Impedance of a sphere oscillating in an elastic medium with and without slip

0.17 MB·English

by  Andrew N. Norris

Impedance of an oscillating sphere February 6, 2008 1
Impedance of a sphere oscillating in an elastic medium
with and without slip
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2 Andrew N. Norris

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a Department of Mechanical & Aerospace Engineering,
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Rutgers University, 98 Brett Road, Piscataway,NJ 08854-8058,USA
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Abstract

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o The dynamic impedance of a sphere oscillating in an elastic medium is considered. Oestreicher’s [1]
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. formula for the impedance of a sphere bonded to the surroundingmedium canbe expressedsimply in terms
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m of three lumped impedances associated with the displaced mass and the longitudinal and transverse waves.
- If the surface of the sphere slips while the normal velocity remains continuous, the impedance formula is
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n modified by adjusting the definition of the transverse impedance to include the interfacial impedance.
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6 Short title: Thermoelastic thin plates
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0 Contact:
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t ph. 1.732.445.3818
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Impedance of an oscillating sphere February 6, 2008 2
1 Introduction
The dynamic impedance of a spherical particle embedded in a medium is important for acousticalmeasure-
ment and imaging. The impedance is used, for instance, in measurement of the mechanical properties of
tissue, e.g. [2], and is intimately related to the radiation forcing on particles [3]. The latter is the basis
for imaging techniques such as vibro-acoustography which has considerable potential in mammography for
detection of microcalcifications in breast tissue [4]. Oestreicher [1] derived the impedance for a rigid sphere
oscillating in a viscoelastic medium over 50 years ago. Although it was derived for a sphere in an infinite
medium, Oestreicher’s formula is also applicable, with minor modification [5], to dynamical indentation
techniques where the particle is incontactwith the surfaceofthe specimen, seeZhang etal. [6] fora review
of related work. Chen et al. [7] recently validated Oestreicher’s formula experimentally by measuring the
dynamicradiationforceonasphereinafluid. Theimpedance formulais basedonperfectnoslipconditions
between the spherical inclusion and its surroundings. This may not always be a valid assumption, e.g., in
circumstanceswhereaforeignobjectisembeddedinsoftmaterial. This waspreciselythesituationinrecent
measurementsof the viscosityof DNA cross-linkedgels by magnetic forcingona smallsteel sphere[8]. This
paper generalizes the impedance formula to include the possibility of dynamic slip.
Tworelatedresultsare derivedin this paper. The first is a modifiedformofOestreicher’sformulawhich
enables it to be interpreted in terms of lumped parameter impedances. This leads to a simple means to
consider the more general case of a sphere oscillating in a viscoelastic medium which is permitted to slip
relative to its surroundings. The slip is characterized by an interfacial impedance which relates the shear
stresstothediscontinuityintangentialvelocities. ThisgeneralizationincludesOestreicher’soriginalformula
as the limit of infinite interfacial impedance, and agrees with previous results for the static stiffness of a
spherical inclusion with and without slip [9].
2 Summary of results
A sphere undergoes time harmonic oscillatory motion of amplitude u in the direction xˆ,
0
usphere =u e−iωtxˆ. (1)
0
The time harmonic factor e−iωt is omitted but understood in future expressions. The sphere, which is
assumedto be rigidandofradiusa, isembedded inanelastic mediumofinfinite extentwithmassdensity ρ
and Lam´e moduli λ and µ. The moduli may be real or complex, corresponding to an elastic or viscoelastic
solid. We will later consider complex shearmodulus µ=µ −iωµ , where the imaginary termdominates in
1 2
a viscous medium. The force exerted on the sphere by the surrounding medium acts in the xˆ-direction, and
Impedance of an oscillating sphere February 6, 2008 3
is defined by
Fxˆ = Tds, (2)
Z
r=a
where T is the traction vector on the surface. The sphere impedance is defined
F
Z = . (3)
−iωu
0
Oestreicher’s expression for the impedance of a sphere that does not slip relative to the elastic medium is1
[1]
4 3i 3 −i 1 a2k2
Z = πa3ρiω 1+ − −2 + 3−
3 ah a2h2 ah a2h2 1−iak
(cid:20) (cid:21)(cid:30)
(cid:0) −i 1 (cid:1) a2k(cid:0)2 (cid:1)a(cid:0)2k2 (cid:1)
+ + 2− . (4)
ah a2h2 1−iak 1−iak
(cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
Here k and h are, respectively, the longitudinal and transverse wavenumbers, k = ω/c , h = ω/c with
L T
c = (λ+2µ)/ρ and c = µ/ρ.
