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A free-vibration analysis for the human cranial system
A FREE-VIBRATION ANALYSIS FOR THE HUMAN CRANIAL SYSTEM J. C. MISR.-\ and SAS.T;\HKATA CHGG;RVAKT~ School of Biomechanics, Department of Mathematics. lnd!an Institute of Technology. Kharagpur 711302. India
Abstract-The problem of free-vibration of the cranial vault is studied in the paper by taking into account the dissipative material behaviour of both the skull and the brain. Two mathematical models are considered for this purpose. In one model, the layered structure (inner table, diplti and outer table) of the skull is considered the outer surface of the skull being assumed to be traction-free while the second model assumes the homogeneity of the skull but takes into account the mechanical influence of the scalp. Results of numerlcal computation show that the dissipative material behaviour of the skull as well as the brain and also the mechanical influence of the scalp have a significant effect on the frequency spectrum of the freely vibrating cranial system.
As a consequence of the observation that in most of the accidents leading to fatalities, the head is the most vulnerable part of the human body, cranial biomechanics has drawn the serious attention of a large number of investigators. Various aspects of this domain of applied mechanics have already been studied -a good many of these studies were made through the use of analytical as well as experimental models. Due to obvious reasons, analytical studies together with the illustrations on their applicability through parametric studies by making use of experimentally determined material constants (or relaxation functions) are of much importance, especially for studies on the mechanics of the human cranial vault. One of the important problems of cranial biomechanics which deserves special attention is that of vibration of the cranial vault. By using various mathematical models the problem of forced vibration was studied by several investigators by considering the geometry of the braincase as spherical. The results of many of these investigations imolving loads and motions of the cervical region have been systematically reported by McElhaney and Roberts (1976), King and Chou (1976). Marangoni 01 (I/. (1978) and Goldsmith (1972, 1979). Roberts et ~1. (1967) performed experiments on fluid-filled human skulls subjected to different loads and examined the intracranial pressure distribution, while the effect of a global axisymmetric impulse was investigated by Engin and Roberts (1971). Another problem of a similar nature was treated by Terry and Roberts (1968) using the finite difference approach. The effect of the eccentricity of the braincase was considered by Misra l197Xa. b). hlisra CI trl. (1977, 1978). Talhouni er nl. I 1975) and Merchant t’l 01. (1973). 635
A free-vibration analysis was put forward by Engin and Liu (1970) for a model consisting of an elastic spherical shell filled with an inviscid and irrotational fluid, by using the thin shell theory. But the dissipative material behaviour of the skull as well as the brain is now very much confirmed as per the experimental observations of McElhaney er 01. (1970), Jamison et (I/. (1968) and Galford and McElhaney (1970). Hickling and Wenner (1973). while studying analytically the effect of an axisymmetric impact on a head, Incorporated the (linear) viscoelastic behaviour of the skull as well as the brain. Further, Engin and Liu (1970), like most of the earlier investigators. considered the skull to be thin and homogeneous. But Evans (1957) pointed out that the braincase consists of those cranial bones that touch the dura, a membrane covering the brain ; also the composition as well as the thickness of the eight bones (one frontal, t\vo parietal. one occipital, two temporal, one sphenoid and one ethmoid) which form the skull is variable. There is a central layer of spongy bone called the diplot! between two layers of cortical (compact) bone -- the outer and inner tables. The diplot? layer is transversely isotropic and reduces the weight of the skull without affecting its strength. A cross-section of this type of structure is somewhat similar to an engineering sandwich structure. Ofcourse, the thickness of the three layers vary at random from one region to another and also from individual to individual. The properties of the skull bone in general and the diplo@ in particular were studied by Melvin et LI/. (1970). By using a sandwich spherical shell model, a dynamic problem of cranial mechanics was analysed by Akkas (1975) through the use of a numerical method. He considered the shell to be thin and elastic, the fluid contained in it representing the brain being taken to be inviscid and compressible.
J. C.
636
and
MISRA
S.A&TTABR;\T.A CHAKRA\..ART’I
The present paper isessentially devoted to a study of the free-vibration problem of a cranial system (treated earlier by Engin and Liu. 1970) by taking into account the thickness of the skull (thin shell theory is not assumed) and the viscoelastic properties of the skull and the brain. Two models are considered for this purpose. In the first one, the three-layered sandwich structure of the skull (as mentioned above) is taken care of. but the outer skull surface is assumed to be free. The second model assumes the skull to be homogeneous but takes into account the mechanical infuence of the scalp overlaying the skull. A parametric study is also made for both the models. The ellects of the damping material behaviour of the skull and the brain. and the mechanical influence of the scalp on the frequency spectra of the freely vibrating human cranial system are illustrated through graphs and a table. BASIC EQUATIOSS
1
(.U,,
t)
=
2
(Iri,j
+
ICj,i).
(1)
The corresponding stress field oij (x,, t) has to satisfy the equation of motion Oij,j = p iii.
the equation 1 ?
(1) may be rewritten
-I+
_,S@
r- ?F --Lr
non-dimensional
--
iii=
1
?
F’ sin 0 (20 [
sine-
variables
as ?@ so
1
+ RiG=O
and
where
AND THEIR SOLUTIONS
For an analysis under the purview of the infinitesimal strain theory, the strain field f:ij (x,, f)can be written in terms of the displacement field tli(x,. I) as
‘;j
In terms of the following
(neglected
body forces),
(2)
11 being the mass-density and dots denoting timederivatives. In conformity to the experimental observations with regard to the mechanical behaviour of the various components of the cranial vault, the following constitutive relation of the linear theory of viscoelasticity will be used in our subsequent analysis:
Ri = R ?i
(i = I, 2).
