A MODEL
OF BRAIN SHEAR UNDER TORSIONAL LOADS*+
IMPULSIVE
K. K. FIROOZBAKHSHS and C. N. DE.%LVA$~ Abstract
This investigation is concerned uith the theoretical determination of the behavior of the bram ~rhcn the human head is sub.jected to torsional loadings. The mathematical model consists of a linear ~iscocla~tic sphere N hich is bounded b>; and bonded to a rigid spherical shell. Two problems of torsional \\ave propagation from the rigid shell mto the viscoelastic medium are solved in closed form. The first problem deals H ith dynamic response of the viscoelastic sphere when the angle of twist.of the outer rigid shell ;Ihout ~1Lcrtical axis is a given function of time. The second problem is to determine the motion of the shell and viscorlastic sphere when the former is acted upon bq a time dependent torque.
I. INTRODL CTION Mechanisms
of in.@! in the head have long been a subject of investigation and discussion. A clear and enlightening review of the hypotheses explaining brain trauma was given bj Goldsmith (1966) who also discussed mathematical modelling. Rather than attempt to summarize
this monograph
review, of Roberts
of Goldsmith.
or the
or ul. (1966). we
In this paper. we treat the brain and cerebrospinal fluid (CSF) as a sphere consisting of a homogeneous isotropic linear viscoelastic solid material with fading memory. This sphere is bounded by and bonded to a rigid spherical shell which models the skull. Two problems of torsional wave propagation in the viscoelastic medium (brain and CSF) are solved in closed form. In the first problem. the angle of twist of the outer shell (skull) about a vertical axis is a given function of time ; in the second problem. a specified time dependent torque acts on the skull and hence the angle of twist is unknown. For both loadings the dilatational response is neglected as compared with the distortional response. based on the experimental results mentioned in the previous paragraph. These problems are representative of the brain shear theory of Holbourn. In this regard we mention the simplified model of Hayashi (1970): two concentric rigid cylinders connected by an elastic layer with shear resistance. The steady state response of a linear viscoelastic sphere undergoing translational and rotational excitation wasexamined bychristensenand Gottenberg (1964). Valanis and Sun (1967) gave the axisymmetric transient response of a linear viscoelastic sphere. with Poisson’s ratio a constant, subjected to meridional surface loading. Lee and Advani (1970) solved the problem of an elastic sphere responding to a step acceleration about a diametrical axis and gave the extension to a viscoelastic sphere. It is shown that the result of Lee and Advani is a special case of the general solution presented in this paper.
will restrict ourselves to the rotational theory of brain trauma proposed by Holbourn (1943). Holbourn obtained photoelastic patterns in gelatine models to demonstrate the importance of shear stresses arising from brain rotation in the cranial cavity: He observed that the bulk modulus of the brain tissue was considerably larger than its shearing modulus and. as a consequence. reasoned that shearing effects dominated compressional effects in causing brain trauma. Some confirmation of this hypothesis was given by Gurdjian and Lissner (1961). A series of experiments by Ommaya (1966) and Ommaya et ~11.(1966) tended to confirm Holbourn’sconjectureabout the importance of the rotational components of the inertial loading even though the experiment also had translational components. Later Ommaya and Hirsch (1970) and Gennarelli cr rrl. (1971. 1971) were able to devise a special head holding device which constrained the head to move in such a way that either the rotational or translational loading could be minimized at choice. These experiments supported Holbourn’s hypothesis. The timeandrate dependent properties of brain. dura. and scalp have been studied b> several investigators. Here we mention Jamison cr trl. (1968). Fallenstein c’rLL/. (1969) and Estes and McElhaney (1970) who have 2. FORMULATION OF THE PROBLEM established the viscoelastic properties of brain material. Moreover. McElhaney cf L/I.(1968) indicate that the The displacement equations of motion for a viscobrain is relative]! incompressible. elastic medium read: C’Ui
* Rcc~l~rr~crl 12Sc/mw/w~197.1. + This work was supported in part b) the Foundation under Grant GK 31081. $ DepartmentofMechanical Engineering. sit!. Shiraz. Iran. \\ lnstitut fir Biomedizinischc Technik. to: iI .Iddresh 311 correspondcncc Mechanical Enyneering Sciences. Wayne College of Enginccrmg. Detroit. Michigan
National
(/. + /I) *
Science
Pahlavi UniverETH Ziirlch. Department of State University. 4820’. L.S.A
Ui.ji
+
P * ‘i.1,
=
I’i-r’.
(Ii
The initial conditions are: at t = 0: “2=0.
?r
(21
K. K. FROOZBAKHSH
66
The boundary conditions are: atr=h w(h, 6. f) = h sin f?@(r) v(h. 0, t) = u(h, 0, t) = 0
(3)
Or
and C. N. DE&VA
brain material indicate that the ratio of its bulk modulus K to its shear modulus n is at least of the order 10’. As a result, the dilatational response induced by the defined loadings can be neglected as compared with the distortional response. With this consideration. and in view of symmetry with respect to the angle 4. the static problem is defined by
atr=h
2 aus -_2v’cote=0 - ;I 80 r2
2~s
v2u=- 7 = T(t) + T,(t)
V2a’
+ f
g
- r2
w(h, 8. t) = h sin O@(r) Ljh, 0. t) = u(h, 8, t) = 0.
(4)
In equation (1). u, are the components of the displacement vector II, ( ).j denoted partial differentiation with respect to Xj and for convenience we define
ir
=
sin’ 8
0
WS
V=w’ -
1
r-
sin- tI
=
(9)
0.
with the boundary condition, w’(b. 0) = b sin 0 v”(b,0) = u”(b, 0) = 0
In equation (3) u. v and w denote the radial (r), meridional (0) and circumferential (4) displacements, respectively. and @(t) is the specified angle of twist of the rigid shell about the c-axis. In equation (4), I, is the mass moment of inertia of the spherical shell about the c-axis, T(t) is the applied torque, and T,(t) is the resisting torque applied to the shell by the viscoelastic sphere. Note that in this problem the angle @(t) is unknown and must be determined from equation (4). In order to solve the mathematical problem formulated by equations (f-4), we will use the superposition principle of Valanis (1966). Valanis had shown that the solution of a viscoelastic problem can be reduced to the solution of an elastic problem (which includes a static solution plus an eigen value solution) and an istegrodifferential equation of Volterra type involving time only. It is easy to show that for a specified displacement boundary condition of the form uxx, t) = u#+(x)@t).
where
v&f%
II
( > ar
r2 sin
* (11)
d; >
a4r, 8) = h,(r) + $ f&(r)P&os 0) k'=l
fir. 0) =
f&(r) sin 0 P&OS
2
e)
k=I
wqr, e) = k$, JIkbY%cOs
(12)
e)
where P&OS 0) is the Legendre polynomial of the first kind and of order k, prime indicates differentiation with respect to the argument cos 0. and P,‘(cos 0) designates the associated Legendre function of the first kind of order k and degree one. Substituting (12) in (9) yields:
d'f,, dfo+ 2.r z
r2 -
dr2
(7) r’ ‘2
+ 2r ‘$
r
-
I.2
2)fk-
(i.k +
z d”+l, ~+2rz
Ml,
-
&it
2f0
= 0
24$.,
= 0
-
= 0
2f,
d’ll/l,
03)
where H(t) is the Heaviside unit step function. Thus, to get the solution to the proposed problems we first solve forrry’and thenuse theinverseofequation(7). Following the technique ofvalanis. we proceed with the analysis of the static problem.
a - sin 8 e de (
The solution of the system (9) is taken in the following series form
(6)
where s is the Laplace transform parameter, f denotes the Laplace transform off with respect to time, and uy’ is defined to be the displacement solution corresponding to a specified displacement boundary condition of the form LIAXt) = ui(x)H(t).
