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Impact on a head-neck structure
IMPACT
ON A HEAlSNECK
STRUCTURE*-?
BENJAMINLANDKOF$and WERNER GOLDSMITH Department of Mechanical Engineering
and J. L. SACIiMAN Department of Civil Engineering. University of California, Berkeley. CA 94720 U.S..4
Abstract-An
analytical and experimental study was executed involving non-destructive. axisqmmetric impact on a fluid-filled shell constrained by a viscoelastic. artificial neck-a system intended to simulate a human head-neck structure. The shell material was considered to be homogeneous. isotropic and elastic, while the fluid was regarded as inviscid and compressible with its motion irrotational. The artificial neck was represented analytically by a linear viscoelastic cantilever beam that was rigidly connected to the shell. The physical model consisted of a water-filled uniform spherical thin-walled shell composed of lucite attached to an artificial neck developed by the Biomedical Laboratories of the General Motors Corporation. The system was loaded by the impact of a spherical aluminum shell with an o.d. of 7.4 in. that formed part of a simple pendulum. The impact force in both the analytical and experimental studies consisted of a half-sine pulse with a duration of 23 msec. The analytical solution first required the determination of the interaction forces and moments at the head-neck junction, which were ascertained from continuity conditions and an assumed rigid shell. These quantities, in turn, were applied simultaneously with the impact force to the deformable fluid-filled shell. yielding a boundary value problem that was solved on a digital computer. The resultant predictions of circumferential shell strain, fluid pressure, and displacement of the junction were found to be in good agreement with the experimental data. Implications of the results for head injury are discussed.
INTRODUCTION The field of head injury has been the subject of
numerous investigations and a variety of mathematical and experimental models purporting to represent the phenomena incident to such an event have been examined. It is generally recognized that different injuries will result from various types of loading, differing orders of magnitude of the duration, alternate directions of a blow, or disparate contact characteristics. Recent comprehensive reviews of the topic have been given by Goldsmith (1972) and Kenner (1971); these include classifications of the most frequently encountered types of head trauma and the nature of the loading causing such dysfunction. This paper is concerned with impact loading in the millisecond range on a model head attached to an artificial neck. This situation resembles that frequently encountered in head-on vehicular collisions when the head of a front-seated occupant strikes the windshield or the instrument panel. Since the duration of the applied force is large compared to the time required for a disturbance to traverse the head, which is of the order of 10&2OOmsec (Goldsmith, 1966),the properties of the neck and its junction to the head will play a vital role in establishing the ensuing motion. This is in * Receiwd 12 March 1975. t This work is based upon a portion
of a dissertation submitted in partial fulfillment of the requirement for the Ph.D. degree at the University of California, Berkeley. CA. U.S.A. 1 Current address: Armament Development Authority. Israel Ministry of Defense, Haifa, Israel. 141
contrast to the situation when the load duration is of the same order or shorter than the transit time; here, wave effects predominate, and the relative motion between the head and torso can be disregarded in establishing the mechanical head injury parameters. The latter domain has been investigated by Engin (1968), Benedict et a[. (1970). Lee and Advani (1970), Liu et al. (1971), Chan and Liu (1974). and Kenner and Goldsmith (1972, 1973). While some recent publications are concerned with the analysis of a fluid-filled spherical shell under longduration loading, no significant attempt has been made to correlate the induced intracranial pressure with the properties and constraining effect of a neck. As was indicated by Liu et ul. (1971) for long duration impact, the motion of the head can be decomposed into (a) translational and rotational motion, with corresponding accelerations, and (b) breathing motion resulting from stress wave propagation. While the former are strongly dependent on the properties of the neck, the breathing motion is virtually unaffected by the presence of this junction. Both types of motion will contribute to the intracranial pressure and both must be considered in the analysis of head injury when the loading period is of the order of milliseconds or longer. FORMULATION AND ANALYSIS OF
THE PROBLEM
The axisymmetric impact on a human head-neck system with a millisecond duration will be modeled
BENJAMINLANDKOF, WERNER GOLDSMITH and J. L. SACKMAN
Elastic
thin spherical head1
I
shell
viscid compeuible fluid brain and cersbrospinal fluid)
I
Head-neck
junctmn
-1
,,,,,
!i;
[,k),&Tnd
I neck
support
)
, torso,
Fig. 1. System schematic.
analytically by the representation shown in Fig. 1. The structure consists of a homogeneous, isotropic, elastic, uniform thin-walled spherical shell filled with a compressible, inviscid fluid that executes solely irrotational motion; the shell is rigidly constrained at its bottom point A by a linearly viscoelastic beam of circular cross section and length 1 that is clamped at its distal end B to a rigid support. The shell replicates the skull, the fluid purports to represent the contents of the cranial cavity, the viscoelastic beam is to model the artificial neck, joint A is regarded as the counterpart of the occipital condyles, and position B denotes the attachment of the neck to the torso. This unit is subjected to a radial pressure pulse of appropriate length, uniformly distributed over a spherical cap, with a resultant F, that is initially orthogonal to the centerline of the neck. Although the magnitude of the impact force is time-dependent, the spatial pressure distribution over a spherical cap of solid angle & will be considered as invariant. However, even though the external load passes through the center of gravity of the shell. the boundary value problem for this container will still be asymmetric due to the constraint exerted by the beam at position A. The analytical procedure requires the solution of two separate problems: (a) The interaction forces and moments at the junction A are computed first, and (b) these forces and moments are then applied together with the impact force to the fluid-filled shell and its response is obtained from the solution of this boundary value problem. In order to achieve the first part of the analytical objective, the following assumptions have been employed : (i) The total motion of the shell and its contents is assumed to be planar, occurring in the z - x, or sagittal plane (corresponding to back and forth motion); this is a consequence of the restrictions placed on the direction and point of application of the resultant impact force. (ii) The interaction forces and moments also lie in the sagittal plane, as implied by assumption (i).
(iii) The shell is considered to be rigid only for the purpose of determining the interaction components at the head-neck junction. This is justified since the contribution of the elastic elements to these forces and moments is negligible compared to the contribution of the rigid shell when the ratio of the impact duration to the fundamental period of free vibration of the fluid-filled system is greater than ten (Moyseyev and Rumyantsev, 1968). (iv) The angle of rotation of the shell y with respect to the beam is assumed to be small. It has been reported by a number of investigators (Huelke et al., 1967) that vehicle occupants anticipating a collision contract their neck muscles which would support the validityofthis assumption with respect to the prototype. (v) The motion of the spherical shell in the xdirection will be neglected relative to that in the zdirection. Figure 2 portrays the forces F and moments M acting on the components of the system; the equations of motion in the x - z (sagittal) plane are given by CF, = m,a,;
CF, = m,az;
CM, =
IodG,(1)
where m,, and I,, are the mass and moment of inertia with respect to the mass center 0 of the fluid-shell system which is regarded as rigid, y is the angle of rotation of the shell about 0 and a, and aZ are the accelerations of 0. From Fig. 2, these relations become 2
F&t)
-F,(t)
=m,
d2y
$(l.t)
+Rdtz
(2)
RF,(t)
-MA(t)
=Io$
and
F,=m,g,
where w is the deflection of the beam in the z-direction and R is the mean shell radius. The last of equations (2) is the consequence of assumption (v). Next, the quantities F,, FH and M, are applied to the viscoelastic cantilever beam. The differential equation for the transverse vibration of a simple linear viscoelastic Euler beam in the Laplace transformed space takes the form E(p)1 ‘2
+ pAp2 W - pw(x, 0) - GJ(x, 0) = 0, (3)
where p is the transform parameter, a bar denotes a transformed quantity, E(p) is p times the transform of the relaxation function of the material, still to be determined, and p, A and I are the mass density, cross-sectional area and moment of inertia of the beam, respectively. After applying the Laplace transform to the first two of equations (2) and employing the initial conditions y(0) w(x,O)
=2(O)=o; =
a;(x.o)=0,
(4)
143
Impact on a head-neck structure
and F
INF,(r) ___---.
(p) = (1 +
ff
(10)
A
where &I, N and A are explicit
functions of the transform variable p and viscoelastic function E(P) (Landkof, 1974). An initial requirement for the solution of the boundary value problem is the establishment and evaluation of the function E(p). In this endeavor, attention will not be focused on the precise distribution of the stresses along the neck itself, but only on the specification of a model that will permit the ready calculation of the interaction forces and moments. On
MA
FH
T
l?%llL the basis of several previous investigations involving the neck, (Melvin et a/.. 1972; Becker. 1973; Melvin et n2., 1973) a three-parameter one-dimensional linear model consisting of a spring of modulus E, in series
F”
e
A
with two parallel elements, another spring of constant E2 and a dashpot with coefficient q2 was chosen as
a reasonable representation for this component. For such a system, it can be readily shown (Landkof, 1974) that
Fig. 2. Free-body diagram of the head-neck system.
(11)
the following set of equations is obtained:
‘&)R
- GA(P) = p21,Xp> The next task is the determination of the parameters
F&p) - F,(p) = r~~o~~~W(I,p) + R?(P); &)I
‘; +pAp"k =0.
