Frequency response analysis of the in-vivo human skull
by Gerald Martin Grammens
A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE
in Mechanical Engineering
Montana State University
© Copyright by Gerald Martin Grammens (1977)
Abstract:
In this study, a finite element model of the human skull is presented. The intent is to explore the
feasibility of determining in-vivo material properties of the human skull by frequency response
analysis.
Qualitative results are obtained by approximating material and geometrical parameters for the skull.
Initially, boundary conditions were chosen to prevent rigid body translation by securing nodes about
the mid-plane. The axisymmetric mode shapes obtained reveal resonance of the fundamental bending
mode at 3700 Hz.
To account for this disparity of results compared with those obtained by previous investigators, an
end-mounted support at a single node was imposed to replace the mid-mount boundary condition. This
forced boundary condition was of the type employed in these previous studies. For the same skull with
the end-mount boundary condition, the first resonance occurred at 440 Hz. The corresponding
fundamental bending mode shape can best be described as a rigid body motion of the bulk of the skull
with a local deformation at the fixed node. Recognition of the nature of this mode shape raises serious
implications concerning past interpretations of such results.
The qualitative results of this study suggest that either of the two mode shapes may prove helpful in
determination of skull material properties, but not without major experimental and analytical
difficulties. STATEMENT OF PERMISSION TO COPY
In p r e s e n t i n g t h i s , t h e s i s in p a r t i a l f u l f i l l m e n t o f th e r e q u i r e ­
ments f o r an advanced degree a t Montana S t a t e U n i v e r s i t y , I agree
t h a t t h e l i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r i n s p e c t i o n .
I
f u r t h e r agre e t h a t pe rm is si on f o r e x t e n s i v e copying o f t h i s t h e s i s
f o r s c h o l a r l y purposes may be g r a n t e d by my major p r o f e s s o r , o r , in
h i s a bs e nc e, by t h e D i r e c t o r o f L i b r a r i e s .
I t i s unde rs too d t h a t
any copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l
no t be al low ed w i t h o u t my w r i t t e n p e rm is si on.
Signature
Date
3
FREQUENCY RESPONSE ANALYSIS OF THE IN-VIVO HUMAN SKULL
by
GERALD MARTIN GRAMMENS
A t h e s i s su bmitted in p a r t i a l f u l f i l l m e n t
o f th e require me nts f o r t h e degree
of
MASTER OF SCIENCE
in
Mechanical Engineering
Approved:
Chairman, Graduate Committee
Head, Major Department
Graduate Dean
MONTANA STATE UNIVERSITY
Bozeman, Montana
March, 1977
-Mi
ACKNOWLEDGMENT
The a u t h o r wishes t o e x p re ss h is a p p r e c i a t i o n t o Dr. Dennis
B l a c k k e t t e r f o r t h e s u g g e s t i o n o f t h i s t h e s i s t o p i c and f o r his
a s s i s t a n c e th ro u g h o u t i t s e v o l u t i o n .
iv
TABLE OF CONTENTS
Page
V I T A ................................................ ..............................' .................................... .... . i i
ACKNOWLEDGMENT............................................ ............................................... .... . . i i i
TABLE OF CONTENTS. . ............................................................................................... iv
LIST OF TABLES....................................................
vi
LIST OF FIGURES................................................
vii
ABSTRACT......................................................................................................................... ix
INTRODUCTION . . . . .
............................................................................... . .
I
CHAPTER I — LITERATURE REVIEW ......................................................................
4
CHAPTER 2 — THE HUMAN SKULL...............................................
„.13
Bone.............................................................
13
Macroscopic P r o p e r t i e s ...............................I . . .................................. 14
Bone D y n a m i c s ............................................................................................ . 15
The S k u l l ................................................ .... . . . ...................... ....
16
CHAPTER 3 — MODELING CONCEPTS.......................... .... . . ...........................21
Geometry G en eration ................................................
21
C oord ina te R e d uc tio n.......................................................................... .... . 23
Inpu t P a r a m e t e r s ...............................................................................................27
Boundary Co nditi ons ................................................................................... 28
Other A s p e c t s .......................................
29
CHAPTER 4 — FINITE ELEMENT THEORY . . . . .
.......................................
30
CHAPTER 5 — FREQUENCY RESPONSE......................................................................... 33
CHAPTER 6 — RESULTS................................................................................... .... . 36
T e s t Geometry R e s u lt s ............................................................................... 36
Skull R e s u l t s ................................................................................................... 39
V
Page
CHAPTER 7 — DISCUSSION.....................................................................
47
Comparison t o Previous S t u d i e s .......................... .............................. . 47
A n a l y t i c a l Comparison ............................................................................... 49
Impl i c a t i o n s ........................................................................................................54
F e a s i b i l i t y f o r D et ermination o f
Rheological P r o p e r t i e s ................................................... 56
Summary.............................................................................................................. . 5 9
REFERENCES.....................................................................................................................60
APPENDIX I
— SHELL ELEMENT .............................................................
64
The I n - p l a n e Element..............................
64
The Bending E l e m e n t ...................................
71
She ll E l e m e n t ................................................................................................... 74
APPENDIX I I — FLOW DIAGRAM FOR REDUCED MASS
AND STIFFNESS GENERATION ............................... 75
vi
LIST OF TABLES
Table
6-1
Page
Comparison o f F i n i t e Element R e s u lt s to Closed .
Form S o l u t i o n f o r a C i r c u l a r R i n g ............................................. 3 7
vii
LIST OF FIGURES
Figure
Page
2-1
Typical Skull
3-1
S p h e ri c a l C oordi na te s . .
3-2
Nodal Nomenclature. . ..............................................................................24
6-1
Grid P a t t e r n f o r C i r c u l a r Ring Te s t Geometry..........................38
6-2
Comparison o f F i n i t e Element t o Closed Form
R e s u lt s f o r th e Fundamental Bending Mode.
o f a Thin S p h e ri c a l S h e l l . . . . . . . . . . .
6-3
6-4
.................................... ................................... .... , . 18
................................... . . . . . .
,22
. . . 38.
Calvarium to be M o d e l e d ..................................................................... 40
Frequency Response f o r Mid-mounted Sphere ...................... .... . 41
6-5
Mode Shape f o r Mid-mounted Sphere a t 3700 Hz. . . .
...
. 41
6-6
Frequency Response f o r End-mounted S p h e r e ................... ...
6-7
Mode Shape f o r End-mounted Sphere a t 440 Hz........................ . 43
6-8
Frequency Response f o r Mid-mounted E l l i p s o i d . . . .
6-9
Mode Shape f o r Mid-mounted E l l i p s o i d a t .4160 Hz..................... 44
43
. .. 4 4
6-10 Frequency Respopse f o r End-mounted E l l i p s o i d ..........................45
6 - 11 Mode Shape f o r End-mounted E l l i p s o i d a t 530 Hz..................... 45
7 - 1 Nickel! & M ar c a l' s Rear-mounted Skull ( 1 s t Mode)................ 51
7-2
Nick ell & M ar ca l1s Rear-mounted Skull (2nd Mode).
7-3
Nick ell & M ar ca l1s Rear-mounted Skull (3rd Mode). . . . .
I-I
1-2
I n - p l a n e Deformations At Node j f o r a Typical
Element Shown in Local Co ordinates . . . .
. . .. 5 2
53
.
...
. . . .6 5
I n - p l a n e loads At Node j f o r a Typical Element
Shown in Local C oordi na te s ...................... . . . . . . .
65
viii
Figure
1-3
1-4
Page
Bending Deformations a t Node j for. a Typical
Element Shown in Local C oor din at es . . . . . . . .
. 71
Bending Loads a t Node j f o r a Typical
Element Shown in Local Coordina tes . . . . . . . .
. 71
ix
ABSTRACT
In t h i s s t u d y , a f i n i t e element model o f the human s k u ll i s p r e ­
s e n t e d . The i n t e n t is, t o e x p lo re th e f e a s i b i l i t y o f det ermi nin g i n vivo m a t e r i a l p r o p e r t i e s o f t h e human s k u l l by fr equency response
analysis.
'
Q u a l i t a t i v e r e s u l t s a r e o b ta in e d by approximating m a t e r i a l and
ge om et ric al pa rame te rs f o r th e s k u l l .
I n i t i a l l y , bouridary c o n d i t i o n s
were chosen t o p r e v e n t r i g i d body t r a n s l a t i o n by s e c u r i n g nodes about
t h e mi d-p lan e. The axisymmetric mode shapes o b ta in e d reveal r e s o ­
nance o f t h e fundamental bending mode a t 3700 Hz.
To ac c ount f o r t h i s d i s p a r i t y o f r e s u l t s compared with those ob­
t a i n e d by pre v io u s i n v e s t i g a t o r s , ah end-mounted s u p p o r t a t a s i n g l e
node was imposed t o r e p l a c e the mid-mount boundary c o n d i t i o n . This
fo r ce d boundary c o n d i t i o n was o f t h e type employed in t h e s e pre vious
s t u d i e s . For t h e same s k u l l with t h e end-mount boundary c o n d i t i o n ,
th e f i r s t resonance o c cu r re d a t 440 Hz; The co rre sp ondin g fundamen­
t a l bending mode shape can b e s t be d e s c r i b e d as a r i g i d body motion
o f th e bulk o f t h e s k u l l with a l o c a l ddfrirmation a t t h e f i x e d node.
Recognition o f t h e n a t u r e o f t h i s mode shape r a i s e s s e r i o u s i m p li c a ­
t i o n s con cer nin g p a s t i n t e r p r e t a t i o n s o f such r e s u l t s .
The q u a l i t a t i v e r e s u l t s o f t h i s stud y su g g es t t h a t e i t h e r of
t h e two mode shapes may prove h e l p f u l in d e t e r m i n a t i o n o f s ku ll ma­
t e r i a l p r o p e r t i e s , but no t w i t h o u t major experimental and a n a l y t i c a l
difficulties.
-
INTRODUCTION
A ho s t o f i n v e s t i g a t i o n s have been c a r r i e d out to determine the
resp on se o f th e human head to e x t e r n a l e x c i t a t i o n .
Most o f th e s e
i n v e s t i g a t i o n s have as t h e i r o b j e c t i v e advancing the unde rs ta ndin g
of
th e e v en ts which ta k e pla ce duri ng th e s h o r t period subsequent to impact
of the s k u ll.
The impetus f r e q u e n t l y c i t e d f o r t h e s e s t u d i e s i s the
s t a t i s t i c a l l y im pr es si ve r a t e o f o c cu r re nc e o f death or i n j u r y due to
impact s u s t a i n e d by th e head.
This i s a r e s u l t , p r i m a r i l y , from a u t o ­
motive and r e l a t e d a c c i d e n t s .
Such s t u d i e s assume an element o f u n c e r t a i n t y when a p a r t i c u l a r s e t
o f r h e o l o g i c a l para mete rs i s as s ig n e d to the a n a l y t i c a l models o f i n vivo ( l i v e ) human bone and a s s o c i a t e d t i s s u e s .
The n o n - a v a i l a b i l i t y o f
t h i s i n - v i v o in fo r m a t io n c o n s t i t u t e s an o b s t a c l e to o b t a i n i n g meaningful
analytical re s u lts.
In previo us s t u d i e s t h i s impasse has been r e a d i l y
circumvented by t a k i n g as i n - v i v o th o s e p r o p e r t i e s determined from, or
e x t r a p o l a t e d from, i n - v i t r o (or e x-v iv o) studies..
The importance of
t h e s e para mete rs in q u a n t i t a t i v e p r e d i c t i o n of i n - v i v o beh avi or of the
human head, along with th e emergence o f a wide range o f val ues in r e c e n t
l i t e r a t u r e , combine to s u b s t a n t i a t e t h e need f o r f u r t h e r re fin em en t of
in-vivo rheological p ro p e rtie s.
A method to deter mine in - v i v o m a t e r i a l p r o p e r t i e s o f human bone has
been advocated by Garner and B l a c k k e t t e r [1 3] . This c a l l s f o r an •
> ,
.
ex perimental procedure t o measure th e response of th e system to an
e x t e r n a l l y a p p l i e d harmonic f o r c e over a rangq of f r e q u e n c i e s . S i m i l a r l y ,
2
a fr equ en cy re s pons e a n a l y s i s is performed by u t i l i z i n g an a n a l y t i c a l
model o f the p h y s i o l o g i c a l s t r u c t u r e under i n v e s t i g a t i o n . ' The i n p u t
m a t e r i a l pa ra m et e rs f o r th e a n a l y t i c a l model a re a l t e r e d so t h a t the
ex perimental and a n a l y t i c a l res ponse s most c l o s e l y c orre sp ond.
Those
optimized m a t e r i a l pa ra m et e rs a re then s a i d t o be t h e i n - v i v o m a t e r i a l
properties.
The long bones, such as t h e t i b i a and u l n a , a r e t y p i c a l l y chosen
f o r such s t u d i e s because th e y can be c h a r a c t e r i z e d as having r e l a t i v e l y
simple geometry.
However, experimental r e s u l t s and a n a l y t i c a l modeling
become complicated with th e p resen ce o f a t t a c h e d and s urro und ing s o f t
tissues.