I
).‘C + -
the displacement solution is given by tj.I = rj!C’$ I -
(10)
+ 2r d’t+, - &I), = 0 &2 dr
2, = k(k + 1); k = 1.2....
(13)
Solutions to ( 13) are found to be
fdr) =
A,,(r/b)
+ B&/h)-
z
lLk(r)=
&r/b)”
+ D,(r/b)_‘-
fk(r)
=
-kAdr/b~-
=
A,,(r/bF-
f
3. ANALYSIS
(i) Atlalysis
of the static
problem
Published values by McElhaney et al. (1968) and Fallenstein et ul. (1969) for the physical properties of
ddr)
k Z 1,2.....
’ + (k + l)Ck(r/b)L’ I
1 + C&,‘b)‘+ 1 (14)
67
A model of brain shear under impulstve torsional loads
The coefficients .4,,. B,, . A,. BP. C, and Dk are integration constants to be determined by the static boundary conditions. Further. thecoefficientsB,,and D,arechosen to be zero because of the requirement of boundedness at I’ = 0. Equations (12) now read: c/71’.(1)= A,,(r b) + i
[ -kA,(r/h)‘-
and the initial condition $ 2&l” = u;. I= I Further.f,(t)
(23)
satisfies the integrodifferential equation
l
(24)
A.= 1
+ (I, + l)C&ihY’ ‘]P,(cos I P(1..0) = x [A&0$’
with the initial conditions
0)
f”(0) = 1; d+ + C,(r,‘b)‘+ ‘1
I= 1
(25)
We take the solutions to (21) in the form:
sin tlP;(cos 0)
u’“‘(r, 0) = T”(r)P”(cos 0)
7
M-Yr..ff) = 1 B&r %)‘lPi(cos 0).
r’“)(r.0) = $.(r)Y$os
(15)
I= 1
0) sin II
M”“)(T.0) = ~.(r)P~(cos 0)
In order to satisfv the static boundary condition (IO). equations ( 15) imply
n = 1.2.....
(26)
where the unknown functionsJ,,(r). $&r) and @. must satisfy
.4k= -c, 1
.-In = +
i -I
d’_?,, + 2r
P,,(cos O)P,(COS@d(COS0) Ii=
B = (2/i + I )b !.
_ = 0.
1-0
Nk+l)
I
’
_,
r2 dr?
dfH dr
.
+
(
, Qj# r- 7 -
C;
;., -
2 7”
1.2
sin 0 P;(cos O)d(cos 0)
which yields .4,, = 0: A,=
B,
-CA=0
B, = 0
=
-b k=l,2....
.j.” = n(n + 1); n = 1,2,. .
!i = 2.3. .
.
(16)
Thus the static solution is fP(r. 0) = ryr. 0) = 0 w’(r, 0) = r sin 0.
C,,is the propagation speed of shear waves in an elastic medium with shear modulus h and density p. i.e. c
(17)
(ii) Anulysis qf the reduced problems
(27)
_
.,-
“2 ,~ i‘) 3
(28) \ I’/ The bounded solutions to (27) can be simplified to read, following Arutiunian and Babloian (1965):
The reduced problem is defined by Lianis as:
uR=O
at
r=b
+B,n(n+ 1)r-32Jn+,2
(18) &r)
G(r) is the shear relaxation function given by
=
?,rm3
2J,+,
0” -r ( C,
i
2
/1(r) = p,,G(r). /I,, = constant. The reduced solution has the form: $(r. 0. t) = z: Z,uj”‘(r. U)“r”(f). n where the orthonormal
(20)
n= 1.2.... (29) where we have used the definition of the spherical Bessel function of the first kind
eigenfunctions uj”) satisfy
/l,, u;“:A+ [I!&]“’ = 0.
n = 1, 2. .
(21) given by
with the homogeneous boundary condition rr;“‘(b.0) = 0.
68
K. K. FIR~~ZBAKHSH and C. N. DESILVA The only undetermined coefficient ,I. will be found later by satisfying initial condition (23).
Equations (26) now read U(nV..O)= { -+-3;‘J.+,;*(~r)
iii Summary of the solutions Having found the static and reduced solutions, one can write in the manner of Lianis (1966) wlc)(r,0, t) = ws - wR = r sin 0 l.wf(r, R,)Pf(cos @f,(t).
- d,
(n + l)r-3;‘J,+I,Z
Qn >I rr
(
(39)
As a consequence of (7). the transform of the solution w(r. 0, t) of the system (l-3) is: . ii@. 0, s) = r sin 0 i(s)
>
-
G&s)
i
x=1
sin 8
&w.R(r. 0.) f,(s)PPf(cos 0).
(40)
The resulting transform of the stress distribution is (41)
(31) ?., b, and E, are constants of integration to be determined from the boundary conditions (22). This will imply for n = 1.2,. . . .
where we have used (24 and 25). Further. it is easy to show &) = & = 0
n = 1, L.. .
(42)
Pf(cos 8)
(32) n=
l$ (33)
1,2....
(43)
= -frm3’*Jn+ ,,* + i,, P,(cos e)]
E., = n(n + I);
(34) Since& + O.equation(34)yieldsthefrequencyequation =o
J n+l?
n = 1,2,....
[2 cot 0 PA(cos 0)
n = 1,2,...
thus tT,&r,8, s) = -s2G$po f
n
7
“=,s+R;G
(35)
C (n-
1)
Since no two Bessel functions whose orders are the halves of odd integers can have a positive root in common, we can conclude from (35) n=
J n+3 2
1,2,....
(36)
The inequation (36) implies that the coefficients ?” and B. in (32 and 33) are both zero. Thus, the only nonvanishing component of the displacement solution of the reduced problem will be
w%.u,t) =
i
II=t
Zp.R(r, R,)P,‘(cos@f-“(t)
(37)
Hhere ~.f(r., Q,) = r- ’ ‘J,, , ?
n=lT .*....
(38)
(45) d,(r, 0. s) = S’Ci&
A=
t nr .,,s+RfC
- 3‘2
R” J “+I 2 F, r [2 cot 0 Pi(cos 0) ( h >
+ i., P,(cos O)].
