[IEEE 2011 18th Iranian Conference of Biomedical Engineering (ICBME) - Tehran, Iran...

Abstract — In bioengineering, the majority of the hard tissues may be modeled as viscoelastic solids with time-dependent responses. However, influences of the viscoelastic constitutive equations of the biological tissues on damping the vibration have been seldom considered rigorously during the simulation stage. Present paper develops a mathematical approach for the analysis of free vibration of human skull, considering its viscoelastic behavior. A better understanding of ranges of the natural frequencies can be employed to provide new medical facilities or technology to avoid the brain injuries. In this regard, three-dimensional theory of viscoelasticity is used in terms of the Navier eigenvectors. The fundamental equation was solved numerically and the natural frequencies are presented in the diagrams and are discussed.

I. INTRODUCTION umerous researchers have studied and investigated

natural frequencies of vibration of the human skull both theoretically and experimentally. Obviously these studies provide a better understanding about the head injuries and known diseases. Vibration analysis of human skull is among the subjects that have drawn attention of some researchers. Based on the determined frequencies, new medical facilities may be developed to prevent the relevant cause of brain injury [1, 2]. Misra [3] studied response of axisymmetric vibration of the human skull-brain system using linear viscoelastic material properties containing of viscoelastic fluid due to changes in axial time-dependent load. Khalil et al. [4] studied experimentally vibrational characteristics of the human skull for measurement of critical frequencies in cadaver skull, and compared them with living human skull. Hakansson et al. [5] studied the resonance frequencies of the human skull in vivo and found the damping ratios for the first times. Charalambopoulos and Dassios [6-8] studied the dynamic characteristics of the human skull using different parameters. In these studies, they investigated the influence of the material properties, i.e. linear isotropic elastic and viscoelastic on the human skull

Prof. M. Shariyat is with Mechanical Engineering Department, K. N.

Toosi University of Technology, Tehran, Iran (e-mail: [email protected].) Hamed Band Band (M.Sc.) is with Mechanical Engineering Department,

K. N. Toosi University of Technology, Tehran, Iran (corresponding author to provide phone: +98-912-3624055; e-mail: [email protected]).

Dr. Hosein Ashrafi is with Mechanical Engineering Department, K. N. Toosi University of Technology, Tehran, Iran (e-mail: [email protected]).

frequencies. Many researchers studied the skull system by choosing

other material behaviors. Misra and Chakravarty [9] proved that different materials related to skull such as brain have significant effects on the human skull free vibration. Sahay and Mehrotra [10] considered the mechanical properties of the brain tissues as hyper elastic by modeling a hollow spherical part and using experimental data to drive the governing equations. Wang et al. [11] analyzed effects of the head size and morphology on the dynamic responses due to impact loading by a numerical finite element method. In this study, the absorption of energy under dynamic loading was considered. Gong et al. [12] considered the response of human head to non-contact impact under dynamic loads using a numerical finite element modeling. In that study, they considered two important factors: sudden motion of human head relative to the body, and deformation and stress distribution of head under such conditions using a new finite element method.

Some authors have used three-dimensional linear viscoelastic constitutive models to study dynamic response of human skull [13-14] by finite element software. In this work, the dynamic mechanical characteristics of the human skull system are studied. In this study, we consider influence of viscoelastic time-independent constitutive model on human skull natural frequencies. The developed mathematical modeling is based on the three-dimensional theory of elasticity and approximation of the skull by a linear viscoelastic material occupying the region bounded by two concentric spheres. Due to existence of complex terms, this leads to two functions in the complex plane. Frequencies for the real and complex Poisson’s ratios have been analyzed.

