Acoustics, Aeroacoustics and Vibrations

Preface

This book is an introduction to the theories of vibroacoustics and aeroacoustics. It cannot therefore be seen as exhaustive. It only presents one of the many ways of presenting the bases of acoustics, the science of vibrations of fluid and solid continua. The scope of applications is extremely large and varied, from musical instruments to ultrasound echography including the active control of vibrations or the transmission of vibrations between different propagation media. The basic equations of acoustics are very simple (in appearance at least), small in number (less than half a dozen) and the underlying hypotheses elementary. The bulk of the difficulties lie in solving these equations.

Whereas numerous books exist in acoustics and in fluid mechanics, a much smaller number exists about the field that is the subject of this book and, to our knowledge, none about this form in which each chapter, which although may be used independently, is based on knowledge that was developed in several others. We hope that the curious or the simply interested readers will find food to satisfy their appetite before indulging by reading more specialized books whose references can be found in the bibliography presented at the end of this publication and that will enable them to deepen their knowledge.

In this book, we did not use the duality system of theory and exercises in a systematical manner. Thus, the reader is strongly encouraged to carry out the demonstrations again and to finish those left aside. Since we have chosen to demonstrate the simplest results only, this method allows for the refining of the understanding of simple ideas without cluttering the text with too cumbersome results.

Fabien ANSELMET

Pierre-Olivier MATTEI

November 2015

1

A Bit of History

In the beginning, there was music... So could begin a history of acoustics, a very old science that has played a primordial role in the process of scientific development. As a matter of fact, it appears that music was one of the first elaborate means of communication. The oldest known instruments date back to 35,000 years ago (lower Paleolithic era called aurignacian) and were simple flutes carved in bones. We had to wait until Greek antiquity to understand the fact that sound is produced by a deformation movement of the body that creates it. However, the process of transmission of sound from the instrument to the ear was unknown to these people. The Theory of Sound, published in 1895 by Lord Rayleigh [STR 45] and which inspired this short introductory chapter, established what is known as classical acoustics.

Conventionally, acoustics is split into three parts: the production of sound, its propagation and reception. For each of them, it can be useful to present a brief historical primer, focusing particularly on the period which witnessed the birth of the basic concepts, between the 16th and 19th Century.

Finally, we will conclude this chapter by mentioning aeroacoustics, a very recent science that has gained momentum only with the pioneering works of Sir James Lighthill in the 1950s.

1.1.The production of sound

It is commonly accepted that, as early as the 6th Century BC, Pythagoras was the first Greek to study the origin of musical sounds. He showed that the highest pitches are produced by the shortest strings and that a string half as long as another emits a pitch an octave above. The method of plucking the strings was gradually developed, although without its relationship to the concept of frequency being established. It does not appear that this approximation had been made before Galileo (1564–1642). In his treatise, he discusses the influence of the length, tension and density of the string. He further observes that those sounds whose frequencies are integers multiple of the lowest frequency combine pleasantly to the ear.

In 1636, the Franciscan friar Mersenne (1588–1648) carried out, in Paris, the first serious publication about the vibration of strings. He was the first to measure the frequency of a musical sound.

The pioneer of experimentation on the connection between the frequency of the sound and the way the string is plucked was Sauveur (1653–1716), who incidentally suggested the term acoustics for the science of sounds. Sauveur, as well as Wallis (1616–1705) at the same time in England, observed that vibrating strings showed motionless points which he named nodes, while others were animated by a maximal amplitude motion called today antinodes. The English mathematician Taylor (1685–1731) built the first strictly dynamic solution to the problem of the vibration of strings. His calculations were in adequate agreement with Galileo and Mersenne’s experiments. Although he had focused on a specific problem, Taylor has paved the way for Bernoulli (1700–1782), d’Alembert (1717–1783) and Euler’s (1707–1783) more elaborate mathematical techniques (partial differential equations).

The phenomenon of the succession of nodes and antinodes on a string characterizes multiple frequencies of the simple vibration frequency of the string. The latter produces a fundamental sound, while the former were baptized harmonics. Experimentally, Sauveur noted that a plucked string emitted a sound with a complex structure in which many harmonics were present. The theoretical explanation of this phenomenon was given by Bernoulli in 1755. The resulting vibration is the algebraic sum of the partial vibrations, an idea that will be later called superposition principle. This drove Fourier, in 1822, to build his famous decomposition theorem whose scope of application is much wider than that of acoustics.

Lagrange (1736–1813) gave an elegant analytical explanation in 1759 to the problem of the strings. He focused on the sounds produced by pipe organs and woodwinds, facing the problem of boundary conditions.