(W
For each n. the frequency equation (35) possesses an infinity of roots Q, and. therefore. the complete solution to the torsional wave oroblem is slinhtiv
69
A model of brain shear under impulsive torsional loads modified
where we have dropped the subscript I in equation (56’ and any subsequent equations. Substitution of(56’ into the Laplace transform of (4), yields
to read:
Mr. 0. r) = r sin O@(r)- x C ~nmwQr.R,,) n “1
J 52 +
(
G”Jl..R,,)P;(cos 0)
(48)
where for n. 1’7= 1. _, 3 bv~Jr,flm)=
5-
$(s) = 2’.
(57)
4. COMMENTS
(49)
.
(50)
r-llJn+,l
h
In equation (57) the only unknown @(r) can be determined when the material property. G(r). and the applied torque, r(t). are specified.
W”Jl.. Q”,‘[2 cot 0 P;(cos 8) + i, P,(cos O,]
\!I
It is easy to show that our general solution. (47-52). of the problem with boundary condition (3) contains the solution of Lee and Advani (1970) as a special case. Consider the angle of twist Q(t) given by: @(t) = &)PH(t):
4(t) = n,H(t’.
(58’
The displacement solution (47) for the case of elastic material reads-see Valanis (1968): (5’)
Em(r,R,,)
= r-3’2
J,,
(52)
1,2
Finally, the coefficient relation
z.,
is determined
from me
’ c%?,,(r.!&,bS(r,
ii-W,0, s) = r sin H(s)
+ sin fI g 2,wE(r. Q m) .F(-&ps’. RI=,
B)p,‘(cos B)]r’
(59’
sin f3dr d0 (53’ /
Jo Jo
w’a(r. %,,)w!,,,@.?,,) [P&OS 0’1’ r2 sin Bdr dr3
which is a consequence of equation (23) and mode normalization condition. Equation (53) after integration and simplification yields
The inverse of (59) upon the use of (58) becomes
.Y., = On+ l;m = 1,2.. . .
.2,m=
!%n !
2C,,b’ 2
)
Q,mJ’s ‘2 7
\
111 =
1. 2. . .
(54)
tJ
The prime denotes differentiation with respect to ([(R,,)/C,Jr) . We conclude this section by evaluating the twistingangle@(t),whichis,asmentionedpreviously, unknown for the second problem. In (4), the applied torque T(r) is a given function and the resisting torque r,(t) can be expressed in terms of displacement by n2 T,(t) = 4nh-’ ~(h. 0. r) sin’ 0 do. (55) J0 Introduction of equation (48) into (55), and use of equations (51. 54 and 35’ yields
I
.cLI
Bfi2J; Gn.)
(1 - cosR,t)
. (60)
Imposing the condition (58) and noting that the displacement solution. ul. introduced by Lee and Advani (1970) is given in a rotating coordinate system, it is easy to show ic - ii, = rr,, sin e - 26 AA@.
(6”
Lee and Advani were only concerned with the transient response and the term &r’u~ is considered to be negligible as compared with rxr, sin 0 for small displacements. Thus. neglecting rd r’rr, in (61’ we obtain: ,
7.n
,^
70
K. K. FMXGBAKH~H and C. N. DESILVA
and from equation (60) u$.(r.0. t) = -
““‘l;:”
i(l
_
r-2)
[
.
m
+ 2q2/np2
Qm
( >
I’ c,’
1 1 jyf,,,(jj,) .m=
*mf (63) 1
‘OS
7
where F = r/a and we have satisfied the initial condition u&, 0, 0) = 0. Equation (63) is the result of Lee and Advani. We have solved in closed form torsional wave propagation from a rigid shell into the interior of a linear viscoelastic sphere with boundary conditions given either by equations (3) or (4). The general expressions for the displacement and stress fields are ‘derived. It is now straighrforward to obtain the shearing effects for specific cases of the loadings given by equations (3 and 4). S. NUMERICAL
RESULTS
The general expressions for stresses developed can now be evaluated, using material properties which most closely represent brain tissue and a representative head rotation. The pertinent data used in modeling the brain are listed in Hickling and Wenner (1973): h = 3G in. p = 9.38 x 10-s lb s&/in’ p = 32.5 lb/in2 z,, = (1 - 100) x lo* rad/sec2. We first consider the simple case of an elastic model. The only non-vanishing component of shear stress (48),
Fig. I. The coordinate system used to describe the model.
reduces to
(1 - cos/!Im?) (64) where we have assumed qf) = fpaOtiH(t):Figures 2 and 3 represent the shear stresses versus time and radius respectively. Although evaluation of brain injury is beyond the scope of an elastic model without coupling of physiological system variables. the rapid buildup of shear stresses indicate that shear stress waves are a possible mechanism of brain injury. This maximum shear stress is induced in the neighborhood of the outer layer of the brain close to the inner surface of skull and occurs at 2.6 and I@8 msec after the impact. We have already referred in the Introduction to the investigators who have established the viscoelastic properties of the brain. For our purpose, we will use the particular data developed by Shuck and Haynes (1970) and Shuck andAdvani( 1972)Theseauthorsmeasured valuesofthe complex impedance of human brain tissue and fitted a Kelvin model having q, = 1, = OQO5psi set; q. = E,
0.2 t 0.16-
Fig. 2. Shear stress vs time, 0 = n/2. elastic solution.
-0
04
I 0.2
0
I 0.4
I D6
I 0.8
F
Fig. 3. Shear stress vs radius.
U = n/2. elastic solution. T,
10-6 2.8
dIzF
-
0
05
I.0
15
7
Fig. 4. Shear stress vs time, 0 = n/2. Kelvin model.
P
Fig. 5. Shear stress vs radius.
f? = n/2. Kelvin model.
I I.0
72
K. K. FIR(K)ZBAKHSH and C. N. DESILVA
. For this model equation (48) reduces to:
[A,
+ AzemX*’
+ .43e-s1L
+ A+e-““I
(66)
where A,
=
A2 = 7 -0.2
0.2
0
0.4
0.6
0.8
10
I.2
I.4
A, 1.6
A4 =
7 Fig.
6. Shear
stress
vs time. 0 = n/2. model.
Maxwell-Kelvin
2psi(as indicated in Fig. 4) to the experimental data. Employing G(t) = ~7”+ qrS(t) for this case, equation (48) reduces to: =
-AI
(emAt - ebBI) + qn
B(A -1B)11
The resulting shear stresses versus time and versus radius are shown in Figs. 4 and 5 respectively. These figures again show that the maximum shearing stress. although highly damped is induced in the vicinity of the outer layer of brain shortly after impact. Finally a Maxwell-Kelvin or four-parameter fluid mode) proposed by Galford and McElhaney (1970) with the following data has been considered (see Fig. 6).
In
El = 3.4 psi;
E2 =
‘I, = 275.4 psi set;
‘I? = 86 psi sec.