II. MODELING AND THEORETICAL FORMULATION We consider a hollow spheres with concentric surfaces s0

and s1 with their radiuses are r0 , r1, respectively (as shown in Fig.1). The region between two spheres is assumed to be filled with a viscoelastic material with following constitutive equation:

Modeling and Vibration Response Analysis of a Human Skull System with a Viscoelastic Nature M. Shariyat, Hamed Band Band and Hosein Ashrafi

N

18th Iranian Conference on BioMedical Engineering, 14-16 December 2011, Tehran, Iran MOD1:1569505863

978-1-4673-1005-5/11/$26.00 ©2011 IEEE 54

( ) ( ) ( ) ( )( )

( ) ( ){ }

( )( )

1

2 1

1τ , , ,2

13

. . ,

t Tr t G t u r u r

G t G t

u r I d

τ τ ττ

τ τ

τ ττ

−∞

∂⎧ ⎡ ⎤= − ∇ + ∇⎨ ⎢ ⎥∂ ⎣ ⎦⎩

+ − − −

∂ ⎫∇ ⎬∂ ⎭

∫

(1)

In which u is displacement vector of middle plane of the sphere, ( )τ ,r t is the stress tensor, I is the identity tensor and G1(t) and G2(t) are independent functions [17-18].

Fig .1 Geometry of the considered model.

Motion of the system is governed by the equation:

, iτ ρ i, j 1, 2,3ij j u= =&& (2)

Indices indicate the components of corresponding tensors and the comma symbol denotes partial differentiation with respect to the corresponding Cartesian coordinate.

For steady state vibration, Fourier transform analysis can be used for the considered problem using the following functions [19]:

( ) ( ) iωtr,ω u r, t eu dt+∞

−

−∞= ∫ (3)

( ) ( ) iωtτ r,ω τ r,ω e dt+∞

−

−∞= ∫ (4)

( ) iωtj j

0g ω G (t)e dt j 1, 2

+∞−= =∫ (5)

where ω equals 1 2iω ω ω= + . Combining equations 1-5 and using properties of the

Fourier transform one has

( ) ( ) ( )( )

( ) ( ) ( )( )( )

21

2 1

2

ˆ ˆ1 , . ,

ˆ

ˆ

21 . ,3

, 0

i g u r u r

i g g u r

u r

ω ω ω ω

ω ω ω ω

ρω ω

⎡ ⎤∇ + ∇ ∇⎣ ⎦

⎡ ⎤+ − ∇ ∇⎣ ⎦

+ =

(6)

or

( ) ( ) ( ) ( )* 2 * * 2.ˆ ˆ ˆ 0G u G u uω ω λ ω ρω⎡ ⎤∇ + + ∇ ∇ + =⎣ ⎦ (7)

where

( ) ( )*1

12

G i gω ω ω=

(8) ( ) ( ) ( )*

2 113

i g gλ ω ω ω ω⎡ ⎤= −⎣ ⎦

are frequency functions that may be used for replacing Lame's constant in case of elastic materials.

( )*G ω and ( )*λ ω may be readily determined in terms of Young's modulus:

( ) ( )( )( )

* E ωG ω

2 1 υ ω=

+ (9)

( ) ( ) ( )( )( ) ( )( )

* E ω υ ωλ ω

1 υ ω 1 2υ ω=

+ − (10)

where

( ) ( ) ( )( ) ( )

21

1 2

3g ωE ω iωg ω

g ω 2g ω=

+ (11)

( ) ( ) ( )( ) ( )

2 1

1 2

g ω g ωυ ω

g ω 2g ω−

=+

(12)

Therefore, ( )E ω and ( )υ ω may be used to define the viscoelastic properties, in the frequency domain.

