Basic Equations of Structural Acoustics and Vibration

Chapter 2 Basic equations of structural acoustics and vibration

2.1 Introduction Copyrighted Material – Taylor & Francis As mentioned in Chapter 1, this book addresses classical numerical tech- niques to solve various vibroacoustics problems. In practice, a typical problem involves a structural domain coupled to bounded or unbounded fluid domains and sound-absorbing materials. We then seek to predict the coupled vibratory/acoustic response of the whole system subjected to the given excitations and boundary conditions. In this chapter, we introduce the fundamental equations governing the linear sound wave propagation in fluid domains (acoustics), in solid domains (elastodynamics), and in porous sound-absorbing materials (poroelasticity). We also establish the coupling equations between these domains. All these partial differential equations will be then solved using specific numerical methods, which will be pre- sented in Chapters 3 and 4.

2.2 Linear acoustics

The acoustic pressure disturbance px(,t) in a perfect fluid volumeΩ f at rest with sound speed c0, and density ρ0, due to an acoustic source distribution Q(x,t), satisfies the inhomogeneous wave equation:

1 2 ∇−px(,t)(2 px,)tQ=− (,xt) (2.1) c0

In the case of a mass source, Q(x,t) represents the rate of mass injection

and is related to the volume velocity Qs(x,t) by

dQs Qx(,t) =ρ0 (2.2) dt

K20584_C002.indd 9 2/9/2015 7:07:16 PM 10 Finite element and boundary methods in structural acoustics and vibration

Mass injection is the typical model of monopoles. In this case, the volume velocity, also called source strength, is the product of the surface area and the normal surface velocity of the monopole. For a proof of Equation 2.1, the reader is invited to look at the excellent book of Pierce (1989). To the pressure disturbance, we can associate a particle velocity Vx(,t) given by Euler’s equation:

∂Vx(,t)(∇px,)t =− (2.3) ∂t ρ0

A particle displacement Ux(,t) written as

∂Ux(,t) Vx(,t) =Copyrighted Material – Taylor & Francis (2.4) ∂t

A density fluctuationρ (,xt)

px(,t) ρ(,xt) = 2 (2.5) c0

Equation 2.1 must be associated to the boundary and initial conditions (Figure 2.1):

• Neumann boundary condition: Normal acoustic displacement is

specified on the part∂Ω f,N of boundary ∂Ωf:

(2.6) Un⋅=Un⋅

r → ∞ ∂Ω f,N n

x ∂Ω ∂Ω f,N f,R Ω f

∂Ω ∂Ω ∪ ∂Ω ∪ ∂Ω f = f,D f,N f,R

Figure 2.1 Fluid domain and boundary conditions.

K20584_C002.indd 10 2/9/2015 7:07:18 PM Basic equations of structural acoustics and vibration 11

• Dirichlet boundary condition: Acoustic pressure is specified on the

part ∂Ωf,D of boundary ∂Ωf:

pp= (2.7)

• Mixed, Robin, or impedance boundary condition:* Specific acoustic

admittance is specified on the part∂Ω f,R of boundary ∂Ωf:

ρ00c Vn. β ==ρ00c (2.8) Zn p

where Zn is the specific acoustic impedance applied on∂Ω f,R. In addition, for problems involving acoustic radiation in unbounded media (exteriorCopyrighted problem, see Material Chapter – 7), Taylor we must & Francis specify a condition to ensure that the wave amplitude vanishes at infinity.† This is the Sommerfeld radiation condition, which in the 3-D case, reads

 ∂p 1 ∂p lim r  +  = 0 (2.9) r →∞  rct  ∂ 0 ∂

Finally, the previous equations must be completed with initial conditions ∂px(,t) px(,t) and at all points of . t =0 Ωf ∂t t =0 In the following equation, we are interested in harmonic problems, that is, dynamic problems for which the temporal dependency of all variables is sinusoidal with circular frequency ω.‡ A convenient mathematical descrip- tion of the variables is to use the complex formalism

px(,tp)[=ℜ()xiexp( ωϕtp)] =ℜ[(xi)exp( ()xi)exp( ωt)] (2.10)

Solving for px(,t) is equivalent to solving for the complex-valued function pxˆ(). There is an amplitude pxˆ() and a phase ϕ()x associated to pxˆ(). In the following equation, we systematically omit the factor exp(iωt) in all the equations. However, we have to keep in mind that the physical value of the acoustic pressure at point x and time t is recovered by multiplying pxˆ() by exp(iωt) and taking the real part.