9.3 psi;
this case G(r) = =+@= I
CYl- VW”
- (4, - s42k-"'l
2
where
p1 = $$ = 70 sec2 I ? 4, = ql = 257.4 psi set = 238 psi si& 2 ct = 1.48; fi = 0.01.
2.4371, + 3.665
-x,
+ x2)( -.Y,
+ .u,)
-2.437.~~ + 3.665 -x2(
-.x2
+ .u,)( -x2
-‘437x, _X,(_.Kj
+ + x*)(-x,
+ .K,)
3.665 + x1)
and x, , x1, x3 are the roots of .x’ + 1.49~’ + CO.0148- 2.437R;].u - 3.665Q; = 0. The shear stress curve vs time is given by Fig. 6 and reconfirms the conclusions drawn from the previous two cases. As a final comment. we point out that equation (64) shows that two different sized brains of the same material and in the same condition experience shear stresses proportional to the square of the brain radius. The square of the brain radius is proportional to the 2/3 power of the’brain mass and consequently we have established the “Scaling law” proposed by Holbourn (1943) for an elastic brain. Equations (65 and 66) show, however, that this is not the case for a viscoelastic model of the brain. A similar result stemming from a very different approach was obtained by Bycroft ( 1973).
REFERENCES Arutiunian. N. K. H. and Babloian. A. i\. (1965) Two dynamical contact problems for an elastic sphere. J. ,lppl. Mtrrh. Mrch. 29. 6X-627. Bycroft. G. N. (1973) Mathematicul model of a head subjected to an angular acceleration. J. Biotmvhtrr~ics 6. 487-495. Christensen. R. and Gottenberg. W. (1964) The dynamic response ofa solid viscoelastic sphere to translational and rotational excitation. J. .-lppl. ,Ifcc/~. 31. 373. 278. Estes. M. S. and McElhaney. J. H. (1970) Rcsponsc of brain tissue to compressive loading. ASME Paper No. 70-BHF-13. Fallenstein. G. T.. Huelke. V. D. and Melvin. J. W. (1969) Dynamic mechanical properties of human brain tissue.
J. Biomvhctrric.s
y2 = y’
-.Y,(
(65)
_ _eeEr where
=
3.665
.Y, x2x,
2. 2 17-226.
Galford. J. E. and McElhaney. J. H. (1970) A viscoelastic study of scalp. brain and dura. J. Eio~nrclrtrrrics 3.11 I-21 i. Genndreih. T.. Thibault. L. and Ommaya. A. K. (1971) Compression of translational and rotational head motions in rxperimcntul cerebral concussion. Plr>c. (!I’ I .%/ISrupp CcrrCrd1 Cor@?rlcc~. pp. 797 803. Gennarelii, T.. Thibault. L. and Ommaya. A. K. (1972) f’athophysiologicresponsestorotational and translational acceleration of the head. Pi-oc. qf 16th Sttrpp Ctrr CIW.X/I CotEfiwrtcc. pp. 296308. Goldsmith. W.( 1966)The physical processes producinp head injury. Hrad 1tljur.r Co+~rr~cr Procdi~rys (Edited by
A model of brain shear under impulsive Cavcness. W. F. and Walker. A. E.). pp. 3.50-382. Lippincott. Philadelphia. Gurdjlan. E. W. and Lissner. H. R. (1961) Photoelastic confirmation of the presence of shear strains at the craniospinal junction in closed head injury. J. Nruroaur~. 18. 58-60. Hayashi. T. I 1970) Brain shear theory of head injury due to rotational impact. J. fuc,ultJ EII(/II(/. I’rlir. Tok!fi 30, 307-313. Hickling. R. and Wenner. M. (1973) Mathematical model of a head subJected to an axisymmetric impact. J. Bio,llrchrrrlic,.s 6, I l5- 132. Holbourn. .4. H. E. ( 1943) Mechanics of head injuries. Lurwt 2. 438 441. Jamison. C. E.. Marangoni. R. D. and Glaser. L. (1968) Viscoelastic properties of soft tissue b! discrete model characterization. .I. Biorwchunics 1. 33- 46. Lee. Y. C. and Advani. S. H. (1970)Transient response of a sphere to torsional loading-A head injury model. J. Afurh. Biosci.6. 473-486. McElhaney. J. H.. Rose. L. S. and Estes. M. S. (1968) Dynamic mechanical propertiesofhuman scalp and brain. Biomechanics Laboratories. West Virginia Llniversity. Progress Report. Morgantown. West Virginia. Ommaya. A. K. (1966) Experimental head injury in the monkr?. Neud I~ljur~~CO~I@UC~CPm-wdinqs. pp. 2f%-275. Ommaya. A. K. and Hirsch. A. E. (1970) Protection from brain injury: The relative significance of translational and rotational motions of the head after impact. Proc. of l4rh Sltrpp Cur Crush CO+I~f~CC, pp. 144- I5 I. Ommaya. A. K.. Hirsch. A. E. and Martinez. J. L. (1966) The role of the whiplash in cerebral concussion. Proc. C$ 10th .Qtr\,/ (‘(II. Crtrsh Cor!tc~,‘c,rIcr. pp. 3 14-324. Roberts. V. L.. Stech. E. L. and Terry. C. T. 11966) Review ofmathematical modelswhich de&be human response to acceleration. ASME Paper No. 66WA ‘BH.Fl3. Shuck. L. Z. and Advani. S. H. 11972) Rheological response of human bram tissue. ASME Paper No. 72,‘WABHF-2. Shuck. L. Z. and Havnes. R. R. (1970) Determination of viscoelastic prop&es of human brain tissue. ASME Paper No. 70-BHF-12. Valanis. K. C. (1966) Exact and variational solutions to a general vlscoelasto-kinetic problem. J. uppl. Me& 33. 88% 892. Valanis. K. C. and Sun. C. T. (1967) Axisymmetric wave propagation in a solid viscoelastic sphere. Inr. J. Engnq Sci. 5. 939. 956.
torsional
loads NOMENCLATl’RE
.A,,. B,,. 4,. C,. D,. T,. B,. E, constants h inner radius of skull elastic shear wave speed shear relaxation function Heaviside unit step function II-kassmoment inertia of skull about the z-axis spherical Bessel function of the first kind bulk modulus Legendrepolynomialofthefirst kindoforder A. associated Legendre function of the first kind of order X and degree one spherical coordinates nondlmensional Laplace time
transform
nondimensional
radial coordinate parameter
time
applied torque resisting torque radial (r). meridional (0) and circumferential (4) displacements. respective11 eigen functions constant angular acceleration nondimensional
frequency
parameters
shear modulus constant Lamt constant mass densit) angle of twist of skull about the r-axis Cartesian components of stress tensor shear stresses Cartesian components of strain tensor frequencies gradient operator in spherical coordinates
Stthscriprs
i. i. k Cartesian components of a vector or tensor. k. I?. 111non-neganvc Integer. Supw.wriprs S. R refer to the “Static” and “Reduced” problems respectivel!.