(5)
As indicated in Fig. 2, the last of equations (5) must also satisfy the following boundary conditions: W(O,p) = d;(o.p)
= 0 = AJ,i,(p)
-E(P)I
d$(I.P>
=
F,(p)
Finally, in order to conform to the requirements of a well-posed problem, a continuity condition must be established between the shell and the viscoelastic beam at point A. It will be assumed that the angle of deflection of the beam equals the angle of rotation of the shell with respect to the beam, y. Then
g(1.1,
= tarry-
y(t),
(7)
or. in the Laplace space,
E,, El and q2 in such a manner that the analytically computed interaction forces and moments would fit the experimental results. This was accomplished with the aid of experimental data supplied by General Motors Corporation that also provided the artificial
neck used in the experimental portion of the present study. The data had been collected from a series of tests in which the neck was used as a part of a Sierra 1050 dummy (Culver et al.. 1972; Culver, 1974); the results were subjected to an identification procedure for the evaluation of these constants by repeatedly correcting the values of an initial guess, using a computer program based on the method of least squares(Landkof, 1974). until a reasonabte tit with the data was obtained. The magnitudes derived for the parameters were as follows: E, = 2986 psi. E, = 2501 psi, v2 = 1294 psi-sec. The moment M, and force F, at the junction could now be evaluated from equations (9 and 10). A spherical coordinate system r, 4. 0 was utilized in the analysis of the response of the shell whose undeformed midsurface radius is given by r = R. Since the impact force was assumed to be uniformly distributed over a spherical cap of angle (ba,
(8) F, =
With the aid of conditions (6 and 8). the system of equations (5) yields M
(p>
A
=
(R - WF,(P) A
(9)
F,(t) 0
forO<~<~,.OiC)<27t otherwise
).
(12)
In a similar manner, it will be assumed that the timedependent interaction force F, and moment M, are also uniformly deployed over some spherical cap
BENJAMIN LANDKOF,WERNERGOLDSMITH and J. L. SACKMAN
I44 around
the junction. F
M
=
H,
A
Thus, F,(t),M,(t)
for 1127~- q& c C$< 1/2x + q&,z - e1 -c 6 c n + t+
0
otherwise
The vertical force F, which balances the weight of shell and contents, does not contribute to the dynamic strains and pressures and does not appear in the subsequent analysis. The equations of motion for the shell are chosen as linear and incorporate membrane and bending effects, but neglect rotatory inertia and transverse shear. The fluid is characterized by a linear wave equation that includes compressibility, but neglects viscosity. The governing relations are (1 + m2) -u,sin
1 --u sin3~
3-v - 2uB,Bcot4
sin4(cot2$
cot Cp 1 lJ 8,888- sin %,e, + ~ sin 6 B*Bbo
+ 2(1 + v)w - cot 4 (2 - v + cot%$)w, + (1 + v + cot+$)w,&, - 2w,,, - WM!M - & 1 - sw*eee
4 (cot24 + v) + u~,# cos f$
+U&&Sin~+
l-v 1+v ~ %,ee + -%,&J 2 sin I$ 2
2cosC$ + .3W,ee# sm d,
+ (1 + v)w,#sin4
‘$+ ps ;
‘ee@I
(R, r$,e, t) -
and
a
1 2 a@ -_ r Z +r2sin~&$
( >
+
(14)
a
.
sm+g
(
a4
>
1 a% --=$$ r sin I$ a82
(17)
where c = (K/p,)“’ is the compressional wave speed in the fluid, v is Poisson’s ratio for the shell, p, and ps are the densities of the fluid and shell, @ is the fluid potential, K is the bulk modulus of the liquid, S and S1 are the areas of application of the impact and restraint forces, and u+, u, and w are the displacements of the shell mid-surface in the 4, 0, and r-directions, respectively. It will also be assumed that no separation of the fluid from the shell will occur, i.e. that normal velocities across the boundary will be continuous. Thus,
;<4,&9 = $ CR,4>& 0. Finally, the initial conditions ~(6
cos I#) + 2Ub& cot 4 + U&,& + -u sin3~ ~,es 1
+ .2 %l,et@+ sin 4
2 sin24 + 1 sin34
Ue,e
2
& =hZ 12R2
- 1) w) - W~~COSC$
sin C$
2 - TW an 4
with
-r2 ar
Eh
cot 4
(3 - V + 4 cot24)W*e
psh
1
X-
(13) 1.
60)
are given by
= a; (4,8,0)
= 0
um(4,e,oj
= a~(s,e,o)
=o
2.4(4,e,o)
=~(4,e,o)
= o
Wr, he, 0) = g Cr. 44 e, 0) = 0. The set of partial differential
(18)
equations
(19) together
with
Fig. 3. Photograph
(Fmitz,q p. 144 )
of the target.
Impact
on a head
the boundary conditions (18) and initial conditions (19) constitutes the boundary value problem. In order to simplify the relations, one of the equations cited above can be uncoupled by means of a substitution first used by Van der Neut (1932) ug = U,,,/sin C#I : w = W. (70)
lid, = u., - I/ sin 4,: Upon the dimensional
additional variables $ = U/R:
employment < = V/R;
the following set of equations
of
the
< = WJR
non-
neck structure
14.5
in spherical coordinates. It is evident that equation (23) is uncoupled and can thus be solved for variable < since FH and MA are known. The solution of the set of equations (17 and 22- 24) is accomplished in the usual manner by first applying a Laplace transform with respect to time t and then representing the result as a sum of spherical harmonics. yielding
(31)
is obtained
x Pf( cos 4) cos 010, (29)
(22) (1 + sc’)(l - \I)@2 + 2)
Fl(h 0,P)
1’2
1 -
= -
psR’
-------+
2
EhS(Md
t- F,R)cscd
1
[xZV4 + V”[2r’ z2 sin4
h-V2 + 4cosdV2 @J
i[ x (1 + a’)(1
+ v) i -
1 -
=
I
”
n=”
m=O
c c
~“H(&e,P>csc$=i
- (1 + a2)(1 + \I)]) lj
. -
where Pr(
E,,(p)
P;(cos
c#i,cos }?d
(30)
f: R,,(p)
n=o
m=O
x P:: (cos 4) cos mf? (31)
2cos~P
M,(&
0.1’) csc$
=
sin& [(l + “~‘)(l + 1’)
i i F,,(p> II=1 m=O
i
x P!( cos 45) cos mfl.
(I - v)V’]
(32)
The uncoupled equations (17 and 23) are now solved separately. yielding in terms of spherical Bessel functions ,j, with the aid of equation (28)
(24) with initial and boundary !Kb. 0.0) = $
conditions
given by
(4. 0,O) = 0
and
^IY
@(P,4.0,0)
= $Y#Au,o,
= 0 where the recurrence
and
v2{P;(cosf$)
cosmfl)
formula = -?t”(P~(cos~)cosrllH~
(25) Equation operators
(17) remains unaltered. The Laplacian V2 and V4 in equations (22-24) are expressed
with 1, = n(n + 1) (35) and the boundary conditions have also been utilized.
between
fluid and shell
46
BENJAMIN LANDKOF, WERNER GOLDSMITH and J. L. SACKMAN
Upon substitution of equations (2629) the other differential equations (22 and 24) are transformed into two algebraic equations in the two unknowns B,,,, and B,,, with I?,,,, given by equation (34). The solution of this set is obtained by Cramer’s rule yielding, with
&o(p)
= RM
EhS
x E,,
The numerical values of the desired strains and pressures were obtained in the following manner: (i) The experimentally determined input pulse F,(t)
.io
2(1 + r) + j),R’ !+p2
(i~R/d
- ’
.i;l (@R/c)
(36)
, was approximated by a sinusoidal function so that
F,(t) =
(lb)
286sin St 0
0 5 t 5 2.5 (msec)
t > 2.5
(47)
with time t in msec. (ii) The inversion of the Laplace transformed quantities $, [. l, 3 was performed numerically by means of the Gaussian quadrature procedure (Bellman and Kalaba, 1964). (iii) The mid-surface strains es (($4, t) and Q, (0, 4. t) were obtained by means of the following relations c = $,+4 - icos4
_R1-v2_
-E El&
Eg= $,&,cot 4 - < cos 4 + $,&+csc’4 + 5.