For an axisymmetric problem, the azimuthal displacement lib is zero, while the radial and tangential displacements I(, and ti,are independent of the angle 4, so that one can write (cf. Hickling and Wenner, 1973)
Expressing Vz in terms of spherical polar coordinates (r. 0. (b) and employing the method of separation of variables, the solutions of equations (4) are obtained in the form
6(< 0) =
i
n=Cl
[A,j,(R,r3
+ B,n,,(R2r7]
P,(cos
0) (5)
and oij (S”, t) =
G,(&j -, [ s’
1 +j:GI(~-~)-G,(t-~):~~ij
IX
1 ds,
Y(r;U) =
in which the relaxation functions G, (t - T) and G, (r - 5) are independent and assume zero values for f < 5, T qXeSenting the NkiXatiOn tkfie. It can be shown (cf. Fliigge, 1967; Lockett, 1972) that corresponding to the consecutive equation (3), the equation of motion (2) for the steadily vibrating cranial vault can be expressed in terms of a scalar potential 0 and an irrotational vector potential Y in the form (VZ + K:)@
= 0,
(V2 + K$Y
= 0.
(4)
Let N represent the average skull radius and c the compressional wave speed in the medium under consideration. If c, and c, represent respectively the compressional wave speed in the skull and the brain, then in the skull region c = c, while in the region occupying the brain matter, c = cb.
i
O),
[C.j,(R,fl+D.n.(R,i)]-&P.(cos
“=O
(3)
(6) where A,, B,, C, and D, are arbitrary constants, j, and n, denote spherical Bessel functions of the first and second kinds respectively, and P, denotes the Legendre polynomial. FIRST MODEL
Here we consider the skull to consist of three different concentric layers: the outer table, the diplo and the inner table. In the following, these will be respectively referred to as layers 1, 2 and 3. The potential functions for the three different layers of the skull and the brain are taken as
6” = 1 [A,,i,(R,,rJ n=O
+ B,,,n,,(K2,~]P,(cos 0).
A free-vibration
6’2 = 1 [dn2jn(KlfJ +
B,,rl,(R,,r~]
analysis
for ihe human
sranial
s>stcm
P, (cos U).
“=”
Fig. 1. Schematic
x
@-‘s = n
[.-ln3j,,(K-,,rJ +
B,,n,(K,,rl]
P, (cos U),
=o
representation of the la)ercd the cranium. First model.
1
E,,j,(R3,r~Pn(cns 0).
A, +iAz
Y* =
1 F,,jn(K,hr)~p,,lcos 0).
n =o
r = r.
(upper surface of the outer table), (8)
The matching conditions for the interfaces of the various layers are : On r = riz (interface between the diplo6 and the outer table).
r,,
(interface
between
can be determined
A, = 0 and A? = 0.
112) from
(13)
Here we propose a mathematical model which is suitable for taking into account the mechanical influence of the scalp. Although the scalp material is also viscoelastic. only its linear elastic behaviour will be studied here. In this part of our study, the skull will be treated as homogeneous but viscoelasticity of both the skull and the brain will be taken into account as in the first model. The quantities &, and l$,,,representing the clrcumferential strains in the scalp and the skull respectiveI;<. in a state of plane strain. may be expressed as
I: ;,,, =
On r = diploe).
= 0.
Thus the frequency spectrum the pair of equations
a”’ r, = 0 = (7;;‘.
1,2,3.-l).
(7)
in which the superscript si denotes the ith layer (i = I, 2, 3) of the skull, while the superscript b refers to quantities for the brain material. The function n,,(kr) being unbounded at r = 0, such a function cannot appear in 6* and Y *. A,,i, B,,i, C,,, Dmi(i = 1, 2, 3) and E,. F,, are arbitrary functions of rl; R,,, KZi are the values of K, and Rz in the ith layer of the skull. The boundary conditions are taken as follows: On
=
) denotes the derivative where i = J( - I) and S:( determinant can be split up into real and imaginar? parts in the form
=o
n
Iof
the frequency equation, The elements of the determinant are all complex. On the assumption of a small loss tangent for viscoelastic materials. \ve can write s,(Kjr) = s,,(K;r - iK;r) + ,Y,(K;r) - iK;(r)S,(Kjr)(j
@ =
structure
the inner table and the
1 F,, E' /I
~
III)
and 1: i,”
On r table).
=
r,
(interface
between
the brain and the inner
By using the fourteen boundary conditions listed in equations (8)-(1 I ). together with the matching conditions. one gets a set of fourteen linear homogeneous equations (as shown in Appendix A) involving fourteen unknowns. viz. A,,. B,,. Cnj. Dnj (j = 1.2. 3). E, and F,. The coefficient determinant when equated to zero gives
= ;
[G -
Yrk7:, + GybJ]
(15)
where F,, and II stand respectively for the tangential force per unit length and the thickness of the scalp. Here and in the sequel, the superscripts c, s and b refer to quantities for the scalp. the skull and the brain respectively; I’ and E denote Poisson’s ratio and Young’s modulus. For the continuity of the circumferential strain on the scalpskull interface defined by r = ro, equating ( 14) and (I 5) we find
F,, = g
Lo;,,,- \,s(G:,+ G&)1
SASTTABRATA CH.AKRAL.ART\I
J. C. &~ISRAand
638 This expression - r,a:,. yields
of FU when
further
equated
0--s r, = T(G:,,~- ~‘8:~) on i = F. (lvritten
in terms of dimensionless
VUhlERICAL
to
(16)
RECLTS
ASD
DISCL’SSIOS
Writing ((t = w, + ic!~? (w, denotes the actual frequency and CI): the attenuation factor). the frequency equations for both the models are expressed in the form
quantities) A, (IU,. w2) + i A2 (co]. w2) = 0. (i =
ivith
(17)
For the validity of this equation.
one must have
A, (w,, w,) = 0 and AZ (u,. For the study of the present model of the cranial vault, equation (16) together with equation (I 7) serves as the mechanical boundary conditions on the scalp-skull interface, while the conditions on the skull-brain interface. viz. the continuity of the radial displacement and the radial stress, can be put in the form 6’
=
lb
and a:, = #;f, on r = <.