The extension of the methods previously described requires knowledge of the relation of the behavior of the body that relates its deformation to the stress that is imposed on it. This issue had been experimentally addressed on solid bodies, between 1660 and 1676 by Hooke who derived on this occasion the concept of elasticity. This law, more comprehensive than that introduced by Hooke, forms the current basis of the concept of linear elasticity, either in the static (strength of materials), dynamic (viscoelasticity) or quasi-static (vibration) fields. This last field is the basis for the study of noise emissions. Hooke’s law was used for theoretical means for the first time in 1744 by Euler, and then in 1751 by Bernoulli in cantilevers or supported beam vibration problems. They based their studies on the deformation energy which later led Rayleigh (1842–1919) to the well-known fourth-order spatial differential equation. This equation, which governs the vibration behavior of the beams, is known as Euler’s equation.

The vibration of elastic plates had been studied by Chladni (1756–1824). The results, published in 1787, show the existence of nodal lines that are the two-dimensional equivalent of the vibration nodes of strings. The experiment consists of sprinkling very fine sand on a plate. When it is subjected to vibrations, for example using a violin bow, the sand gathers at the vibration nodes of the plate, the nodal lines. A few examples of one of the plates proposed by Chladni in his book [CHL 09] are presented in Figure 1.1. Following these works, Napoleon in 1802 granted the Institut de France 3,000 F with the subject of the prize being to present the mathematical theory of vibrations of elastic surfaces, and compare them to the experiment [CHL 09]. The winner, Sophie Germain (1776–1831), who was the first French self-taught female mathematician, gave an exact fourth-order equation in 1816.

Figure 1.1. A few examples of Chladni’s figures (engraving from [CHL 09])

Currently, only the structures with a simple form can possibly be solved analytically. This mainly concerns plates (circular, elliptical and rectangular) and shells (cylindrical and spherical), which is sufficient in most industrial applications. All these models are originating from the three-dimensional elasticity equations in which a simplification is introduced assuming that a dimension is small compared with the others, usually the thickness. The complexity of the obtained models suggests that it is unrealistic to expect that geometrical structures of complicated form be addressed in a simple way. It is necessary to resort to approximate numerical methods such as finite elements, and there again, models do not guarantee entire satisfaction. Needless to say that nonlinear elasticity, anisotropic or coupling (interaction) problems between the vibrating structure and the fluid that surrounds it are still the focus of research.

1.2.The propagation of sound

As has already been stressed, the Greeks were aware of the importance of the air in the propagation of sound, without, however, understanding by which mechanism fluids intervened. Since, during observations, the air remains motionless during the transportation of acoustic waves, some philosophers did not admit this point of view. For example, Gassendi from France (1592–1655) thought that sound spreads using beams of fine particles capable of impressing the ear. During this time, experiments related to the propagation of sound in a rarefied atmosphere did not invalidate the ideas of Gassendi. These latter were only challenged by Boyle (1627–1691) who carried out a significant experiment that allowed the observation of the decay of the sound intensity transmitted with the intensification of the vacuum. He concluded that the air conveys sound, without being the only material to exhibit this property. The question was to know how fast the air conveys the sound. By using firearms, Gassendi came up with a speed of 480 m/s. Better experiments driven by Mersenne gave 450 m/s. While Aristotle argued that treble pitches were more rapidly transmitted than low ones, Gassendi achieved an important observation by highlighting the fact that the speed of sound is independent of its tonal pitch. In 1656, the Italians Borelli (1608–1679) and Viviani (1622–1703), without specifying the air temperature nor its dampness, found a value of 350 m/s. In 1740, the Italian Branconi showed that the speed of sound increases with the temperature. The first significant measurement in open air was without any doubt conducted in 1822 by the members of the Bureau des Longitudes in Paris by means of a cannon. The observers were divided into two groups situated at two stations distant from each other by 18,700 m. Each group determined the time interval between the perceptions of light and sound of the cannon fired by the other group and then fired a cannon shot in turn so that the first group could perform a similar measurement. Measurements invariably gave 55 s at 15 degree C (therefore, a speed of 340 m/s). Brought down to zero degrees Celsius, the results yielded 332 m/s. During the 18th and the first half of 19th Century, a very large number of experiments confirmed this value. The accepted value is 331.36 ± 0.08 m/s with air at rest at 0 C and 1013.25 HPa.

In 1808, the physicist Biot (1774–1862) made the first experiments on the speed of sound inside a solid medium by using a 951-m long cast-iron pipe which had to be used to provide drinking water in the city of Paris. The observer measured a difference of 2.5 s in the travel times taken by sound through the metal and air. If c0 and a refer to the velocities of the waves in the air and metal, Biot immediately obtained 951/c0 −951/a = 2.5 or a = 10.5c0 that is a wave velocity in cast iron used to make the pipe 10 times greater than that of the air.