L T
Nopting that Oestreicher’spformula can be rewritten
−1
Z = 4πa3ρiω −1+ 1 1− 3(1−ika) −1+ 2 1− 3(1−iha) −1 , (5)
3 3 k2a2 3 h2a2
(cid:26) (cid:20) (cid:21) (cid:27)
(cid:0) (cid:1) (cid:0) (cid:1)
implies our first result, that the impedance satisfies
3 1 2
= + , (6)
Z+Z Z +Z Z +Z
m L m T m
where the three additional impedances are defined as
4
Z =iω πa3ρ, (7a)
m
3
Z =(iω)−14πa(λ+2µ)(1−ika), (7b)
L
Z =(iω)−14πaµ(1−iha). (7c)
T
The second result is that if the sphere is allowed to slip relative to the elastic medium then the general
formofeq. (6)ispreservedwithZ modified. Specifically,supposethetangentialcomponentofthetraction
T
satisfies
T·ˆt=z (u˙sphere−v)·ˆt, r=a, (8)
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where ˆt is a unit tangent vector, v the velocity of the elastic medium adjacent to the sphere, and z is an
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interfacial impedance2, discussed later. Equation (8) holds at each point on the interface r = a. We find
that Z now satisfies
3 1 2
= + , (9)
Z+Z Z +Z Z +Z
m L m S m
1Equation(4)isOestreicher’s[1]eq. (18)withireplacedby−isinceheusedtimedependence eiωt.
2CapitalZ andlowercasez areusedtodistinguishimpedancesdefinedbyforceandstress,respectively.
Impedance of an oscillating sphere February 6, 2008 4
where the new impedance Z is given by
S
1 1 1
= + . (10)
Z Z 4πa2z +(iω)−18πaµ
S T I
These results are derived in the next Section and discussed in Section 4.
3 Analysis
We use Oestreicher’s [1] representation for the elastic field outside the sphere,
h (kr) x
u=−A grad 1 x +B 2h (hr)gradx−h (hr)r3grad , r ≥a, (11)
1 kr 1 0 2 r3
(cid:0) (cid:1) (cid:2) (cid:3)
where r =|r| is the sphericalradius and x is the componentof r in the xˆ−direction,both with originat the
center of the sphere. Also, h are spherical Hankel functions of the first kind [10]. Let ˆr=r−1r denote the
n
unit radial vector, then
h (kr) x x
u=−A 1 xˆ−h (kr) ˆr +B 2h (hr)xˆ−h (hr)(xˆ−3 ˆr) . (12)
1 2 1 0 2
kr r r
(cid:2) (cid:3) (cid:2) (cid:3)
The surface traction is T = σˆr where σ is the stress tensor. The traction can be calculated from (12) and
the following identity [1] for an isotropic solid,
µ ∂ 1
T=ˆrλdivu+ gradr·u+µ − u. (13)
r ∂r r
(cid:0) (cid:1)
Thus, referring to (2), we have
x2 x2 A
T·xˆ = 2µh (kr) 1−3 +(λ+2µ)krh (kr) 1
2 r2 1 r2 r
(cid:2) (cid:0) x(cid:1)2 x2 (cid:3) B
1
+ 2h (hr) 1−3 −hrh (hr) 1− 3µ . (14)
2 r2 1 r2 r
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
Integrating over the sphere surface, the resultant is
4 h (ka) h (ha)
F = πa3ρω2 A 1 −6B 1 . (15)
1 1
3 ka ha
(cid:2) (cid:3)
The coefficients A and B follow from the conditions describing the interaction of the sphere with its
1 1
surroundings. These are the general slip condition (8) plus the requirement that the normal velocity is
continuous. The conditions at the surface of the sphere are
u·ˆr=u xˆ·ˆr
0
r =a. (16)
T·ˆt=iωz (u−u xˆ)·ˆt )
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By symmetry, the only non-zero tangential component is in the plane of ˆr and xˆ, and we therefore set