where Ati = ~1nP2n - ~nfjin
(39)
(1 + a*) (1 - v - A,) - l$p.R’p2
(40)
and z 1”
=
2,=cr2R,Z-a1,120r2-(f+a2)(f
CI
+
I’))
pi” = (1 + N2)(1 + V)- Y2(2 - 2,)
(41) (42)
P2n = -2(1 + v) - s12[A,”- a,(1 - v))
- psR’
Ylmn
=
(43)
=E{(n-
I)[-3L,a2+(1+cr2)
(44)
(1 + 1’)- 3x’] - 4i,,r2 - 2(1 + x2)(1 + v)) 6 InIll=
(1 + a2)(t1 - nr + 1) : 4 - (n + 2) (1 + 1)) 2(2n + 3) (45)
6
II +
2mn=
111 +
211+ 3
1
(48) (49)
The differentiation was performed with the aid of well-known recurrence formulas. (iv) The pressure P was obtained by numerical differentiation of the fluid potential and employment of the relation p=
-p,g
(50)
(v) In the computational process, evaluation of the spherical Bessel functions’ and associated Legendre polynomials was required. The higher order Bessel and Legendre functions were obtained by utilizing appropriate recurrence relations. When required, integration of the associated Legendre polynomials was performed by means of a library routine that used Simpson’s rule of integration. EXPERIMENT’
(1 + x2) (n - 111) j4+(n--l)(l+V)j 2(2n - 1)
Yam
- i,+ sin4 + 5
[(n + 2) [&x2 - (1 + a2)(1 + v)
+ 2a*] - 4&x2 - 2(1 + x2)(1 + v)). (46)
The target for the impact tests. shown in Fig. 3, consisted of a thin-walled spherical lucite shell filled with distilled water that had been used in previous investigations (Kenner and Goldsmith, 1973), an aluminum adapter connecting the shell with the artificial neck, and a heavy steel plate serving as the base support. The input to the system was measured by (l/32)-in. thick X-cut quartz transducers of (l/2)-in. dia. and sandwiched between short stubs of aluminum, also of the same diameter; the units were assembled with conducting epoxy and cemented to a point on the equatorial plane of the shell with structural epoxy. The pertinent mechanical and geometric properties of the shell are as follows: Weight density psg = 0043 lb/in3; Young’s Modulus E = 0.67 x lo6 lb/in2, Poisson’s
137
Impact on a head-neck struch~re ratio v = 0.38. mean midsurface radius R = 3.636 in.; wall thickness h = 0.146 in.; thus R/h = 24.9 and the loading angle Q!Q,= tan-’ (0,25/R) = 3.94”. The adapter was constructed in the form of a boss with an upper, scooped-out surface of 1.0 in. dia. to which the shell was glued. and a lower surface with a dia. of 2.5 in. featuring four holes through which (l/4)-in. dia. bolts were fastened to the neck. The latter consisted of a steel and molded polyurethane structure exhibiting four ball-jointed components and one upper. pin-connected ‘nodding’ segment; the plastic material connecting the articulated units provided both bending resistance and energy dissipation. The ball joint used a split-ball concept permitting both neck compression and elongation. Rotation about the longitudinal axis of the neck was also permitted, but opposed by friction between the resistive elements and the loading disks. The unit weighs 3.1 lb. has a total length of 5-5 in. and exhibits an outer diameter of 3.4 in. The lower end of the neck was attached to a 22 lb steel plate with dimensions of 3,‘4 x IO x 10 in., serving as a model of the torso. by means of four (l/4)-in. dia. steel bolts. In a few tests, the artificial neck was removed, and the shell was either ‘freely suspended’ by placement upon a circular support of sheet aluminum (Kenner and Goldsmith, 1973) or else directly bolted to the steel plate serving as the base for the system. Strain histories were measured both on the inside and outside surfaces of the shell by means of epoxymounted BLH SPB2-12-12 semiconductor gages with gage factors of I 16.7 and resistances of 121 Q both in the meridional (or hoop) direction H at (I) d, = 30. 0 = -60 and (2) (11= 150’. 0 = 60. and in the circumferential direction 4 at (1) 4 = 30-, 0 = 270’. (2) 4 = 150’. If = 90 and (3) Q, = 180”. These gages were incorporated in a potentiometric circuit so as to give average strain b! connecting corresponding inner and outer transducers in series. Pressures in the water were measured by means of O.lOO-in. dia. gold-plated Z-cut tourmaline crystals, OWI in. thick. located in one shell hemisphere alt nominal non-dimensional radii ? = r;‘R of 0. 0.25, 0.5, 0.75 and 0.875 from the shell center. To obtain the pressure records for the second hemisphere. the impact position was moved 180’ and the experiment was repeated. The piezoelectric constant for the tourmaline crystals was found to be 10.75 x 1OY” C;lb and that for the input quartz crystal was determined as 10.X x 10-l’ C/lb. Details of the calibration procedure for the transducers are given in Kenner and Goldsmith (1973). The output signals from all detecting devices were displayed and photographed on Tektronix 565 oscilloscopes using amplifiers that exhibited a bandpass with a drop of 3 dB at a frequency of 1 MHz. The lowfrequency response of the piezoelectric transducers was improved by adding a 10,000 ppf capacitance in parallel to the Input crystal and a 500 /L/If capacitance in parallel to the pressure gages. Transient phenomena were initiated by the impact on the structure of a spherical aluminum shell with an equatorial o.d. of
z Y)
5 ; + 1000
w”-
-3000
_ $=30” e.-600
0
1
2
3
ANALYSIS
I
"0
fl
Ll
EXPERIMENT
oooooo
L
5
1
6
-J
7
Fig. 4. Comparison of predicted and measured hoop strain E,+in the shell: 4 = 30 1 fl = 770 7.40 in. and a wall thickness of 0141 in. that was incorporated in a simple pendulum arrangement. The drop height was adjusted so as to yield a peak force of 286 lb that corresponded to a duration of 2.5 msec; triggering was accomplished by the signal from the input transducer. In a few tests, the displacement of the shell-neck joint was determined by means of a dial gage located at that position whose excursion was photographed by a Fastax intermediate speed movie camera operating at a nominal framing rate of XOO/sec. the actual speed being indicated by timing marks.
Figures 4 and 5 provide representative comparisons between the experimental average strain results and analyticalmidsurfacepredictionsfor thecircumferential strain E# and hoop strain E, at two different positions. The agreement between analysis and data is reasonable during the loading interval, but deteriorates beyond. Possible reasons for such discrepancies include the approximation of the average strain as being equivalent to the midsurface strain (exaggerated when the inner and outer surface strains are of different signs). the
-
EXPERIMENT
oaoo
ANALYSIS
1 6
7
Fig. 5. Comparison of predicted and measured cn-cumferentlal strain Ed,in the shell: Q, = 30 I! = - 60
BENJAMIN LANDKOF,WERNERGOLDSMITHand J. L. SACKMAN (iii) Generally, the loading
_
EXPERIMENT
____.
ANALYSIS
, n-at la)
10 0 -10
II-20
0
1
2
3 t
L ,Timc
5
6
7
, msec
6
9
pressure develops
during
up to 25 msec in the hemisphere nearest the impact point. while the pressure in the distal hemisphere is slightly negative. At the end of the loading process, the pressure in the nearer hemisphere also becomes negative. Figure 9 shows a crossplot of the pressure distribution along the impact axis for various times after contact. It is evident that the maximum positive and negative pressures in the system develop near the instant of the maximum impact force, at t = 1.25 msec, when the acceleration in the z-direction is also near its maximum. This circumstance would provide the explanation for the extreme magnitudes of these pressures at that particular time. At the end of the loading period, for t > 2.5 msec, deceleration of the fluid-filled shell will occur that is entirely controlled by the mechanical properties of the artificial neck. At this instant, a negative pressure is initiated in the frontal hemisphere. The maximum value of this negative pressure is of the order of 24 psi and
t , Time
:
a positive
period
10
(b) ._
206
k
10
::
F.0.19
,
,
(
,
,
,
,
z 2 0 h ;
20
-10 -20
t
‘j; P 0123L5676
msec
t , Time,
9
10
(c) Fig, 6. Comparison of predicted and measured pressures in the fluid: (a) i = 0.84, (b) i = 0.71, (c) i = 0.49.
behavior
computational
and
of the shell that was neglected
numerical
procedure.
approximations
e :,
0
rk
-10
o
-20
in
in the
In general, the strain data conform to the results found previously (Kenner and Goldsmith, 1973) in that the records from the hemisphere around the impact point closely resemble the shape of the input pulse with a rapidly attenuated tail, but no such correlation exists in the distal hemisphere where the levels are much lower. A similar comparison of theoretical predictions and experimental results for the fluid pressure at nine different stations along the impact axis is presented in Figs. 6-8; agreement is considered to be reasonable. The general characteristics of the pressure history in the fluid are as follows: (i) The pressure records obtained from the experiment show a general pattern of low frequency vibrations due to rigid-body shell motion with superposed high frequency fluctuations resulting from wave propagation in the fluid. The latter disturbances disappear after a few milliseconds as would be expected. (ii) The high-frequency oscillations are not present in the analytical pressure curves due to the use of a time increment of excessive size.
ANALYSIS
iz0.27
1
2
3
5
t , Time,
viscoelastic
EXPERIMENT
--__
10
0
the analysis.
-
6
7
6
7
6
7
rnsc (a)
9
10
6
9
10
6
9
10
i=O 20 .g
10
2 r‘
0
E k
-10
Ii -20
0
1
2
3
_L t , Time,
5
msec
(b) F=-0.27
20 ‘5 =
10
t :
0
1 k
-10
a -20 I 1
II 2
3
I 4 t ,Twne
II 5
I msec (cl
Fig. 7. Comparison of predicted and measured pressures in the fluid: (a) 7 = 0.27, (b) i = 0. (c) i = -027.
Impact
t .Time
,
on a head-neck
-
EXPERIMENT
__--
ANALYSIS
I49
structure
msec
la1 i,
Nondlmensmnol
Radws
Fig. 9. Pressure distribution along the impact dimensional radius i: for various times after impact.