In 1827, Colladon and Sturm studied the speed of sound in Lake Geneva by measuring the time difference between the image of an inflammation produced by the shock of a hammer on a bell and the sound perceived by an observer located on the shore as indicated in the engraving reproduced in Figure 1.2. They found 1,435 m/s at 8º.

A first theorization test was proposed by Newton in 1687. Circa 1760, Lagrange proposed another model which yielded once again Newton’s results. However, the numerical value that they obtained is significantly different from that obtained experimentally. No significant progress was made before 1816, date at which Laplace criticized the isotherm hypothesis of his predecessors. He replaced it by an adiabaticity hypothesis that seemed to him more appropriate. The two velocities, obtained by Laplace and Newton, are in the ratio of specific heats. They showed the existence of a specific heat at constant volume and constant pressure whose values were known only with a very relative accuracy. Laplace used the numerical value of 1.5 for their ratio γ obtained by experimenters Laroche and Bernard. He obtained a value of the speed of sound of 346 m/s at 6 C. He estimated this value as almost identical to the experimental value of 337 m/s. A few years later, γ was again measured and the value still accepted nowadays of γ = 1.41 was obtained. This corresponds to a speed of sound in perfect agreement with the experimental values. It should be noted that if Lagrange had had full confidence in his theory, he could have indirectly measured γ very precisely.

Figure 1.2. Device setup for measuring sound in Lake Geneva (engraving from [DRI 73])

Tests for solving the equation of propagation of sound followed d’Alembert’s works on the vibration of bodies. In particular, at the end of the 18th Century, the problem of the propagation of sound in pipes started to be well known. Kundt (1839–1894) developed a method that enabled the measurement of the speed of stationary waves. Poisson derived in 1820 a theory of propagation in pipes for almost all cases. This work was finished by Helmholtz (1821–1894) in 1860. The particular case of the abrupt change in cross-section was studied by Poisson, as well as the reflection and transmission of sound under normal incidence at the boundary of two different fluids. The case of oblique incidence was treated by Green (1793–1841) in 1838. This had the purpose of bringing forward the similarities and differences between sound and light.

Nowadays, the interest is toward problems of propagation in limited (shallowwater propagation and cavity resonance) inhomogeneous media as well as in random media (turbulence).

1.3.The reception of sound

During the historical development of acoustics, the primary receiver of the studied sound was the human ear. However, no satisfactory or complete theory of hearing has been elaborated, leaving this psychoacoustics problem open, as are also most aspects that relate to the brain and its ability to interpret its environment.

After establishing the evidence of the relationship between the frequency and the plucking of strings, an important task was the determining of the frequency limits of human hearing. Savart (1791–1841) placed these boundaries between 8 and 24,000 Hz. Later Seebeck (1770–1831), using tuning forks, Biot, Koënig and Helmholtz obtained values between 16 and 32 Hz for low frequencies. Nowadays, an audible range between 20 and 20,000 Hz is commonly accepted.

In 1843, Ohm showed that the ear is capable of achieving a spectral analysis of complex sounds. This created a renewed interest in physiologic acoustics. Helmholtz, in 1862, proposed a theoretical model of the mechanism of the ear. It was during this work, called Theory of Resonance, that he invented the resonator that now bears his name and which is presented in Figure 1.3. He developed the theory of summation and difference of pitches and laid the foundations for all the future research in this area [HEL 68].

From the middle of the 19th Century, acoustics has undergone considerable development and here it is impossible to continue our history of acoustics which would otherwise take us in too many different directions. The interested readers can refer to the historical book (and still actually relevant in many aspects) written by Strutt and Rayleigh [STR 45], as well as to the summary book edited by Rossing [ROS 07] which contains a more complete historical introduction than this in addition to a list of historical references.

Figure 1.3. Helmholtz resonator (engraving from [HEL 68])

1.4. Aeroacoustics

Aeroacoustics is a discipline that involves varied aerodynamic and acoustic phenomena, which are in addition tightly coupled. Its real development as a special branch of physics is fairly recent, since it dates back to the founding works published by Lighthill in 1952. In general, two stages can be distinguished in the sound radiation by flows: the generation of noise in the areas of turbulence where nonlinear effects are very important, and the linear propagation of acoustic waves in a medium at rest until far-field. In order to identify the sources of noise, it is, therefore, necessary to have a good knowledge of turbulent flows; this explains why the progress of aeroacoustics was directly related to that of fluid mechanics, both at the experimental and the numerical level. However, in order to take into account the physical characteristics of acoustic fluctuations, a number of original techniques had to be also developed.