ˆt = θˆ ≡ (ˆrcosθ −xˆ)/sinθ where θ = arccosˆr·xˆ is the spherical polar angle. Using polar coordinates,
u=u ˆr+u θˆ and T=σ ˆr+σ θˆ, and (16) becomes
r θ rr rθ
u =u cosθ
r 0
r =a, 0≤θ ≤π. (17)
σ −iωz u =iωz u sinθ )
rθ I θ I o
Impedance of an oscillating sphere February 6, 2008 5
The shear stress follows from the identity
∂u 1∂u u
θ r θ
σ =µ + − , (18)
rθ
∂r r ∂θ r
(cid:0) (cid:1)
and the interface conditions (17) then imply, respectively,
h (ka) h (ha)
1 1
h (ka)− A +6 B =u . (19a)
2 1 1 0
ka ha
h (ka) 2µh (ka) µh2a2(cid:2) h (ha) (cid:3)2µ
1 2 1
− + A + (2+ ) −(1+ )h (ha) 3B =u . (19b)
1 2 1 0
ka iωaz iωaz ha iωaz
I I I
(cid:2) (cid:3) (cid:2) (cid:3)
Solving for A and B , then substituting them into eqs. (15) and (3), and using the known forms for the
1 1
spherical Hankel functions, yields
Z =−Z +3/ 1/(Z +Z )+2/(Z +Z ) . (20)
m L m S m
(cid:2) (cid:3)
Equation (20) is identical to (9), which completes the derivation of the generalized impedance formula.
4 Discussion
It is useful to recall some basic properties of lumped parameter impedances. The impedance of a spring
mass damper system of stiffness K, mass M and damping C is
Z =(iω)−1K−C+iωM. (21)
Two impedances Z and Z combinedin serieshave aneffective impedance (Z−1+Z−1)−1, while the result
1 2 1 2
for the same pair in parallel is (Z +Z ).
1 2
Referring to the definitions of eq. (7), it is clear that Z is the impedance of the mass of the volume
m
removed from the elastic medium. The impedance of a longitudinal or transverse plane wave is defined as
the ratio of the stress (normal or shear) to particle velocity, and equals z , z , where
L T
z =ρc , z =ρc . (22)
L L T T
Thus, both Z and Z have the form
L T
1
Z = −1 4πa2z, (23)
iκa
(cid:0) (cid:1)
where κ is the wavenumber (k or h). In particular, the impedances Z and Z have stiffness and damping,
L T
but no mass contribution. The damping can be ascribed to the radiation of longitudinal and transverse
waves from the sphere.
The impedance Z of eq. (10) corresponds to Z in series with an impedance Z , where
S T I
8πa2z
Z =4πa2z + T . (24)
I I
iha
Impedance of an oscillating sphere February 6, 2008 6
Thus, Z can be interpreted as the total interfacial impedance for the surface area of the sphere in parallel
I
with twice the stiffness part of Z .
T
The limit of a purely acoustic fluid is obtained by letting the shearmodulus µ tend to zero with λ finite,
while an incompressible elastic or viscous medium is obtained in the limit as the bulk modulus λ+ 2µ
3
becomes infinite with µ finite. The acoustic and incompressible limits follow from (20) as
2 + 3 −1, acoustic medium,
Zm ZL
Z = (25)
(cid:0) (cid:1)
1Z + 3Z , incompressible medium.
2 m 2 S
Thus, Z for the acoustic fluid comprises 1Z in series with 1Z . Note that, as expected, the interfacial
2 m 3 L
impedance z is redundantin the acoustic limit. The impedance for the incompressible medium is 1Z and
I 2 m
3Z in parallel, and it depends upon the interfacial impedance.