,
msec
4 5 t ,Time,
nmc
t ,lime
(b)
-10
0
I
2
3
6
7
e
9
IO
(c) Fig. 8 Comparison of predicted and measured pressures the fluid: (a) T = -049. (b)? = -0.71. (c)i = -0.84.
in
occurs in the distal hemisphere here at two different locations: initially at t = 15 msec at a non-dimensional radius i- = -0.27. and then at 6.4 msec at r = -0.49 where its duration is considerably greater than at the first location. The tests involving the three different boundary conditions for the shell were recorded by means of a triple exposure so that the response of each of the transducers for the three different conditions appeared on the same photograph. The results indicated only a very slight variation in both amplitude and duration of the input pulse. Furthermore. the pressures measured during the first two or three milliseconds in the shellneck and freely suspended shell systems were of relatively low amplitude, while that in the rigidlyattached shell system rose rapidly and to higher magnitudes, as expected. Figure 10 presents a comparison of the experimentally determined displacement of the head-neck junction and the corresponding analytical prediction. The latter was obtained from the solution of the last of equations (5) and the appropriate boundary conditions, again using the Laplace transform and a Gaussian quadrature inversion. The values employed for the required material and geometric quantities were: p.4 = 1.46 x 10e3 lb- sec’/in’. I = 0.0688 in”. 1 = 53 in. The correspondence appears to be quite good; the
axis vs nonbeginning 01
smoothness of the experimental curve is probably due to the inertia of the dial gage employed. The present effort was primarily directed towards the determination of the intracranial pressure variation under conditions of impact to the model. so that its significance with respect to possible brain injury mechanisms might be appraised. The group adhering to the cavitation theory of brain trauma have claimed that there exist locations within the cranial cavity where the negative pressure developed in the cerebra spinal fluid would be sufficient to rupture the capillary walls. According to the damage criterion for closed brain injuries postulated by Liu er (I/. (1971). the severity of the brain injury at a given location is proportional to the average time of negative pressure exertion beyond a critical level at a given position. Based on this criterion, it appears that the potential location of the most severe brain injury in the present model will be at J = -0.49. at an intermediate coup position. This result agrees qualitatively with the predictions of several other investigators (Ommaya and Corrao. 1969: Ommaya, 1973; Hickling and Wenner, 1973). Comparisons of the results of the present investigation with those for the free shell (Kenner. 1971) indicates that the shape of the strain curves at b, = 30 produced by the impact of a 2-in. dia. solid steel sphere closely resemble the present data. The peak magnitudes found -ANALYSIS 6
____
4
t , Time,
Fig. 10. Predicted
EXPERIMENT
msec
and measured displacement head-neck junction.
history
of the
50
BENJAMINLANDKOF,WERNER GOLDSMITHand J. L. SACKMAN
in the previous generated pared
tests are lower since the impact
was smaller. The pressures
since the current
trailsient
disturbances
investigation and rigid-body
the previous work was concerned
force
can not be comdeals with both motion,
whereas
only with the former
event.
CONCLUSIONS
The principal conclusions of the present study involving impact on a model human head-neck system are: (i) Reasonable agreement was obtained between theoreticalpredictionsandexperimentaldataofcircumferential and hoop strains and fluid pressures in a waterfilled lucite shell attached to an artificial neck under conditions of axisymmetric impact for a loading period of 2.5 msec. Poorer correlation at later times may be due to the lack of equivalence of measured average strain and calculated midsurface strain, neglect of the viscoelastic properties of the shell in the analysis, or numerical errors in the computational procedure. (ii) The experimental and corresponding numerical results of the pressure histories indicate that the peak negative pressures in the fluid are observed mainly in the distal hemisphere during the impact period, while positive pressures develop concurrently in the frontal hemisphere. Beyond the end of impact, negative pressures are also generated at the front. (iii) The maximum negative pressures appear at two locations in the distal hemisphere, each at a different time; the one nearer to the center (F = -0.27) occurs during loading, while the other (F = -049) is encountered during the deceleration period controlled by the artificial neck. The duration at the second position is much longer and hence more serious for potential brain injury based on a cavitation hypothesis. (iv) The mathematical description of the artificial neck could be reasonably accomplished by means of a three parameter elastic-viscous model. (v) Good correlation was obtained between the analytical and measured displacements of the headneck junction for a period cu. 15 times the contact duration. The smoothness of the experimental curve is attributed to the inertia of the dial gage employed. (vi) The results indicate that the neck constraint has considerable influence on the pressure distribution in the fluid modelling the brain and can not be ignored when impact forces with durations of the order of a few milliseconds or longer are applied to the fluidshell system examined.
Acknowledgements-The senior author was the recipient of a Government of Israel scholarship during the period of the present investigation. The financial support of JTCG/ME and of the Office of Naval Research for the conduct of the research is sincerely appreciated. The authors are also grateful to the General Motors Research Laboratories, Warren, Michigan, for the gift of the artificial neck
REFERENCES Becker, E. B. (1973) Preliminary discussion of an approach to modeling living human head and neck to -G, impact acceleration. In Human Impact Response (Edited by King, W. F. and Mertz, M. J.), pp. 289-320. Plenum Press, NY. Bellman, R. E. and Kalaba, R. E. (1964) Modern Analytic and Computational Methods in Science and Mathematics. Elsevier, NY. Benedict, J. V., Harris. E. H. and von Roseberg, D. U. (1970) An analytical investigation of the cavitation hypothesis of brain damage, 1. Basic Enana. ” ” Trans. ASME. 920. 597-603. Chan, H. S. and Liu, Y. K. (1974) The asymmetric response ofa fluid-filled spherical shell-a mathematical simulation of a glancing blow to the head. J. Biomechanics 7, 43-60. Culver. C. C., Neathery, R. F. and Mertz, M. J. (1972) Mechanical necks with human like responses. In Proc. 16th SrapP Car Crash Corlf, pp. 61-72. Detroit, MI. Culver, C. C. (1974) Personal communication. Engin. A. E. (1969) The axisymmetric response of a fluidfilled spherical shell to a local radial impulse-a model for head injury. J. Biomechanics 2, 324341. Goldsmith, W. (1966) The physical process in producing head injury. In Head Injury (Edited by Caveness, W. F. and Walker, A. E.) pp. 35&382. Lippincott, PA. Goldsmith, W. (1972) Biomechanics of head injury, In Biomechanics: Its Foundations and Objectives (Edited by Fung, Y. C., Perrone, N. and Anliker, M.), pp. 585-634. Prentice-Hall, NJ. Huelke, D. F. rt al. (1967) Automobile occupant injuries from striking the windshield. Ann Arbor. University of Michigan, Highway Safety Res. Inst. Rept. Kenner, V. H. (1971) On the dynamic loading of fluidfilled spherical shells-A head injury model. Ph.D. Dissertation. University of California, Berkeley. Kenner. V. H. and Goldsmith, W. (1972) On dynamic loading of fluid-filled spherical shells. Int. J. Mech. Sci. 14. pp. 557-568. Kenner, V. H. and Goldsmith, W. (1973) Impact on a simple physical model of the head. J. Biomechanics 6. l-l 1. Kraus, H. (1967) Thin Elastic Shells. Wiley, NY. Landkof. B. (1974) Impact on a head-neck structure. Ph.D. Dissertation, University of California, Berkeley, CA. Lee, Y. C. and Advani, S. H. (1970) Transient response of a head to torsional loading-A head injury model. Math. Biosci., 6, 473-486. Liu, Y. K., Chan. H. S. and Nelson, J. (1971) Intracranial pressure wave propagation in closed head impact. In Proc. Summer Computer Simulation Conf. pp. 984-994. Boston, MA. Melvin, J. W., McElhaney. J. H. and Roberts, V. L. (1972) Improved neck simulation for anthropometric dummies. In Proc. 16th Stapp Car Crash Co& pp. 4560. Detroit, MI. Melvin, J. W., McElhaney, J. H. and Roberts, V. L. (1973) Evaluationofdummy neck performance. In Human Impact Response (Edited by King, W. F. and Mertz, M. J.), pp. ?!89_3Z?O. Plenum Press, NY. Moisevev. N. N. and Rumvantsev. V. V. (1968) Dvnamic Stability ofBodies Containing Fluid (Edited by Abramson, N. H.) Springer. Berlin. Novozhilov, V. V. (1964) Thin Shell Theory. 2nd Edn. Noordhoff. Groningen. Ommava. A. K. and Corrao. P. (1969) Patholoaic biome&a& of central nervous system injbry in heai impact and whiplash trauma. In Proc. Int. Conf Accident Pathology, Government Printing Office, Washington, DC. Ommaya, A. K. (1973) Head Injury Mechanisms, NIH Tech. Rept. DOT HS-800959, October, 1973; National Institute of Neurological Diseases and Stroke, NIH, Bethesda. MD. Van der Neut, A. (1932) De Elastische Stahiliteit van der Dunwandigen Bol, H. J. Paris, Amsterdam, Holland.
Impact
on a head-neck
YOMENCLATURE N
A ‘4,, c E E,. El
F, Y
h
I0 I” K ,?I 171”
M
P Pf: R S S,
acceleration cross-sectional area coefficients for spherical harmonic series wave speed Young’s modulus elasticconstants for i-element viscoelastic beam model shell-fluid parameter, f = Rpr :hps force impact force acceleration of gravity shell thickness moment of inertia of neck centroidal moment of inertia of fluid-filled shell spherical Bessel function of order II bulk modulus length of viscoelastic beam index mass of fluid-filled shell moment order of spherical Bessel function Laplace transform parameter pressure associated Legendre function spherical coordinate midsurface shell radius area of application of impact force area of application of restraint force time
151
structure
11.N
c:. I. 11 9. z % ;’ t ‘I2 $0 4, 1.”
1’
li:...; @ V
displacement components displacement variables Cartesian coordinates shell parameter. ^/ = II’ 17R’ angle of rotation strain \ iscous constant in 3-element viscoelastic beam model capangleofuniform presxux distribution distributed cap an& of uniformI> constramt loadmg 11(11+ I I Poisson’s ratio dcnsit) nondimensional displacement5 lluid potential Laplacian.
Slrhsc~rip,t,s : H II 5 ,I .I. : 0. 4
at pomt -1. the shell- neck junction Iluid horirontal orderofLegendrePolynomialorspherica1 Bcsscl function shell vertical in directions .x. I in directions H. 4.
A comma after a variable denotes differentiation respect to the following quantity. A bar over ;I quantity denotes its Laplace transform,
with
ON A HEAlSNECK
STRUCTURE*-?