Also, s l i g h t changes in o r i e n t a t i o n and s u p p o r t c o n d i t i o n s
can s i g n i f i c a n t l y e f f e c t experimental re s pon se c h a r a c t e r i s t i c s , which
in t u r n a d v e r s e l y e f f e c t s r e p r o d u c i b i l i t y . .
Because th e p r o p e r t i e s d i f f e r s i g n i f i c a n t l y between th e long bones
and th o s e o f th e s k u l l , i t i s im p e r a ti v e t h a t s i m i l a r d e t e r m i n a t i o n s
/
be atte m pt ed f o r t h e s k u l l i t s e l f .
In a d d i t i o n to t h e d e s i r a b i l i t y of
de te r m in in g t h e i n - v i v o p r o p e r t i e s o f th e s k u l l , such a s tu d y w ill p r o ­
vi de f u r t h e r i n d i c a t i o n o f th e u s e f u l n e s s o f freque ncy resp ons e a n a l y ­
ses f o r t h e s e pu rp os e s. .
One can surmise t h a t t h e s k u l l i s l e s s s u b j e c t to t h e d i f f i c u l t i e s
encoun tere d in o b t a i n i n g r e p r o d u c i b l e experimental r e s u l t s .
Damping due
to th e b r a i n and i t s s urro undin g t i s s u e can be c o n s id e re d c o n s t a n t over
p e r i o d s of time such as might e l a p s e between one exper imen ta l sampling
3
and th e n e x t.
In a d d i t i o n , one might e xpect l i t t l e v a r i a t i o n in b r a in
p r o p e r t i e s from one i n d i v i d u a l to a n o t h e r . P o s i t i o n i n g o f th e s kull and
a c c e l e r o m e t e r t r a n s d u c e r s can be e a s i l y reproduced, and, s k u ll su pp ort
c o n d i t i o n s have minimal e f f e c t s f o r p r o p e r l y chosen modes.
In he re nt
with th e s k u l l i s a g r e a t e r complexity o f geometry, which tends to
d e t r a c t from i t s d e s i r a b i l i t y as t h e s u b j e c t of dynamic a n a l y s i s .
Because c r a n i a l bone p r o p e r t i e s cannot be equated to th o s e of bone
s e r v in g o t h e r f u n c t i o n s [ 3 1 ] , a method i s needed to deter mine in -v i v o
bone p r o p e r t i e s s p e c i f i c a l l y f o r c a r n i a l bone.
The q u e s t i o n a t hand,
t h e n , a d d r e s s e s th e f e a s i b i l i t y o f using th e human s k u l l to determine
th e m a t e r i a l p r o p e r t i e s (or changes in m a t e r i a l p r o p e r t i e s ) of i n - v i v o
bone.
Success in t h i s ve n tu re i s c o n t i n g e n t upon th e a b i l i t y of both
th e ex perimental and a n a l y t i c a l i n v e s t i g a t i o n s to a c c u r a t e l y d e t e c t
and p r e d i c t a t l e a s t one bending mode o f deformation e x c i t e d by a
s p e c ific driving force.
I t i s a l s o e s s e n t i a l to c o n v in c i n g ly demon­
s t r a t e t h a t indeed th e same modes a r e being d e s c r ib e d by both segments
o f th e p ro c e du re.
The emphasis in t h i s stud y i s pla ced on an a n a l y t i c a l
d e t e r m i n a t i o n o f th e freque ncy resp ons e o f th e human s k u l l , and, an
a t t e m p t to deter mine which mode i s l i k e l y to aid in th e e s t a b l i s h m e n t o f
i n - v i v o s k u ll
(bone) p r o p e r t i e s .
Chapter I
LITERATURE REVIEW
The bulk o f work done in de te rm in in g th e mechanical p r o p e r t i e s of
bone has been c a r r i e d out in th e l a s t t e n y e a r s .
For e a r l i e r work th e
emphasis had been placed on de te rm in in g parameters which d e s c r i b e bone
as an e l a s t i c m a t e r i a l .
As e a r l y as 1965 S e d lin [27] publi sh ed the
r e s u l t s o f ex per im ent al work in which he demonstrated t h a t , f o r small,
d e f o r m a t i o n s , bone behaves as a l i n e a r v i s c o e l a s t i c m a t e r i a l .
Further­
more, S e d l i n ' s ex perimental r e s u l t s were b e s t modeled by t h e 3-parameter
s o l i d model, a l s o r e f e r r e d to as th e s ta n d a r d l i n e a r s o l i d model.
S e d l i n ' s work was conducted using i n - v i t r o compact femoral bone.
■
Curry [5 ] conducted exper imen ta l i n v e s t i g a t i o n s on t h e microscopic
l e v e l d e a l i n g with th e e f f e c t o f mineral c o n t e n t o f bone and i t s r e l a ­
t i o n s h i p to measured e n g i n e e r i n g p r o p e r t i e s .
Curry s t u d i e d the e f f e c t
o f m i n e r a l i z a t i o n or c a l c i f i c a t i o n on t h e e l a s t i c p a ra m e te r—Young's
Modulus.
As w i l l be seen in t h e fo ll o w i n g c h a p t e r on bone s t r u c t u r e ,
one might have expected t h a t C u r r y ' s work would v e r i f y t h e high c a u s a l e f f e c t r e l a t i o n s h i p between mineral c o n t e n t and Modulus o f E l a s t i c i t y .
Because o f re q u ir e m e n ts f o r n o n - d e s t r u c t i v e t e s t i n g , most i n f o r ­
mation on bone, p a r t i c u l a r l y human, i s th e d i r e c t r e s u l t o f i n - v i t r o
experimentation.
Values used in s t u d i e s f o r which i n - v i v o bone p r o p e r t i e s
a r e n e c e s s a r y i n p u t para mete rs a r e u s u a l l y taken d i r e c t l y from o r , a r e
e x t r a p o l a t e d from, a v a i l a b l e i n - v i t r o d a t a .
The v a l i d i t y of using
5
.
i n - v i t r o p r o p e r t i e s t o d e s c r i b e i n - v i v o beh avi or i s dependent upon th e
changes i n c u r r e d by bone fo ll o w i n g d e a th .
Pas t s t u d i e s a p p a r e n t l y a re hot in concurrence as t o th e e f f e c t o f
post-mortem age on bone p r o p e r t i e s .
The fol lo wi ng s t u d i e s demonstrate
th is point.
McElhaney [2 2] i n d i c a t e s t h a t th e mechanical p r o p e r t i e s , o f embalmed
bone do no t d i f f e r " s i g n i f i c a n t l y " from-immediate post-mortem p ro p e r­
t i e s . McElhaney's p r o p e r t i e s r e f e r t o t h e e l a s t i c p r o p e r t i e s —Young's
Modulus and P o i s s o n !s R a t io .
Evans [ 8 ] modeled bone as a v i s c o e l a s t i c m a t e r i a l and s t a t e s t h a t
t h e e l a s t i c parameter i s not e f f e c t e d by embalming.
Tennyson.[28] i n v e s t i g a t e d th e s t r e s s - s t r a i n c h a r a c t e r i s t i c s of
b e e f femur bone as a f u n c t i o n o f post-mortem age (PMA).
Tennyson
i n i t i a l l y modeled bone usi ng th e 3 -p a ra m e te r l i n e a r v i s c o e l a s t i c . m o d e l .
His r e s u l t s i n d i c a t e d t h a t one o f th e s t i f f n e s s pa rameters was small
enough t o al low him to n e g l e c t i t s e f f e c t .
th e 2 - p a ra m e te r or Voight model.
This reduced t h e model to
F u r t h e r r e s u l t s based on th e Voight
model produced l i t t l e change in th e s t i f f n e s s parameter with i n c re a s e d
PMA, however, th e vis co u s parameter showed r a d i c a l change during e a r l y
PMA.
Here, e a r l y PMA r e f e r s to th e f i r s t 24 hours o f post-mortem time.
I t i s n e c e s s a r y t o e x t r a p o l a t e Tennyson's d a ta to s p e c u l a t e on immediate
post-mortem e f f e c t on bone p r o p e r t i e s .
With d e c r e a s i n g PMA, approaching
z e r o , t h e vis co u s parameter appears t o i n c r e a s e e x p o n e n t i a l l y f o r a t
6
l e a s t as f a r as Tennyson's d a ta i s shown (beginning approxim at el y a t
.5 days PMA) .
Tennyson obt a in e d h is experimental d a ta from th e s p l i t -
Hopkins on -b a r t e c h n i q u e .
Garner [13] modeled the human forearm using the Se d li n or 3 - p a r a ­
meter l i n e a r v i s c o e l a s t i c s o l i d .
data
Comparison to i n - v i v o experimental
i n d i c a t e s l a r g e d i f f e r e n c e s in i n - v i v o and i n - v i t r o r h e o l o g i c a l
bone p r o p e r t i e s .
At t h i s p o i n t t h e l i t e r a t u r e seems to i n d i c a t e t h a t bo n e 's t r a n s ­
i t i o n from the i n - v i v o s t a t e to an i n - v i t r o s t a t e in v o lv e s small changes
in the e l a s t i c c h a r a c t e r i s t i c s and marked changes in th e vis cous or
damping c h a r a c t e r i s t i c s .
The i n c o n c l u s i v e n e s s of pre vio us work
e s t a b l i s h e s th e need f o r meaningful i n - v i v o r e s u l t s .
Furthermore, in
th e l i t e r a t u r e reviewed h e r e , va lu es taken f o r Young's modulus f o r th e
s k u ll range from 6 .0 x IO^
psi to 2.29 x IO^ p s i , w hile th e m a j o r i t y of
i n v e s t i g a t o r s use an i n t e r m e d i a t e value o f 2.0 x IO^ p s i .
The fol lo wi ng
d i s c u s s i o n can account f o r a p o r t i o n o f th e v a r i a t i o n in t h i s one
p a ra m et e r.
.
Bone p r o p e r t i e s vary from s p e c i e s to s p e c i e s , from one in d iv id u a l
t o a n o t h e r , and from one f u n c t i o n a l p h y s i o l o g i c a l l o c a t i o n to an oth er
w i t h i n t h e same specimen.
These f a c t s a r e borne out by t h e following
studies.
Wood [31] t e s t e d i n - v i t r o c r a n i a l bone samples from 30 s u b j e c t s
ranging in age from 25 to 95 y e a r s , and performed t e n s i l e t e s t i n g vs.
strain rate.
For c r a n i a l bone, no r e g io n a l v a r i a t i o n in th e p r o p e r t i e s
7.
measured was n o t e d , and no v a r i a t i o n o f p r o p e r t i e s with r e s p e c t to
t a n g e n t i a l d i r e c t i o n was found.
Wood a l s o s t a t e s t h a t c r a n i a l bones
have p r o p e r t i e s i n t e r m e d i a t e o f th o s e f o r th e l o n g i t u d i n a l and t r a n s v e r s e
p r o p e r t i e s o f long bones.
Hubbard [18] used beam t e s t i n g te c h n i q u e s to deter mine th e f l e x u r a l
s t i f f n e s s and s t r e n g t h o f la yer ed c r a n i a l bone.
Tes t specimens were .
o b ta in e d from embalmed c a l v a r i a (s k u l l l e s s jaw and f a c i a l bones).
From /
t h e s e , t h e e l a s t i c p r o p e r t i e s were determined using a t h r e e - p o i n t
flexure t e s t .
An e q u i v a l e n t modulus o f e l a s t i c i t y was determined f o r
.
th e a c t u a l c r o s s - s e c t i o n .
Hubbard noted no s i g n i f i c a n t e f f e c t . r e s u l t i n g
from t h e t e s t sample e x t r a c t i o n s i t e , nor due to o r i e n t a t i o n , normal or
i n v e r t e d ( i . e . convex o r concave) o f th e t e s t specimen in th e t e s t
apparatus.
McElhaney [21] s e t ou t to determine th e mechanical, p r o p e r t i e s o f
c r a n i a l bone which a r e r e l e v a n t to th e stu dy o f biomechanics o f head
injury.
Bone specimens were o b ta in e d from embalmed c a d a v e r s , c r a n i o ­
to m ie s , and au to psy .
In a d d i t i o n to th e human c r a n i a l bone t e s t e d ,
specimens were a l s o o b ta in e d from monkeys and t e s t e d . .
I t is. i n t e r e s t i n g
t o note t h a t t h e r e were s i g n i f i c a n t d i f f e r e n c e s in r e s u l t i n g e l a s t i c
para mete rs between t h e two s p e c i e s .
T e s t i n g was done on t h e Tin ius
01 sen E l e c t r o m a t i c T e s t i n g Machine with the bone samples main ta ine d in
"wet" c o n d i t i o n .
McElhaney noted t h a t v a r i a t i o n s in t h e t h i c k n e s s o f
t h e porous middle l a y e r o f bone ( d i p l o e ) r e s u l t e d in a high stan da rd
8
d e v i a t i o n f o r m a t e r i a l p r o p e r t i e s such as energy a b s o r p t i o n and gros s '
stiffness.
I t was a l s o shown t h a t t h e t e n s i l e p r o p e r t i e s , f o r the i n n e r
and o u t e r l a y e r s ( t a b l e s ) o f compact c o r t i c a l bone d i s p l a y no s i g n i f i - .
cant d ifferen ces.