* Also called radiation or absorption condition. † This condition assumes the absence of acoustic sources at infinity. ‡ Note that if the disturbance spectrum is broadband, the temporal signal can be recon- 1 +∞ structed using Fourier transform: px(,tp)(= ∫−∞ xi,)ωωexp( td),ω ω being the conjugate variable of time t. 2π

K20584_C002.indd 11 2/9/2015 7:07:25 PM 12 Finite element and boundary methods in structural acoustics and vibration

For harmonic temporal dependence, Equation 2.1 can be rewritten as

22ˆ ˆ ˆ (2.11) ∇+px() kp()xQ=− ()x

where k = ω/c0 is the wave number. Equation 2.11 is known as Helmholtz equation. Dirichlet boundary condition rewrites

ˆ (2.12) pp= Neumann boundary condition can be rewritten using Euler’s equation (2.3):

∂pˆ 2 =⋅ρω0 Un (2.13) ∂n Copyrighted Material – Taylor & Francis The impedance boundary condition rewrites

∂pˆ +=ikβˆ pˆ 0 (2.14) ∂n Finally, for exterior problems, Sommerfeld radiation condition becomes

 ∂p  lim r  + ikp = 0 (2.15) r →∞  ∂r 

2.3 Linear elastodynamics

Let us consider a linear elastic solid occupying volume Ωs (see Figure 2.2). The fundamental equations governing its dynamic behavior are given by the conservation of mass equation, the conservation of momentum equation, and the behavioral law of the solid (the model defining how the solid deforms in response to an applied stress). In the framework of small dis- turbances around the solid equilibrium position, we can linearize these equations to come up with the following linear elastodynamics equations, relating the linearized displacement field u, strain tensor ε, and stress ten-

sor σ at all points of Ωs

2u ∂ (2.16) ρss2 =∇⋅+σρFb ∂t

σε= C (2.17)

K20584_C002.indd 12 2/9/2015 7:07:27 PM Basic equations of structural acoustics and vibration 13

1  ∂ui ∂uj  εij = + (2.18) 2  ∂x ∂x   j i 

where ρs is the density of the solid, ∇ is the nabla operator (see Appendix 3B), Fb is the body force vector per unit volume, C is the fourth-order stiff-

ness (elasticity) tensor, εij and ui, i,j = 1,2,3 are the strain tensor and displacement components along the 3 directions of a cartesian coordinate system. Equation 2.17 represents Hooke’s law.

In Equation 2.16, the body force distribution per unit mass Fb can include various effects: gravity, thermal effects, initial deformation or pre- stress, etc. It is common to rewrite this equation according to d’Alembert

as ∇⋅σρ+=sbF 0 where the body force ρsbF accounts for the inertial 22 pseudo-forceCopyrighted Iu=−ρs()∂∂/ tMaterial. – Taylor & Francis The boundary ∂Ωs of the solid is subjected to two types of boundary conditions (Figure 2.2):

• Specified contact forces F per unit area applied on ∂Ωs,N:

σ⋅nF= (2.19)

• Specified displacement over∂Ω s,D = ∂Ωs/∂Ωs,N:

uu= (2.20)

Finally, the previous equations are supplemented by two initial conditions providing the values of ux(,t) and ((ux,)tt/ ) at all points of . t =0 ∂∂t =0 Ωs For harmonic problems, the linear elastodynamics equation writes

∇⋅σρˆ ++Fuˆ ρω2 ˆ = 0 (2.21) sb s

∂Ω s,D n F

x ∂Ω s,N Ω s

∂Ω ∂Ω ∪ ∂Ω s = s,N s,D

Figure 2.2 Solid domain and boundary conditions.

K20584_C002.indd 13 2/9/2015 7:07:30 PM 14 Finite element and boundary methods in structural acoustics and vibration

The boundary conditions of Equations 2.19 and 2.20 remain unchanged. For more details, the reader is invited to refer, for example, to Reddy’s book (Reddy 2010).