OF BRAIN SHEAR UNDER TORSIONAL LOADS*+
IMPULSIVE
K. K. FIROOZBAKHSHS and C. N. DE.%LVA$~ Abstract
This investigation is concerned uith the theoretical determination of the behavior of the bram ~rhcn the human head is sub.jected to torsional loadings. The mathematical model consists of a linear ~iscocla~tic sphere N hich is bounded b>; and bonded to a rigid spherical shell. Two problems of torsional \\ave propagation from the rigid shell mto the viscoelastic medium are solved in closed form. The first problem deals H ith dynamic response of the viscoelastic sphere when the angle of twist.of the outer rigid shell ;Ihout ~1Lcrtical axis is a given function of time. The second problem is to determine the motion of the shell and viscorlastic sphere when the former is acted upon bq a time dependent torque.
I. INTRODL CTION Mechanisms
of in.@! in the head have long been a subject of investigation and discussion. A clear and enlightening review of the hypotheses explaining brain trauma was given bj Goldsmith (1966) who also discussed mathematical modelling. Rather than attempt to summarize
this monograph
review, of Roberts
of Goldsmith.
or the
or ul. (1966). we
In this paper. we treat the brain and cerebrospinal fluid (CSF) as a sphere consisting of a homogeneous isotropic linear viscoelastic solid material with fading memory. This sphere is bounded by and bonded to a rigid spherical shell which models the skull. Two problems of torsional wave propagation in the viscoelastic medium (brain and CSF) are solved in closed form. In the first problem. the angle of twist of the outer shell (skull) about a vertical axis is a given function of time ; in the second problem. a specified time dependent torque acts on the skull and hence the angle of twist is unknown. For both loadings the dilatational response is neglected as compared with the distortional response. based on the experimental results mentioned in the previous paragraph. These problems are representative of the brain shear theory of Holbourn. In this regard we mention the simplified model of Hayashi (1970): two concentric rigid cylinders connected by an elastic layer with shear resistance. The steady state response of a linear viscoelastic sphere undergoing translational and rotational excitation wasexamined bychristensenand Gottenberg (1964). Valanis and Sun (1967) gave the axisymmetric transient response of a linear viscoelastic sphere. with Poisson’s ratio a constant, subjected to meridional surface loading. Lee and Advani (1970) solved the problem of an elastic sphere responding to a step acceleration about a diametrical axis and gave the extension to a viscoelastic sphere. It is shown that the result of Lee and Advani is a special case of the general solution presented in this paper.
will restrict ourselves to the rotational theory of brain trauma proposed by Holbourn (1943). Holbourn obtained photoelastic patterns in gelatine models to demonstrate the importance of shear stresses arising from brain rotation in the cranial cavity: He observed that the bulk modulus of the brain tissue was considerably larger than its shearing modulus and. as a consequence. reasoned that shearing effects dominated compressional effects in causing brain trauma. Some confirmation of this hypothesis was given by Gurdjian and Lissner (1961). A series of experiments by Ommaya (1966) and Ommaya et ~11.(1966) tended to confirm Holbourn’sconjectureabout the importance of the rotational components of the inertial loading even though the experiment also had translational components. Later Ommaya and Hirsch (1970) and Gennarelli cr rrl. (1971. 1971) were able to devise a special head holding device which constrained the head to move in such a way that either the rotational or translational loading could be minimized at choice. These experiments supported Holbourn’s hypothesis. The timeandrate dependent properties of brain. dura. and scalp have been studied b> several investigators. Here we mention Jamison cr trl. (1968). Fallenstein c’rLL/. (1969) and Estes and McElhaney (1970) who have 2. FORMULATION OF THE PROBLEM established the viscoelastic properties of brain material. Moreover. McElhaney cf L/I.(1968) indicate that the The displacement equations of motion for a viscobrain is relative]! incompressible. elastic medium read: C’Ui
* Rcc~l~rr~crl 12Sc/mw/w~197.1. + This work was supported in part b) the Foundation under Grant GK 31081. $ DepartmentofMechanical Engineering. sit!. Shiraz. Iran. \\ lnstitut fir Biomedizinischc Technik. to: iI .Iddresh 311 correspondcncc Mechanical Enyneering Sciences. Wayne College of Enginccrmg. Detroit. Michigan
National
(/. + /I) *
Science
Pahlavi UniverETH Ziirlch. Department of State University. 4820’. L.S.A
Ui.ji
+
P * ‘i.1,
=
I’i-r’.
(Ii
The initial conditions are: at t = 0: “2=0.
?r
(21
K. K. FROOZBAKHSH
66
The boundary conditions are: atr=h w(h, 6. f) = h sin f?@(r) v(h. 0, t) = u(h, 0, t) = 0
(3)
Or
and C. N. DE&VA
brain material indicate that the ratio of its bulk modulus K to its shear modulus n is at least of the order 10’. As a result, the dilatational response induced by the defined loadings can be neglected as compared with the distortional response. With this consideration. and in view of symmetry with respect to the angle 4. the static problem is defined by
atr=h
2 aus -_2v’cote=0 - ;I 80 r2
2~s
v2u=- 7 = T(t) + T,(t)
V2a’
+ f
g
- r2
w(h, 8. t) = h sin O@(r) Ljh, 0. t) = u(h, 8, t) = 0.
(4)
In equation (1). u, are the components of the displacement vector II, ( ).j denoted partial differentiation with respect to Xj and for convenience we define
ir
=
sin’ 8
0
WS
V=w’ -
1
r-
sin- tI
=
(9)
0.
with the boundary condition, w’(b. 0) = b sin 0 v”(b,0) = u”(b, 0) = 0
In equation (3) u. v and w denote the radial (r), meridional (0) and circumferential (4) displacements, respectively. and @(t) is the specified angle of twist of the rigid shell about the c-axis. In equation (4), I, is the mass moment of inertia of the spherical shell about the c-axis, T(t) is the applied torque, and T,(t) is the resisting torque applied to the shell by the viscoelastic sphere. Note that in this problem the angle @(t) is unknown and must be determined from equation (4). In order to solve the mathematical problem formulated by equations (f-4), we will use the superposition principle of Valanis (1966). Valanis had shown that the solution of a viscoelastic problem can be reduced to the solution of an elastic problem (which includes a static solution plus an eigen value solution) and an istegrodifferential equation of Volterra type involving time only. It is easy to show that for a specified displacement boundary condition of the form uxx, t) = u#+(x)@t).
where
v&f%
II
( > ar
r2 sin
* (11)
d; >
a4r, 8) = h,(r) + $ f&(r)P&os 0) k'=l
fir. 0) =
f&(r) sin 0 P&OS
2
e)
k=I
wqr, e) = k$, JIkbY%cOs
(12)
e)
where P&OS 0) is the Legendre polynomial of the first kind and of order k, prime indicates differentiation with respect to the argument cos 0. and P,‘(cos 0) designates the associated Legendre function of the first kind of order k and degree one. Substituting (12) in (9) yields:
d'f,, dfo+ 2.r z
r2 -
dr2
(7) r’ ‘2
+ 2r ‘$
r
-
I.2
2)fk-
(i.k +
z d”+l, ~+2rz
Ml,
-
&it
2f0
= 0
24$.,
= 0
-
= 0
2f,
d’ll/l,
03)
where H(t) is the Heaviside unit step function. Thus, to get the solution to the proposed problems we first solve forrry’and thenuse theinverseofequation(7). Following the technique ofvalanis. we proceed with the analysis of the static problem.
a - sin 8 e de (
The solution of the system (9) is taken in the following series form
(6)
where s is the Laplace transform parameter, f denotes the Laplace transform off with respect to time, and uy’ is defined to be the displacement solution corresponding to a specified displacement boundary condition of the form LIAXt) = ui(x)H(t).