Again, McElhaney i s modeling c r a n i a l bone as an
e la stic m aterial.
Research i n s p i r e d by a high occurance r a t e o f head i n j u r i e s l a r g e l y
due t o automotive a c c i d e n t s has been conducted in which v a ri o u s
approaches have been undertaken to a t t e m p t to i n c r e a s e u nde rs ta ndin g o f .
th e biomechanics o f t h e human s k u l l .
Gordon e t . a l . [14] used a 2-dimensional model o f t h e human head to
de ter mi ne th e resp onse to impact.
The r e s u l t s were in terms o f a
p r e s s u r e d i s t r i b u t i o n through t h e b r a i n .
The work employed a somewhat
more s o p h i s t i c a t e d model o f th e head than t h a t used in pre vio us a n a l y s e s ,
in t h a t a f i n i t e wave speed was used f o r th e b r a i n . a n d s k u l l .
Engin [6 ] determined th e re sp onse f o r a t h i n e l a s t i c , s p h e r i c a l
s h e l l f i l l e d with an . i n v i s c i d c om pre ss ib le f l u i d .
In a s i m i l a r a n a l y s i s
Kenner and Goldsmith [20] worked with, wave pro pag at io n through a l i q u i d
f i l l e d , sp her e f o r impacts, o f g r e a t e r d u r a t i o n than t h o s e which can be
r e p r e s e n t e d by D i r a c ' s d e l t a f u n c t i o n .
The above three, a n a l y s e s r e p r e s e n t an approach t o modeling th e
human s k u l l and b r a i n which in volv e s t h e use of c lo s e d form s o l u t i o n s .
These methods a r e usef ul in de m on st rati ng q u a l i t a t i v e l y phenomena such
as wave p r o p a g a t i o n , however, th e c l o s e d form s o l u t i o n s r e q u i r e
i.
; .
9
s i m p l i f y i n g assumptions such, as a s p h e r i c a l geometry f o r t h e s k u l l ,
c o n s t a n t t h i c k n e s s o f th e cranium, and an i n v i s c i d i r r o t a t i o n a l f l u i d
fo r the brain.
An a l t e r n a t e approach i n c o r p o r a t e s t h e use o f f i n i t e
element t e c h n i q u e s , which allows c o n s i d e r a t i o n o f i r r e g u l a r geometries
and v a r i a t i o n in s k u l l t h i c k n e s s .
Hardy and Marcal [16] hypothe si zed in t h e i r i n v e s t i g a t i o n o f b ra in
damage ( p a r t i c u l a r l y t h a t r e s u l t i n g from automotive a c c i d e n t s ) t h a t ,
even though i t i s t h e b r a i n i t s e l f which l i m i t s th e impact a b s o r p t i o n
c a p a c i t y o f t h e human head, t h i s l i m i t may be b e s t measured in terms o f
s k u ll re s p o n se .
This prompted Hardy and Marcal to model t h e s kull using
doubly curved t r i a n g u l a r s h e l l f i n i t e el em en ts .
The s k u ll was modeled
as an e l a s t i c m a t e r i a l , and, only s t a t i c d e f l e c t i o n s were s t u d i e d .
Nickel! and Marcal [24] c a r r i e d th e pre ceding a n a l y s i s one s te p
further.
The same f i n i t e element model was used, and t h e e s s e n t i a l
a s p e c t f o r any impact s t u d y - - t h e dynamic response o f t h e s k u l l - - w a s
considered.
Three d i f f e r e n t ty pe s o f su p p o rt boundary c o n d i t i o n s
( f r o n t a l , r e a r , and base mounted) were used to determine n a t u r a l f r e ­
quencies and a s s o c i a t e d mode shapes which ( i t was hoped) might be
r e l a t i v e l y independent o f th e a r b i t r a r y su p p o rt c o n d i t i o n s .
The lower
mode shapes a r e t h e r e s u l t o f th e p a r t i c u l a r imposed boundary c o n d i t i o n s .
In g e n e r a l , t h e s e can be d e s c r i b e d a s . r o t a t i o n (mostly r i g i d body) about
th e p re s crib ed ,
support.
The t h i r d mode in each case a ppea rs to. e x h i b i t
r e l a t i v e motion o f p o i n t s a t t h e s u p p o rt and th os e d i a m e t r i c a l l y o p p o s i t e
10
and occurs in t h e fr equency range o f 400-700 Hz.
Nickel! and Marcal
hy p o th e si ze t h a t t h i s mode shape i s a r e s u l t o f deforma tio ns due to the
v a r i a t i o n s in t h i c k n e s s and c u r v a t u r e s o f th e p a r t i c u l a r s k u ll modeled.
The above stu dy used a modulus o f e l a s t i c i t y approxim at el y one
t h i r d o f t h a t found in l i t e r a t u r e o f s i m i l a r i n v e s t i g a t i o n s .
This would
cause th e observance of lower f r e q u e n c i e s f o r given mode shapes.
No
r e f e r e n c e i s given to su p p o rt th e value taken f o r t h i s pa rameter.
Hick lin g [ 1 7 ] , as was th e case with most o t h e r i n v e s t i g a t o r s , had
as hi s goal t h e d e t e r m i n a t i o n o f th e mechanics o f b r a i n and s ku ll damage
du rin g and a f t e r im pact, p a r t i c u l a r l y t h a t a s s o c i a t e d with automotive
accidents.
He modeled th e s k u ll and b r a i n as l i n e a r v i s c o e l a s t i c and
i s o t r o p i c m a t e r i a l s and used a s p h e r i c a l geometry f o r th e head.
For h i s
stu dy and indeed a p p l i c a b l e to o t h e r such s t u d i e s , Hick lin g laments,
" U n f o r t u n a t e l y , t h e a b i l i t y of th e p r e s e n t model to
p r e d i c t t o l e r a n c e s i s g r e a t l y hampered because adequa te
d a t a on th e m a t e r i a l p r o p e r t i e s o f the s kull and b r a i n
a r e l a c k i n g and because a p p r o p r i a t e damage c r i t e r i a
a r e not known".
Having examined some o f th e models o f the s kull and no tin g i n h e r e n t
d i f f i c u l t i e s and i n a d e q u a c i e s , our a t t e n t i o n i s now d i r e c t e d to some o f
th e ex per im ent al work which has been performed on t h e s k u l l .
>
Ear ly exper imen ta l work on th e dynamic response o f t h e sku ll was
.
performed by Franke [10,] and updated by Gurdjian e t . al,. [1 5] .
Franke f r o n t a l l y loaded a dry p r e p a r a t i o n ( s k u l l ) with a small
s u p p o rt a t. t h e o c c i p u t ( r e a r o f s k u l l ) and found resonance a t 820 Hz.
n
The same l o a d i n g , when a p p l i e d to a l i v e human s k u l l , produced no
measurable re s p o n s e .
This f a i l u r e was d e s c r i b e d as t h e r e s u l t of
i n s u f f i c i e n t c oupl ing between t h e f o r c e p i s t o n and t h e s k u l l due to the
layer of skin.
Franke then performed h i s experiment on cadavers with
t h e skin, removed a t t h e f o r c i n g l o c a t i o n .
found a t 610 Hz.
The f i r s t f l e x u r a l mode was
Without s u b s t a n t i a t i o n , t h i s mode was taken as the
dampened v e r s i o n o f t h a t found in th e dry p r e p a r a t i o n a t 820 Hz.
Gurdjian f r o n t a l l y mounted cadaver c a l v a r i a to a so u rc e o f a h a r ­
monic f o r c e and measured t h e re s p o n se a t t h e l e f t s i d e , v e r t e x , and o c ­
c ip u t of the s k u ll.
wit h s i l i c o n g e l .
300 and 900 Hz.
put acc e leratio n
The c a l v a r i a were t e s t e d both empty and f i l l e d
Resonant f r e q u e n c i e s were d e t e c t e d a t approxim at el y
The f i r s t , an a n t i - r e s o n a n t mode, r e s u l t e d in an o c c i ­
amp litude t h r e e to f o u r times g r e a t e r than t h a t of
t h e f r o n t a l l o c a t i o n where t h e harmonic load was a p p l i e d .
The second
re so nan ce e x h i b i t e d r e l a t i v e l y l a r g e f r o n t a l ex cu r si o n as opposed to
t h a t of the occiput.
With re s pon se monitored a t only t h r e e p o i n t s ,
l i t t l e i n fo r m a t io n i s p r e s e n t e d t o giv e i n d i c a t i o n o f a c t u a l mode shape.
In a d d i t i o n to t h e cadave r s t u d i e s , human v o l u n t e e r s were used in
a s i m i l a r exper imen ta l s e t up, and re s p o n se to a f r o n t a l f o r c e was mea­
s u re d .
Gurdjian s t a t e s t h a t f o r f r e q u e n c i e s below 150 Hz. only lo cal
d e f l e c t i o n s a t t h e p o i n t o f a p p l i c a t i o n of t h e f o r c e a r e d e t e c t ! b l e .
P r i o r and su bsequent work has been done to deter mine resonance o f
biological stru c tu re s.
Those most a p p l i c a b l e to t h i s s tu d y a r e work
12
done by J u r i s t £19] and Garner and B l a c k k e t t e r £13].
J u r i s t ' s work i s
m ost ly c l i n i c a l in n a t u r e , and c o n c e n t r a t e s on t h e long bones o f the
o u t e r e x t r e m i t i e s ( u l n a , r a d i u s , t i b i a , and f i b u l a ) .
J u r i s t modeled
t h e s e as hollow c y l i n d e r s o f c o n s t a n t c r o s s s e c t i o n and proceeded to
determine t h e f i r s t bending f r e q u e n c i e s .
G a r n e r ' s work, which i s th e a n t e c e d e n t t o t h i s study, r e p r e s e n t s a
more s o p h i s t i c a t e d a n a l y t i c a l - e x p e r i m e n t a l approach t o c o r r e l a t e low
frequency resp on se t o para mete r d e t e r m i n a t i o n .
Garner used a f i n i t e
element model o f t h e forearm in which th e bones ( r a d i u s and u ln a ) along
with th e s o f t t i s s u e s were modeled as v i s c o e l a s t i c m a t e r i a l s .
Garner
v a r i e d t h e r h e o l o g i c a l pa rameters in h i s a n a l y t i c a l model u n t i l the
resp onse most c l o s e l y c o in c i d e d with t h a t obta ine d from h i s experimental
work, and th u s d e fi n e d t h e r h e o l o g i c a l p r o p e r t i e s .
Chapter 2
THE HUMAN SKULL.
In o r d e r to more f u l l y j u s t i f y t h e use o f s i m p l i f y i n g assumptions
n e c e s s i t a t e d by our a n a l y t i c a l model o f the s k u l l , t h e p r e s e n t c h a p t e r
is included.
We w i l l f i r s t examine t h e m a t e r i a l o f which t h e s k u l l ,
t h e s u b j e c t o f our i n v e s t i g a t i o n , i s ■composed--bone.■ Our n e x t concern
w i l l be with t h e b i o l o g i c a l s t r u c t u r e , th e s ku ll i t s e l f .
Bone
Bone i s a h i g h l y s p e c i a l i z e d form o f co n n ec ti v e t i s s u e . • On a
mi c ro s co pic l e v e l , bone i s composed o f a s o f t branching m a t r i x of c e l l s
arround which a r e d e p o s i t e d mineral s a l t s .
These s a l t s a r e composed
p r i m a r i l y of calcium and ph osp ha te , which r e s u l t s in b o n e 's c h a r a c t e r ­
i s t i c h a rd n e s s .
I t is. t h i s hardness which most d i f f e r e n t i a t e s bone
from o t h e r c o n n e c ti v e t i s s u e s such as c a r t i l a g e and l i g a m e n t s . ■ In ad ­
d i t i o n to t h e i n o r g a n i c bone s a l t s and w a t e r , the i n t e r s t i t i a l sub­
s t a n c e i n c l u d e s a f i b r o u s s t r u c t u r e comprised o f c o l l a g e n .
These c o l ­
lagenous f i b e r s tend to o r i e n t themselves in a l o n g i t u d i n a l f a s h io n in
long bones so as t o maximize s t r e n g t h in d i r e c t i o n s o f g r e a t e s t t e n s i l e
stress.
Most bones ( p a r t i c u l a r l y th e long bones) a r e modeled during e a r l y
development as c a r t i l a g e , and only in l a t e r de vel op m en t.a re the y r e ­
modeled as c o r t i c a l bone.
I t i s t h i s remodeling pro c e ss which r e s u l t s
in t h e pr e s e n c e o f h a v e r s i a n systems ( o r o s t e o n s ) .
Consider a hollow
1.4
c or e (many times g r e a t e r in l e n g th than dia m e te r) c a l l e d t h e ha v e r si a n
canal.
Around t h i s h a v e r s i a n c a n a l , t h e bone m a tr ix and i n t e r s t i t i a l
m a t e r i a l s a r e d e p o s i t e d in c o n c e n t r i c l a y e r s or l a m e l l a e . ' Lamellae of
a h a v e r s i a n system and t h e o u t e r l a m e l l a e o f a d j o i n i n g h a v e r s ia n s y s ­
tems become connected along what a r e termed cement l i n e s . . Thus the
t h i c k n e s s of say a long bone i s composed o f many h a v e r s i a n systems.