2.4 Linear poroelasticity

From a qualitative point of view, a porous material is made up of a solid phase (the matrix or skeleton) and a fluid phase (pores) saturating its net- work of pores. The matrix can be continuous (e.g., plastic foams, porous ceramics) or not (fibrous or granular materials). The complexity of the microscopic geometry of such a medium makes it difficult to model it at this scale. The modeling is rather done at a macroscopic scale (defined by the wavelength in the medium), wherein this heterogeneous medium is seen as the superpositionCopyrighted in time Material and space – ofTaylor two continuous & Francis coupled media, a solid and a fluid. This is the basis of the Biot theory. An extension of this theory, the Biot–Allard theory, is dedicated to the acoustics of porous media. It establishes partial differential equations involving macroscopic solid and fluid displacements(, uUsf) averaged over a representative elemen- tary volume. Alternatively, these equations can be rewritten in terms of solid-phase displacement and interstitial pressure (,ups f ). Thus, the wave propagation in poroelastic materials is commonly described using either the classic displacement form (,uUsf) or a mixed- displacement pressure (,ups f ) form. These are the most popular forms that have been implemented in the context of the FEM over the years. It is not the purpose of this book to cover all the various forms of poroelasticity equations. The reader can refer to Allard and Atalla’s book (Allard and Atalla 2009) for details about the modeling of poroelastic materials. Instead, it has been chosen to focus on the mixed (,ups f ) displacement pressure form, which proves to be pretty efficient from the numerical point of view. Other forms can be implemented in a similar way (Allard and Atalla 2009). In addition, these equations will only be written in the frequency domain. Equations in the time domain can be found in Gorog et al. (1997) and Fellah et al. (2013). The governing equations* of a poroelastic material in the framework of the mixed (,ups f ) read as (Allard and Atalla 2009)

∇⋅σωs ++2ργupˆ sf∇=ˆ 0 (2.22)

ss2 f * These equations can be written in vector form ∇⋅σω++ργup∇= 0 and

ρ 22 ρ 22 s ∇+2 pˆˆffω2 pu−∇γ ⋅=ˆ 0. 2 R φp

K20584_C002.indd 14 2/9/2015 7:07:34 PM Basic equations of structural acoustics and vibration 15

ρ 22 ρ 22 ∇+2pˆˆffω2pu−∇γ ⋅=ˆ s 0 (2.23) R 2 φp

where σ s is the in vacuo solid-phase stress tensor

s νp s s ση =+21Ni()p ∇⋅uI ++21Ni()ηεp (2.24) 12 − νp

where N, ηp, and νp are the solid-phase shear modulus, damping loss factor, and Poisson’s ratio, respectively. εs is the solid-phase strain tensor. γ is a coupling coefficient given by

Copyrightedρ12 Q Material – Taylor & Francis γφ =−p (2.25)  ρ R   22 

where ϕp denotes the porosity, Q can be interpreted as a coupling coefficient between the deformation of the solid phase and the fluid phase, and R

is the dynamic bulk modulus of the fluid phase occupying a fractionϕ p of a unit volume of the porous material. Q and R are related to the dynamic

bulk modulus of the air in the pores K e. K e accounts for the dissipation due to the thermal exchanges between the two phases. ρ is a dynamic density given by

2 ρ12 ρρ=−11 (2.26) ρ 22

where ρ11, ρ12, and ρ 22 are Biot’s complex-valued dynamic densities given by

ρφ11 =−()11psρφkp+−ρα0((ω))

ρφ12 =− pρα0((ω))− 1 (2.27) (() ρφ22 = pρα0 ω

where ρ0 is the density of the fluid in the pores,ρ sk is the density of the material of the skeleton, and αω() is the dynamic tortuosity. This coefficient accounts for the dissipation due to the viscous effects in the fluid phase.