BENJAMINLANDKOF$and WERNER GOLDSMITH Department of Mechanical Engineering
and J. L. SACIiMAN Department of Civil Engineering. University of California, Berkeley. CA 94720 U.S..4
Abstract-An
analytical and experimental study was executed involving non-destructive. axisqmmetric impact on a fluid-filled shell constrained by a viscoelastic. artificial neck-a system intended to simulate a human head-neck structure. The shell material was considered to be homogeneous. isotropic and elastic, while the fluid was regarded as inviscid and compressible with its motion irrotational. The artificial neck was represented analytically by a linear viscoelastic cantilever beam that was rigidly connected to the shell. The physical model consisted of a water-filled uniform spherical thin-walled shell composed of lucite attached to an artificial neck developed by the Biomedical Laboratories of the General Motors Corporation. The system was loaded by the impact of a spherical aluminum shell with an o.d. of 7.4 in. that formed part of a simple pendulum. The impact force in both the analytical and experimental studies consisted of a half-sine pulse with a duration of 23 msec. The analytical solution first required the determination of the interaction forces and moments at the head-neck junction, which were ascertained from continuity conditions and an assumed rigid shell. These quantities, in turn, were applied simultaneously with the impact force to the deformable fluid-filled shell. yielding a boundary value problem that was solved on a digital computer. The resultant predictions of circumferential shell strain, fluid pressure, and displacement of the junction were found to be in good agreement with the experimental data. Implications of the results for head injury are discussed.
INTRODUCTION The field of head injury has been the subject of
numerous investigations and a variety of mathematical and experimental models purporting to represent the phenomena incident to such an event have been examined. It is generally recognized that different injuries will result from various types of loading, differing orders of magnitude of the duration, alternate directions of a blow, or disparate contact characteristics. Recent comprehensive reviews of the topic have been given by Goldsmith (1972) and Kenner (1971); these include classifications of the most frequently encountered types of head trauma and the nature of the loading causing such dysfunction. This paper is concerned with impact loading in the millisecond range on a model head attached to an artificial neck. This situation resembles that frequently encountered in head-on vehicular collisions when the head of a front-seated occupant strikes the windshield or the instrument panel. Since the duration of the applied force is large compared to the time required for a disturbance to traverse the head, which is of the order of 10&2OOmsec (Goldsmith, 1966),the properties of the neck and its junction to the head will play a vital role in establishing the ensuing motion. This is in * Receiwd 12 March 1975. t This work is based upon a portion
of a dissertation submitted in partial fulfillment of the requirement for the Ph.D. degree at the University of California, Berkeley. CA. U.S.A. 1 Current address: Armament Development Authority. Israel Ministry of Defense, Haifa, Israel. 141
contrast to the situation when the load duration is of the same order or shorter than the transit time; here, wave effects predominate, and the relative motion between the head and torso can be disregarded in establishing the mechanical head injury parameters. The latter domain has been investigated by Engin (1968), Benedict et a[. (1970). Lee and Advani (1970), Liu et al. (1971), Chan and Liu (1974). and Kenner and Goldsmith (1972, 1973). While some recent publications are concerned with the analysis of a fluid-filled spherical shell under longduration loading, no significant attempt has been made to correlate the induced intracranial pressure with the properties and constraining effect of a neck. As was indicated by Liu et ul. (1971) for long duration impact, the motion of the head can be decomposed into (a) translational and rotational motion, with corresponding accelerations, and (b) breathing motion resulting from stress wave propagation. While the former are strongly dependent on the properties of the neck, the breathing motion is virtually unaffected by the presence of this junction. Both types of motion will contribute to the intracranial pressure and both must be considered in the analysis of head injury when the loading period is of the order of milliseconds or longer. FORMULATION AND ANALYSIS OF
THE PROBLEM
The axisymmetric impact on a human head-neck system with a millisecond duration will be modeled
BENJAMINLANDKOF, WERNER GOLDSMITH and J. L. SACKMAN
Elastic
thin spherical head1
I
shell
viscid compeuible fluid brain and cersbrospinal fluid)
I
Head-neck
junctmn
-1
,,,,,
!i;
[,k),&Tnd
I neck
support
)
, torso,
Fig. 1. System schematic.
analytically by the representation shown in Fig. 1. The structure consists of a homogeneous, isotropic, elastic, uniform thin-walled spherical shell filled with a compressible, inviscid fluid that executes solely irrotational motion; the shell is rigidly constrained at its bottom point A by a linearly viscoelastic beam of circular cross section and length 1 that is clamped at its distal end B to a rigid support. The shell replicates the skull, the fluid purports to represent the contents of the cranial cavity, the viscoelastic beam is to model the artificial neck, joint A is regarded as the counterpart of the occipital condyles, and position B denotes the attachment of the neck to the torso. This unit is subjected to a radial pressure pulse of appropriate length, uniformly distributed over a spherical cap, with a resultant F, that is initially orthogonal to the centerline of the neck. Although the magnitude of the impact force is time-dependent, the spatial pressure distribution over a spherical cap of solid angle & will be considered as invariant. However, even though the external load passes through the center of gravity of the shell. the boundary value problem for this container will still be asymmetric due to the constraint exerted by the beam at position A. The analytical procedure requires the solution of two separate problems: (a) The interaction forces and moments at the junction A are computed first, and (b) these forces and moments are then applied together with the impact force to the fluid-filled shell and its response is obtained from the solution of this boundary value problem. In order to achieve the first part of the analytical objective, the following assumptions have been employed : (i) The total motion of the shell and its contents is assumed to be planar, occurring in the z - x, or sagittal plane (corresponding to back and forth motion); this is a consequence of the restrictions placed on the direction and point of application of the resultant impact force. (ii) The interaction forces and moments also lie in the sagittal plane, as implied by assumption (i).
(iii) The shell is considered to be rigid only for the purpose of determining the interaction components at the head-neck junction. This is justified since the contribution of the elastic elements to these forces and moments is negligible compared to the contribution of the rigid shell when the ratio of the impact duration to the fundamental period of free vibration of the fluid-filled system is greater than ten (Moyseyev and Rumyantsev, 1968). (iv) The angle of rotation of the shell y with respect to the beam is assumed to be small. It has been reported by a number of investigators (Huelke et al., 1967) that vehicle occupants anticipating a collision contract their neck muscles which would support the validityofthis assumption with respect to the prototype. (v) The motion of the spherical shell in the xdirection will be neglected relative to that in the zdirection. Figure 2 portrays the forces F and moments M acting on the components of the system; the equations of motion in the x - z (sagittal) plane are given by CF, = m,a,;
CF, = m,az;
CM, =
IodG,(1)
where m,, and I,, are the mass and moment of inertia with respect to the mass center 0 of the fluid-shell system which is regarded as rigid, y is the angle of rotation of the shell about 0 and a, and aZ are the accelerations of 0. From Fig. 2, these relations become 2
F&t)
-F,(t)
=m,
d2y
$(l.t)
+Rdtz
(2)
RF,(t)
-MA(t)
=Io$
and
F,=m,g,
where w is the deflection of the beam in the z-direction and R is the mean shell radius. The last of equations (2) is the consequence of assumption (v). Next, the quantities F,, FH and M, are applied to the viscoelastic cantilever beam. The differential equation for the transverse vibration of a simple linear viscoelastic Euler beam in the Laplace transformed space takes the form E(p)1 ‘2
+ pAp2 W - pw(x, 0) - GJ(x, 0) = 0, (3)
where p is the transform parameter, a bar denotes a transformed quantity, E(p) is p times the transform of the relaxation function of the material, still to be determined, and p, A and I are the mass density, cross-sectional area and moment of inertia of the beam, respectively. After applying the Laplace transform to the first two of equations (2) and employing the initial conditions y(0) w(x,O)
=2(O)=o; =
a;(x.o)=0,
(4)
143
Impact on a head-neck structure
and F
INF,(r) ___---.
(p) = (1 +
ff
(10)
A
where &I, N and A are explicit
functions of the transform variable p and viscoelastic function E(P) (Landkof, 1974). An initial requirement for the solution of the boundary value problem is the establishment and evaluation of the function E(p). In this endeavor, attention will not be focused on the precise distribution of the stresses along the neck itself, but only on the specification of a model that will permit the ready calculation of the interaction forces and moments. On
MA
FH
T
l?%llL the basis of several previous investigations involving the neck, (Melvin et a/.. 1972; Becker. 1973; Melvin et n2., 1973) a three-parameter one-dimensional linear model consisting of a spring of modulus E, in series
F”
e
A
with two parallel elements, another spring of constant E2 and a dashpot with coefficient q2 was chosen as
a reasonable representation for this component. For such a system, it can be readily shown (Landkof, 1974) that
Fig. 2. Free-body diagram of the head-neck system.
(11)
the following set of equations is obtained:
‘&)R
- GA(P) = p21,Xp> The next task is the determination of the parameters
F&p) - F,(p) = r~~o~~~W(I,p) + R?(P); &)I
‘; +pAp"k =0.