Macroscopic-Properties .
In macroscopic p e r s p e c t i v e t h e r e a r e two b a s i c ty p e s o f bone:
I)
hard compact c o r t i c a l bone such as t h a t found in t h e s h a f t s o f long
bones, and 2) spongy o r cancellous, bone which c o n s i s t s o f many f i n e
p a r t i t i o n s c a l l e d t r a b e c u l a e , which form c a v i t i e s c o n t a i n i n g red or
f a t t y marrow.
Can cellous bone i s found in t h e v e r t e b r a e , m a j o r i t y of
f l a t bone s, and a t t h e end o f long bones.
Thus c a n c e l l o u s bone i s por
ous and i s c h a r a c t e r i z e d by i n t r i c a t e a r c h i t e c t u r e .
The i n o r g a n i c mineral i s t h e major component o f bone.
McLean and
U r i s t [23] c i t e f o r Bovine bone th e f o ll o w i n g weight pe rc e n ta g e s
Mineral
Organic Matter
Water
72%
24%
4%
When t h e f a t t y m a t e r i a l i s removed, 90-96% o f the remaining org a nic
m a t t e r of bone i s c o l l a g e n .
15
Bone Dynamics
Changes oc cur in bone by way o f t h e normal p h y s i o l o g i c a l mechanisms
o f gr o w th , a g i n g , maintenance and r e p a i r , by p a t h o l o g i c a l mechanisms
such as o s t e o m a l a c i a , and by trauma such as impact with a f o r e i g n ob­
j e c t c u l m in a ti n g i n f r a c t u r e .
Three type s o f s p e c i a l i z e d c e l l s account f o r th e p h y s i o l o g i c a l dy­
namics o f bone.
The f i r s t o f t h e s e i s th e o s t e o b l a s t .
Osteoblasfs
appea r on t h e s u r f a c e o f growing o r devel opi ng bone and pro vid e t h e b i o ­
l o g i c a l mechanism for. t h e for mati on o f new bone;
O st eo c yte s a r e o s t e o ­
b l a s t s which have become surrounded by the m a t r i x , which su bs e q u e n tl y
c a l c i f i e s , and pro v id e f o r t h e maintenance of surrou nding bone.
l a s t o f th e t h r e e ty pe s o f c e l l s is, .the o s t e o c l a s t .
The
O steoclasts are
u s u a l l y found on t h e b o n e . s u r f a c e near a r e a s o f a c t i v e remodeling and
perform t h e p h y s i o l o g i c a l f u n c t i o n o f r e s o r p t i o n o f e x i s t i n g bone^
Changes in b o n e . a r e a p p a r e n t d uri ng growth, but even when the geo­
m e t r i c pa ra m et e rs a r e c o n s t a n t , as in p e r io d s during a d u l t h o o d , the
p ro c e ss o f remodeling (r epl ac em en t o f e x i s t i n g bone with new) i s a c ti v e *
By way o f example, McLean and U r i s t [23] s t a t e t h a t f o r a t y p i c a l 57
y e a r ol d human, t h e bone mass t u r n o v e r in one day i s 0.036% f o r the
femur and f i b u l a and 0.012% f o r th e t i b i a .
The p h y s i o l o g i c a l mechanism o f aging and the p a t h o l o g i c a l mechan­
isms can cause an u p s e t in t h e ba la n c e o f o s te o g e n ic a c t i v i t y , r e s u l t i n g
16
in changes in b o n e 's m ic ros cop ic s t r u c t u r e " a n d t h e r e f o r e , th e macro­
s co p ic p r o p e r t i e s .
McLean and U r i s t d i s c u s s the c o n d i t i o n o f a tro phy
which r e f e r s to a l o s s o f s u b st a n c e o f b o n e 'w it h o u t a correspo ndi ng
l o s s in t h e e x t e r n a l dimensions ( o r g r o s s volume) of t h e bone.
This
may advance to a s t a t e where as much as. 75% o f the o r i g i n a l bone mass
has been l o s t .
L i t t l e i s known o f th e e f f e c t o f e a r l y post-mortem time on th e
macroscopic m a t e r i a l p r o p e r t i e s o f bone. r Some r e s e a r c h e r s , such as
McElhaney [21] i n d i c a t e t h a t t h e r e i s T i t t l e e f f e c t on bone p r o p e r ­
t i e s due to post-mortem age.
O t h e r s , such as Garner and BTackketter
[ 1 3 ] , c i t e i n d i c a t i o n s o f marked changes. . Tennyson [ 2 8 ] , using S p T it Hopkin s o n -b a r t e s t i n g t e c h n i q u e s , p l o t s changes in an e l a s t i c and v i s ­
cous pa rame te r f o r bone with r e s p e c t to post-mortem tim e.
Hts r e s u l t s
i n d i c a t e l i t t l e change in t h e e l a s t i c pa rame te r but g r e a t e f f e c t on
th e v is c o u s pa rame te r immediately a f t e r d e at h and through th e f i r s t
few days post-mortem.
The SkUll
The human s k u l l c o n s i s t s o f t h e c a l v a r i u m , t h e j a w ( m a n d i b l e ) ,
and t h e f a c i a l bon es .
The c a l v a r i u m i s an e l l i p s o i d a l
s u r e which s e r v e s t o p r o t e c t t h e b r a i n .
shaped e n c l o ­
The one m a jo r o p en in g i n t h e
17
c a l v a r i u m , through which t h e medula o blo ng a ta ( a n t e r i o r p o r t i o n of
t h e b r a i n which co nn ect s t o th e s p i n a l cord) p a s s e s , i s t h e foramen
magnum l o c a t e d a t th e base of th e s k u l l .
The calva rium i s composed
o f e i g h t d i f f e r e n t bones which a r e j o i n e d by c a r t i l a g e and a f i b r o u s
t i s s u e in e a r l y development.
By .adulthood t h e s e bones a r e j o i n e d in
a much more r i g i d manner a t t h e bound a rie s along c o n n e c t i v e forma­
t i o n s r e f e r r e d to as s u t u r e s .
Examination of cadaver s k u l l s r e v e a l s
g r e a t v a r i a t i o n s in t h e degree to which th e c r a n i a l bones have fused
a t the su tu res.
The r a ng e i s .from i n t e r l o c k i n g , b u t s t i l l
separate,
bones to c as e s in which f u s i o n i s so complete t h a t t h e s u t u r e l i n e s
are hardly d is c e r n ib l e .
The bones which comprise the c al va riu m a r e sandwich s t r u c t u r e d
through the t h i c k n e s s .
The bones a r e composed o f i n n e r and o u t e r
t a b l e s o f compact c o r t i c a l bone s e p a r a t e d by a porous i n n e r l a y e r of
bone c a l l e d t h e d i p l o e .
The t h i c k n e s s o f th e d i p l o e l a y e r v a r i e s with
in t h e c o n s t i t u e n t bones from as much as h a l f of t h e t o t a l s k u ll t h i c k
ness t o no t h i c k n e s s a t t h e s u t u r e s .
v a r i e s with l o c a t i o n .
The th i c k n e s s o f t h e calvarium
Typical v a r i a t i o n s might be from 0.12 inches
in t h e temporal r e g i o n t o 1 0.25 in c h es in th e f r o n t a l r e g i o n .
The r e g io n a l nomenclature f o r a t y p i c a l s kull i s shown in
Fig ur e 2-1.
18
S a g i t t a l s u tu r e
Coronal s u t u r e
Left p a rie ta l
bone
Fr ontal bone
Fac ial bones
Foramen magnum
Mandible
FIGURE 2 - 1 .
Typical S kull .
Having d i s c u s s e d b r i e f l y the p h y s i o l o g i c a l n a t u r e o f bone and the
s k u l l i t s e l f , we have the b a s i s on which e v a l u a t i o n o f the v a l i d i t y of
s i m p l i f y i n g assumptions can be made.
We must a l s o c o n s i d e r the cases
f o r which s p e c i f i c c o n d i t i o n s a re r e q u i s i t e to the a c c e pta nc e of a
p a r t i c u l a r s i m p l i f y i n g assumption.
E a r l i e r i t was noted t h a t , with the ex cep tio n o f the collagenous
f i b e r s , bone appea rs to be homogeneous on a macroscopic l e v e l .
The
e f f e c t o f the c olla ge no us f i b e r s i s minimal in the f l a t bones such as
th ose which comprise the s k u l l .
Along t h e s e same l i n e s , i t was p r e ­
v io u s l y r e p o r t e d from an experimental study by Hubbard [18] t h a t no
s i g n i f i c a n t v a r i a t i o n in bone p r o p e r t i e s was measured in d i r e c t i o n s
19
t a n g e n t i a l t o th e s k u ll s u r f a c e .
Thus, we have the b a s i s f o r assuming
bone t o be homogeneous in i n - p l a n e d i r e c t i o n s .
V a r i a t i o n s o f t h i c k n e s s o f th e i n n e r and o u t e r t a b l e s , and the
••
d i p l o e l a y e r , along with s l i g h t l y a l t e r e d p r o p e r t i e s . i n a normal ( to th e
s u r f a c e ) d i r e c t i o n su gge st t h a t i n c l u s i o n o f t h e s e c h a r a c t e r i s t i c s is
necessary for q u a n tita tiv e r e s u l t s .
This i s o f even g r e a t e r s i g n i f i c a n c e
f o r bending de fo rm a ti o n , where s t i f f n e s s i n c r e a s e s as t h e
d i s t a n c e from t h e c e n t r a l f i b e r .
cube
o f the
A c o n s t a n t t h i c k n e s s and th e use of
gr o s s p r o p e r t i e s t o compensate f o r d i p l o e t h i c k n e s s can only be expected
t o produce q u a l i t a t i v e r e s u l t s .
The homogeneous s h e l l model w il l s u f f i c e t o p r e d i c t bending response,
f o r a s ku ll where s u t u r e s are. s u f f i c i e n t l y fused so as to provide con­
t i n u i t y between component bones.
I t must be cau ti oned t h a t i n - v i v o
s k u l l s with merely i n t e r l o c k i n g or c a r t i l a g e n e o u s s u t u r e s p r e s e n t a
physical, s i t u a t i o n which i s not l i k e l y to be a d e q u a te ly modeled quan­
t i t a t i v e l y by t h e homogeneous s h e l l .
The c a l v a r i u m , being n e a r l y a complete e n c l o s u r e , can be modeled, in
s i m p l e s t form as a s p h e r i c a l s h e l l o f c o n s t a n t t h i c k n e s s .
Though the
sphere i s a f i r s t g e n e r a t i o n model o f th e s k u l l , i t s resp ons e i s i n d i ­
c a t i v e o f t h e r e s o n a n t f r e q u e n c i e s and mode shapes of t h e s k u l l . A more
a c c u r a t e geom et ric al r e p r e s e n t a t i o n i s t h e e l l i p s o i d .
The t y p i c a l
c a lv a ri u m , however, d e v i a t e s somewhat from t h i s geometry.
In some
specimens i r r e g u l a r i t i e s even r e s u l t in a la c k of symmetry about the
sagittal
20
plane ( i . e . non-symmetry between th e l e f t and r i g h t s i d e s of
t h e calvarium)^
A t r u l y q u a n t i t a t i v e model would ta k e i n t o account even
these i r r e g u l a r i t i e s .
Other d i f f i c u l t i e s in modeling th e s ku ll which must be con sid ered
o r f o r which p r o p e r assumptions must be made a r e :
t h e e f f e c t s of
a t t a c h e d or surrou nding f a c i a l bones, jaw bone (mand ibl e), b r a in and
f l e s h y m a t e r i a l s , and the p h y s i o l o g i c a l su pp ort system.
Chapter 3
MODELING CONCEPTS
The a n a l y t i c a l model o f th e s k u ll pre s e n te d in t h i s study was
developed so t h a t , i f n e c e s s a r y , c o n s i d e r a t i o n can be given to the
v i s c o e l a s t i c n a t u r e o f bone, i r r e g u l a r i t i e s in geometry, and region al
v a r i a t i o n s in t h i c k n e s s and m a t e r i a l p r o p e r t i e s .
I n i t i a l l y , however, t h e
s k u ll i s modeled as a t h i n wall homogeneous s h e l l of c o n s t a n t t h i c k n e s s
and o f axisymmetric geometry.
The i n t e n t here i s to o b t a i n q u a l i t a t i v e
i n s i g h t i n t o s k u ll resp onse c h a r a c t e r i s t i c s .
The use o f an axisymmetric geometry permits th e g e n e r a t i o n o f nodal
c o o r d i n a t e s and assumed modes t o be completed with minimal computational
e f f o r t . Of g r e a t e r s i g n i f i c a n c e i s th e c a p a c i t y to e f f e c t a c o o rd in a te
r e d u c t i o n o f th e o r i g i n a l 336 degree o f freedom system t o one de sc ri b e d
by 28 g e n e r a l i z e d c o o r d i n a t e s , while m a in ta in in g th e c a p a b i l i t y of
i d e n t i f y i n g any axisymmetric mode.
Geometry Generation
The s k u l l was modeled in t h i s stu dy as an e l l i p s o i d a l s h e l l f o r ea s e
in g e n e r a t i n g nodal l o c a t i o n s and to f i l l t h e require me nts o f a xia l
symmetry.