Replacing dimensionless variables of equation (7) leads to:

( ) ( ) ( ) ( ) ( )( )( )

* 2 * *

2

ˆ ˆω ' u r' ' '.u r'

Ω u r' 0ˆ

G G ω λ ω⎡ ⎤∇ + + ∇ ∇⎣ ⎦

+ =

%

(13)

where

( )'1

rr α r , ' αα

= = ∇ = ∇ (14)

( ) ( )( )

**

*G ω

G ω 1G ω

= =% (15)

( ) ( ) ( ) ( )( )

* * * 2υ ωλ ω λ ω / G ω

1 2υ ω= =

−% (16)

( )( ) ( )Ω αω 2ρ 1 υ ω / E ω= + (17)

The characteristics of the system may be determined by above the mathematical formulation if proper boundary condition are defined on surfaces s0 and s1. On these surfaces:

( )T u ' 0r = (18) where


55

( ) ( ) ( ) ( )* ' ' * ' *ˆ ˆT 2 ω r . r . ω r 'ˆG Gλ ω= ∇ + ∇ + ×∇% % % (19)

Denotes the dimensionless surface stress in the middle plane V, r is the unit normal vector of surfaces s0 and s1 and

( )( )

*'

*μ 1

G

G

ω

ω= = (20)

( )( )

( )( )

*'

*2

λ1 2G

λ ω υ ωυ ωω

= =−

(21)

A solution approach based on expressing the displacement field ( )ru in terms of Navier’s eigenvectors [20-21] is developed for this problem. Navier eigenvectors are created through a Helmholtz-type decomposition and constituting a complete set of vector functions in the space of solutions of the time-independent Navier equation.

By decomposing the eigenvector:

( ) ( ) ( )

( )( )

m,l ' l mn n n

l 'n m

n'

L r g Ωr' P r

ˆg Ωr

n n 1 B (r)Ωr

=

+ +

&

(22)

( ) ( ) ( )m,l ' l ' ' mn n s nM r n n 1 g k r C (r)ˆ= + (23)

( ) ( )( )

( )

( ) ( ) ( )

l ' 'n sm,l ' m

n n' 's

l ' 'n sl ' ' m

n s n' 's

g k rN r n n 1 P r

k r

g k rn n 1 g k r B (r)

k r

ˆ

ˆ

= +

+ + &

(24)

0,1, 2, ; , 1, , 1, ; 1,2n m n n n n l= … = − + … − =

where

*1/2Ω'sk

G=

% (25)

( ) ( )( )

1 ( ) 2 ( )

nln

n

J x l spherical Beselg x

Y x l spherical Neumann⎧ =⎪= ⎨ =⎪⎩

(26)

where ( ) ( )ˆ ˆ , ( )ˆ , m m mn n nP r B r C r are functions defined on the

unit sphere are spherical harmonic vectors with relations

( ) ( )ˆ ˆ ˆm mn nP r rY r= (27)

( )( )1 ( )

1

ˆˆˆ ˆm mn nB r Y r

sinn nϕϑ

ϑ ϑ ϕ⎡ ⎤∂ ∂

= +⎢ ⎥∂ ∂+ ⎣ ⎦ (28)

( )( )1 ( )

sin

ˆˆˆ ˆ

1m mn nC r Y r

n nϑ ϕϑ ϕ ϕ

⎡ ⎤∂ ∂= −⎢ ⎥

∂ ∂+ ⎢ ⎥⎣ ⎦ (29)

Where ( ) (c )ˆ osm m imn nY r P e ϕϑ= are spherical harmonics and

(cos )mnP ϑ are Legendre functions. Finally the displacement filed is defined as

( ) ( ) ( ){( )}

2 , , , ,

0 1, ,

n m l m l m l m ln n n n

n m nlm l m ln n

r L r M r

N r

α β

γ

+∞

= =− == +

+

∑ ∑ ∑u (30)

Now, the problem of determination ( )ru reduces to determination of coefficients of Navier eigenfunctions in equation (30). Equation (30) satisfies Navier equation. This relation must satisfy boundary condition appeared in equation (18). Therefore, incorporating condition of the traction vector in equation (30), it can be conclude that for each specific pair of integers (n, m), with m n≤ six coefficients in relation (30) satisfy a linear homogeneous system of six equations of the form

m mn n 0=xD (31)

where

,1 ,2 ,1 ,2 ,1 ,2, , , , ,Tm m m m m m m

n n n n n n nα α β β γ γ⎡ ⎤= ⎣ ⎦x (32)

