form:

[2.26]

The hypothesis of a glue thickness which is small compared to the wavelengths involved can be expressed by the conditions:

[2.27]

where we introduced the longitudinal and transverse wave numbers associated with the frequency ω in the bonding material.

Note 2. We might think that, on the hypothesis that has just been made, the problem of the interaction of an acoustic wave with a plane interface with a bonding of very small thickness does not involve any characteristic length, and consequently does not depend on the wave frequency. This is not correct. The thickness d of the glue will not affect the result (due to conditions [2.27]), but another characteristic length appears by comparing the elastic constants of the two bonded solids and the stiffnesses of the rheological model characterizing the bonding.

Indeed, the elasticity constants of the materials that constitute the bonded solids (such as Lamé coefficients or Young modulus) have the dimension of a stress, whereas the parameters of stiffness KN and KT have the dimension of a stress divided by a length. The modeling of bonding with small thickness by a rheological model therefore introduces a characteristic length depending on both the bonding and the elastic nature of the materials that are bonded.

Thus, a problem of interaction between an acoustic wave and a bonding plane interface will be discussed under the assumption of a monochromatic wave.

Note 3. As a consequence of the previous notes, there will be some interest in expressing gluing relationships [2.24], in terms of the complex representations of monochromatic stresses and displacements fields. Under this complex formulation, the stiffness parameters KN and KT may themselves be considered as complex valued (imaginary parts then representing the effects of viscoelasticity of the bonding material as mentioned at the beginning of this section).

2.1.2. Plane interface separating two fluid or isotropic solid media

We return now to the particular case of a plane interface between two fluid or solid propagation media. The incident field will be composed of a monochromatic plane wave whose interaction with the