The eq u at i o n f o r an e l l i p s o i d in c a r t e s i a n c o o r d i n a t e s is
22
where J , Y , and I a r e c a r t e s i a n c o o r d i n a t e valu es in the X, Y, and Z
c o o r d i n a t e d i r e c t i o n s r e s p e c t i v e l y , and A, B, and C a r e c o n s t a n t s which
d e f i n e th e shape of the e l l i p s o i d .
The sphere i s a s p e c i a l case o f an
e l l i p s o i d where th e c o n s t a n t s A, B, and C a r e a l l nu m e ri c a l ly equal to
the radius.
F i n i t e element nodal l o c a t i o n s a r e ex pre sse d in terms of the global
c a r t e s i a n c o o r d i n a t e s X, Y, and Z.
These global c o o r d i n a t e values
become th e geometric in put parameters f o r f i n i t e element mass and
s t i f f n e s s generation.
The g e n e r a t i o n o f node l o c a t i o n s i s most e a s i l y c a r r i e d out in
s p h e r i c a l c o o r d i n a t e s and then trans for me d i n t o c a r t e s i a n c o o r d i n a t e s .
The s p h e r i c a l c o o r d i n a t e s 8, <j>, and p w i l l be employed as shown in
Figure 3-1 below.
FIGURE 3-1.
S phe ric a l Coordin at es.
23
The e x p r e s s i o n s r e l a t i n g c a r t e s i a n c o o r d i n a t e s to s p h e r i c a l c o o rd i nates are
J
= . psin(e)cos(<j>)
T
=
psin(0)sinC<l>)
Z
=
p c os( e)
(3-2)
S u b s t i t u t i n g e q u a t i o n s (3- 2) i n t o (3- 1) and s i m p l i f y i n g g iv e s
s i n 2 ( e ) c o s 2 (<f>)
P
Ad
,
s i n 2 ( e ) s i n 2 (*)
+
a
P
,
c os 2 (e)
+
p
F
For t h e e l l i p s o i d d e f i n e d by c o n s t a n t s A, B, arid C1 p i s t h e d i s t a n c e
from t h e o r i g i n to t h e node p o s i t i o n e d a t a n g u l a r c o o r d i n a t e s e and <f>.
The c a r t e s i a n v a lu es f o r J , T, and I a r e o b ta in e d by i n s e r t i n g t h i s
va lu e f o r p i n t o e q u a ti o n ( 3 - 2 ) .
To o b t a i n an adequa te g r i d s i z e ( i . e . elements small enough so as
t o c l o s e l y approximate s h e l l c u r v a t u r e ) , nodes were g e n e r a t e d f o r each
change in 6, a n d > o f O.lir r a d i a n s .
Taking f u r t h e r advantage of sym­
m e t r y , t h e e l l i p s o i d need only be g e n e r a te d f o r 0 <_ # <_
C oor din at e Reduction
The f i n i t e elements were g e n e r a te d such t h a t r i n g s o f nodes appear
in p la n es which a r e p a r a l l e l to th e glob al XY p la n e .
For axisymmetric
geometry, t h e s e nodes which a r e d e s i g n a t e d by the s u b s c r i p t r have th e
same s p h e r i c a l a ngl e e.
See Figure 3-2 f o r nomenclature.
In a d d i t i o n .
24
th e nodes a r e ge ne ra te d in v e r t i c a l r i n g s which a r e c o p l a n a r along with
th e Z - a x i s .
These plan es a r e s i t u a t e d a t an angle ( <t>) from the X -a xis ,
and, a r e d e s i g n a t e d by th e s u b s c r i p t ( s ) .
P o la r node
FIGURE 3-2.
Nodal Nomenclature.
For a x i sy m m e t ri c d i s p l a c e m e n t s ( a b o u t t h e Z - a x i s ) a l l nodes in r i n g
( r ) w i l l e x p e r i e n c e t h e same e x c u r s i o n in t h e Z - d i r e c t i o n .
The s en s e o f
t h e components o f d i s p l a c e m e n t p a r a l l e l t o t h e XY p l a n e i s d i r e c t e d
25
p e r p e n d i c u l a r t o t h e Z- a x i s , so t h a t f o r a u n i t d is pla ce m e nt of a node
in t h i s d i r e c t i o n
and
• Ur s -
cds(<j>s )
inches
Vrs
Sin(^s )
inches
=
(3-4)
where Ufg and Vrg a r e th e dis p la ce m e n ts o f node ( r s ) in t h e X and Y
coordinate d ire c tio n s re sp ectiv ely .
S i m i l a r l y a n g u l a r d e fo rm a ti o n s , ex pre sse d on a u n i t b a s i s , f o r th e
nodes in r i n g ( r ) a r e
0xr s =
and
eyr s =
' sin^ s )
radians
-cos(<|> )
radians
(3-5)
where Gxrg and Gyr g .a r e th e a n g u la r de for ma tio ns a t node ( r s ) about the
X and Y axes r e s p e c t i v e l y .
For axisymmetric geometry, t h e r e i s no
a n g u l a r di sp la c e m e n t about th e Z - a x i s .
The deformation Urg and Vyig a re
no t mu tu a lly ind ep en de nt , nor a r e Qxr s and Gyy15.
The dis pla cem en ts
f o r t h e nodes in r i n g ( r ) can then be ex pre sse d as
=
pr l ’ 1
Ur s
=
Pr2coS(^s )
and
=
Pr2Sin((J)g )
Pr3SinUs )
and Gyyig=
-Pr3COS((J)s )
Il
X
O
Wrs
S-
LD
Thus, t h r e e reduced c o o r d i n a t e s , Pr l , Pr 2 , and Py13 w i l l completely
d e s c r i b e t h e d is pla ce m e nt o f each r i n g ( r ) .
The t h r e e correspo ndi ng
assumed modes f o r r i n g ( r ) a r e then g e n er at e d such t h a t t h e f i r s t , when
m u l t i p l i e d by Pyi-J, y i e l d s only de for ma tio ns o f a l l nodes in t h a t r i n g
in the Z - d i r e c t i o n .
The second, when m u l t i p l i e d by Pyi^, gives the
26
de fo rm a tio ns o f o nly nodes in r i n g ( r ) in th e X and Y c o o r d i n a t e
d i r e c t i o n , and th e t h i r d assumed mode, when m u l t i p l i e d by p ^ , r e s u l t s
in a n g u l a r de fo rm a tio ns about th e X and Y a xes .
Al I o f t h i s i s in
accordance with e q u a t i o n s ( 3 - 6 ) .
Only one reduced c o o r d i n a t e i s n e c e s s a r y to d e s c r i b e the motion
of each of t h e two p o l a r nodes, which a r e allowed t r a n s l a t o r y motion
along th e Z - ax is only f o r axisymmetric de fo rm a tio n.
According to methods d e s c r i b e d by Anderson [ 1 ] , t h e pre vio us r e s u i t s a r e combined in th e form of
} .
=
[y ] (P>
{&}
=
glob al d is pl a ce m e nt v e c t o r
' (P)
=
reduced c o o r d i n a t e v e c t o r
{ &
(3-7 )
Il
I- - -- - 1
and
I_ _ _ I
where .
t r a n s f o r m a t i o n m a tr ix
The columns of t h e t r a n s f o r m a t i o n m a tr ix [y] a r e not assumed modes
in t h e usual s e n s e .
By l i m i t i n g t h e re sp onse to a x i a l symmetry, th e
c o o r d i n a t e r e d u c t i o n i s executed w i t h o u t l o s s o f g e n e r a l i t y .
Only com­
p u t a t i o n a l r o u n d - o f f e r r o r w i l l produce d i s c r e p a n c i e s from t h e axisym­
m e t r i c re s p o n se had i t been obta in e d i n s t e a d from t h e t o t a l system.
Implementation o f t h e c o o r d i n a t e r e d u c t i o n p ro c e ss i s d is c u s s e d in
the c h a p t e r d e a l i n g with frequ en cy re s ponse methods.
The a c tu a l c o o r ­
d i n a t e r e d u c t i o n was performed on each element upon g e n e r a t i o n and then
27
superimposed in a global reduced mass and s t i f f n e s s m a t r i x .
This was
done so as to avoid g e n e r a t i o n and s t o r a g e o f the e n t i r e mass and s t i f f ­
ness m a t r i c e s .
Inp ut Parameters
Measurements were taken from a cad av er s ku ll to o b t a i n approximate
ge om et ric al p a ra m e te rs .
For th e s p h e r i c a l model th e c o n s t a n t s
A = B = "C = 3.25 in .
which produce a sphere with a r a d i u s o f 3.25 i n c h e s , were chosen to
r e p r e s e n t th e s k u l l .
The e l l i p s o i d de fi n e d by
A
=
3.0 i n .
B
=
3 .0 in .
C
=
3.5 i n .
has a c i r c u l a r c r o s s - s e c t i o n a t th e XY pla ne of 3.0 inc hes r a d i u s , and,
a d is ta n c e , along t h e Z- ax is from pole to pole of 7 .0 i n c h e s .
This
geometry was used f o r the approximation o f th e s k u ll as an e l l i p s o i d .
A constant thickness ( t
=
.15 i n c h e s ) , which in c l u d e s c o n s i d e r ­
a t i o n o f t h e d i p l o e l a y e r , was used as an average o v e r a l l v a lu e.
.
The bone o f the s k u ll was modeled as an e l a s t i c m a t e r i a l , due
p r i m a r i l y to th e absence o f r e l i a b l e v i s c o e l a s t i c p r o p e r t i e s .
The
para mete rs used f o r t h i s q u a l i t a t i v e model a r e taken from Wood [ 3 1 ] ,
and, appea r t o be th e most widely used in c u r r e n t l i t e r a t u r e .
These a r e
28
2 .0 x 10
E
6 lbf
(Young's Modulus)
( P o i s s o n ' s R at io)
V'
I b^-sec
.0002
2
(Mass Density)
Boundary Conditions
For th e dynamic system, i t becomes n e c e s s a ry to impose boundary
c o n d i t i o n s which p re c lu d e t h e p o s s i b i l i t y o f u n r e s t r a i n e d r i g i d body
motion.
Two d i f f e r e n t s e t s o f boundary c o n d i t i o n s were imposed f o r
t h e s k u ll in t h i s st udy a t th e c o o r d i n a t e re d u c t i o n l e v e l .
A mid-
mounted c o n d i t i o n was accomplished by p r o h i b i t i n g t r a n s l a t i o n in the
Z - d i r e c t i on o f a l l th e nodes which l i e in th e f i f t h r i n g ( i . e . in the
XY p l a n e ) .
This i s done by e x c l u s i o n , from the y m a t r i x , o f the assumed
mode which would o th e r w is e have provided t h e only mechanism f o r Zd i r e c t i on t r a n s l a t i o n o f a l l nodes in t h e f i f t h r i n g ( i . e . th os e l y in g
in t h e XY p l a n e ) .
S i m i l a r l y , an end-mounted boundary c o n d i t i o n was
imposed by removing, from th e p a r t i c u l a r s e t of assumed modes d is c u s s e d
e a r l i e r , t h a t assumed mode, whose s o l e purpose was t o p ro vid e th e
degree o f freedom a ll ow in g t r a n s l a t i o n o f one o f th e p o l a r nodes along
the Z-axis.
For th e end-mounted boundary c o n d i t i o n , t h e assumed mode
which i s devoted to Z - d i r e c t i on t r a n s l a t i o n of th e nodes in ri n g 5, must
be r e i n s t a t e d .
In e f f e c t , t h i s procedure r e s u l t s in t h e g e n e r a ti o n of
n o n - s i n g u l a r global reduced mass and s t i f f n e s s m a t r i c e s .
29
Other Aspects
The c o o r d i n a t e t r a n s f o r m a t i o n s from globa l to l o c a l c o o r d i n a t e s ,
f o r f i n i t e element g e n e r a t i o n , and back t o global c o o r d i n a t e s were
e xec ute d in usual fa s h i o n [32] and [2 5 ] .
The c o o r d i n a t e r e d u c t i o n was employed on an element by element
basis.
This produced elemental mass and s t i f f n e s s m a t r i c e s which were
o v e r - 1 ayed to form th e global reduced mass and s t i f f n e s s m a t r i c e s [ 3 2 ] .
Chapter 4
FINITE ELEMENT THEORY
For th e development of f i n i t e element t h e o r y , t h e s t a t e of a small
but f i n i t e p o r t i o n o f t h e system under s tu d y i s c o n s i d e r e d .
This allows
f o r t h e d e s c r i p t i o n o f th e element s t a t e to be approximated in s i m p l i ­
f i e d and us ef ul form.
,The f i n i t e elements u t i l i z e d in t h i s a n a l y s i s a r e
d e ri v e d us in g assumed d i s p l a c e m e n t s .
In a d d i t i o n , t h e s h e l l element i s
t h e r e s u l t o f th e s u p e r p o s t i t i o n o f a p l a t e bending element and a p l a n a r
(o r membrane) element d e r i v e d w i t h i n t h e c o n f in e s o f small disp la cem en t
and t h i n s h e l l th e o r y [ 3 2 ] .