(18)
Applying the boundary conditions, we get a set of six equations involving six unknowns A,, B,, C,, D,. E, and F, (say). When their coefficients are split into real and imaginary parts in the same way as indicated at the end of the preceding section, the resulting equations read .-l,(cr’, , - ia;,)
+ B,(rr;2
- iu’;*) -
C,((t;3
-
iOl;j) + D”(u;,
E,(u’,,
-
in’;,)
.&(uk,
- ia:,)
+ B,(n;,
-
+ Bn(uk2
- in;,)
+ C,(u;, A,(u;,
-
ia’;,)
= 0
+ B,(n;,
F, (a&
-
iu’;, = 0
iu’;z) = 0
-
ia;,)
+ Dn(u;,
- ia;,)
= 0
- i~‘;~)
- iaZl) + B,(u’,,
,!?,(a;, - iu;s) = 0
-
iu&)
+ C,(u&
-
+ D,(rz&
-
ia;,)
- E,(u’,,
- in:,)
-
-
itr &) = 0
Fn(rr&
ia&)
(19)
where the derived expressions for the quantities uij and ai; are shown in Appendix B. Hence the frequency equation in this case is of the form
‘J L I
t **
lUIL
-
,
u;,
-
lN12
0
0 iu’;,
I, (14I - 1fl4, ,, “5 I - Uis, I, 0,. - ‘(1.1
a;,
u>, - iu&
-
0
,I (‘44 - ‘a.&.$
0 u63
The frequency-dependent values of the shear modulus G,* were taken to be the same as in Misra (1978). For scalp, EC = 34.5 x lo6 Pa and h = 6mm (cf. Khalil and Hubbard, 1977). Further we took r. = 83 mm and ri = 71 mm for a human-sized head. In Figs 3-6 the first three roots of the aforementioned frequency equations are exhibited for the first two modes (n = 1, 2) in each case. The frequency spectrum for a human-sized freely vibrating viscoelastic spherical shell filled with a viscoelastic fluid (as a better representative of the human head) has been exhibited in Fig. 7. The corresponding frequency spectrum for an elastic spherical shell (having the same dimension as that of the aforementioned) filled with an inviscid irrotational fluid, studied by Engin and Liu (1970). has been reproduced in Fig. 8. In both Figs 7 and 8 the first nine modes (II = I, 2, 3,. . .9) are taken
0
I I, (143 - 1043 I
0.
it = 0.207 x 10LoN/m2, pb = 0.9774 x IO3 kg/m’.
(I;~ - ia’;
0
- kc’;,
,, “12 - IN?1 I, 052 - IN5L ,, - 1l162
=
If A, = 0 and AZ = 0 are plotted on a wI versus Q2 plane, the coordinates of the points of intersections of the curves represented by A, = 0 and AZ = 0 will give the complex frequencies. The plots for A, = 0 and AZ = 0 on the W,-OJ~ plane can be made by first fixing (?Jz and then finding an w, that satisfies the transcendental equation A, = 0 or AZ = 0 by trial-and-error method. The locations of the first few roots can be graphically determined. In order to be able to compare the results with those reported by Engin and Liu (l970), CJ, and 012 were replaced by Q, = clw,lc, and R, = urozfcb (c,, denotes the compressional wave speed in brain. its value being taken to be 398.78m/s, cf. Engin. 1969). For the purpose of the computational work. assuming the homogeneity of the skull material, the following values of the material constants and other parameters were taken (cf. Misra, 1978).
‘a63
(‘64
-
#I
l”64
-(a’,,
-
-
(46
0
0
0
0
- (4, - ia’;,)
0 I,
0
- ((1;s - iuy5)
0
0
I,
‘1 12 -
C:JJ
i.*’ = (27.6 + 1’0.37) x IO* N/m2 = GtS, p” = 2.0844 x I O3 kg/m3 ;
-
.4(4,
ia;‘,)
- iu’;&) -
.A(41
-
J - 1).
- iu&)
ia’;,)
i =o.
0
- (a& - in&j)
’
(20)
63’1
A free-vibration .malksls for the human cranIaI system
small head (sa). head of an experimental animal) h) taking r,, = 34.30 mm and ri = 31.75 mm and be using the second model of our study. The first three roo[s (non-dimensional) of the frequencv equation. in this case. are found to be 1.55. 5.45 and 7.90 for II = I.
-5colP .Skull .0rc,n
COUCLL Dl\<; RE\I-\RES
Although in the analysis for the first model, the three different layers of the skull were paid duz attention. owing to non-:~v~~ilabilit~ of all the requiGre ekperFig. 7.
Schematic representation of the layered structure the cranium. Second model.
of
imental
data
for
for consideration. These two figures considered together have a strong potential to signify as well as to quantitate the effect of the experimentally established damping material behaviour of both the skull and the brain on the frequency spectrum of the freely vibrating cranial vault. For comparing the values for the two models considered here with the corresponding values obtained by Engin and Liu (1970) (who neglected the dissipative behaviour of the skull and the brain), a table is provided below (Table I ). It can now be readily concluded that the dissipative material behaviour of both the skull and the brain as well as the mechanical influence of the scalp affect the frequencies of the vibrating cranial system quite significantly. We also computed the frequency values for a
3
individual
layers.