2 S
In order to examine the role of z , we first express the impedance Z of eq. (20) in a form similar to (5),
I
Z = 4πa3ρiω −1+ 1 1− 3(1−ika) −1+ 2 1− 3(1−iha) −1 −1 , (26)
3 3 k2a2 3 h2a2[1+ χ(1−iha)]
(cid:26) (cid:20) 2 (cid:21) (cid:27)
(cid:0) (cid:1) (cid:0) (cid:1)
where the influence of the interfacial impedance is represented through the non-dimensional parameter
iωazI −1
χ= 1+ . (27)
2µ
(cid:0) (cid:1)
The form of χ is chosen so that it takes on the values zero or unity in the limit that the sphere is perfectly
bonded or is perfectly lubricated,
0, no slip, z →∞,
I
χ= (28)
(1, slip, zI =0.
The acoustic and incompressible limits of (25) are explicitly
4πa3ρiω (1−ika) , acoustic,
3 2(1−ika)−k2a2
Z = (29)

6πiωaµ 1+χ12−(1i−haiha) − h29a2 , incompressible.
Oestreicher[1]showedthattheoriginal(cid:2)formula(4)provide(cid:3)stheacousticandincompressiblelimitsforperfect
bonding (χ = 0). Ilinskii et al. [11] derived the impedance in the context of incompressible elasticity, also
for the case of no slip.
ThebehaviorofZ atlowandhighfrequenciesdependsuponhowz andhenceχbehavesinthese limits.
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For simplicity, let us consider χ as constant in each limit, equal to χ0 at low frequency, and χ∞ at high
frequency. Then,
(iω)−1 12πaµ 1−iha 2+c3T/c3L +O(h2a2) , |ha|,|ka|≪1,
2+χ0+c2T/c2L 2+χ0+c2T/c2L
Z = (cid:2) (cid:0) (cid:1) (cid:3) (30)
43πa2ρcL − 1+2ccTL(1−χ∞) + ik1a 1−4ccTL[1+(ccTL −1)χ∞]
+O 1 , |ha|,|ka|≫1.
(cid:2) (cid:0) (cid:1) (cid:0) k2a2 (cid:1)
 (cid:0) (cid:1)(cid:3)
Impedance of an oscillating sphere February 6, 2008 7
The leading order term at high frequency is a damping, associated with radiation from the sphere. The
dominant effect at low frequency is, as one might expect, a stiffness, with the second term a damping. The
low frequency stiffness is identical to that previously determined by Lin et al [9] who considered the static
problem of a sphere in an elastic medium with an applied force. They derived the resulting displacement,
and hence stiffness, under slip and no slip conditions. In order to compare with their results, we rewrite the
leading order term as
24πaµ(1−ν)
Z =(iω)−1 1+O 1 , (31)
5−6ν+2(1−ν)χ
0
(cid:2) (cid:0) (cid:1)(cid:3)
where ν is the Poisson’s ratio,
1c2 −c2
ν = 2 L T. (32)
c2 −c2
L T
Equation (31) with χ = 0 and χ = 1 agrees with eqs. (40) and (41) of Lin et al. [9], respectively. In an
0 0
incompressible viscous medium with ν ≈1/2 and µ=−iωµ , (31) becomes
2
6πaµ
2
Z ≈− , (33)
1+ 1χ
2 0
whichreducestotheStokes[12]dragformulaF =−6πaµ v forperfectbonding. Whenthereisslip(χ =1)
2 0
thedragisreducedbyonethird,F =−4πaµ v. Itisinterestingtonotethatonethirdofthecontributionto
2
thedraginStokes’formulaisfrompressure,2πaµ v,theremainedfromshearactingonthesphere. However,
2
under slip conditions, the shear force is absent and the total drag 4πaµ v is caused by the pressure.
2
The simplest example of the interfacial impedance is a constant value, which is necessarily negative and
corresponds to a damping, z =−C. For an elastic medium we have
I
1 2µ
χ= , ω = , elastic medium, z =−C. (34)
c I
1−iω/ω aC
c
Hence, χ0 = 1 and χ∞ = 0, corresponding to slip at low frequency and no slip at high frequency. The
transition from the low to high frequency regime occurs for frequencies in the range of a characteristic
frequency ω . Alternatively, if the medium is purely viscous µ = −iωµ , again with constant z , the
c 2 I
parameter χ becomes
1
χ= , viscous medium, µ=−iωµ , z =−C. (35)
1+ aC 2 I
2µ2
Inthiscaseχisconstantwithavaluebetween0and1thatdepends uponthe ratioofthe interfacialtobulk
viscous damping coefficients, and also upon a. One can define a characteristic particle size a =µ /C, such
c 2
that spheres of radius a≪a (a≫a ) are effectively bonded (lubricated).