The gener al proc ed ure f o r t h i s ty pe o f f i n i t e element development
c a l l s f i r s t o f a l l f o r a s t a t e m e n t o f t h e assumed dis p la c e m e n t f i e l d
in terms o f c o o r d i n a t e l o c a t i o n with y e t u n s p e c i f i e d c o e f f i c i e n t s .
For
th e p l a n e - s t r e s s e lem ent shown in l o c a l c o o r d i n a t e s in Fig ur e I - I , the
assumed di s p la c e m e n t f i e l d i s
and
u(x,y)
=
v(x,y)
=
a, + Ci9X + a , y '
1
2
3
+ c^x + a gy
■'
■
(4-1) .
where (u) i s t h e u n i t d is pla ce m e nt in t h e X c o o r d i n a t e d i r e c t i o n and (v)
i s t h e u n i t d is p la c e m e n t in t h e Y c o o r d i n a t e d i r e c t i o n .
Because t h i s
di s p la c e m e n t f i e l d g iv e s a c o n s t a n t s t r e s s d i s t r i b u t i o n , t h e element i s
sometimes r e f e r r e d t o as t h e c o n s t a n t s t r e s s t r i a n g u l a r element.
six
The
a c o e f f i c i e n t s can be ex pre sse d in terms of t h e s i x degre es o f
freedom f o r each e ! e m e n t- - d is p la c e m e n t in t h e X and Y d i r e c t i o n s a t
31
each node.
By t a k i n g t h e a p p r o p r i a t e d e r i v a t i v e s , th e s t r a i n can be
w r i t t e n in terms o f d i s p l a c e m e n t s , and, using the c o n s t i t u t i v e r e l a t i o n ­
s h i p , a n d in teg ral expression for
s t r a i n energy i s o b t a i n e d .
Mini­
m iz a ti o n o f p o t e n t i a l energy y i e l d s an e x p r e s s io n f o r t h e element
stiffness.
A s i m i l a r procedure in v o lv in g minim iza tion o f k i n e t i c energy
r e s u l t s in t h e e x p r e s s i o n f o r th e c o n s i s t e n t mass m a tr ix [12] and [3 2 ] .
The mass m a tr ix i s " c o n s i s t e n t " in th e sense t h a t i t i s de ri v e d from th e
same assumed dis p la ce m e n ts ( o f t e n r e f e r r e d to as shape f u n c t i o n s ) which
were used in th e s t i f f n e s s d e r i v a t i o n .
The same procedure i s a p p l i c a b l e to t h e development o f the bending
p o r t i o n o f th e s h e l l element [ 3 2 ] .
The assumed normal dis placement
f i e l d o f t h e element shown in Figure 1-3 i s taken as t h e t h i r d ord er
polynomial expansion o f th e independent c o o r d i n a t e l o c a t i o n s x and y.
This expansion in v o lv e s te n te rm s , however, only nine degre es of freedom
( d is pl a ce m e nt in t h e z d i r e c t i o n and a n g u l a r r o t a t i o n about the x and y
axes a t each node) a r e a v a i l a b l e to determine the te n c o e f f i c i e n t s f o r
bending.
This d i f f i c u l t y i s r e a d i l y circumvented by a r b i t r a r i l y ig n o r i n g
one o f t h e polynomial expansion te rm s.
conforming element.
'
The r e s u l t o f t h i s i s a non-
2
By removing th e x y term, we m a i n t a i n conformity o f
d is pla ce m e nt and r o t a t i o n a t c o i n c i d e n t nodes of a d j a c e n t el em ent s, but
a r e not gua ra nt ee d c o n t i n u i t y of sl o p e along one s i d e o f each element.
A v a r i e t y o f a l t e r n a t i v e s a re a v a i l a b l e t o produce conforming elements
which g e n e r a l l y a r e o f much g r e a t e r complexity and r e q u i r e s i g n i f i c a n t
a d d i t i o n a l computation e f f o r t .
32
The a p p a r e n t d e f i c i e n c i e s o f th e non-conforming elements do not
n e c e s s a r i l y m a n i f e s t themselves in a p p l i c a t i o n .
Both Zienkiewicz [32]
and G a ll a g h e r [12] c i t e examples (both s t a t i c and dynamic) in which th e
r e s u l t s from t h e s i m p l i e r non-conforming elements a r e s u p e r i o r to th o s e
o f t h e much more s o p h i s t i c a t e d conforming elem ents .
In any e v e n t, th e
non-conforming element used in t h i s stu dy i s r a p i d l y conv ergen t and
produces e x c e l l e n t r e s u l t s in dis pl a cm e ne t f o r "adequate" g r i d s i z e s .
The complete d e r i v a t i o n f o r th e s t i f f n e s s m a t r i x o f th e t r i a n g u l a r
s h e l l element i s given in Appendix I .
Chapter- 5
FREQUENCY RESPONSE
The dynamic v i s c o e l a s t i c problem [13] can be ex pre sse d in Laplace
tr a n s f o r m space as
( ^ [ M ] + R1 U K K h] + R g ( ^ ) C K j ] ) { 6 } = { F }
(5-1)
where R1 d e s c r i b e s th e h y d r o s t a t i c v i s c o e l a s t i c s t r e s s - s t r a i n r e l a t i o n ­
sh ip such t h a t
s
( a > = R1 ( a ) e ( A )
and Rg s i m i l a r l y d e s c r i b e s th e d e v i a t o r i c v i s c o e l a s t i c s t r e s s - s t r a i n
relationship such.that
5 (4) =
Rg(4) E(4)
-6
=
Laplace v a r i a b l e
s (-6)
=
h y d r o s t a t i c s t r e s s in Laplace space
eU)
=
h y d r o s t a t i c s t r a i n in Laplace space
SU )
=
d e v i a t o r i c s t r e s s in Laplace space
E(4)
=
d e v i a t o r i c s t r a i n in Laplace space
where
nodal dis pla cem en t v e c t o r
6
F
M
=
external force vector
c o n s i s t e n t mass m a tr ix
hydrostatic stiffn e ss
34
K,
d
=
deviatoric stiffn ess
and, t h e o v e r - b a r im pl ie s "the Laplace tr a n s f o r m o f " .
Equation (5-1) can be solved in i t s p r e s e n t form, however, f o r th e
problem a t hand, frequ en cy response d e t e r m i n a t i o n r e q u i r e s i n v e r t i n g a
p r o h i b i t i v e l y l a r g e system o f e q u a t i o n s f o r each fr equency sampled.
For
t h e g r i d s i z e shown in Figure 6 - 3 , th e e n t i r e system would involve a s e t
o f 1092 e q u a t i o n s .
To circumvent t h i s d i f f i c u l t y , a c o o r d i n a t e r e d u c t i o n
i s employed as o u t l i n e d by Anderson [ 1 ] , and d is c u s s e d in a previous
chapter.
We choose t o approximate th e a c t u a l motion o f t h e system with a
reduced s e t o f c o o r d i n a t e s {P} such t h a t
{6} = [ y ]{P}
where th e columns o f [ y ] a r e t h e "assumed modes" f o r t h e system.
I f we
make t h i s s u b s t i t u t i o n p r i o r to t h e min imiza tion o f t o t a l energy we can
w r i t e e q u a ti o n (5-1) in reduced c o o r d i n a t e s as
U 2[RM] + R1 U ) [ H h] + Rg(A) [ R i y )(P} = (FR)
(5-2)
where, th e reduced mass, h y d r o s t a t i c s t i f f n e s s , and d e v i a t o r i c s t i f f n e s s
are
[RKh]=.
[y]T[Kh][y ]
cT
I—
I_ _ _ _ I
II
W 1M
I
[RN] =
[y]
[y]T[Kd][y ]
35
The reduced f o r c e becomes
' (FR) = [ y ] T{F}
To tr a n s f o r m e q u a ti o n (5-2) i n t o th e fr equency domain [26] we e f f e c t
th e s u b s t i t u t i o n
where
J
=
a)
=
/=T
frequ en cy of f o r c i n g f u n c t i o n
The e x p r e s s i o n f o r t h e fr equency resp ons e then becomes
(-032 [ R M ] . + R1 (Jto)CRKh ] + R2 (Joj )IR K d ] ) ( P ( J c o ) ) = { F R ( j o ) ) }
The fr equency r e s p o n s e , being a measure of th e am plitud e r a t i o
(Ar ) o f d is pla ce m e nt to i n p u t f o r c e , can then be ex pre sse d as
Ar
P ( J o 1)
F R ( J o 1)
(-O12 CRM] + R1 ( Jo1)CRKh ] + R2 ( Jo1)CRKd ] )
Chapter 6
RESULTS
In o r d e r t o e s t a b l i s h a le v el o f c onfid en ce in our f i n i t e element
model, th e dynamic performance i s compared to r e s u l t s o b ta in e d from
c l o s e d form s o l u t i o n s f o r geometries where such a s o l u t i o n i s r e a d i l y
available.
T h i s , i t was f e l t , was n e c e s s a r y to pre c lu d e any p o s s i b i l i t y
o f th e f i n i t e element model being t h e source o f a p p a r e n t i n c o n s i s t e n c i e s
with pr e vi ous work.
Two t e s t g e o m e t r i e s r - t h e c i r c u l a r r i n g and the s p h e r i c a l s h e l l - w i l l be modeled using th e t r i a n g u l a r s h e l l f i n i t e element.
The c i r c u l a r
r i n g i s u s ef u l in t h a t t h e two modes o f d e fo rm a tio n, i n - p l a n e and
bending, can be e v a l u a t e d s e p a r a t e l y .
The e x t e n s i o n a l mode o f v i b r a t i o n ,
in which a l l p o i n t s a r e moving uniformly in a r a d i a l d i r e c t i o n , inv olv es
only i n - p l a n e def o rm a ti o n .
The f i r s t bending mode, in which the r i n g
assumes an " e l l i p t i c a l " shape a t t h e i n s t a n t o f maximum e x c u r s i o n , i s
c h a r a c t e r i z e d p r i m a r i l y by bending ( i . e .
i n c re a s e d o r de creased c u r ­
vature).
Comparison o f t h e f i n i t e element r e s u l t s f o r th e c i r c u l a r r in g
shown in Figure 6-1 to th e cl os e d form s o l u t i o n given by Timoshenko
[29] i s p r e s e n t e d in Table 6-1.
37
Table 6 - 1 .
Comparison o f f i n i t e element r e s u l t s to c lo s e d
form s o l u t i o n f o r a c i r c u l a r r i n g .
Resonant Frequency (Hz.)
F i n i t e Element
Model
Extensional
Mode
4908
F i r s t Bending
Mode
243
Timoshenko
% D iff ere nc e
4897
.2
233
4.3
As was p r e v i o u s l y n o te d , th e s k u l l ' s geometry can be approximated
as an e l l i p s o i d a l s h e l l .
However, due to th e n o n - a v a i l a b i l i t y of a
c l o s e d form s o l u t i o n f o r an e l l i p s o i d a l s h e l l , we must r e v e r t to the
t h i n wall s p h e r i c a l s h e l l to t e s t our f i n i t e element model.
Finite
element r e s u l t s f o r t h e low est axisymmetric bending mode o f a sphere a r e
I
compared to c lo s e d form s o l u t i o n s o b ta in e d by Baker [ 2 J .
In Figure 6-2
■the d i f f e r e n c e in r e s o n a n t f r e q u e n c i e s o b ta in e d from f i n i t e elements vs.
th o s e from c l o s e d form s o l u t i o n e xpresse d as a pre c en ta ge o f the c lo se d
form frequ en cy i s p l o t t e d a g a i n s t sphere r a d i u s while t h i c k n e s s i s held
constant.
The
f i n i t e element f r e q u e n c y , as must be t h e c a s e , i s always
g r e a t e r than t h e t h e o r e t i c a l r e s o n a n t fr equency.
A
38
Figure 6-
Grid P a t t e r n f o r C i r c u l a r Ring Te s t Geometry.
Shell t h i c k n e s s = .20 inches
Shell Radius (i n c h e s)
Comparison of F i n i t e Element to Closed-form R es ults
FIGURE 6f o r the Fundamental Bending Mode of a Thin Snhe rica l S h e l l .
39
I t i s i m p o r t a n t to note t h a t B ak e r' s development i s based on
membrane t h e o r y only .
As t h e s h e l l d ia m e te r d e c r e a s e s , th e t h i c k n e s s /
d ia m e te r r a t i o i n c r e a s e s , and, th e e f f e c t o f th e bending s t i f f n e s s
becomes more pronounced.
Thus th e dive rg e nc e o f r e s u l t s a t small
di a m e te r s i s not n e c e s s a r i l y a t t r i b u t a b l e to f i n i t e element d e f i c i e n c i e s ,
but i n s t e a d can be viewed as th e m a n i f e s t a t i o n o f a c t u a l d e v i a t i o n from
t o t a l membrane d e fo rm a tio n.
I t can be .se e n t h a t , a t l a r g e d i a m e t e r s ,
t h e f i n i t e element model f r e q u e n c i e s converge to a l e v e l ap proxim ately
2.5% g r e a t e r than t h e o r e t i c a l . .
Skull R e s u lt s
The g r i d s i z e , along with the geom et ric al and r h e o l o g i c a l parameters
shown in Figure 6-3 a r e used to g e n e r a t e q u a l i t a t i v e in fo r m at io n p e r ­
t a i n i n g t o s k u l l re so nan ce.