Note that (1 − ϕp)ρsk represents the apparent mass of the porous material. Generally, the Johnson–Champoux–Allard (JCA) model (Johnson et al. 1987; Champoux and Allard 1991) is chosen to describe the fluid phase of the porous material. In this case, five input parameters are required, namely the

porosity ϕp, the flow resistivityσ p, the tortuosity α∞, the viscous characteristic

K20584_C002.indd 15 2/9/2015 7:07:38 PM 16 Finite element and boundary methods in structural acoustics and vibration

length Λ, and the thermal characteristic length Λ′. In addition, if the vibration of the skeleton is accounted for and if the solid phase is assumed isotropic,

Young’s modulus Ep, loss factor ηp, and Poisson’s ratio νp are required. Using the JCA model, the dynamic tortuosity can be written as (Allard and Atalla 2009)

φσpp αω()=−α∞ iG()ω (2.28) ωρ0

where G()ω is a viscous correction factor given by

4αη2 ρω Gi()ω =+1 ∞ 0 (2.29) σφ22Λ 2 Copyrightedpp Material – Taylor & Francis

and the dynamic bulk modulus K e()ω reads

γP0 Ke()ω = −1 (2.30)  8η  γγ−−()11+ G ′  iBΛ′22ωρ   0 

where G ′()ω is a thermal correction factor

22 Λ′ B ρω0 Gi ′()ω =+1 (2.31) 16 η

2 where B = ηCp/ktc is Prandtl’s number, η is the fluid dynamic viscosity, Cp is the heat capacity at constant pressure, ktc is the thermal conductivity, and γ = Cp/Cv is the heat capacity ratio. Other expressions of dynamic tortuosity and dynamic bulk modulus can be used if additional porous material parameters are available (Pride et al. 1993; Wilson 1993; Lafarge and Lemarinier 1997). Note that the fluid displacement is obtained from the interstitial pressure and the solid-phase displacement by

f ˆ φp ˆ f ρ12 s Up=∇2 − uˆ (2.32) ρω 22 ρ 22

The normal flux is given by

ˆ f s φp Uu− ˆ ⋅ n (2.33) ()

K20584_C002.indd 16 2/9/2015 7:07:42 PM Basic equations of structural acoustics and vibration 17

and the total displacement of the porous material is written as

s f ()1 −+φφp uU p (2.34)

Also note that the total stress tensor of the porous material is

t  Q  σσˆ =− s φ 1 + pIˆ f (2.35)  R   

where φp((1 + QR/ )) is referred to as Biot–Willis coefficient. An important case of interest is when the porous material skeleton can be considered rigid s and motionlessCopyrighted or limp. In the Material case of a– rigidTaylor motionless & Francis frame, uˆ = 0 and the porous material is completely described by its interstitial pressure pˆ f . Then, Equation 2.23 reduces to

ρ ∇+2pˆˆff22 ω2p = 0 (2.36) R

which resembles Helmholtz equation. Using the analogy with a fluid, Equation2.36 can be rewritten as

ρ 22pˆˆf e pf 0 ∇+ω = (2.37) Ke

where ρρep= 22 /φ and KRep= /φ correspond, respectively, to an effective complex dynamic density and an effective dynamic complex bulk modulus of an equivalent fluid occupying the totality of a unit volume of porous material. Another important case is the one where the elasticity modulus of the matrix is weak. Then, the elastic force in the solid phase is negligible com- pared to the inertial and the pressure forces. The equation of motion of the solid phase reduces to

2 ˆ s ˆ f (2.38) ωρup+∇γ = 0

Taking the divergence of Equation 2.38, the solid-phase dilatation can be substituted in Equation 2.23 to give

ρ′ 22pˆˆf e pf 0 ∇+ω = (2.39) Ke

K20584_C002.indd 17 2/9/2015 7:07:47 PM 18 Finite element and boundary methods in structural acoustics and vibration

where

−1  1 γ 2  ρ′ =+ (2.40) e  ρ φρ   ep

where ρ′e is an apparent dynamic complex density of the fluid phase of the soft material. This equation is similar to the one of a rigid frame motionless porous material but accounts for the mass and the damping added by the solid phase.

2.5 eLasto-acoustic coupling

When an elasticCopyrighted solid vibrates Material in the presence– Taylor of & a Francisfluid, there is an inter- action between the elastic and the acoustic waves. In this case, we must simultaneously solve the structural and the fluid equations subjected to the coupling conditions at the interface between the two domains. For harmonic problems, the condition of stresses-continuity at the interface is written as

σˆ np+=ˆn 0 (2.41)

In addition, the continuity of normal displacements at the interface gives

1 ∂pˆ 2 =⋅unˆ (2.42) ρω0 ∂n Finally, if the solid domain is coupled to an unbounded fluid domain (exterior problem, see Chapter 7), Sommerfeld condition Equation 2.15 must also be fulfilled.