It is, therefore, in 1952 that Lighthill suggested an analogy from which most of the aeroacoustic theories have been developed. By recombining the fluid mechanics equations to show the noise produced by a flow as the solution of an equation of propagation in a medium at rest, Lighthill has enabled the foundation, on rigorous mathematical bases, of aeroacoustics, which then was stumbling a lot: in particular, in this framework, the experimental observation of the variation of noise radiated by a subsonic engine jet as a power eight function of the speed, previously of mysterious origin, is explained in a very simple way. In the context of the Lighthill analogy, the generation of noise is effectively identified by the term on the right of the Lighthill equation, by means of acoustic sources terms constructed from the velocity field and, in particular, from fourth-order correlations of turbulent velocity fluctuations which constitute, in the majority of cases, the primary source term by identifying the source to a quadripole.

The reduction, or even the control (passive or active), of the noise of aerodynamic origin constitutes today a major industrial challenge. Thus, for example, in the nuclear sector, the acoustic radiation produced in the pipes by fast flows is likely to severely damage the structures due to resonant effects being induced. Nonetheless, it is of course in the fields of aviation and land transportation that noises are more directly felt, such that gradually more stringent standards should be respected, and that the concept of acoustic comfort of users must be taken into account from the beginning by the manufacturers. It is, therefore, desirable to be able to accurately predict the sound field generated by turbulent flows if the main purpose is to address it efficiently.

2

Elements of Continuum Mechanics

The subjects covered in this book all base themselves on the approximation of continua, be it to characterize the properties of solid entities under deformation, or those of fluids with sound waves moving though them. This continuum approximation is very simple in the physical concepts it employs, that are nothing more than the principles of conservation of mass, momentum and energy, but transposing them mathematically to use them to solve problems can be slightly arduous and highly complex as it requires tensor calculus. Thus, compared to the relative simplicity of particle mechanics, the formal complexity of the mechanics of deformable bodies is due to it taking into account their deformations and the strain resulting within these bodies (or vice-versa). This chapter therefore presents, in an intentionally succinct fashion, the main elements that will be necessary to move on to the problems in the chapters further on. It also refers to specialist books that will, if necessary, allow the reader to go deeper into continuum mechanics. We will begin this chapter with a general presentation of continuum mechanics and then of the laws of conservation of mass, momentum and energy. We will then end by describing the behaviours of common media that are the elastic solid media, thermoelastic, viscoelastic as well as fluid media

2.1. Mechanics of deformable media

2.1.1. Continuum

A medium is called a continuum in a domain Ω ∈ if its measurable physical properties vary in a continuous manner in Ω. If ρ is the density of the material, the mass of the material that occupies Ω is given by:

[2.1]

We define the notion of particle by the smallest set of Ω for which the density keeps a constant value. A continuum can in some cases present discontinuities of the first kind such as shock waves. More formally, it is desirable to assume that the physical quantities be integrable within Lebesgue’s sense (see, for example, relation [3.4]) which allows zero-measure discontinuities to be addressed (such as cracks in a solid medium and shock waves in a fluid).

2.1.2. Kinematics of deformable media

In order to describe a continuum, it is important to have some way of describing it. Twodifferent visions exist,theLagrange and Euler kinematics. In Lagrange’s kinematics, which addresses the motion from a local perspective, the main focus is on the evolution (transport and deformation) over time of a small area of fluid. In Euler’s kinematics, on the contrary, it is sought to describe the motion of the continuum in its entirety1.

2.1.2.1. Lagrange’s kinematics

This kinematics is based on the concept of fluid particle. Suppose that at an instant t = t0, the position of the particles that constitute the continuum is known. If each of them can be followed over time, the motion of the continuum may be known in its entirety. Given aj, the position of the particle at instant t0, the Lagrange kinematics is equivalent to providing a relation of the form xi = gi (a j, t ). The independent parameters aj are named Lagrange variables. The curve traced in the referential relatively to time is called trajectory. At any time, the following properties are verified:

1) the application which from the initial point allows shifting to the current point is one-to-one;

2) the partial derivatives xi,j = ∂xi /∂xj exist and are continuous;

3)

with Einstein’s summation convention on the repeated indices. This determines a Cramer system. If its determinant is non-zero: dai = (∂ai/∂xj )duj.

Considering the particle which is located in P at time t, d is its displacement during dt. The vector of components is named velocity of the continuum medium. Since this derivative is calculated by following a particle during its motion, this derivative is named material derivative. In this case, the terms aj are constants and it yields vi = dxi/dt = ∂gi (aj, t) /∂t. Similarly, the particle acceleration is defined as

2.1.2.2 . Euler’s kinematics

We will now set aside the idea of following each particle motion to the benefit of looking for the description of the motion of the whole continuum medium. For example, we will replace the notion of the particles trajectory to rather focus on the information given by a velocity field in the set Ω. It is upon this idea that the Euler kinematics relies. Consider a velocity field where xi and t are the Euler variables. The xi are no longer the coordinates of a particle but independent space variables.