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )

1 2 1 2n 1 n 1 n 1 n 1

1 2n 1 n 1

1 2 1 2n 1 n 1 n 1 n 11 1 1 2n 0 n 0 n 0 n 0

1 2n 0 n 0

1 2 1 2n 0 n 0 n 0 n 0

r' r' 0 0 r' r'

0 0 r' r' 0 0

r' r' 0 0 r' r'

r' r' 0 0 r' r'

0 0 r' r' 0 0

r' r' 0 0 r' r'

mn

A A D D

C C

B B E E

A A D D

C C

B B E E

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

D

(33)

and

( ) ( ) ( )

( )

' ''

2 2

4 Ω 'Ω Ω '

( 1)2Ω 1 Ω 'Ω '

l l ln n n

ln

A r g r g rr

n n g rr

λ⎡

= − +⎢⎣

⎤⎛ ⎞++ − ⎥⎜ ⎟

⎝ ⎠ ⎦

&

(34)

( ) ( ) ( )l 'n 2 2

Ω '1r' 2 ( 1) Ω' Ω '

lnl

ng r

B n n g rr r

⎡ ⎤⎢ ⎥= + −⎢ ⎥⎣ ⎦&

(35)

( ) ( ) ( )

( )

1 1l ' ' ' '2 2n

1' '2

( 1) 2 Ω 2 Ω

1 2 Ω'

ln

ln

C r n n g r

g rr

λ λ

λ

⎡ ⎛ ⎞⎢ ⎜ ⎟= + + +⎢ ⎜ ⎟

⎝ ⎠⎣⎤⎛ ⎞⎥⎜ ⎟− +⎥⎜ ⎟

⎝ ⎠⎦

&

(36)


56

( )( ) ( )

( )

1 1' ' ' '2 2

l 'n 1

' '2

2 Ω 2 Ω

2 ( 1)'

2 Ω

l ln ng r g r

D r n nr

r

λ λ

λ

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟+ +⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎢ ⎥= + −⎢ ⎥⎢ ⎥+⎢ ⎥⎢ ⎥⎣ ⎦

&

(37)

( )( )

( ) ( )

( )

( )( )

1' '2

l 'n

1 1' ' '2 2

1' '2

1' '22

2 Ω

( 1) 2'

2 Ω 2 Ω

1 12 2 Ω

2 Ω

ln

ln

ln

g r

E r n nr

g r

n ng r

r

λ

λ λ

λ

λ

⎡ ⎛ ⎞⎢ ⎜ ⎟+⎢ ⎜ ⎟

⎝ ⎠⎢= + −⎢⎢⎢⎢⎣

⎛ ⎞⎜ ⎟− + +⎜ ⎟⎝ ⎠

⎤⎥⎛ ⎞+ − ⎥⎜ ⎟+ +⎥⎜ ⎟

⎝ ⎠⎥+ ⎥⎦

&

(38)

For solving equation (31), the following relation must be satisfied:

( )( )det Ω 0mn =D (39)

After solving equation (39) and determination of the roots, eigenfrequencies of the system are constructed through using the characteristic equation.

III. NUMERICAL SOLUTION Characteristic equation of the frequencies (39) is used for

numerical solution and m m1 2Ω Ω i Ωm = + is found from

solving the mentioned equation. The spherical Bessel functions and their derivatives are used for finding the complex frequencies using backward recurrence equations. The spherical Neumann functions and their derivatives are computed using mixed recurrence equations. Decomposition of relation (39) into two parts, i.e., complex and real parts and setting each part equal to zero, we can derive the results. The following specifications are used for the human skull:

( )( ) ( )

10 3 2

3 30 1

6.895 10 0.092 10 / , 0.25,

ρ 2.132 10 / , 0.076 ( ), 0.082

E i N m v

kg m r m r m

= × + × =

= × = = (40)

Solving equations with real Poisson ratio for the elastic material, results are listed in table 1. Table 2 shows results determined based on roots of equation (39) for 0.25ν = for real Poisson ratio and 0.25 0. 2iν = + for the complex Poisson ratio. Results of Tables 1 and 2 reveal that increasing the imaginary part ( 2ω ) of the firstω for every n of Bessel functions decreases the real part ( 1ω .