(5)
As indicated in Fig. 2, the last of equations (5) must also satisfy the following boundary conditions: W(O,p) = d;(o.p)
= 0 = AJ,i,(p)
-E(P)I
d$(I.P>
=
F,(p)
Finally, in order to conform to the requirements of a well-posed problem, a continuity condition must be established between the shell and the viscoelastic beam at point A. It will be assumed that the angle of deflection of the beam equals the angle of rotation of the shell with respect to the beam, y. Then
g(1.1,
= tarry-
y(t),
(7)
or. in the Laplace space,
E,, El and q2 in such a manner that the analytically computed interaction forces and moments would fit the experimental results. This was accomplished with the aid of experimental data supplied by General Motors Corporation that also provided the artificial
neck used in the experimental portion of the present study. The data had been collected from a series of tests in which the neck was used as a part of a Sierra 1050 dummy (Culver et al.. 1972; Culver, 1974); the results were subjected to an identification procedure for the evaluation of these constants by repeatedly correcting the values of an initial guess, using a computer program based on the method of least squares(Landkof, 1974). until a reasonabte tit with the data was obtained. The magnitudes derived for the parameters were as follows: E, = 2986 psi. E, = 2501 psi, v2 = 1294 psi-sec. The moment M, and force F, at the junction could now be evaluated from equations (9 and 10). A spherical coordinate system r, 4. 0 was utilized in the analysis of the response of the shell whose undeformed midsurface radius is given by r = R. Since the impact force was assumed to be uniformly distributed over a spherical cap of angle (ba,
(8) F, =
With the aid of conditions (6 and 8). the system of equations (5) yields M
(p>
A
=
(R - WF,(P) A
(9)
F,(t) 0
forO<~<~,.OiC)<27t otherwise
).
(12)
In a similar manner, it will be assumed that the timedependent interaction force F, and moment M, are also uniformly deployed over some spherical cap
BENJAMIN LANDKOF,WERNERGOLDSMITH and J. L. SACKMAN
I44 around
the junction. F
M
=
H,
A
Thus, F,(t),M,(t)
for 1127~- q& c C$< 1/2x + q&,z - e1 -c 6 c n + t+
0
otherwise
The vertical force F, which balances the weight of shell and contents, does not contribute to the dynamic strains and pressures and does not appear in the subsequent analysis. The equations of motion for the shell are chosen as linear and incorporate membrane and bending effects, but neglect rotatory inertia and transverse shear. The fluid is characterized by a linear wave equation that includes compressibility, but neglects viscosity. The governing relations are (1 + m2) -u,sin
1 --u sin3~
3-v - 2uB,Bcot4
sin4(cot2$
cot Cp 1 lJ 8,888- sin %,e, + ~ sin 6 B*Bbo
+ 2(1 + v)w - cot 4 (2 - v + cot%$)w, + (1 + v + cot+$)w,&, - 2w,,, - WM!M - & 1 - sw*eee
4 (cot24 + v) + u~,# cos f$
+U&&Sin~+
l-v 1+v ~ %,ee + -%,&J 2 sin I$ 2
2cosC$ + .3W,ee# sm d,
+ (1 + v)w,#sin4
‘$+ ps ;
‘ee@I
(R, r$,e, t) -
and
a
1 2 a@ -_ r Z +r2sin~&$
( >
+
(14)
a
.
sm+g
(
a4
>
1 a% --=$$ r sin I$ a82
(17)
where c = (K/p,)“’ is the compressional wave speed in the fluid, v is Poisson’s ratio for the shell, p, and ps are the densities of the fluid and shell, @ is the fluid potential, K is the bulk modulus of the liquid, S and S1 are the areas of application of the impact and restraint forces, and u+, u, and w are the displacements of the shell mid-surface in the 4, 0, and r-directions, respectively. It will also be assumed that no separation of the fluid from the shell will occur, i.e. that normal velocities across the boundary will be continuous. Thus,
;<4,&9 = $ CR,4>& 0. Finally, the initial conditions ~(6
cos I#) + 2Ub& cot 4 + U&,& + -u sin3~ ~,es 1
+ .2 %l,et@+ sin 4
2 sin24 + 1 sin34
Ue,e
2
& =hZ 12R2
- 1) w) - W~~COSC$
sin C$
2 - TW an 4
with
-r2 ar
Eh
cot 4
(3 - V + 4 cot24)W*e
psh
1
X-
(13) 1.
60)
are given by
= a; (4,8,0)
= 0
um(4,e,oj
= a~(s,e,o)
=o
2.4(4,e,o)
=~(4,e,o)
= o
Wr, he, 0) = g Cr. 44 e, 0) = 0. The set of partial differential
(18)
equations
(19) together
with
Fig. 3. Photograph
(Fmitz,q p. 144 )
of the target.
Impact
on a head
the boundary conditions (18) and initial conditions (19) constitutes the boundary value problem. In order to simplify the relations, one of the equations cited above can be uncoupled by means of a substitution first used by Van der Neut (1932) ug = U,,,/sin C#I : w = W. (70)
lid, = u., - I/ sin 4,: Upon the dimensional
additional variables $ = U/R:
employment < = V/R;
the following set of equations
of
the
< = WJR
non-
neck structure
14.5
in spherical coordinates. It is evident that equation (23) is uncoupled and can thus be solved for variable < since FH and MA are known. The solution of the set of equations (17 and 22- 24) is accomplished in the usual manner by first applying a Laplace transform with respect to time t and then representing the result as a sum of spherical harmonics. yielding
(31)
is obtained
x Pf( cos 4) cos 010, (29)
(22) (1 + sc’)(l - \I)@2 + 2)
Fl(h 0,P)
1’2
1 -
= -
psR’
-------+
2
EhS(Md
t- F,R)cscd
1
[xZV4 + V”[2r’ z2 sin4
h-V2 + 4cosdV2 @J
i[ x (1 + a’)(1
+ v) i -
1 -
=
I
”
n=”
m=O
c c
~“H(&e,P>csc$=i
- (1 + a2)(1 + \I)]) lj
. -
where Pr(
E,,(p)
P;(cos
c#i,cos }?d
(30)
f: R,,(p)
n=o
m=O
x P:: (cos 4) cos mf? (31)
2cos~P
M,(&
0.1’) csc$
=
sin& [(l + “~‘)(l + 1’)
i i F,,(p> II=1 m=O
i
x P!( cos 45) cos mfl.
(I - v)V’]
(32)
The uncoupled equations (17 and 23) are now solved separately. yielding in terms of spherical Bessel functions ,j, with the aid of equation (28)
(24) with initial and boundary !Kb. 0.0) = $
conditions
given by
(4. 0,O) = 0
and
^IY
@(P,4.0,0)
= $Y#Au,o,
= 0 where the recurrence
and
v2{P;(cosf$)
cosmfl)
formula = -?t”(P~(cos~)cosrllH~
(25) Equation operators
(17) remains unaltered. The Laplacian V2 and V4 in equations (22-24) are expressed
with 1, = n(n + 1) (35) and the boundary conditions have also been utilized.
between
fluid and shell
46
BENJAMIN LANDKOF, WERNER GOLDSMITH and J. L. SACKMAN
Upon substitution of equations (2629) the other differential equations (22 and 24) are transformed into two algebraic equations in the two unknowns B,,,, and B,,, with I?,,,, given by equation (34). The solution of this set is obtained by Cramer’s rule yielding, with
&o(p)
= RM
EhS
x E,,
The numerical values of the desired strains and pressures were obtained in the following manner: (i) The experimentally determined input pulse F,(t)
.io
2(1 + r) + j),R’ !+p2
(i~R/d
- ’
.i;l (@R/c)
(36)
, was approximated by a sinusoidal function so that
F,(t) =
(lb)
286sin St 0
0 5 t 5 2.5 (msec)
t > 2.5
(47)
with time t in msec. (ii) The inversion of the Laplace transformed quantities $, [. l, 3 was performed numerically by means of the Gaussian quadrature procedure (Bellman and Kalaba, 1964). (iii) The mid-surface strains es (($4, t) and Q, (0, 4. t) were obtained by means of the following relations c = $,+4 - icos4
_R1-v2_
-E El&
Eg= $,&,cot 4 - < cos 4 + $,&+csc’4 + 5.