The assumed modes a re chosen such t h a t any
axisymmetric mode shape can be s y n t h e s i z e d .
Included
i s th e p o s s i b i l i t y
o f lo c a l de for ma tio ns a t th e "p oles" of our model.
To o b t a i n dynamic r e s u l t s resembling the fundamental bending mode
f o r a s p h e r e , boundary c o n d i t i o n s a r e imposed such t h a t nodes on the
mid-plane a r e r e s t r a i n e d from t r a n s l a t o r y motion p a r a l l e l to the a x i s .
An a n a l y t i c harmonic f o r c e i s a p p l i e d a t t h e p o le s .
Modeled as an
e l a s t i c m a t e r i a l , t h e frequency resp ons e f o r the s p h e r i c a l s ku ll i s
shown in Figure 6-4 .
Resonance occurs a t 3700 Hz. and, t h e mode shape
which r e s u l t s i s shown in Figure 6-5 .
The f i r s t bending mode i s shown
Calvarium to be
modeled.
Grid p a t t e r n f o r the
s p h e r i c a l model.
Grid p a t t e r n f o r the
e l l i p s o i d a l model.
Frequency Response
41
1000
2000
0000
3700
5000
6000
Frequency (Hz.)
FIGURE 6 - 4 .
o
- undeformed
skul I
+
- mode shape
FIGURE 6 - 5 .
Frequency Response f o r Mid-mounted Sph ere..
Mode Shape f o r Mid-mounted Sphere a t 3700 Hz.
42
on ly .
C o n ti n u a ti o n o f th e fr equency resp ons e procedure y i e l d s r e s o n a n t
mode shapes a t h i g h e r f r e q u e n c i e s which c l o s e l y correspond to those
p r e d i c t e d by t h e c lo s e d form s o l u t i o n .
In th e search f o r lower frequ en cy
modes, t h e freque ncy resp onse was computed f o r in p u t fr equency s te p
s i z e s o f 5 Hz.
None was found.
Thus , our mid-mounted s p h e r i c a l model
produces a fundamental bending mode a t a frequency which i s n e a r l y an
o r d e r o f magnitude g r e a t e r than t h a t i n d i c a t e d by pre vio us s t u d i e s .
In an a tt e m p t to s im u la te c o n d i t i o n s which could p l a u s i b l y e x p l a i n
th e ex perimental and a n a l y t i c a l r e s u l t s obta in e d by pre vio us i n v e s t i ­
g a t o r s , an a l t e r n a t e s e t o f b o u n d a r y .c o n d it io n s was imposed on the
s p h e r i c a l model.
The node a t one pole was held s t a t i o n a r y , and, a
harmonic f o r c e , a x i a l l y d i r e c t e d , was a p p l i e d a t th e o p p o s i t e pole.
The
fr equ en cy resp on se i s shown in Figure 6 - 6 , and th e mode shape a t r e ­
sonance (440 Hz.) i s shown in Figure 6-7 .
The mode.shape appears to be
th e s u p e r p o s i t i o n o f a combined l o c a l deformation o f t h e . sphere proximal
to th e p o i n t o f s u p p o r t , along with r i g i d body motion o f t h e remaining
portion of the s k u l l .
The a d d i t i o n a l e f f e c t i v e mass, due to the bulk o f
t h e s k u l l moving in r i g i d body f a s h i o n , accounts f o r a resonance which
i s an o r d e r o f magnitude l e s s than pre v io u s r e s u l t s f o r t h e same s ph er e.
S i m i l a r q u a l i t a t i v e r e s u l t s a r e o b ta in e d f o r the e l l i p s o i d shown in
Figure 6-3.
The freque ncy response f o r mid-plane s u p p o r t and f o r end
su p p o rt a r e shown in Fig ur es 6-8 and 6-10 r e s p e c t i v e l y , and, mode shapes
a t resonance a r e shown in Figur es 6-9 and 6-11.
Again, note t h a t no
43
Frequency Response
1.0 r
100
200
I
- t^ l v—. I
100
440
500
I
800
Frequency (Hz.)
FIGURE 6 - 6 .
Frequency Response f o r End-mounted Sphere.
o - unde fome d
skul I
+
- mode shape
FIGURE 6 - 7 .
Mode Shape f o r End-mounted Sphere a t 440 Hz.
Frequency Response
44
1000
0
3000
4160
5000
6000
Frequency (Hz.)
FIGURE 6 - 3 .
o
- undeformed
skul I
+
- mode shape
FIGURE 6 - 9 .
Frequency Response f o r Mid-mounted E l l i p s o i d .
Mode Shape f o r Mid-mounted E l l i p s o i d a t 4160 Hz.
Frequency Response
45
300
400
5
Frequency (Hz.)
FIGURE 6 - 1 0 .
Frequency Response f o r End-mounted E l l i p s o i d .
o - undeformed
s kul l
+ - mode shape
FIGURE 6 - 1 1 .
Mode Shape f o r End-mounted E l l i p s o i d a t 530 Hz.
46
low fr equ en cy lo c a l de for ma tio ns m a t e r i a l i z e f o r th e mid-su ppo rted
s ph er e.
Comparison r e v e a l s r e s u l t s which a r e s i m i l a r t o t h o s e d e r i v e d
from t h e s p h e r i c a l model.
The mode shape a t 530 Hz. f o r t h e end-
mounted e l l i p s o i d e x h i b i t s th e same r i g i d body-local deformation
c h a r a c t e r i s t i c t h a t was seen in t h e r e s u l t s obta in e d from th e s p h e r i c a l
model.
In a f u r t h e r a t t e m p t t o e xhau s t t h e sea rc h f o r p o s s i b l e lo cal
deformation (w ith ou t r i g i d body m ot io n ) , t h e same a n a l y s i s was employed
f o r an ex tre me ly shallow e l l i p s o i d .
This i s to check c o n j e c t u r e t h a t
low fr eq ue ncy re s p o n se s a r e the r e s u l t o f l o c a l de fo rm a tio ns o f r e l a t i v e l y
f l a t p o r t i o n s o f th e s k u l l .
Again, t h e presence o f a l o c a l deformation
a t a resonance lower than t h a t o f t h e f i r s t bending mode i s not i n d i c a t e d .
Chapter 7
DISCUSSION
A f i n i t e element model o f the human s k u l l has been p r e s e n t e d .
The r e s u l t s f o r a t h i n wall s p h e r i c a l s h e l l c l o s e l y a g re e in both r e s o ­
nan t fr eq u e n cy and co rre sp ondi ng mode shaoe to t h a t p r e d i c t e d by a
c lo s e d form s o l u t i o n .
This model, when su p p li e d with th e geometrical
and r h e o l o g i c a l pa rame te rs approximating t h o s e f o r a human s k u l l , i n d i ­
c a t e s re so nan ce o f the fundamental bending mode a t ap pro xim a te ly 3700
Hz.
F u r t h e r i n s i g h t i n t o t h i s a p p a r e n t i n c o n s i s t e n c y with previous
s t u d i e s can be gained by s u b j e c t i n g t h e same sk ull t o an end mounting
as d e s c r i b e d e a r l i e r .
This r e s u l t s in a s i g n i f i c a n t l y d i f f e r e n t mode
sh ape , b e s t d e s c r i b e d as a r i g i d body-local deformatio n (RBLD) mode.
The f o ll o w i n g d i s c u s s i o n de mo nst rat es t h a t th e RBLD mode i n t e r p r e t a t i o n
i s c o n s i s t e n t with th e r e s u l t s of pre vio us r e la te d , i n v e s t i g a t i o n s , and
the r a m i f i c a t i o n s o f t h i s i n t e r p r e t a t i o n a r e c o n s i d e r e d .
Comparison to Pr evious S tu d i e s
We w i l l f i r s t examine an ex perimental study in which a cadaver
s k u l l i s s u b j e c t e d t o an e x t e r n a l harmonic f o r c e .
The i n t e n t of the
p a r t i c u l a r stu dy i s to i n f e r t h e re s ponse of the human head s u b je c te d
to sim ila r external stimulus.
G urdjian [15] f r o n t a l l y , mounted a cadaver sk ull to a f o r c e g e n e r ­
ator.
A f o r c e of c o n s t a n t amplitude was imparted t o t h e s ku ll over a
s p e c i f i e d fr equ en cy ra n g e .
A c c e l e r a t i o n s were measured a t and dia m e tr i
48
c a l I y o p p o s i t e the l o a d i n g , a t t h e ap ex, and a t th e s i d e o f the s k u l l .
G urdjian r e p o r t s t h a t an a n t i - r e s o n a n t and r e s o n a n t c o n d i t i o n occur a t
a pp ro xi m a te ly 300 Hz. and 900 Hz. r e s p e c t i v e l y .
The a n t i - r e s o n a n t
mode a t 300 Hz. is o f p a r t i c u l a r i n t e r e s t in t h a t i t s d e s c r i p t i o n c l o s e ­
l y c or re sp onds to t h a t o f th e RBLD mode d e s c r i b e d in t h i s st udy .
The
measurement o f a c t i v i t y a t o nly f o u r l o c a t i o n s over t h e e n t i r e s k u ll in
G u r d j i a n ' s s tu d y makes c o n c l u s i v e comparison o f th e two modes u n l i k e l y .
The a n t i - r e s o n a n t mode i s d e s c r i b e d by s t a t i n g , " th e mode of
v i b r a t o r y motion of t h e head i s such t h a t very l i t t l e movement occurs
a t t h e d r i v e n p o i n t compared to the o c c i p u t . "
Viewed in a s l i g h t l y
d i f f e r e n t p e r s p e c t i v e , t h i s a n t i - r e s o n a n t mode c orre spo nds to the RBLD
mode shown in Fig ur e 6 - 7 , where th e r e s u l t i n g lo a di ng a t th e f i x e d
node i s analogous to G u r d j i a n ' s e x c i t a t i o n .
As noted p r e v i o u s l y , th e
RBLD mode e x h i b i t s l a r g e r e l a t i v e de fo rm a tio ns of the o c c i p u t vs. th e
fro n ta l region.
A l a r g e mechanical impedance ( a l s o ap pea rin g in G u r d j i a n ' s meas­
urements) i s th e consequence o f th e RBLD mode.
This impedance occurs
as t h e r e s u l t of a lar^ge p o r t i o n of t h e s ku ll t r a n s l a t i n g in r i g i d body
fashion.
The s t r a i n energy becomes c o n c e n t r a t e d a t t h e f i x e d node--a
c o n d i t i o n which i s t h e r e s u l t o f th e p a r t i c u l a r b o u n d a ry ,c o n d it io n
chosen.
Because of t h e absence o f complete experimental i n f o r m a t i o n ,
v e r i f i c a t i o n of the c o i n c i d e n c e of th e RBLD mode and G u r d j i a n ' s a n t i ­
r e s o n a n t mode must be t h e p ro du c t o f f u r t h e r e x p e r i m e n t a l . work.
The
49
determination
of
such a correspondence would mandate r e - e xam in at io n
and p o s s i b l e r e - i n t e r p r e t a t i o n o f t h e re s p o s e o f the human head to
external disturbance.
S p e c i f i c a r e a s o f r e - e v a l u a t i o n w i l l be ad dressed
l a t e r in t h i s c h a p t e r .
A n a l y t i c a l Comparison
The fo ll o w i n g d i s c u s s i o n w i l l c o n c e n t r a t e on th e a n a l y t i c behavior
o f th e p r e s e n t axisymmetric model compared t o t h a t d e s c r i b e d in a
s i m i l a r study by Nickel! and Marcal [ 2 4 ] .
In t h a t s t u d y , t h e s kull i s
modeled using doubly curved t r i a n g u l a r f i n i t e e le m en ts , and, the lowest
mode shapes were o b ta in e d f o r t h r e e d i f f e r e n t s e t s o f p r e s c r i b e d
boundary c o n d i t i o n s .
I t i s th e c hoice o f t h e s e boundary c o n d i t i o n s and
t h e manner in which th e y i n f l u e n c e t h e e f f e c t i v e system which r a i s e s t h e
ca se f o r an a l t e r n a t i v e and perhaps more p r e c i s e i n t e r p r e t a t i o n of
r e s u l t a n t mode shapes.
In each case one node ( l o c a t e d a t t h e f r o n t ,
r e a r , or a t th e base) i s r e s t r a i n e d so as to exclude th e p o s s i b i l i t y o f
t r a n s l a t o r y motion.
A second a r b i t r a r y s t i f f n e s s pro vid e s " s u f f i c i e n t
r e s t r a i n t a t an a d j a c e n t p o i n t to p r e v e n t r o t a t i o n . "
For each s e t o f boundary c o n d i t i o n s , fo u r mode shapes were obta in e d
The f i r s t two w i l l be d i s c u s s e d only b r i e f l y because t h e y a r e i n . e s s e n c e
r i g i d body r o t a t i o n ( r o c k in g ) about t h e f i x e d p o i n t .
These modes are
no t axisymmetric and, t h e r e f o r e , do no t a f f o r d comparison to the r e s u l t s
o f our axisymmetric s tu d y . The f i r s t and second mode shapes are
50
reproduced in Fig ur es 7-1 and 7-2.