C C
Figures 1 and 2 show the reactance and resistance of a sphere of radius 0.01 m in a medium with
the parameters considered by Oestreicher [1] based on measurements of human tissue, ρ = 1100 kg/m3,
µ =2.5×103Pa, µ =15Pasec, λ=2.6×109 Pa. The perfectly bonded (χ=0) and perfect slip (χ=1)
1 2
conditionsarecompared. Figure1indicatesthatthemass-likereactanceisgenerallyreducedbytheslipping,
Impedance of an oscillating sphere February 6, 2008 8
and it also shows that the low frequency stiffness is two-thirds that of the bonded case, eq. (31). Interfacial
slipleads to a significantdecreaseinthe resistance,as evidentfromFigure2 whichshowsa reductionforall
frequencies.
Impedance of an oscillating sphere February 6, 2008 9
References
[1] H. L. Oestreicher. Field and impedance of an oscillating sphere in a viscoelastic medium with an
application to biophysics. J. Acoust. Soc. Am., 23:707–714,1951.
[2] Y.L. Yin, S. F. Ling, and Y. Liu. A dynamic indentation method for characterizingsoft incompressible
viscoelastic materials. Mat. Sci. Engin. A- Struct. Mat. Prop. Microstruct. Proc., 379:334–340,2004.
[3] S. Chen, M. Fatemi, and J. F. Greenleaf. Remote measurement of material properties from radiation
force induced vibration of an embedded sphere. J. Acoust. Soc. Am., 112:884–889,2002.
[4] M. Fatemi, L. E. Wold, A. Alizad, and J. F. Greenleaf. Vibro-acoustic tissue mammography. IEEE
Trans. Med. Imag., 21:1–8, 2002.
[5] H. E. von Gierke, H. L. Oestreicher, E. K. Franke, H. O. Parrack,and W. W. von Wittern. Physics of
vibrations in living tissues. J. Appl. Physiol., 4:886–900,1952.
[6] X.Zhang,T.J.Royston,H.A.Mansy,andR.H.Sandler. Radiationimpedanceofafinitecircularpiston
on a viscoelastic half-space with application to medical diagnosis. J. Acoust. Soc. Am., 109:795–802,
2002.
[7] S. Chen, G. T. Silva, R. R. Kinnick, M. Fatemi, and J. F. Greenleaf. Measurement of dynamic and
static radiation force on a sphere. 71:056618,2005.
[8] D. C. Lin, B. Yurke, and N. A. Langrana. Mechanical properties of a reversible, DNA-crosslinked
polyacrylamide hydrogel. J. Biomech. Eng., 126:104–110,2004.
[9] D. C. Lin, N. A. Langrana, and B. Yurke. Force-displacement relationships for spherical inclusions in
finite elastic media. J. Appl. Phys., 97:043510,2005.
[10] M.AbramowitzandI.Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Math-
ematical Tables. Dover, New York, 1974.
[11] Y. A. Ilinskii, G. D. Meegan, E. A. Zabolotskaya, and S. Y. Emelianov. Gas bubble and solid sphere
motion in elastic media in response to acoustic radiation force. J. Acoust. Soc. Am., 117:2338–2346,
2005.
[12] G.G.Stokes.Ontheeffectoftheinternalfrictionoffluidsonthemotionofpendulums. Proc.Cambridge
Philos. Soc., 9:8–106,1851.
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Figure 1: The reactance (real part of Z) for an oscillating sphere in a tissue-like material [1]. The solid and
dashed curves correspond to a bonded (χ = 0) and slipping (χ = 1) spherical interface, respectively. The
reactance is positive (mass-like) except for frequency below 30 c.p.s. (50 c.p.s. for the dashed curve) where
it is negative (stiffness-like).