I
).‘C + -
the displacement solution is given by tj.I = rj!C’$ I -
(10)
+ 2r d’t+, - &I), = 0 &2 dr
2, = k(k + 1); k = 1.2....
(13)
Solutions to ( 13) are found to be
fdr) =
A,,(r/b)
+ B&/h)-
z
lLk(r)=
&r/b)”
+ D,(r/b)_‘-
fk(r)
=
-kAdr/b~-
=
A,,(r/bF-
f
3. ANALYSIS
(i) Atlalysis
of the static
problem
Published values by McElhaney et al. (1968) and Fallenstein et ul. (1969) for the physical properties of
ddr)
k Z 1,2.....
’ + (k + l)Ck(r/b)L’ I
1 + C&,‘b)‘+ 1 (14)
67
A model of brain shear under impulstve torsional loads
The coefficients .4,,. B,, . A,. BP. C, and Dk are integration constants to be determined by the static boundary conditions. Further. thecoefficientsB,,and D,arechosen to be zero because of the requirement of boundedness at I’ = 0. Equations (12) now read: c/71’.(1)= A,,(r b) + i
[ -kA,(r/h)‘-
and the initial condition $ 2&l” = u;. I= I Further.f,(t)
(23)
satisfies the integrodifferential equation
l
(24)
A.= 1
+ (I, + l)C&ihY’ ‘]P,(cos I P(1..0) = x [A&0$’
with the initial conditions
0)
f”(0) = 1; d+ + C,(r,‘b)‘+ ‘1
I= 1
(25)
We take the solutions to (21) in the form:
sin tlP;(cos 0)
u’“‘(r, 0) = T”(r)P”(cos 0)
7
M-Yr..ff) = 1 B&r %)‘lPi(cos 0).
r’“)(r.0) = $.(r)Y$os
(15)
I= 1
0) sin II
M”“)(T.0) = ~.(r)P~(cos 0)
In order to satisfv the static boundary condition (IO). equations ( 15) imply
n = 1.2.....
(26)
where the unknown functionsJ,,(r). $&r) and @. must satisfy
.4k= -c, 1
.-In = +
i -I
d’_?,, + 2r
P,,(cos O)P,(COS@d(COS0) Ii=
B = (2/i + I )b !.
_ = 0.
1-0
Nk+l)
I
’
_,
r2 dr?
dfH dr
.
+
(
, Qj# r- 7 -
C;
;., -
2 7”
1.2
sin 0 P;(cos O)d(cos 0)
which yields .4,, = 0: A,=
B,
-CA=0
B, = 0
=
-b k=l,2....
.j.” = n(n + 1); n = 1,2,. .
!i = 2.3. .
.
(16)
Thus the static solution is fP(r. 0) = ryr. 0) = 0 w’(r, 0) = r sin 0.
C,,is the propagation speed of shear waves in an elastic medium with shear modulus h and density p. i.e. c
(17)
(ii) Anulysis qf the reduced problems
(27)
_
.,-
“2 ,~ i‘) 3
(28) \ I’/ The bounded solutions to (27) can be simplified to read, following Arutiunian and Babloian (1965):
The reduced problem is defined by Lianis as:
uR=O
at
r=b
+B,n(n+ 1)r-32Jn+,2
(18) &r)
G(r) is the shear relaxation function given by
=
?,rm3
2J,+,
0” -r ( C,
i
2
/1(r) = p,,G(r). /I,, = constant. The reduced solution has the form: $(r. 0. t) = z: Z,uj”‘(r. U)“r”(f). n where the orthonormal
(20)
n= 1.2.... (29) where we have used the definition of the spherical Bessel function of the first kind
eigenfunctions uj”) satisfy
/l,, u;“:A+ [I!&]“’ = 0.
n = 1, 2. .
(21) given by
with the homogeneous boundary condition rr;“‘(b.0) = 0.
68
K. K. FIR~~ZBAKHSH and C. N. DESILVA The only undetermined coefficient ,I. will be found later by satisfying initial condition (23).
Equations (26) now read U(nV..O)= { -+-3;‘J.+,;*(~r)
iii Summary of the solutions Having found the static and reduced solutions, one can write in the manner of Lianis (1966) wlc)(r,0, t) = ws - wR = r sin 0 l.wf(r, R,)Pf(cos @f,(t).
- d,
(n + l)r-3;‘J,+I,Z
Qn >I rr
(
(39)
As a consequence of (7). the transform of the solution w(r. 0, t) of the system (l-3) is: . ii@. 0, s) = r sin 0 i(s)
>
-
G&s)
i
x=1
sin 8
&w.R(r. 0.) f,(s)PPf(cos 0).
(40)
The resulting transform of the stress distribution is (41)
(31) ?., b, and E, are constants of integration to be determined from the boundary conditions (22). This will imply for n = 1.2,. . . .
where we have used (24 and 25). Further. it is easy to show &) = & = 0
n = 1, L.. .
(42)
Pf(cos 8)
(32) n=
l$ (33)
1,2....
(43)
= -frm3’*Jn+ ,,* + i,, P,(cos e)]
E., = n(n + I);
(34) Since& + O.equation(34)yieldsthefrequencyequation =o
J n+l?
n = 1,2,....
[2 cot 0 PA(cos 0)
n = 1,2,...
thus tT,&r,8, s) = -s2G$po f
n
7
“=,s+R;G
(35)
C (n-
1)
Since no two Bessel functions whose orders are the halves of odd integers can have a positive root in common, we can conclude from (35) n=
J n+3 2
1,2,....
(36)
The inequation (36) implies that the coefficients ?” and B. in (32 and 33) are both zero. Thus, the only nonvanishing component of the displacement solution of the reduced problem will be
w%.u,t) =
i
II=t
Zp.R(r, R,)P,‘(cos@f-“(t)
(37)
Hhere ~.f(r., Q,) = r- ’ ‘J,, , ?
n=lT .*....
(38)
(45) d,(r, 0. s) = S’Ci&
A=
t nr .,,s+RfC
- 3‘2
R” J “+I 2 F, r [2 cot 0 Pi(cos 0) ( h >
+ i., P,(cos O)].