The problem that arises is the representation into Euler variables of the motion in Lagrange variables. Let by hypothesis xi = xi (aj, t) and vi = vi (aj, t ). Thus, with xi = xi (aj,t ), we have aj = aj (xι, t ) and therefore vi = vi (aj (xι,t ), t) =vi (xl,t) where the xl are the independent space variables. The inverse problem which is solved by integration is much more demanding.

The trajectories are defined by the three relations dxi/dt = vi (xi, t) = 0 which determine a set of curves with three parameters. A streamline is the curve tangent to the velocity vector at each of the considered points of the space. Consider a differential motion on the streamline, . In addition, v1v2, v3 are the components of and dx1, dx2, dx3 those of (for example, in ). The streamlines equation is given by dx1/v1 = dx2/v2 = dx3/v3 = λ, for an arbitrary λ. A streamtube is the set of streamlines with regard to a closed surface. In the case of a permanent or a stationary flow, streamlines and pathlines are indistinct.

2.1.2.3. Kinematics of a surface

We will follow the motion of a surface in Eulerian coordinates. g (xi, t) = 0 is the equation of the surface. We name H the norm of g and the normal external to the surface. Between t and t + τ, the function g remains zero. If is the velocity of the surface, we have:

2.1.2.4. Material derivatives

The material derivative is the derivative with respect to time when following the motion of the particle. The variation of a scalar function f (M, t) is expressed in the form df (M, t) = (∂f (M, t )/∂t )dt + (∂f (M, t )/∂xi )dxi, or:

[2.2]

This last formula applies to vector functions. It then yields:

[2.3]

Note that since taking into account the anti-symmetry of the vector product the following formulation is thus obtained for the material derivative:

[2.4]

2.1.3. Deformation tensor (or Green’s tensor)

Consider a small volume element of length dxi placed in A, of Eulerian coordinates xi (see Figure 2.1)

This small volume element is transported and deformed during the motion of the continuum (fluid or solid). During the time interval δt, A moves to A + δA, δA has for coordinates δxi = ∂xi/∂tδt. Since the elementary volume underwent a deformation in addition to the translation δA, B is not found in B + δA but in B'. To estimatethe deformation of this volume, it is necessary to calculate the variation of the length

Figure 2.1 . Elementary volume

We will characterize the vector B' – (B + δA ). Before deformation, the distance was After deformation, it is . Consider ui, the displacement vector. In Lagrange coordinates, ui = xi – ai. Since , then Considering dui = ∂ui /∂xj dxj, it appears by replacing:

[2.5]

since 2(∂ui/∂xj )dxjdxi = (∂ui /∂xj +∂uj/∂xi) dxidxj and (∂ui/∂xj )× (∂ui /∂xk)dxjdxk= (∂uj/∂xi) × (∂ uj/∂xk)dxidxk. Thus, the length after deformation dl' given by relation [2.5] is written dl'² = dl² + 2dikdxidxk, where we have defined dik by:

[2.6]

The tensor of components dij is named the deformation tensor or Green’s tensor. In the case of small deformations, the quadratic term is ignored: (∂uj/∂xi) × (∂uj/∂xk) ∂ui/∂xk and (∂uj/∂xi) × (∂uj/∂xk) ∂uk/∂xi. Then, it gives:

[2.7]

A synthetic notation is often used:

[2.8]

where the gradient vector is a tensor of components ∂ui/∂xj. This notation is advantageous since it allows the deformation tensor to be easily written in any coordinate system, while everything that has been covered so far was done in Cartesian coordinates.

2.2. Conservation laws

Here, we introduce the three fundamental equations of mechanics, the law of conservation of mass, the fundamental principle of dynamics or law of conservation of momentum and the equation of conservation of energy. As we will see later, the principle of conservation of energy is indirectly introduced in the form of a condition at infinity (Sommerfeld’s condition or principle of limit absorption in harmonic regime and outgoing wave condition in temporal regime).