TABLE I

FREQUENCIES FOR 0.25v = AND 1 0.082 r m= , 0 0.076r m=

Elastic Skull Viscoelastic Skull

Ω ω 1ω 2ω

0.708 2709.282 2709.291 18.159

0.8522 3259.638 3259.711 21.849

0.9486 2628.364 2628.446 24.320

1.063 4065.941 4065.033 27.253

1.1971 4578.870 4578.972 30.691

1.2184 4660.341 4660.446 31.237

1.4210 5435.280 5435.402 36.431

1.5479 5920.669 5920.802 39.685

1.6695 6385.785 6385.928 42.802

1.8924 7238.370 7238.533 48.517

2.5324 9686.350 9686.567 64.925

2.6253 10041.689 10041.915 67.307

TABLE II

FREQUENCIES FOR 0.25 0. 2iν = + AND 1 0.082 r m= , 0 0.076r m=

Elastic Skull

ω Ω

2784.727 0.7083 (n=2)

3355.030 0.08522 (n=3)

3735.108 0.9486 (n=4)

4184.679 1.0631 (n=5)

4692.286 1.1971 (n=2)

4793.594 1.2184 (n=6)

5588.646 1.4210 (n=7)

6096.023 1.5479 (n=0)

6565.997 1.6695 (n=8)

7452.738 1.8924 (n=1)

7419.119 1.8924 (n=3)

9953.767 2.5324 (n=4)

10323.095 2.6253 (n=2)


57

Viscoelastic Skull

1Ω 1ω 2Ω 2ω 60.662 -0.0792 2711.926 0.7243

110.256 -0.0853 3262.434 08776

148.535 -0.0879 3628.634 0.9770

189.571 -0.0922 4062.345 1.0946

30.675 -0.1530 4579.019 1.2274

239.799 -0.0995 4650.483 1.2539

300.132 -0.0110 5419.097 1.4619

667.104 -0.0278 5866.431 1.5946

369.188 -0.1252 6364.619 1.7175

813.968 -0.0345 7172.260 1.9495

48.526 -0.2418 7240.025 1.9407

65.107 -0.3245 9713.487 2.6037

970.888 -0.0904 9955.172 2.7003

The dependence of 1ω and 2ω on 2ν ( 2ν is a constant) for

the fourth eigenfrequency are shown in figures 2 and 3, respectively. Variations of 1ω are small but the variations of

1ω are substantially large.

IV. CONCLUSIONS In this paper, we investigated and analyzed free vibration

behavior of human skull system, taking into account the viscoelastic behavior of the solid tissues materials. A better understanding of viscoelastic effects on the frequencies and order of the natural frequencies can be employed to develop the new biomedical facilities for preventing the brain injuries due to vibration resonances. The analysis is carried out based on the three dimensional linear viscoelasticity and expressing the displacement field based on Navier’s eigen vectors. Results related to eigenfrequencies are reported in diagrams and tables for both real and complex Poisson ratios and Young's moduli, and eigenfrequencies reveal that a little sensitivity to 1ω and significant sensitivity to 2ω .

0 0.05 0.1 0.15 0.24050

4055

4060

4065

4070

Poisson's Ratio

An

gula

r F

requ

ency

Fig 2. Variation of frequencies 1

mω , with 2ν for

1 0.082 r m= and 0 0.076r m=

0 0.05 0.1 0.15 0.20

50

100

150

200

250

300

350

Poisson's Ratio

Att

enu

atio

n

Fig 3. Variation of frequencies 2

mω , with 2ν for

1 0.082 r m= and 0 0.076r m=

REFERENCES [1] A.W. Toga, and J.C. Mazziota, Brain Mapping: The Systems.