I
5
7
9
II
13
n7 Fig. 3. Plots of the curves A t = 0 and A,
the
com-
rltl,tout~rtclenrrrr-The authors are highly grarsful to the reviewers for their valuable comments and suggesrlons. the second author further acknowledges the assistance provided by CSIR.
I
1
the
putational work was based on available informations of the average values for the skull parameters. The effect of the influence of the scalp on the frequencies of vibration can be estimated from the values sit-en in the first and the second columns ofTable 1. while the effect of the skull and brain damping can be read from the first and the third columns of the same table. It is worthwhile to mention here that the results presented above, being based on an exact method of solution, possesses a potential to provide an analytical check for finite element modelers on the soundness of their numerical program. Moreover the present stud, represents a much improved version of the problem considered by Engin and Lit] (1970).
= 0 in the R, R, plane In =
1). First
model
__--
(- - _ ---_
---___
\ !i
2
!i I
i!
$2
i/
!i
(I
2
-z
b4
t i
J. C. M~SRA and SASTABRATA CHAKRAV.ART>
1
z
3
Fig. 8. Frequency
spectrum
4 5 Mode number of fluid-filled
REFERENCES
8, 275-284.
Engin, A. E. (1969)The axisymmetric response of a fluid-filled spherical shell to a local radial impulse-a model for head injury. J. Biomechakx 2, 325-341. Engin, A. E. and Liu, Y. K. (1970) Axisymmetric response of a fluid-filled spherical shell in free vibrations. J. Biontecltanics 3, 11-22. Engin, A. E. and Roberts, V. L. (1971) A mathematical model to determine the brain damage when the human head is subjected to impulsive loads. Proc. Symp. on Biodynamic Models and Their Applications. AMRL-TR-71-29, pp. 877-903. Evans, F. G. (1957) Stress and struin itI bones, p. 149. Springfield, Illinois.
Table Mode n
spherical
1. Values for the non-dimensional
First model
7
8
9
shell (cf. Engin and Liu, 1970).
Galford, J. E. and McElhaney, J. H. (1970) A visccelastic study of scalp, brain and dura. J. Biontechunics 3,21 l-222. Fliigge, W. (1967) Viscoehsricity. Blaisdell, Waltham, MA. Goldsmith, W. (1972) Biomechanics of head injury. In: Biotnechurtics. irs Fountl~rions tmd Objecriws, (Edi ted by Fung, Y. C., Perrone, N. and Anliker, M.), pp. 585-634. Prentice Hall, Englewood Cliffs, NJ. Goldsmith, W. (1979) Some aspects of head and neck injury and protection. Proyress it: Biornechmics; Proc. NATO Advanced Study Institute. Noordhoff, Amsterdam. Hickling. R. and Wenner, M. (1973) Mathematical model of the head subjected to an axisymmetric impact. J. Biomechanics 6, 115-132. Jamison, C. E., Marangoni. R. D. and Glaser, A. A. (1968) Viscoelastic properties of soft tissues by discrete model charactization. J. Biomechmics 1, 33-46. Khalil, B. T. and Hubbard (1977) Parametric study of head
Akkas, N. (1975) Dynamic analysis of a fluid-filled spherical sandwich shell - a model of the human head. J. Biomechanics
6 n -
Second
model
frequencies
(Q,
1
Values of Engin and Liu (1970)
1
1.30 2.65 3.90
2.02 4.30 5.30
0.00 2.50 4.40
2
1.09 2.06 4.65
2.20 3.10 3.90
1.00 3.90 5.50
643
Ihe human cramal system Xlerchunt. anJl>jis
H. C. 2nd Crispino. A. J. (19731 A dynamic of an elastic model of the human head. J.
B~~~rw
_‘Y?-311. &lisr,~. 1. C hl;lrangoni. R. D.. Saez. C. A.. Wcycl. D. A. and Polosk~. R. A. I 1978)Impact stresses in human head-neck model. J. Em/a/ .\/w/I. Div. 104, IS?- 176. XIcEihaney. J. H. (1966) Dyumic response of bone and muscle tissue. J. crppl./‘/I) sird 20, 1231- 1136. Xlc Elhaney. J. H.. Fogle, J. L.. Mehin. J. W., Haynes. R. R.. Roberts. V L. and Alrn. N. hf. (1970) Klechanical properta of cranial bone. J. Birl,ttt,c,lrlr,~ics3. 495 -5 12. XlcElhaney. J. H.. Koberts. V. L. and Hilyard. J. (19761 RI,J)~I~‘(./I‘/)I~(.~ ~$Tr~~nrtr. Duke University Press, Durham.
(197831 Axlsymmetric
0. (19771 Effect of Ra. C~~~IL~I 4.
vibration
of human
skull-brain s\stcm. /!t!q. .+lr&. 47, 1 I-IV. Xl1sra. J C Il$iSb) Stress in a human skull dus to pulsr Iwdlny. /!I+ .4w/i. 17. 339-337. >lisr;t. 1 C. (1976~1 Response of a human head tL>.m Impact. .\/n/. Lilr Sci. Erkj,zy 4, l12- 152. Roberts. i’.. Hodpson, V. and Thomas. L. ht. 11967) Fluid pressure gradients caused b> impact to the humm skull. Biomechanics \lonograph. pp. 223-225. ASLIE. ?!e\t York. Talhouni. 0. ,nd Dimawio. F. ( 1975 I Dvnamic response of :I lluid tilled sphoroida? shell ~ an improved model for stud! ins head injury. J. Bi,,,,r~,~,h~rnic,s 8, 219-228. Terry. C. T. and Roberts. V. L. Il968) A viscoelaws model af the hum;m spins subjected to Gz accelerations. J. Bi<,wc~~/~lo~i[~r 1. 161 168.
u’;, = (i:
+ ZG3)[R;,f?‘,,,
‘2n;(R;,Fi)
+ r;:R;,n;‘(R’*,<)j]
+ yR;,jn.(R;,i.) 1
+
Abstract-The problem of free-vibration of the cranial vault is studied in the paper by taking into account the dissipative material behaviour of both the skull and the brain. Two mathematical models are considered for this purpose. In one model, the layered structure (inner table, diplti and outer table) of the skull is considered the outer surface of the skull being assumed to be traction-free while the second model assumes the homogeneity of the skull but takes into account the mechanical influence of the scalp. Results of numerlcal computation show that the dissipative material behaviour of the skull as well as the brain and also the mechanical influence of the scalp have a significant effect on the frequency spectrum of the freely vibrating cranial system.