where Ati = ~1nP2n - ~nfjin
(39)
(1 + a*) (1 - v - A,) - l$p.R’p2
(40)
and z 1”
=
2,=cr2R,Z-a1,120r2-(f+a2)(f
CI
+
I’))
pi” = (1 + N2)(1 + V)- Y2(2 - 2,)
(41) (42)
P2n = -2(1 + v) - s12[A,”- a,(1 - v))
- psR’
Ylmn
=
(43)
=E{(n-
I)[-3L,a2+(1+cr2)
(44)
(1 + 1’)- 3x’] - 4i,,r2 - 2(1 + x2)(1 + v)) 6 InIll=
(1 + a2)(t1 - nr + 1) : 4 - (n + 2) (1 + 1)) 2(2n + 3) (45)
6
II +
2mn=
111 +
211+ 3
1
(48) (49)
The differentiation was performed with the aid of well-known recurrence formulas. (iv) The pressure P was obtained by numerical differentiation of the fluid potential and employment of the relation p=
-p,g
(50)
(v) In the computational process, evaluation of the spherical Bessel functions’ and associated Legendre polynomials was required. The higher order Bessel and Legendre functions were obtained by utilizing appropriate recurrence relations. When required, integration of the associated Legendre polynomials was performed by means of a library routine that used Simpson’s rule of integration. EXPERIMENT’
(1 + x2) (n - 111) j4+(n--l)(l+V)j 2(2n - 1)
Yam
- i,+ sin4 + 5
[(n + 2) [&x2 - (1 + a2)(1 + v)
+ 2a*] - 4&x2 - 2(1 + x2)(1 + v)). (46)
The target for the impact tests. shown in Fig. 3, consisted of a thin-walled spherical lucite shell filled with distilled water that had been used in previous investigations (Kenner and Goldsmith, 1973), an aluminum adapter connecting the shell with the artificial neck, and a heavy steel plate serving as the base support. The input to the system was measured by (l/32)-in. thick X-cut quartz transducers of (l/2)-in. dia. and sandwiched between short stubs of aluminum, also of the same diameter; the units were assembled with conducting epoxy and cemented to a point on the equatorial plane of the shell with structural epoxy. The pertinent mechanical and geometric properties of the shell are as follows: Weight density psg = 0043 lb/in3; Young’s Modulus E = 0.67 x lo6 lb/in2, Poisson’s
137
Impact on a head-neck struch~re ratio v = 0.38. mean midsurface radius R = 3.636 in.; wall thickness h = 0.146 in.; thus R/h = 24.9 and the loading angle Q!Q,= tan-’ (0,25/R) = 3.94”. The adapter was constructed in the form of a boss with an upper, scooped-out surface of 1.0 in. dia. to which the shell was glued. and a lower surface with a dia. of 2.5 in. featuring four holes through which (l/4)-in. dia. bolts were fastened to the neck. The latter consisted of a steel and molded polyurethane structure exhibiting four ball-jointed components and one upper. pin-connected ‘nodding’ segment; the plastic material connecting the articulated units provided both bending resistance and energy dissipation. The ball joint used a split-ball concept permitting both neck compression and elongation. Rotation about the longitudinal axis of the neck was also permitted, but opposed by friction between the resistive elements and the loading disks. The unit weighs 3.1 lb. has a total length of 5-5 in. and exhibits an outer diameter of 3.4 in. The lower end of the neck was attached to a 22 lb steel plate with dimensions of 3,‘4 x IO x 10 in., serving as a model of the torso. by means of four (l/4)-in. dia. steel bolts. In a few tests, the artificial neck was removed, and the shell was either ‘freely suspended’ by placement upon a circular support of sheet aluminum (Kenner and Goldsmith, 1973) or else directly bolted to the steel plate serving as the base for the system. Strain histories were measured both on the inside and outside surfaces of the shell by means of epoxymounted BLH SPB2-12-12 semiconductor gages with gage factors of I 16.7 and resistances of 121 Q both in the meridional (or hoop) direction H at (I) d, = 30. 0 = -60 and (2) (11= 150’. 0 = 60. and in the circumferential direction 4 at (1) 4 = 30-, 0 = 270’. (2) 4 = 150’. If = 90 and (3) Q, = 180”. These gages were incorporated in a potentiometric circuit so as to give average strain b! connecting corresponding inner and outer transducers in series. Pressures in the water were measured by means of O.lOO-in. dia. gold-plated Z-cut tourmaline crystals, OWI in. thick. located in one shell hemisphere alt nominal non-dimensional radii ? = r;‘R of 0. 0.25, 0.5, 0.75 and 0.875 from the shell center. To obtain the pressure records for the second hemisphere. the impact position was moved 180’ and the experiment was repeated. The piezoelectric constant for the tourmaline crystals was found to be 10.75 x 1OY” C;lb and that for the input quartz crystal was determined as 10.X x 10-l’ C/lb. Details of the calibration procedure for the transducers are given in Kenner and Goldsmith (1973). The output signals from all detecting devices were displayed and photographed on Tektronix 565 oscilloscopes using amplifiers that exhibited a bandpass with a drop of 3 dB at a frequency of 1 MHz. The lowfrequency response of the piezoelectric transducers was improved by adding a 10,000 ppf capacitance in parallel to the Input crystal and a 500 /L/If capacitance in parallel to the pressure gages. Transient phenomena were initiated by the impact on the structure of a spherical aluminum shell with an equatorial o.d. of
z Y)
5 ; + 1000
w”-
-3000
_ $=30” e.-600
0
1
2
3
ANALYSIS
I
"0
fl
Ll
EXPERIMENT
oooooo
L
5
1
6
-J
7
Fig. 4. Comparison of predicted and measured hoop strain E,+in the shell: 4 = 30 1 fl = 770 7.40 in. and a wall thickness of 0141 in. that was incorporated in a simple pendulum arrangement. The drop height was adjusted so as to yield a peak force of 286 lb that corresponded to a duration of 2.5 msec; triggering was accomplished by the signal from the input transducer. In a few tests, the displacement of the shell-neck joint was determined by means of a dial gage located at that position whose excursion was photographed by a Fastax intermediate speed movie camera operating at a nominal framing rate of XOO/sec. the actual speed being indicated by timing marks.
Figures 4 and 5 provide representative comparisons between the experimental average strain results and analyticalmidsurfacepredictionsfor thecircumferential strain E# and hoop strain E, at two different positions. The agreement between analysis and data is reasonable during the loading interval, but deteriorates beyond. Possible reasons for such discrepancies include the approximation of the average strain as being equivalent to the midsurface strain (exaggerated when the inner and outer surface strains are of different signs). the
-
EXPERIMENT
oaoo
ANALYSIS
1 6
7
Fig. 5. Comparison of predicted and measured cn-cumferentlal strain Ed,in the shell: Q, = 30 I! = - 60
BENJAMIN LANDKOF,WERNERGOLDSMITHand J. L. SACKMAN (iii) Generally, the loading
_
EXPERIMENT
____.
ANALYSIS
, n-at la)
10 0 -10
II-20
0
1
2
3 t
L ,Timc
5
6
7
, msec
6
9
pressure develops
during
up to 25 msec in the hemisphere nearest the impact point. while the pressure in the distal hemisphere is slightly negative. At the end of the loading process, the pressure in the nearer hemisphere also becomes negative. Figure 9 shows a crossplot of the pressure distribution along the impact axis for various times after contact. It is evident that the maximum positive and negative pressures in the system develop near the instant of the maximum impact force, at t = 1.25 msec, when the acceleration in the z-direction is also near its maximum. This circumstance would provide the explanation for the extreme magnitudes of these pressures at that particular time. At the end of the loading period, for t > 2.5 msec, deceleration of the fluid-filled shell will occur that is entirely controlled by the mechanical properties of the artificial neck. At this instant, a negative pressure is initiated in the frontal hemisphere. The maximum value of this negative pressure is of the order of 24 psi and
t , Time
:
a positive
period
10
(b) ._
206
k
10
::
F.0.19
,
,
(
,
,
,
,
z 2 0 h ;
20
-10 -20
t
‘j; P 0123L5676
msec
t , Time,
9
10
(c) Fig, 6. Comparison of predicted and measured pressures in the fluid: (a) i = 0.84, (b) i = 0.71, (c) i = 0.49.
behavior
computational
and
of the shell that was neglected
numerical
procedure.
approximations
e :,
0
rk
-10
o
-20
in
in the
In general, the strain data conform to the results found previously (Kenner and Goldsmith, 1973) in that the records from the hemisphere around the impact point closely resemble the shape of the input pulse with a rapidly attenuated tail, but no such correlation exists in the distal hemisphere where the levels are much lower. A similar comparison of theoretical predictions and experimental results for the fluid pressure at nine different stations along the impact axis is presented in Figs. 6-8; agreement is considered to be reasonable. The general characteristics of the pressure history in the fluid are as follows: (i) The pressure records obtained from the experiment show a general pattern of low frequency vibrations due to rigid-body shell motion with superposed high frequency fluctuations resulting from wave propagation in the fluid. The latter disturbances disappear after a few milliseconds as would be expected. (ii) The high-frequency oscillations are not present in the analytical pressure curves due to the use of a time increment of excessive size.
ANALYSIS
iz0.27
1
2
3
5
t , Time,
viscoelastic
EXPERIMENT
--__
10
0
the analysis.
-
6
7
6
7
6
7
rnsc (a)
9
10
6
9
10
6
9
10
i=O 20 .g
10
2 r‘
0
E k
-10
Ii -20
0
1
2
3
_L t , Time,
5
msec
(b) F=-0.27
20 ‘5 =
10
t :
0
1 k
-10
a -20 I 1
II 2
3
I 4 t ,Twne
II 5
I msec (cl
Fig. 7. Comparison of predicted and measured pressures in the fluid: (a) 7 = 0.27, (b) i = 0. (c) i = -027.
Impact
t .Time
,
on a head-neck
-
EXPERIMENT
__--
ANALYSIS
I49
structure
msec
la1 i,
Nondlmensmnol
Radws
Fig. 9. Pressure distribution along the impact dimensional radius i: for various times after impact.