2.6 poro-elasto-acoustic coupling

Let us again consider the harmonic problems. At the interface between an elastic domain and a poroelastic domain, there is continuity of both the displacement vector and the total stress vector. Moreover, there is no relative displacement between the two phases (the flux is null). Thus

unˆˆs ⋅=un⋅   t σσˆˆ⋅=nn⋅ (2.43)  f φ (Uuˆ −⋅ˆ s) n = 0  p

K20584_C002.indd 18 2/9/2015 7:07:49 PM Basic equations of structural acoustics and vibration 19

At the interface between a fluid domain and a poroelastic domain, there is continuity of the total normal displacement, the total stress vector, and the pressure. Thus

 s f 1 ∂pˆ ˆ ˆ (−1 φφp)⋅un+⋅Un= 2  ρω0 ∂n (2.44)  ˆ t ˆ σ ⋅=np− n ppˆˆf  =

Finally, at the interface between two poroelastic domains (described by superscripts (1) and (2)), there is continuity of the total stress vector, the interstitial pressure,Copyrighted the solid-phase Material displacement, – Taylor & andFrancis the fluxes. This can be written as

ppˆˆff,(12),= ()  tt,(12),() σσˆˆ⋅=nn⋅  unˆˆss,(12),⋅=un()⋅ (2.45)  f ,(1 ) f ,(2 )  ()1 Uuˆˆ ˆ s,(1 ) nU()2 ˆ unˆ s,(2 ) φp − ⋅=φp − ⋅  ()()

2.7 Conclusion

This chapter described the fundamental governing equations for three classic problems of mechanics: linear acoustics, linear elastodynamics, and linear poroelasticity. The next chapter will introduce the associated integral forms necessary for their resolution using the finite and BE methods.

REFERENCES

Allard, J. F. and N. Atalla. 2009. Propagation of Sound in Porous Media, Modelling Sound Absorbing Materials. 2nd ed. Chichester, UK: Wiley-Blackwell. Champoux, Y. and J.-F. Allard. 1991. Dynamic tortuosity and bulk modulus in air- saturated porous media. Journal of Applied Physics 70 (4): 1975–79. Fellah, Z. E. A., M. Fellah, and C. Depollier. 2013. Transient acoustic wave propagation in porous media:Chapter 6. In Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices. Vol. 621. InTech. http://hal. archives-ouvertes.fr/docs/00/86/82/18/PDF/InTech-Transient_acoustic_wave_ propagation_in_porous_media.pdf.

K20584_C002.indd 19 2/9/2015 7:07:50 PM 20 Finite element and boundary methods in structural acoustics and vibration

Gorog, S., R. Panneton, and N. Atalla. 1997. Mixed displacement–pressure formula- tion for acoustic anisotropic open porous media. Journal of Applied Physics 82 (9): 4192. Johnson, D. L., J. Koplik, and R. Dashen. 1987. Theory of dynamic permeability and tortuosity in fluid-saturated porous media. Journal of Fluid Mechanics 176: 379–402. Lafarge, D. and P. Lemarinier. 1997. Dynamic compressibility of air in porous struc- tures at audible frequencies. Journal of the Acoustical Society of America 102: 1995–2006. Pierce, A. D. 1989. Acoustics, an Introduction to Its Physical Principles and Applications. New York, USA: McGraw-Hill. Pride, S. R., F. D. Morgan, and A. F. Gangi. 1993. Drag forces of porous-medium acoustics. Physical Review B, Condensed Matter 47 (9): 4964–78. Reddy, J. N. 2010. Principles of Continuum Mechanics: A study of Conservation Principles with Applications. Cambridge, UK: Cambridge University Press. Wilson, D. K. Copyrighted1993. Relaxation-matched Material modeling – Taylor of &propagation Francis through porous media, including fractal pore structure. The Journal of the Acoustical Society of America 94 (2): 1136–45.

K20584_C002.indd 20 2/9/2015 7:07:50 PM