As a consequence of the observation that in most of the accidents leading to fatalities, the head is the most vulnerable part of the human body, cranial biomechanics has drawn the serious attention of a large number of investigators. Various aspects of this domain of applied mechanics have already been studied -a good many of these studies were made through the use of analytical as well as experimental models. Due to obvious reasons, analytical studies together with the illustrations on their applicability through parametric studies by making use of experimentally determined material constants (or relaxation functions) are of much importance, especially for studies on the mechanics of the human cranial vault. One of the important problems of cranial biomechanics which deserves special attention is that of vibration of the cranial vault. By using various mathematical models the problem of forced vibration was studied by several investigators by considering the geometry of the braincase as spherical. The results of many of these investigations imolving loads and motions of the cervical region have been systematically reported by McElhaney and Roberts (1976), King and Chou (1976). Marangoni 01 (I/. (1978) and Goldsmith (1972, 1979). Roberts et ~1. (1967) performed experiments on fluid-filled human skulls subjected to different loads and examined the intracranial pressure distribution, while the effect of a global axisymmetric impulse was investigated by Engin and Roberts (1971). Another problem of a similar nature was treated by Terry and Roberts (1968) using the finite difference approach. The effect of the eccentricity of the braincase was considered by Misra l197Xa. b). hlisra CI trl. (1977, 1978). Talhouni er nl. I 1975) and Merchant t’l 01. (1973). 635
A free-vibration analysis was put forward by Engin and Liu (1970) for a model consisting of an elastic spherical shell filled with an inviscid and irrotational fluid, by using the thin shell theory. But the dissipative material behaviour of the skull as well as the brain is now very much confirmed as per the experimental observations of McElhaney er 01. (1970), Jamison et (I/. (1968) and Galford and McElhaney (1970). Hickling and Wenner (1973). while studying analytically the effect of an axisymmetric impact on a head, Incorporated the (linear) viscoelastic behaviour of the skull as well as the brain. Further, Engin and Liu (1970), like most of the earlier investigators. considered the skull to be thin and homogeneous. But Evans (1957) pointed out that the braincase consists of those cranial bones that touch the dura, a membrane covering the brain ; also the composition as well as the thickness of the eight bones (one frontal, t\vo parietal. one occipital, two temporal, one sphenoid and one ethmoid) which form the skull is variable. There is a central layer of spongy bone called the diplot! between two layers of cortical (compact) bone -- the outer and inner tables. The diplot? layer is transversely isotropic and reduces the weight of the skull without affecting its strength. A cross-section of this type of structure is somewhat similar to an engineering sandwich structure. Ofcourse, the thickness of the three layers vary at random from one region to another and also from individual to individual. The properties of the skull bone in general and the diplo@ in particular were studied by Melvin et LI/. (1970). By using a sandwich spherical shell model, a dynamic problem of cranial mechanics was analysed by Akkas (1975) through the use of a numerical method. He considered the shell to be thin and elastic, the fluid contained in it representing the brain being taken to be inviscid and compressible.
J. C.
636
and
MISRA
S.A&TTABR;\T.A CHAKRA\..ART’I
The present paper isessentially devoted to a study of the free-vibration problem of a cranial system (treated earlier by Engin and Liu. 1970) by taking into account the thickness of the skull (thin shell theory is not assumed) and the viscoelastic properties of the skull and the brain. Two models are considered for this purpose. In the first one, the three-layered sandwich structure of the skull (as mentioned above) is taken care of. but the outer skull surface is assumed to be free. The second model assumes the skull to be homogeneous but takes into account the mechanical infuence of the scalp overlaying the skull. A parametric study is also made for both the models. The ellects of the damping material behaviour of the skull and the brain. and the mechanical influence of the scalp on the frequency spectra of the freely vibrating human cranial system are illustrated through graphs and a table. BASIC EQUATIOSS
1
(.U,,
t)
=
2
(Iri,j
+
ICj,i).
(1)
The corresponding stress field oij (x,, t) has to satisfy the equation of motion Oij,j = p iii.
the equation 1 ?
(1) may be rewritten
-I+
_,S@
r- ?F --Lr
non-dimensional
--
iii=
1
?
F’ sin 0 (20 [
sine-
variables
as ?@ so
1
+ RiG=O
and
where
AND THEIR SOLUTIONS
For an analysis under the purview of the infinitesimal strain theory, the strain field f:ij (x,, f)can be written in terms of the displacement field tli(x,. I) as
‘;j
In terms of the following
(neglected
body forces),
(2)
11 being the mass-density and dots denoting timederivatives. In conformity to the experimental observations with regard to the mechanical behaviour of the various components of the cranial vault, the following constitutive relation of the linear theory of viscoelasticity will be used in our subsequent analysis:
Ri = R ?i
(i = I, 2).