,
msec
4 5 t ,Time,
nmc
t ,lime
(b)
-10
0
I
2
3
6
7
e
9
IO
(c) Fig. 8 Comparison of predicted and measured pressures the fluid: (a) T = -049. (b)? = -0.71. (c)i = -0.84.
in
occurs in the distal hemisphere here at two different locations: initially at t = 15 msec at a non-dimensional radius i- = -0.27. and then at 6.4 msec at r = -0.49 where its duration is considerably greater than at the first location. The tests involving the three different boundary conditions for the shell were recorded by means of a triple exposure so that the response of each of the transducers for the three different conditions appeared on the same photograph. The results indicated only a very slight variation in both amplitude and duration of the input pulse. Furthermore. the pressures measured during the first two or three milliseconds in the shellneck and freely suspended shell systems were of relatively low amplitude, while that in the rigidlyattached shell system rose rapidly and to higher magnitudes, as expected. Figure 10 presents a comparison of the experimentally determined displacement of the head-neck junction and the corresponding analytical prediction. The latter was obtained from the solution of the last of equations (5) and the appropriate boundary conditions, again using the Laplace transform and a Gaussian quadrature inversion. The values employed for the required material and geometric quantities were: p.4 = 1.46 x 10e3 lb- sec’/in’. I = 0.0688 in”. 1 = 53 in. The correspondence appears to be quite good; the
axis vs nonbeginning 01
smoothness of the experimental curve is probably due to the inertia of the dial gage employed. The present effort was primarily directed towards the determination of the intracranial pressure variation under conditions of impact to the model. so that its significance with respect to possible brain injury mechanisms might be appraised. The group adhering to the cavitation theory of brain trauma have claimed that there exist locations within the cranial cavity where the negative pressure developed in the cerebra spinal fluid would be sufficient to rupture the capillary walls. According to the damage criterion for closed brain injuries postulated by Liu er (I/. (1971). the severity of the brain injury at a given location is proportional to the average time of negative pressure exertion beyond a critical level at a given position. Based on this criterion, it appears that the potential location of the most severe brain injury in the present model will be at J = -0.49. at an intermediate coup position. This result agrees qualitatively with the predictions of several other investigators (Ommaya and Corrao. 1969: Ommaya, 1973; Hickling and Wenner, 1973). Comparisons of the results of the present investigation with those for the free shell (Kenner. 1971) indicates that the shape of the strain curves at b, = 30 produced by the impact of a 2-in. dia. solid steel sphere closely resemble the present data. The peak magnitudes found -ANALYSIS 6
____
4
t , Time,
Fig. 10. Predicted
EXPERIMENT
msec
and measured displacement head-neck junction.
history
of the
50
BENJAMINLANDKOF,WERNER GOLDSMITHand J. L. SACKMAN
in the previous generated pared
tests are lower since the impact
was smaller. The pressures
since the current
trailsient
disturbances
investigation and rigid-body
the previous work was concerned
force
can not be comdeals with both motion,
whereas
only with the former
event.
CONCLUSIONS
The principal conclusions of the present study involving impact on a model human head-neck system are: (i) Reasonable agreement was obtained between theoreticalpredictionsandexperimentaldataofcircumferential and hoop strains and fluid pressures in a waterfilled lucite shell attached to an artificial neck under conditions of axisymmetric impact for a loading period of 2.5 msec. Poorer correlation at later times may be due to the lack of equivalence of measured average strain and calculated midsurface strain, neglect of the viscoelastic properties of the shell in the analysis, or numerical errors in the computational procedure. (ii) The experimental and corresponding numerical results of the pressure histories indicate that the peak negative pressures in the fluid are observed mainly in the distal hemisphere during the impact period, while positive pressures develop concurrently in the frontal hemisphere. Beyond the end of impact, negative pressures are also generated at the front. (iii) The maximum negative pressures appear at two locations in the distal hemisphere, each at a different time; the one nearer to the center (F = -0.27) occurs during loading, while the other (F = -049) is encountered during the deceleration period controlled by the artificial neck. The duration at the second position is much longer and hence more serious for potential brain injury based on a cavitation hypothesis. (iv) The mathematical description of the artificial neck could be reasonably accomplished by means of a three parameter elastic-viscous model. (v) Good correlation was obtained between the analytical and measured displacements of the headneck junction for a period cu. 15 times the contact duration. The smoothness of the experimental curve is attributed to the inertia of the dial gage employed. (vi) The results indicate that the neck constraint has considerable influence on the pressure distribution in the fluid modelling the brain and can not be ignored when impact forces with durations of the order of a few milliseconds or longer are applied to the fluidshell system examined.
Acknowledgements-The senior author was the recipient of a Government of Israel scholarship during the period of the present investigation. The financial support of JTCG/ME and of the Office of Naval Research for the conduct of the research is sincerely appreciated. The authors are also grateful to the General Motors Research Laboratories, Warren, Michigan, for the gift of the artificial neck
REFERENCES Becker, E. B. (1973) Preliminary discussion of an approach to modeling living human head and neck to -G, impact acceleration. In Human Impact Response (Edited by King, W. F. and Mertz, M. J.), pp. 289-320. Plenum Press, NY. Bellman, R. E. and Kalaba, R. E. (1964) Modern Analytic and Computational Methods in Science and Mathematics. Elsevier, NY. Benedict, J. V., Harris. E. H. and von Roseberg, D. U. (1970) An analytical investigation of the cavitation hypothesis of brain damage, 1. Basic Enana. ” ” Trans. ASME. 920. 597-603. Chan, H. S. and Liu, Y. K. (1974) The asymmetric response ofa fluid-filled spherical shell-a mathematical simulation of a glancing blow to the head. J. Biomechanics 7, 43-60. Culver. C. C., Neathery, R. F. and Mertz, M. J. (1972) Mechanical necks with human like responses. In Proc. 16th SrapP Car Crash Corlf, pp. 61-72. Detroit, MI. Culver, C. C. (1974) Personal communication. Engin. A. E. (1969) The axisymmetric response of a fluidfilled spherical shell to a local radial impulse-a model for head injury. J. Biomechanics 2, 324341. Goldsmith, W. (1966) The physical process in producing head injury. In Head Injury (Edited by Caveness, W. F. and Walker, A. E.) pp. 35&382. Lippincott, PA. Goldsmith, W. (1972) Biomechanics of head injury, In Biomechanics: Its Foundations and Objectives (Edited by Fung, Y. C., Perrone, N. and Anliker, M.), pp. 585-634. Prentice-Hall, NJ. Huelke, D. F. rt al. (1967) Automobile occupant injuries from striking the windshield. Ann Arbor. University of Michigan, Highway Safety Res. Inst. Rept. Kenner, V. H. (1971) On the dynamic loading of fluidfilled spherical shells-A head injury model. Ph.D. Dissertation. University of California, Berkeley. Kenner. V. H. and Goldsmith, W. (1972) On dynamic loading of fluid-filled spherical shells. Int. J. Mech. Sci. 14. pp. 557-568. Kenner, V. H. and Goldsmith, W. (1973) Impact on a simple physical model of the head. J. Biomechanics 6. l-l 1. Kraus, H. (1967) Thin Elastic Shells. Wiley, NY. Landkof. B. (1974) Impact on a head-neck structure. Ph.D. Dissertation, University of California, Berkeley, CA. Lee, Y. C. and Advani, S. H. (1970) Transient response of a head to torsional loading-A head injury model. Math. Biosci., 6, 473-486. Liu, Y. K., Chan. H. S. and Nelson, J. (1971) Intracranial pressure wave propagation in closed head impact. In Proc. Summer Computer Simulation Conf. pp. 984-994. Boston, MA. Melvin, J. W., McElhaney. J. H. and Roberts, V. L. (1972) Improved neck simulation for anthropometric dummies. In Proc. 16th Stapp Car Crash Co& pp. 4560. Detroit, MI. Melvin, J. W., McElhaney, J. H. and Roberts, V. L. (1973) Evaluationofdummy neck performance. In Human Impact Response (Edited by King, W. F. and Mertz, M. J.), pp. ?!89_3Z?O. Plenum Press, NY. Moisevev. N. N. and Rumvantsev. V. V. (1968) Dvnamic Stability ofBodies Containing Fluid (Edited by Abramson, N. H.) Springer. Berlin. Novozhilov, V. V. (1964) Thin Shell Theory. 2nd Edn. Noordhoff. Groningen. Ommava. A. K. and Corrao. P. (1969) Patholoaic biome&a& of central nervous system injbry in heai impact and whiplash trauma. In Proc. Int. Conf Accident Pathology, Government Printing Office, Washington, DC. Ommaya, A. K. (1973) Head Injury Mechanisms, NIH Tech. Rept. DOT HS-800959, October, 1973; National Institute of Neurological Diseases and Stroke, NIH, Bethesda. MD. Van der Neut, A. (1932) De Elastische Stahiliteit van der Dunwandigen Bol, H. J. Paris, Amsterdam, Holland.
Impact
on a head-neck
YOMENCLATURE N
A ‘4,, c E E,. El
F, Y
h
I0 I” K ,?I 171”
M
P Pf: R S S,
acceleration cross-sectional area coefficients for spherical harmonic series wave speed Young’s modulus elasticconstants for i-element viscoelastic beam model shell-fluid parameter, f = Rpr :hps force impact force acceleration of gravity shell thickness moment of inertia of neck centroidal moment of inertia of fluid-filled shell spherical Bessel function of order II bulk modulus length of viscoelastic beam index mass of fluid-filled shell moment order of spherical Bessel function Laplace transform parameter pressure associated Legendre function spherical coordinate midsurface shell radius area of application of impact force area of application of restraint force time
151
structure
11.N
c:. I. 11 9. z % ;’ t ‘I2 $0 4, 1.”
1’
li:...; @ V
displacement components displacement variables Cartesian coordinates shell parameter. ^/ = II’ 17R’ angle of rotation strain \ iscous constant in 3-element viscoelastic beam model capangleofuniform presxux distribution distributed cap an& of uniformI> constramt loadmg 11(11+ I I Poisson’s ratio dcnsit) nondimensional displacement5 lluid potential Laplacian.
Slrhsc~rip,t,s : H II 5 ,I .I. : 0. 4
at pomt -1. the shell- neck junction Iluid horirontal orderofLegendrePolynomialorspherica1 Bcsscl function shell vertical in directions .x. I in directions H. 4.
A comma after a variable denotes differentiation respect to the following quantity. A bar over ;I quantity denotes its Laplace transform,
with