Academic Press, 2000.

[2] J.D. Bronzino, Medical Devices and Systems. C RC Press, 2006.

[3] J.C. Misra, “Axisymmetric vibration of human skull-brain system,” Ingenieur Archiv, Vol. 47, pp. 11-19, 1978.

[4] T.B. Khalil, D.C. Viano, and D.L. Smith, “Experimental analysis of the vibrational characteristics of the human skull,” Journal of Sound and Vibration, Vol. 63, pp. 351-376, 1979.

[5] B. Hakansson, A. Brandt, and P. Carlsson, “Resonance frequencies of the human skull in vivo”, J. Acoust. Soc. Am., Vol. 95, No. 3, pp. 1474-1481, 1994.

[6] A. Charalambopoulos, G. Dassios, D.I. Fotiadis, V. Kostopoulos, and C.V. Massalas, “On the dynamic characteristics of the human skull,” International Journal of Engineering Science, Vol. 34, No. 12, pp. 1339-1348, 1996.


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[7] A. Charalambopoulos, G. Dassios, D.I. Fotiadis, and C.V. Massalas, “Dynamic characteristics of the human skull-brain system,” Mathematical Computer Modelling, Vol. 27, No. 2, pp. 81-101, 1998.

[8] A. Charalambopoulos, D.I. Fotiadis, and C.V. Massalas, “Free vibrations of the viscoelastic human skull,” International Journal of Engineering Science, Vol. 36, No. 5, pp. 565-576, 1998.

[9] J.C. Misra, and S. Chakravarty, “A free-vibration analysis for the human cranial system,” Journal of Biomechanics, Vol. 15, pp. 635-6, 1982.

[10] K.B. Sahay, R. Mehrotra, U. Sachdera, and A.K. Bonerji, “Elastomechanical characterization of brain tissues,” Journal of Biomechanics, Vol. 25, pp. 319-326, 1992.

[11] F. Wang, H.P. Lee, and C. Lu, “Effects of head size and morphology on dynamic responses to impact loading,” Medical & Biological Engineering & Computing, Vol. 45, pp. 747-757, 2007.

[12] S.W. Gong, H.P. Lee, and C. Lu, “Computational simulation of the human head response to non-contact impact”, Computers & Structures, Vol. 86, No. 7, pp. 758-770, 2008.

[13] O. Jirousek, J. Jirova, and J. Jira, “Multiresolutional finite element models of human skull based on data from computer tomography,” Acta of Bioengineering and Biomechanics, Vol. 5, pp. 214–217, 2003.

[14] D.W.A. Brands, G.W.M. Peters, and P.H.M. Bovendeerd, “Design and numerical implementation of a 3-D non-linear viscoelastic constitutive model for brain tissue during impact,” Journal of Biomechanics, Vol. 37, pp. 127–134, 2004.

[15] Y.C. Fung, Biomechanics: Motion, Flow, Stress and Growth. Springer-Velag, 1990.

[16] M. Kuchařová, S. Ďoubal, P. Klemera, P. Rejchrt, and M. Navrátil, “Viscoelasticity of biological materials,” Physiological Research, Vol. 56, pp. S33-S37, 2007.

[17] R.S. Lakes, Viscoelastic Materials. Cambridge University Press, 2009.

[18] W. Flügge, The Viscoelasticity. Springer-Verlag, 1975.

[19] W. Brown, and R.V. Churchill, Fourier Series and Boundary Value Problems. 5th ed., McGraw-Hill, 1993.

[20] A.C. Eringen, and S.S. Suhubi, Elastodynamics. Academic Press, 1975.

[21] S.S. Rao, Vibration of Continuous Systems. John Wiley & Sons, 2007.


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