For an axisymmetric problem, the azimuthal displacement lib is zero, while the radial and tangential displacements I(, and ti,are independent of the angle 4, so that one can write (cf. Hickling and Wenner, 1973)
Expressing Vz in terms of spherical polar coordinates (r. 0. (b) and employing the method of separation of variables, the solutions of equations (4) are obtained in the form
6(< 0) =
i
n=Cl
[A,j,(R,r3
+ B,n,,(R2r7]
P,(cos
0) (5)
and oij (S”, t) =
G,(&j -, [ s’
1 +j:GI(~-~)-G,(t-~):~~ij
IX
1 ds,
Y(r;U) =
in which the relaxation functions G, (t - T) and G, (r - 5) are independent and assume zero values for f < 5, T qXeSenting the NkiXatiOn tkfie. It can be shown (cf. Fliigge, 1967; Lockett, 1972) that corresponding to the consecutive equation (3), the equation of motion (2) for the steadily vibrating cranial vault can be expressed in terms of a scalar potential 0 and an irrotational vector potential Y in the form (VZ + K:)@
= 0,
(V2 + K$Y
= 0.
(4)
Let N represent the average skull radius and c the compressional wave speed in the medium under consideration. If c, and c, represent respectively the compressional wave speed in the skull and the brain, then in the skull region c = c, while in the region occupying the brain matter, c = cb.
i
O),
[C.j,(R,fl+D.n.(R,i)]-&P.(cos
“=O
(3)
(6) where A,, B,, C, and D, are arbitrary constants, j, and n, denote spherical Bessel functions of the first and second kinds respectively, and P, denotes the Legendre polynomial. FIRST MODEL
Here we consider the skull to consist of three different concentric layers: the outer table, the diplo and the inner table. In the following, these will be respectively referred to as layers 1, 2 and 3. The potential functions for the three different layers of the skull and the brain are taken as
6” = 1 [A,,i,(R,,rJ n=O
+ B,,,n,,(K2,~]P,(cos 0).
A free-vibration
6’2 = 1 [dn2jn(KlfJ +
B,,rl,(R,,r~]
analysis
for ihe human
sranial
s>stcm
P, (cos U).
“=”
Fig. 1. Schematic
x
@-‘s = n
[.-ln3j,,(K-,,rJ +
B,,n,(K,,rl]
P, (cos U),
=o
representation of the la)ercd the cranium. First model.
1
E,,j,(R3,r~Pn(cns 0).
A, +iAz
Y* =
1 F,,jn(K,hr)~p,,lcos 0).
n =o
r = r.
(upper surface of the outer table), (8)
The matching conditions for the interfaces of the various layers are : On r = riz (interface between the diplo6 and the outer table).
r,,
(interface
between
can be determined
A, = 0 and A? = 0.
112) from
(13)
Here we propose a mathematical model which is suitable for taking into account the mechanical influence of the scalp. Although the scalp material is also viscoelastic. only its linear elastic behaviour will be studied here. In this part of our study, the skull will be treated as homogeneous but viscoelasticity of both the skull and the brain will be taken into account as in the first model. The quantities &, and l$,,,representing the clrcumferential strains in the scalp and the skull respectiveI;<. in a state of plane strain. may be expressed as
I: ;,,, =
On r = diploe).
= 0.
Thus the frequency spectrum the pair of equations
a”’ r, = 0 = (7;;‘.
1,2,3.-l).
(7)
in which the superscript si denotes the ith layer (i = I, 2, 3) of the skull, while the superscript b refers to quantities for the brain material. The function n,,(kr) being unbounded at r = 0, such a function cannot appear in 6* and Y *. A,,i, B,,i, C,,, Dmi(i = 1, 2, 3) and E,. F,, are arbitrary functions of rl; R,,, KZi are the values of K, and Rz in the ith layer of the skull. The boundary conditions are taken as follows: On
=
) denotes the derivative where i = J( - I) and S:( determinant can be split up into real and imaginar? parts in the form
=o
n
Iof
the frequency equation, The elements of the determinant are all complex. On the assumption of a small loss tangent for viscoelastic materials. \ve can write s,(Kjr) = s,,(K;r - iK;r) + ,Y,(K;r) - iK;(r)S,(Kjr)(j
@ =
structure
the inner table and the
1 F,, E' /I
~
III)
and 1: i,”
On r table).
=
r,
(interface
between
the brain and the inner
By using the fourteen boundary conditions listed in equations (8)-(1 I ). together with the matching conditions. one gets a set of fourteen linear homogeneous equations (as shown in Appendix A) involving fourteen unknowns. viz. A,,. B,,. Cnj. Dnj (j = 1.2. 3). E, and F,. The coefficient determinant when equated to zero gives
= ;
[G -
Yrk7:, + GybJ]
(15)
where F,, and II stand respectively for the tangential force per unit length and the thickness of the scalp. Here and in the sequel, the superscripts c, s and b refer to quantities for the scalp. the skull and the brain respectively; I’ and E denote Poisson’s ratio and Young’s modulus. For the continuity of the circumferential strain on the scalpskull interface defined by r = ro, equating ( 14) and (I 5) we find
F,, = g
Lo;,,,- \,s(G:,+ G&)1
SASTTABRATA CH.AKRAL.ART\I
J. C. &~ISRAand
638 This expression - r,a:,. yields
of FU when
further
equated
0--s r, = T(G:,,~- ~‘8:~) on i = F. (lvritten
in terms of dimensionless
VUhlERICAL
to
(16)
RECLTS
ASD
DISCL’SSIOS
Writing ((t = w, + ic!~? (w, denotes the actual frequency and CI): the attenuation factor). the frequency equations for both the models are expressed in the form
quantities) A, (IU,. w2) + i A2 (co]. w2) = 0. (i =
ivith
(17)
For the validity of this equation.
one must have
A, (w,, w,) = 0 and AZ (u,. For the study of the present model of the cranial vault, equation (16) together with equation (I 7) serves as the mechanical boundary conditions on the scalp-skull interface, while the conditions on the skull-brain interface. viz. the continuity of the radial displacement and the radial stress, can be put in the form 6’
=
lb
and a:, = #;f, on r = <.