2.2.1. Conservation of mass

Let Ω, an arbitrary fixed area with boundary ∂Ω. Ω is named control volume. At instant t, the control volume has a mass m given by relation [2.1] m(Ω, t) = . Over time, with the exception of the sources of mass, there is no change in the mass of the medium , where d/dt is the material time derivative. The lemma on the derivatives of the integrals of physical volumes can be used to give outside the surfaces of discontinuity where is the velocity of the volume Ω. If is the normal external to

is obtained, where further, since the volume control is arbitrary:

[2.9]

By introducing the material derivative in the expression above and taking into account the relation div gradp, the equation of conservation of mass yields:

[2.10]

2.2.2. Conservation of momentum

We are going to apply the fundamental principle of dynamics. It is necessary to make an evaluation of the surface and volume forces acting on the various volume elements. Considering again our control volume Ω, with boundaries ∂ Ω, the momentum = contained in Ω is given by the integral The variation of this quantity over time is balanced by the constraints applied on the boundaries of the domain ∂Ω and by the volume forces applied to the volume

is the stress tensor which relates the stresses within a body2 to the deformations it undergoes. We obtain

where ⊗ is the tensor product. If we use Ostrogradsky’s theorem for the boundary integral:

Since the control volume is arbitrary stop, the equality only takes place by equating the integrands . In scalar form, this relation is written for all indices i (with Einstein’s convention):

[2.11]

In order to write this law in matrix form, it should be noted that div

If the equation of conservation of mass [2.10] is introduced, we obtain the equation of conservation of momentum:

[2.12]

2.2.3. Conservation of energy

We are going to apply the first law of thermodynamics which states that the variation of internal energy e specific to a control volume Ω over time depends only on the variations of energy corresponding to the work exchanged with the external continuum and the amount of energy being considered in the form of heat. We will call the heat flux3 and pqe the heat sources and the normal external to the control volume Ω. The first law of thermodynamics is written as [FIL 99]

With the relation Ostrogradsky’s theorem and by noting that the relation is true for an arbitrary control volume Ω, it gives

Considering the equations of conservation of mass [2.10] and momentum (or impulse) [2.12], the equation of conservation of energy is obtained:

[2.13]

where the Green tensor has been defined by relation [2.8]. The notation : characterizes the product twice contracted of tensors a and ; as these tensors are of order 2, the contracted product corresponds to a scalar.

2.3. Constitutive laws

We have briefly seen in previous sections how a continuum medium is deformed when it is subjected to a displacement or a set of forces. In order to complete the description of this continuum, it should be characterized more specifically by seeking how it will react to this deformation. For more clarity, we will consider two extremely simple examples. The first example is that of a metal bar. When it is deformed, bent, for example, it will resist the deformation and return to its initial position if the applied force is not too strong. The second example is that of a volume of liquid. It is general experience to know that liquids have no proper shape4 and that the concept of deformation has little meaning. More specifically, according to Boussinesq’s definition, a fluid is a homogeneous and an isotropic medium that can restore its isotropy after all the possible deformations and keep it during these deformations as long as these are carried out sufficiently slowly. These simple examples remind us that continuum media can show radically opposite behaviors. Of course, there are continua that have intermediate behaviors, or even completely different ones. For example, invar, which is a magnetostrictive alloy of iron and nickel that has practically invariable dimensions (hence its name) when it is subjected to temperature variations. It is widely used in the construction of liquid gas tanks (at –160oC). Any other metal that would suffer such thermal contraction would certainly cause the rupture of the tanks. Some polymers with a solid consistency can behave like fluids, while certain fluids present a constraint threshold below which they behave like solids. There are heterogeneous fluid continua such as smokes or mists, alloys with form memory5, not to mention metamaterials which possess amazing properties (with negative refractive light index, or with a negative Poisson ratio).

Practically, the only means of description of the constitutive laws of continuum media originate from experience. The objective is to establish a relationship that relates deformation (or the velocity of deformation for liquids) to the stress.

2.3.1. Elasticity

We will subject a test tube of steel of initial length l0 to tractions applied at both ends in parallel to its axis and we are going to plot the elongation curve l – l0.

Three zones, very schematically represented in Figure 2.2, can be distinguished. Region I is the elastic region where the elongation remains proportional to the applied force. This constitutes the definition of linear elasticity and is called Hooke’s law. The mathematical writing of this law is = Cijkldkl where σij are the components of the stress tensor (or Kirchhoff’s tensor), Cijkl are the components of the fourth-order tensor that characterizes the material and dkl are the components of the Green deformation tensor. Region II is the plastic region in which the body shows a residual deformation. In this region, despite a force that increases only very lightly, the material significantly elongates. Region III or necking region is the region in which the body breaks. The behavior of the material is no longer uniform. These regions have characteristics which vary considerably from one body to another and are shown here only as mnemonics. Their study largely falls beyond the scope of this book.