(W
For each n. the frequency equation (35) possesses an infinity of roots Q, and. therefore. the complete solution to the torsional wave oroblem is slinhtiv
69
A model of brain shear under impulsive torsional loads modified
where we have dropped the subscript I in equation (56’ and any subsequent equations. Substitution of(56’ into the Laplace transform of (4), yields
to read:
Mr. 0. r) = r sin O@(r)- x C ~nmwQr.R,,) n “1
J 52 +
(
G”Jl..R,,)P;(cos 0)
(48)
where for n. 1’7= 1. _, 3 bv~Jr,flm)=
5-
$(s) = 2’.
(57)
4. COMMENTS
(49)
.
(50)
r-llJn+,l
h
In equation (57) the only unknown @(r) can be determined when the material property. G(r). and the applied torque, r(t). are specified.
W”Jl.. Q”,‘[2 cot 0 P;(cos 8) + i, P,(cos O,]
\!I
It is easy to show that our general solution. (47-52). of the problem with boundary condition (3) contains the solution of Lee and Advani (1970) as a special case. Consider the angle of twist Q(t) given by: @(t) = &)PH(t):
4(t) = n,H(t’.
(58’
The displacement solution (47) for the case of elastic material reads-see Valanis (1968): (5’)
Em(r,R,,)
= r-3’2
J,,
(52)
1,2
Finally, the coefficient relation
z.,
is determined
from me
’ c%?,,(r.!&,bS(r,
ii-W,0, s) = r sin H(s)
+ sin fI g 2,wE(r. Q m) .F(-&ps’. RI=,
B)p,‘(cos B)]r’
(59’
sin f3dr d0 (53’ /
Jo Jo
w’a(r. %,,)w!,,,@.?,,) [P&OS 0’1’ r2 sin Bdr dr3
which is a consequence of equation (23) and mode normalization condition. Equation (53) after integration and simplification yields
The inverse of (59) upon the use of (58) becomes
.Y., = On+ l;m = 1,2.. . .
.2,m=
!%n !
2C,,b’ 2
)
Q,mJ’s ‘2 7
\
111 =
1. 2. . .
(54)
tJ
The prime denotes differentiation with respect to ([(R,,)/C,Jr) . We conclude this section by evaluating the twistingangle@(t),whichis,asmentionedpreviously, unknown for the second problem. In (4), the applied torque T(r) is a given function and the resisting torque r,(t) can be expressed in terms of displacement by n2 T,(t) = 4nh-’ ~(h. 0. r) sin’ 0 do. (55) J0 Introduction of equation (48) into (55), and use of equations (51. 54 and 35’ yields
I
.cLI
Bfi2J; Gn.)
(1 - cosR,t)
. (60)
Imposing the condition (58) and noting that the displacement solution. ul. introduced by Lee and Advani (1970) is given in a rotating coordinate system, it is easy to show ic - ii, = rr,, sin e - 26 AA@.
(6”
Lee and Advani were only concerned with the transient response and the term &r’u~ is considered to be negligible as compared with rxr, sin 0 for small displacements. Thus. neglecting rd r’rr, in (61’ we obtain: ,
7.n
,^
70
K. K. FMXGBAKH~H and C. N. DESILVA
and from equation (60) u$.(r.0. t) = -
““‘l;:”
i(l
_
r-2)
[
.
m
+ 2q2/np2
Qm
( >
I’ c,’
1 1 jyf,,,(jj,) .m=
*mf (63) 1
‘OS
7
where F = r/a and we have satisfied the initial condition u&, 0, 0) = 0. Equation (63) is the result of Lee and Advani. We have solved in closed form torsional wave propagation from a rigid shell into the interior of a linear viscoelastic sphere with boundary conditions given either by equations (3) or (4). The general expressions for the displacement and stress fields are ‘derived. It is now straighrforward to obtain the shearing effects for specific cases of the loadings given by equations (3 and 4). S. NUMERICAL
RESULTS
The general expressions for stresses developed can now be evaluated, using material properties which most closely represent brain tissue and a representative head rotation. The pertinent data used in modeling the brain are listed in Hickling and Wenner (1973): h = 3G in. p = 9.38 x 10-s lb s&/in’ p = 32.5 lb/in2 z,, = (1 - 100) x lo* rad/sec2. We first consider the simple case of an elastic model. The only non-vanishing component of shear stress (48),
Fig. I. The coordinate system used to describe the model.
reduces to
(1 - cos/!Im?) (64) where we have assumed qf) = fpaOtiH(t):Figures 2 and 3 represent the shear stresses versus time and radius respectively. Although evaluation of brain injury is beyond the scope of an elastic model without coupling of physiological system variables. the rapid buildup of shear stresses indicate that shear stress waves are a possible mechanism of brain injury. This maximum shear stress is induced in the neighborhood of the outer layer of the brain close to the inner surface of skull and occurs at 2.6 and I@8 msec after the impact. We have already referred in the Introduction to the investigators who have established the viscoelastic properties of the brain. For our purpose, we will use the particular data developed by Shuck and Haynes (1970) and Shuck andAdvani( 1972)Theseauthorsmeasured valuesofthe complex impedance of human brain tissue and fitted a Kelvin model having q, = 1, = OQO5psi set; q. = E,
0.2 t 0.16-
Fig. 2. Shear stress vs time, 0 = n/2. elastic solution.
-0
04
I 0.2
0
I 0.4
I D6
I 0.8
F
Fig. 3. Shear stress vs radius.
U = n/2. elastic solution. T,
10-6 2.8
dIzF
-
0
05
I.0
15
7
Fig. 4. Shear stress vs time, 0 = n/2. Kelvin model.
P
Fig. 5. Shear stress vs radius.
f? = n/2. Kelvin model.
I I.0
72
K. K. FIR(K)ZBAKHSH and C. N. DESILVA
. For this model equation (48) reduces to:
[A,
+ AzemX*’
+ .43e-s1L
+ A+e-““I
(66)
where A,
=
A2 = 7 -0.2
0.2
0
0.4
0.6
0.8
10
I.2
I.4
A, 1.6
A4 =
7 Fig.
6. Shear
stress
vs time. 0 = n/2. model.
Maxwell-Kelvin
2psi(as indicated in Fig. 4) to the experimental data. Employing G(t) = ~7”+ qrS(t) for this case, equation (48) reduces to: =
-AI
(emAt - ebBI) + qn
B(A -1B)11
The resulting shear stresses versus time and versus radius are shown in Figs. 4 and 5 respectively. These figures again show that the maximum shearing stress. although highly damped is induced in the vicinity of the outer layer of brain shortly after impact. Finally a Maxwell-Kelvin or four-parameter fluid mode) proposed by Galford and McElhaney (1970) with the following data has been considered (see Fig. 6).
In
El = 3.4 psi;
E2 =
‘I, = 275.4 psi set;
‘I? = 86 psi sec.