(18)
Applying the boundary conditions, we get a set of six equations involving six unknowns A,, B,, C,, D,. E, and F, (say). When their coefficients are split into real and imaginary parts in the same way as indicated at the end of the preceding section, the resulting equations read .-l,(cr’, , - ia;,)
+ B,(rr;2
- iu’;*) -
C,((t;3
-
iOl;j) + D”(u;,
E,(u’,,
-
in’;,)
.&(uk,
- ia:,)
+ B,(n;,
-
+ Bn(uk2
- in;,)
+ C,(u;, A,(u;,
-
ia’;,)
= 0
+ B,(n;,
F, (a&
-
iu’;, = 0
iu’;z) = 0
-
ia;,)
+ Dn(u;,
- ia;,)
= 0
- i~‘;~)
- iaZl) + B,(u’,,
,!?,(a;, - iu;s) = 0
-
iu&)
+ C,(u&
-
+ D,(rz&
-
ia;,)
- E,(u’,,
- in:,)
-
-
itr &) = 0
Fn(rr&
ia&)
(19)
where the derived expressions for the quantities uij and ai; are shown in Appendix B. Hence the frequency equation in this case is of the form
‘J L I
t **
lUIL
-
,
u;,
-
lN12
0
0 iu’;,
I, (14I - 1fl4, ,, “5 I - Uis, I, 0,. - ‘(1.1
a;,
u>, - iu&
-
0
,I (‘44 - ‘a.&.$
0 u63
The frequency-dependent values of the shear modulus G,* were taken to be the same as in Misra (1978). For scalp, EC = 34.5 x lo6 Pa and h = 6mm (cf. Khalil and Hubbard, 1977). Further we took r. = 83 mm and ri = 71 mm for a human-sized head. In Figs 3-6 the first three roots of the aforementioned frequency equations are exhibited for the first two modes (n = 1, 2) in each case. The frequency spectrum for a human-sized freely vibrating viscoelastic spherical shell filled with a viscoelastic fluid (as a better representative of the human head) has been exhibited in Fig. 7. The corresponding frequency spectrum for an elastic spherical shell (having the same dimension as that of the aforementioned) filled with an inviscid irrotational fluid, studied by Engin and Liu (1970). has been reproduced in Fig. 8. In both Figs 7 and 8 the first nine modes (II = I, 2, 3,. . .9) are taken
0
I I, (143 - 1043 I
0.
it = 0.207 x 10LoN/m2, pb = 0.9774 x IO3 kg/m’.
(I;~ - ia’;
0
- kc’;,
,, “12 - IN?1 I, 052 - IN5L ,, - 1l162
=
If A, = 0 and AZ = 0 are plotted on a wI versus Q2 plane, the coordinates of the points of intersections of the curves represented by A, = 0 and AZ = 0 will give the complex frequencies. The plots for A, = 0 and AZ = 0 on the W,-OJ~ plane can be made by first fixing (?Jz and then finding an w, that satisfies the transcendental equation A, = 0 or AZ = 0 by trial-and-error method. The locations of the first few roots can be graphically determined. In order to be able to compare the results with those reported by Engin and Liu (l970), CJ, and 012 were replaced by Q, = clw,lc, and R, = urozfcb (c,, denotes the compressional wave speed in brain. its value being taken to be 398.78m/s, cf. Engin. 1969). For the purpose of the computational work. assuming the homogeneity of the skull material, the following values of the material constants and other parameters were taken (cf. Misra, 1978).
‘a63
(‘64
-
#I
l”64
-(a’,,
-
-
(46
0
0
0
0
- (4, - ia’;,)
0 I,
0
- ((1;s - iuy5)
0
0
I,
‘1 12 -
C:JJ
i.*’ = (27.6 + 1’0.37) x IO* N/m2 = GtS, p” = 2.0844 x I O3 kg/m3 ;
-
.4(4,
ia;‘,)
- iu’;&) -
.A(41
-
J - 1).
- iu&)
ia’;,)
i =o.
0
- (a& - in&j)
’
(20)
63’1
A free-vibration .malksls for the human cranIaI system
small head (sa). head of an experimental animal) h) taking r,, = 34.30 mm and ri = 31.75 mm and be using the second model of our study. The first three roo[s (non-dimensional) of the frequencv equation. in this case. are found to be 1.55. 5.45 and 7.90 for II = I.
-5colP .Skull .0rc,n
COUCLL Dl\<; RE\I-\RES
Although in the analysis for the first model, the three different layers of the skull were paid duz attention. owing to non-:~v~~ilabilit~ of all the requiGre ekperFig. 7.
Schematic representation of the layered structure the cranium. Second model.
of
imental
data
for
for consideration. These two figures considered together have a strong potential to signify as well as to quantitate the effect of the experimentally established damping material behaviour of both the skull and the brain on the frequency spectrum of the freely vibrating cranial vault. For comparing the values for the two models considered here with the corresponding values obtained by Engin and Liu (1970) (who neglected the dissipative behaviour of the skull and the brain), a table is provided below (Table I ). It can now be readily concluded that the dissipative material behaviour of both the skull and the brain as well as the mechanical influence of the scalp affect the frequencies of the vibrating cranial system quite significantly. We also computed the frequency values for a
3
individual
layers.
I
5
7
9
II
13
n7 Fig. 3. Plots of the curves A t = 0 and A,
the
com-
rltl,tout~rtclenrrrr-The authors are highly grarsful to the reviewers for their valuable comments and suggesrlons. the second author further acknowledges the assistance provided by CSIR.