Figure 2.2. Typical traction curve

Note that the 81 coefficients Cijkl are reduced in the case of homogeneous (the material consists of a single component) and isotropic bodies (the properties of the medium are independent of the direction) to two independent coefficients. This pair of coefficients takes several expressions: (λ,μ ): Lamé’s coefficients, (E, v ): the elasticity modulus or Young’s modulus (E) and Poisson’s ratio (v ) or even (K, μ ): the compression modulus K and the shear modulus or Lamé’s modulus μ. These coefficients are related by the relations:

[2.14]

[2.15]

Note that since K and μ are positive, we easily derive that if K = 0, v = –1, and that if μ = 0, v = 0.5 or –1 < v < 0.5. Young’s modulus (or elasticity modulus), which represents the ability of the continuum to withstand pressure, and Lamé’s coefficients are expressed in Pascals. Poisson’s ratio (or lateral contraction coefficient) is the ratio of the lateral contraction to the elongation. As we have just seen, v is comprised between -1 and 1/2; but in practice, no natural material (with the exception of pyrite [ZEN 48]) presents a Poisson ratio v < 0 at the macroscopic level (it is then referred to as auxetic material) that is to say whose elongation is accompanied by a transversal dilatation; however, it is commonly known how to manufacture wire mesh-based materials that exhibit a negative Poisson ratio. In practice, 0 < v < 1/2. For most metals, v ≈ 1/3. It should also be noted that all the coefficients K, E, λ and µ have a pressure dimension. Hooke’s law is written in the case of a homogeneous and isotropic body as:

[2.16]

where is the Kronecker’s symbol that equals 1 if i = j and 0 otherwise. The term Tr is the trace of the tensor which is the sum of the termsTr which is the sum of the terms of the diagonal of the tensor. Hooke’s law is easily expressed relatively to Young’s modulus and to Poisson’s ratio as σij = E (1 + ν) (dij + ν /(1 − 2ν )dllδij ). The inverse formula,

known as Kirchhoff–St Venant’s law. Expresses the deformation based on the stress dij = ((1 + ν)σij − νσu δij) /E.

2.3.1.1. Stress-deformation tensor

The stress tensor components, given by Hooke’s law, are written in expanded form:

[2.17]

2.3.1.2. Infinitesimal strain tensor

In order to ease the writings, we use the following synthetic notation:ui,j = ∂ui /∂xj. In all cases, the displacement field of the solid is defined by

CARTESIAN COORDINATES.– in the Cartesian coordinate system (O, x1, x2, x3 ), we have:

[2.18]

CYLINDRICAL COORDINATES.– in the cylindrical coordinate system (O, x1, x2, x3)(compared to the usual notations we have:

[2.19]

SPHERICAL COORDINATES.– in the spherical coordinate system (compared to the usual notations we have:

[2.20]

2.3.2. Thermoelasticity and effects of temperature variations

Although it is a common experience to observe that a heated body expands, a solid that is deformed suffers a change of temperature. Reversible elastic processes are coupled to the irreversible processes produced by the thermal conduction of the warmer areas toward the colder areas. As a result of the second law of thermodynamics, by creation of entropy, a part of the mechanical energy is converted into heat. In a number of cases, this energy loss although small is not negligible. When a material is stressed in a reversible adiabatic process, the heterogeneity of the strain field creates a temperature distribution. Local gradients and the thermal conductivity of the material then initiate a flux, described by Fourier’s law, which allows the solid to regain its equilibrium state. Coupled thermoelastic equations, using the isotherm constants of the material, are derived from the free energy as a function of the temperature and the infinitesimal strain tensor around a reference state.

The classical elasticity and heat equations when they are decoupled have very damped and dispersive elastic waves and heat waves as solution. The coupled equations have for solutions dispersive quasi-elastic and damped waves and quasi-thermal waves. Helmholtz theorem of the decomposition of the displacement field in the sum of an irrotational term with a term without divergence makes it possible to show that transverse waves are decoupled from thermal phenomena. Therefore, only bending and compression stresses cause a coupling between the mechanical and thermal fields.6

The concept of usual relaxation time in viscoelasticity appears naturally in the thermoelastic theory and justifies the a posteriori usage of an equivalent rheological model. This thermoelastic relaxation time defines a crucial region in which the conduction process manages to balance the inequalities of temperature (isothermal or relaxed regime) or not (adiabatic or non-relaxed regime) during a vibration period. In the adiabatic regime, the thermal and the mechanical fields are in phase while in an isothermal regime, they are out of phase by 90°. In the transition area, these fields are out of phase by 45° and the damping is maximum.