9.3 psi;
this case G(r) = =+@= I
CYl- VW”
- (4, - s42k-"'l
2
where
p1 = $$ = 70 sec2 I ? 4, = ql = 257.4 psi set = 238 psi si& 2 ct = 1.48; fi = 0.01.
2.4371, + 3.665
-x,
+ x2)( -.Y,
+ .u,)
-2.437.~~ + 3.665 -x2(
-.x2
+ .u,)( -x2
-‘437x, _X,(_.Kj
+ + x*)(-x,
+ .K,)
3.665 + x1)
and x, , x1, x3 are the roots of .x’ + 1.49~’ + CO.0148- 2.437R;].u - 3.665Q; = 0. The shear stress curve vs time is given by Fig. 6 and reconfirms the conclusions drawn from the previous two cases. As a final comment. we point out that equation (64) shows that two different sized brains of the same material and in the same condition experience shear stresses proportional to the square of the brain radius. The square of the brain radius is proportional to the 2/3 power of the’brain mass and consequently we have established the “Scaling law” proposed by Holbourn (1943) for an elastic brain. Equations (65 and 66) show, however, that this is not the case for a viscoelastic model of the brain. A similar result stemming from a very different approach was obtained by Bycroft ( 1973).
REFERENCES Arutiunian. N. K. H. and Babloian. A. i\. (1965) Two dynamical contact problems for an elastic sphere. J. ,lppl. Mtrrh. Mrch. 29. 6X-627. Bycroft. G. N. (1973) Mathematicul model of a head subjected to an angular acceleration. J. Biotmvhtrr~ics 6. 487-495. Christensen. R. and Gottenberg. W. (1964) The dynamic response ofa solid viscoelastic sphere to translational and rotational excitation. J. .-lppl. ,Ifcc/~. 31. 373. 278. Estes. M. S. and McElhaney. J. H. (1970) Rcsponsc of brain tissue to compressive loading. ASME Paper No. 70-BHF-13. Fallenstein. G. T.. Huelke. V. D. and Melvin. J. W. (1969) Dynamic mechanical properties of human brain tissue.
J. Biomvhctrric.s
y2 = y’
-.Y,(
(65)
_ _eeEr where
=
3.665
.Y, x2x,
2. 2 17-226.
Galford. J. E. and McElhaney. J. H. (1970) A viscoelastic study of scalp. brain and dura. J. Eio~nrclrtrrrics 3.11 I-21 i. Genndreih. T.. Thibault. L. and Ommaya. A. K. (1971) Compression of translational and rotational head motions in rxperimcntul cerebral concussion. Plr>c. (!I’ I .%/ISrupp CcrrCrd1 Cor@?rlcc~. pp. 797 803. Gennarelii, T.. Thibault. L. and Ommaya. A. K. (1972) f’athophysiologicresponsestorotational and translational acceleration of the head. Pi-oc. qf 16th Sttrpp Ctrr CIW.X/I CotEfiwrtcc. pp. 296308. Goldsmith. W.( 1966)The physical processes producinp head injury. Hrad 1tljur.r Co+~rr~cr Procdi~rys (Edited by
A model of brain shear under impulsive Cavcness. W. F. and Walker. A. E.). pp. 3.50-382. Lippincott. Philadelphia. Gurdjlan. E. W. and Lissner. H. R. (1961) Photoelastic confirmation of the presence of shear strains at the craniospinal junction in closed head injury. J. Nruroaur~. 18. 58-60. Hayashi. T. I 1970) Brain shear theory of head injury due to rotational impact. J. fuc,ultJ EII(/II(/. I’rlir. Tok!fi 30, 307-313. Hickling. R. and Wenner. M. (1973) Mathematical model of a head subJected to an axisymmetric impact. J. Bio,llrchrrrlic,.s 6, I l5- 132. Holbourn. .4. H. E. ( 1943) Mechanics of head injuries. Lurwt 2. 438 441. Jamison. C. E.. Marangoni. R. D. and Glaser. L. (1968) Viscoelastic properties of soft tissue b! discrete model characterization. .I. Biorwchunics 1. 33- 46. Lee. Y. C. and Advani. S. H. (1970)Transient response of a sphere to torsional loading-A head injury model. J. Afurh. Biosci.6. 473-486. McElhaney. J. H.. Rose. L. S. and Estes. M. S. (1968) Dynamic mechanical propertiesofhuman scalp and brain. Biomechanics Laboratories. West Virginia Llniversity. Progress Report. Morgantown. West Virginia. Ommaya. A. K. (1966) Experimental head injury in the monkr?. Neud I~ljur~~CO~I@UC~CPm-wdinqs. pp. 2f%-275. Ommaya. A. K. and Hirsch. A. E. (1970) Protection from brain injury: The relative significance of translational and rotational motions of the head after impact. Proc. of l4rh Sltrpp Cur Crush CO+I~f~CC, pp. 144- I5 I. Ommaya. A. K.. Hirsch. A. E. and Martinez. J. L. (1966) The role of the whiplash in cerebral concussion. Proc. C$ 10th .Qtr\,/ (‘(II. Crtrsh Cor!tc~,‘c,rIcr. pp. 3 14-324. Roberts. V. L.. Stech. E. L. and Terry. C. T. 11966) Review ofmathematical modelswhich de&be human response to acceleration. ASME Paper No. 66WA ‘BH.Fl3. Shuck. L. Z. and Advani. S. H. 11972) Rheological response of human bram tissue. ASME Paper No. 72,‘WABHF-2. Shuck. L. Z. and Havnes. R. R. (1970) Determination of viscoelastic prop&es of human brain tissue. ASME Paper No. 70-BHF-12. Valanis. K. C. (1966) Exact and variational solutions to a general vlscoelasto-kinetic problem. J. uppl. Me& 33. 88% 892. Valanis. K. C. and Sun. C. T. (1967) Axisymmetric wave propagation in a solid viscoelastic sphere. Inr. J. Engnq Sci. 5. 939. 956.
torsional
loads NOMENCLATl’RE
.A,,. B,,. 4,. C,. D,. T,. B,. E, constants h inner radius of skull elastic shear wave speed shear relaxation function Heaviside unit step function II-kassmoment inertia of skull about the z-axis spherical Bessel function of the first kind bulk modulus Legendrepolynomialofthefirst kindoforder A. associated Legendre function of the first kind of order X and degree one spherical coordinates nondlmensional Laplace time
transform
nondimensional
radial coordinate parameter
time
applied torque resisting torque radial (r). meridional (0) and circumferential (4) displacements. respective11 eigen functions constant angular acceleration nondimensional
frequency
parameters
shear modulus constant Lamt constant mass densit) angle of twist of skull about the r-axis Cartesian components of stress tensor shear stresses Cartesian components of strain tensor frequencies gradient operator in spherical coordinates
Stthscriprs
i. i. k Cartesian components of a vector or tensor. k. I?. 111non-neganvc Integer. Supw.wriprs S. R refer to the “Static” and “Reduced” problems respectivel!.