I
1
the
putational work was based on available informations of the average values for the skull parameters. The effect of the influence of the scalp on the frequencies of vibration can be estimated from the values sit-en in the first and the second columns ofTable 1. while the effect of the skull and brain damping can be read from the first and the third columns of the same table. It is worthwhile to mention here that the results presented above, being based on an exact method of solution, possesses a potential to provide an analytical check for finite element modelers on the soundness of their numerical program. Moreover the present stud, represents a much improved version of the problem considered by Engin and Lit] (1970).
= 0 in the R, R, plane In =
1). First
model
__--
(- - _ ---_
---___
\ !i
2
!i I
i!
$2
i/
!i
(I
2
-z
b4
t i
J. C. M~SRA and SASTABRATA CHAKRAV.ART>
1
z
3
Fig. 8. Frequency
spectrum
4 5 Mode number of fluid-filled
REFERENCES
8, 275-284.
Engin, A. E. (1969)The axisymmetric response of a fluid-filled spherical shell to a local radial impulse-a model for head injury. J. Biomechakx 2, 325-341. Engin, A. E. and Liu, Y. K. (1970) Axisymmetric response of a fluid-filled spherical shell in free vibrations. J. Biontecltanics 3, 11-22. Engin, A. E. and Roberts, V. L. (1971) A mathematical model to determine the brain damage when the human head is subjected to impulsive loads. Proc. Symp. on Biodynamic Models and Their Applications. AMRL-TR-71-29, pp. 877-903. Evans, F. G. (1957) Stress and struin itI bones, p. 149. Springfield, Illinois.
Table Mode n
spherical
1. Values for the non-dimensional
First model
7
8
9
shell (cf. Engin and Liu, 1970).
Galford, J. E. and McElhaney, J. H. (1970) A visccelastic study of scalp, brain and dura. J. Biontechunics 3,21 l-222. Fliigge, W. (1967) Viscoehsricity. Blaisdell, Waltham, MA. Goldsmith, W. (1972) Biomechanics of head injury. In: Biotnechurtics. irs Fountl~rions tmd Objecriws, (Edi ted by Fung, Y. C., Perrone, N. and Anliker, M.), pp. 585-634. Prentice Hall, Englewood Cliffs, NJ. Goldsmith, W. (1979) Some aspects of head and neck injury and protection. Proyress it: Biornechmics; Proc. NATO Advanced Study Institute. Noordhoff, Amsterdam. Hickling. R. and Wenner, M. (1973) Mathematical model of the head subjected to an axisymmetric impact. J. Biomechanics 6, 115-132. Jamison, C. E., Marangoni. R. D. and Glaser, A. A. (1968) Viscoelastic properties of soft tissues by discrete model charactization. J. Biomechmics 1, 33-46. Khalil, B. T. and Hubbard (1977) Parametric study of head
Akkas, N. (1975) Dynamic analysis of a fluid-filled spherical sandwich shell - a model of the human head. J. Biomechanics
6 n -
Second
model
frequencies
(Q,
1
Values of Engin and Liu (1970)
1
1.30 2.65 3.90
2.02 4.30 5.30
0.00 2.50 4.40
2
1.09 2.06 4.65
2.20 3.10 3.90
1.00 3.90 5.50
643
Ihe human cramal system Xlerchunt. anJl>jis
H. C. 2nd Crispino. A. J. (19731 A dynamic of an elastic model of the human head. J.
B~~~rw
_‘Y?-311. &lisr,~. 1. C hl;lrangoni. R. D.. Saez. C. A.. Wcycl. D. A. and Polosk~. R. A. I 1978)Impact stresses in human head-neck model. J. Em/a/ .\/w/I. Div. 104, IS?- 176. XIcEihaney. J. H. (1966) Dyumic response of bone and muscle tissue. J. crppl./‘/I) sird 20, 1231- 1136. Xlc Elhaney. J. H.. Fogle, J. L.. Mehin. J. W., Haynes. R. R.. Roberts. V L. and Alrn. N. hf. (1970) Klechanical properta of cranial bone. J. Birl,ttt,c,lrlr,~ics3. 495 -5 12. XlcElhaney. J. H.. Koberts. V. L. and Hilyard. J. (19761 RI,J)~I~‘(./I‘/)I~(.~ ~$Tr~~nrtr. Duke University Press, Durham.
(197831 Axlsymmetric
0. (19771 Effect of Ra. C~~~IL~I 4.
vibration
of human
skull-brain s\stcm. /!t!q. .+lr&. 47, 1 I-IV. Xl1sra. J C Il$iSb) Stress in a human skull dus to pulsr Iwdlny. /!I+ .4w/i. 17. 339-337. >lisr;t. 1 C. (1976~1 Response of a human head tL>.m Impact. .\/n/. Lilr Sci. Erkj,zy 4, l12- 152. Roberts. i’.. Hodpson, V. and Thomas. L. ht. 11967) Fluid pressure gradients caused b> impact to the humm skull. Biomechanics \lonograph. pp. 223-225. ASLIE. ?!e\t York. Talhouni. 0. ,nd Dimawio. F. ( 1975 I Dvnamic response of :I lluid tilled sphoroida? shell ~ an improved model for stud! ins head injury. J. Bi,,,,r~,~,h~rnic,s 8, 219-228. Terry. C. T. and Roberts. V. L. Il968) A viscoelaws model af the hum;m spins subjected to Gz accelerations. J. Bi<,wc~~/~lo~i[~r 1. 161 168.
u’;, = (i:
+ ZG3)[R;,f?‘,,,
‘2n;(R;,Fi)
+ r;:R;,n;‘(R’*,<)j]
+ yR;,jn.(R;,i.) 1
+