Even though the process of dissipation is similar, the nature of the coupling depends a lot on the studied elastic wave. The thermoelastic coupling introduced by longitudinal waves and bending waves is qualitatively different. The distance between cold and warm regions of a structure greatly differs depending on the considered wave [LIF 99]. It is connected as a first approximation to the cross-sectional dimension for bending waves7, the characteristic distance is proportional to a half wavelength in the case of a compressional wave. A significant consequence in the latter case is that the high-frequency thermal relaxation time is very short and that the process is isothermal (or relaxed), whereas at low frequencies the process is adiabatic: it is exactly the reverse for bending waves. A certain number of distances and relaxation times related to thermoelastic coupling characterize the propagation of a wave in an infinite plate. The discreet relaxation spectrum resulting therefrom is altered in the presence of boundary conditions. In fact, most of the models are based on a simplified geometric description. The Zener model [ZEN 48], commonly used in order to describe the thermal field, retains only the first transverse mode, to which a single and unique distance is associated, the transverse distance. As a result, only the heat transfers relatively to the thickness are taken into account; transfers in the plane of the plate such as the influence of the boundary conditions on the edges of the plate are ignored.

Suppose that a body undergoes a temperature variation8 T – T0, where T0 is the temperature at rest and that this temperature variation is assumed to be sufficiently low to only induce low-intensity distortions (by thermal expansion). The relative variation in volume during the deformation is given by dll = α (T – T 0) where α is the coefficient of thermal expansion. The relations which express the deformation based on the stress in the presence of thermal deformation generalize Hooke’s law and are called Duhamel–Neumann’s law [PAR 84]:

[2.21]

And therefore:

[2.22]

These given equations introduce an additional variable, the temperature T. Either it is a quantity of the problem (it is the case for external heat input), or it is an unknown of the problem as in the case of heat produced by the deformation. In the latter case, an additional equation is necessary to calculate it. To this end, we use the law of conservation of energy [2.13]. The internal energy for thermoelastic solids is given

by e = e (s, )[FIL 99], the first differential of the state equation is :d , where was defined the temperature T =(∂e /∂s) and the stress tensor . The equation of energy conservation [2.13] written on the entropy s becomes:

[2.23]

If in this equation, Fourier’s law is introduced that characterizes the thermal conduction as well as the expression of the entropy [LAN 89b] ρs = ρs0 + ρcv (T − T0)/T0 + 3ασu, where s0 is the entropy at rest and cv is the specific heat per unit volume at constant strain, the linearized heat conduction equation is obtained:

[2.24]

The third term of the left member of equation [2.24] represents the thermomechanical coupling. The Duhamel–Neumann law coupled with the thermal conduction equation allows thermoelastic losses to be characterized in structures. To solve these coupled equations, it is assumed that fluctuations around the equilibrium quantities are very small and dT/dt is replaced by ∂T/∂t and dσu/dt by ∂σu/∂t and, in the case of thin structures, it will be considered that the thermal field is one-dimensional and leads to a transverse gradient (Zener hypothesis for the bending of thin structures [ZEN 48]).

2.3.3. Viscoelasticity

Viscoelasticity characterizes the ability of certain bodies to store some of the energy and dissipate another part. These two combined behaviors put any material between the two ideals that are the perfectly elastic material and the Newtonian viscous fluid. Viscoelasticity phenomena are various (frequency dependence, creep and dissipation) and are extensively addressed in many reference books [CHR 82, PER 60]. In mechanics, the viscoelastic equations comprise a time dependency other than inertial and model damping when the local dissipation9 can be characterized by a constitutive law introducing a temporal dependency between stress and strain. As we have mentioned in the previous section, since thermoelastic dissipation emerges from a local process in the structure plane, thermoelastic damping can be represented by viscoelastic operators in the case of thin structures.

The approximation obtained by the Zener model is such that the homogeneous damping model includes thickness as a geometric parameter; it is, therefore, not intrinsic and does only model damping during bending at low frequency . This model reflects, however, how extrapolations are easily permitted by the viscoelastic models.

Linear viscoelasticity is modeled after two formalisms that allow us to account for the viscoelastic effects (memory effect, frequency dependence, phase shift between dual quantities and dissipation). The first is based on the introduction in the constitutive equations of temporal differential operators ∑i αi ∂i / ∂t i in combination with conventional operators such as the rigidity operator. The second is based on integral convolution operators whose kernel is called delay function or relaxation. This last formulation consists of describing the dissipation by the introduction in the partial differential equations of motion10 of a convolution kernel (t,τ) which leads to an integro-differential equation the particular case of the viscosity η (x,t )being taken into account by this representation if it is assumed that the dissipation kernel takes the form (t,τ) =η δτ(t ), where δτ(t )is the delayed Dirac distribution at time t of time τ. These formalisms characterize differently a behavior which can be equivalent, and then symbolized by a same rheological model [CHR 82]. These operators are used with both integer and fractional partial derivative (or