In each f i g u r e t h e mode shapes a re
a l s o shown superimposed on th e undeformed s k u ll to i l l u s t r a t e d i s t o r t i o n .
The f i r s t i s n e a r l y a pure r i g i d body r o t a t i o n about t h e f i x e d node.
Only a s l i g h t d i s t o r t i o n o f th e s k u ll i s d e t e c t a b l e , t h i s o c c u r r in g a t
t h e f r o n t a l base o f th e s k u l l .
The fr equency a s s o c i a t e d with t h i s mode
shape i s then p r i m a r i l y dependent upon th e su pp ort s t i f f n e s s p r e s c r i b e d
by the i n v e s t i g a t o r .
The second mode i s s i m i l a r to t h e f i r s t in t h a t the predominant
c h a r a c t e r i s t i c i s a r i g i d body or rocking movement about t h e fi x e d node.
In a d d i t i o n to t h e r i g i d body motion, a s i g n i f i c a n t d i s t o r t i o n occurs a t
th e r e a r and th e f r o n t a l base o f th e s k u l l .
Again, t h e r e s o n a n t f r e ­
quency w i l l be g r e a t l y i n f l u e n c e d by t h e n a t u r e and magnitude of the
a r t i f i c i a l r e s t r a i n i n g s t i f f n e s s which has been a p p li e d as a boundary
condition.
Nickell and M a r c a l' s t h i r d mode i s d e s c r i b e d as e x h i b i t i n g " la rg e
lo c a l s k u l l d i s t o r t i o n in th e f r o n t a l and o c c i p i t a l r e g i o n s . "
3a shows th e
Figure 7-
t h i r d mode and th e undeformed sk ull taken from t h e study
by Nickel! and M ar ca l.
In Figure 7-3a th e d is pla cem en ts a r e shown such
t h a t t h e f i x e d node i s c o i n c i d e n t in t h e undeformed and t h i r d mode
r e p r e s e n t a t i o n . Figure 7-3b shows t h e same two shapes superimposed so as
to i l l u s t r a t e r e la tiv e d is to r tio n .
With t h e ex cep tio n o f th e deformation
o c c u r r i n g a t th e f r o n t a l base o f th e s k u l l , th e mode shape appears to be
th e r i g i d body-local deformation mode o b ta in e d from t h e axisymmetric
s tu dy p r e s e n t e d in t h i s pap er.
See Figure 6-7.
51
— ------------
undeformed
skul I
------------------ mode shape
FIGURE 7 - 1.
Nickel I & M a r c a l 1s r e a r mounted s k u l l ( a . ) with
t h e 1 s t mode & undeformed shap e c o i n c i d e n t a t t h e f i x e d
node, and ( b . ) w it h t h e undeformed s k u l l and mode shape
sup erimp osed t o i l l u s t r a t e r e l a t i v e d i s t o r t i o n .
52
- - undeformed
skul I
- - mode shape
FIGURE 7 - 2 .
Mickell & M a r c a l 1s r e a r mounted s k u l l ( a . ) with
t h e 2nd mode & undeformed shape c o i n c i d e n t a t t h e f i x e d
node, and ( b . ) w it h t h e undeformed s k u l l and mode shape
superimposed t o i l l u s t r a t e r e l a t i v e d i s t o r t i o n .
53
--
undeformed
skul I
--
mode shape
FIGURE 7 - 3 .
Nickel I & M a r c a I 1s r e a r mounted s k u l l ( a . ) with
t h e 3rd mode & undeformed shape c o i n c i d e n t a t t h e f i x e d
node, and ( b . ) w it h t h e undeformed s k u l l and mode shape
sup erimp osed to i l l u s t r a t e r e l a t i v e d i s t o r t i o n .
54
The a re a o f l o c a l deformation shown in Figure 7-3 i s somewhat
l a r g e r than t h a t i n d i c a t e d by th e axisymmetric model (F igu re 6 - 7 ) .
V a r i a t i o n s in s k u ll t h i c k n e s s and geom et ric al anomalies which were
inc lu de d in th e N i c k e l ! -Marcal model c a n . a t l e a s t p a r t i a l l y account f o r
d i f f e r e n c e s in t h e a re a o f l o c a l d e f o r m a t i o n .
The g r e a t e s t t h i c k n e s s
was a s c r i b e d to t h e . o c c i p u t while a s u b s t a n t i a l l y t h i n n e r zone was
modeled in t h e proximal regio n where maximum bending a p p e a rs .
The boundary c o n d i t i o n s imposed by Nickel I and Marcal a r e the same
as th o s e employed in t h e axisymmetric model with th e a d d i t i o n a l freedom
to allow rocking about th e f i x e d node.
lowe st modes o b t a i n e d .
This op tio n give s l i c e n s e to th e
The t h i r d mode d i s p l a y s th e e s s e n t i a l c h a r a c ­
t e r i s t i c s which p o i n t to i n t e r p r e t a t i o n as a RBLD mode, b u t , not w ith ou t
t h e i r r e g u l a r i t i e s emanating from a la ck o f geometrical and s t r u c t u r a l
symmetry.
Implications
Some immediate q u e s t i o n s a r i s e co ncerning previo us i n t e r p r e t a t i o n s
o f s k u ll resp onse in t h e s e and s i m i l a r i n v e s t i g a t i o n s .
t i o n in such s k u ll
A common assump­
(in - v ac u o ) s t u d i e s i s t h a t response o f th e sk ull to
e x t e r n a l e x c i t a t i o n i s i n d i c a t i v e o f t h a t f o r the human head s u b je c te d
to t h e same e x c i t a t i o n .
How t h i s premise
w it h s ta n d s t h e RBLD mode
i n t e r p r e t a t i o n i s th e s u b j e c t o f th e f i r s t query.
RBLD mode be e x c i t e d in th e i n - v i v o human head?
Secondly, can the
The i m p l i c a t i o n here i s
55
t h a t t h i s p a r t i c u l a r mode may be i n h e r e n t to t h e system which has been
modif ied by t h e use o f a r t i f i c i a l ex perimental and a n a l y t i c boundary
conditions.
And, t h i r d , how does t h e above
e f f e c t p re v io u s c o n c lu s io n s
p e r t a i n i n g t o t h e mechanisms o f b r a i n damage?
Use o f t h e re s p o n se o f t h e s k u ll a lo n e to imply t h e n a t u r e o f head
i n j u r y n e c e s s i t a t e s th e assumption t h a t t h e b r a i n a c t s in a p a s s iv e
m a n n e r - - c o n t r i b u t i n g to damping and producing only sec ondary e f f e c t s in
re s p o n se c h a r a c t e r i s t i c s .
In our axisymmetric s t u d y , i t has been noted
t h a t t h e change in boundary c o n d i t i o n s from mid-mounted to end-mounted
r e s u l t s in a r e d u c t i o n o f th e f i r s t bending mode fr eq u e n cy by n e a r l y an
o r d e r of magnitude.
This i s t h e consequence o f an i n c r e a s e d e f f e c t i v e
mass producing l a r g e r i n e r t i a l f o r c e s which must be c o un te r ed a t t h e
end s u p p o r t .
I t f o ll o w s t h a t i n c l u s i o n of t h e mass o f t h e b r a i n w i l l
f u r t h e r i n c r e a s e t h e e f f e c t i v e mass o f t h e system, p a r t i c u l a r l y with
r i g i d body r e l a t e d modes.
Somewhat l e s s e v i d e n t i s th e e f f e c t t h a t
t h e b r a i n has on t h e o v e r a l l s t i f f n e s s of th e head.
F a i l u r e to i n c o r ­
p o r a t e th e e f f e c t s o f t h e b r a i n where predomin at el y r i g i d body motion
o f t h e s k u l l p r e v a i l s , w i l l produce r e s u l t s o f q u e s t i o n a b l e v a l i d i t y .
The RBLD mode a ppear s in a n a l y t i c a l and experimental s t u d i e s in
which a node a t ( o r d i a m e t r i c a l l y o p p o s i t e t o ) th e e x t e r n a l loading
p o i n t i s he ld s t a t i o n a r y .
To m a i n t a i n t h i s su p p o rt d u r in g frequency
re s po ns e t e s t i n g , l a r g e f o r c e s a r e n e c e s s a r y to c o u n t e r t h e i n e r t i a l
e f f e c t s o f r i g i d body motion.
For i n - v i t r o experimental work, the
56
specimen i s r i g i d l y a t t a c h e d to th e t e s t a p p a r a t u s .
Non-invasive i n - v i v o
t e s t i n g does not al low so c o n ve nie nt a s o l u t i o n . A p r e s s u r e sup port can
m a in ta in t h e boundary c o n d i t i o n f o r only t h e compression h a l f o f the
cycle.
Whether t h i s su p p o rt c o n d i t i o n can a d e q u a te ly be s y n th e s iz e d f o r
i n - v i v o e x p e r i m e n t a t i o n emerges as a q u e s t i o n which must be r e s o l v e d .
An in - d e p t h d i s c u s s i o n o f t h e mechanisms o f b r a i n damage i s beyond
t h e scope o f t h i s p a p er , so t h a t t h e t h i r d qu e st i o n w i l l go l a r g e l y
unanswered.
Let i t s u f f i c e t o s t a t e t h a t extreme c a u t i o n must be
p r a c t i c e d in t h e i n t e r p r e t a t i o n o f r e s u l t s from s t u d i e s s i m i l a r to th o s e
d i s c u s s e d h e re .
The case f o r th e e x i s t e n c e and the r o l e o f th e RBLD
mode must be c o n s id e re d f u l l y .
F e a s i b i l i t y f o r D et ermination o f Rheological P r o p e r t i e s
Retu rnin g our d i s c u s s i o n t o th e o r i g i n a l purpose o f t h i s paper we
c o n s i d e r t h e f e a s i b i l i t y o f d e te r m in in g t h e r h e o l o g i c a l p r o p e r t i e s of
\
t h e human s k u l l using low fr equency v i b r a t i o n s . Based on t h e r e s u l t s o f
th e axisymmetric model, two approaches a r e a v a i l a b l e to a ch ie ve our
objective.
The f i r s t in vo lv e s t h e use o f th e fundamental bending mode
which i s a resp onse c h a r a c t e r i s t i c
skull.
o f th e fr ee -b ody (o r mid-mounted)
Because th e r e s o n a n t fr equency a s s o c i a t e d with t h i s mode shape
i s q u i t e high (3700 Hz. f o r th e axisymmetric s kull modeled in t h i s
s t u d y ) , t h e fo ll o w i n g enumeration o f r e q u i r e m e n t s , upon which su ccess i s
contingent, is presented.
57
1.
Tr ans ducer s and experimental equipment must be a v a i l a b l e with
i n t r i n s i c s e n s i t i v i t i e s c ap a ble o f measuring e x tr e m e l y small am­
p l i t u d e motion o f t h e s k u ll a t high f r e q u e n c i e s (2 t o 5 KHz.).
Along t h e s e same l i n e s , a system to impart t h e n e c e s s a r y harmonic
f o r c e t o t h e s k u l l must a l s o be a v a i l a b l e .
2.
An e f f e c t i v e means t o contend with t h e damping e f f e c t o f the l a y e r
o f s k in c ove r in g the. s kull must be developed.
The d i f f i c u l t y i s
p a r t i c u l a r l y troublesome a t t h e l o c a t i o n s o f a p p l i e d load and r e ­
sponse measurements.
3.
At e l e v a t e d f r e q u e n c i e s t h e i n t e r n a l damping o f t h e bone and th e
c o n t e n t s o f t h e s k u l l w i l l f u r t h e r reduce th e p r o b a b i l i t y of
measuring r e s p o n s e .
4.
The s u c c e s s i o n o f r e s o n a n t modes f o r a s p h e r i c a l s h e l l a r e s e p a r ­
a t e d by a fr eq ue ncy d i f f e r e n c e which i s small r e l a t i v e to the
fr eq ue ncy i t s e l f .
When allowed i n s i g h t i n t o t h e re s ponse o f only
a few nodes, t h e c h a r a c t e r i s t i c s which d i f f e r e n t i a t e one mode
from a n o t h e r may become ex tre m el y s u b t l e .
As t h e geometry i n -
.
c r e a s e s in c o m p le x it y , so does t h e t a s k o f d e f i n i n g with a r e a s o n ­
a b l e degre e o f ac c ura cy r e s u l t a n t mode shapes.
The concern here
i s f o r t h e q u a n t i t y and q u a l i t y of. i n s t r u m e n t a t i o n r e q u i r e d f o r
such a d e t e r m i n a t i o n .
58
An a l t e r n a t i v e approach i s to u t i l i z e
mental bending mode.
the RBLD mode as th e funda­
The lower f r e q u e n c i e s a l l e v i a t e some o f the
d i s t r e s s i n g re qui re m e nts posed above, b u t , in t u r n , t h e use o f t h i s mode
w a r r a n t s i t s own l i s t o f re q u ir e m e n ts .
1.
These fol lo w.
The q u e s t i o n r e l a t i n g to th e c a p a b i l i t y or need to reproduce the
end-mount boundary c o n d i t i o n f o r i n - v i v o s t u d i e s must be r e s o l v e d .
One would e xp ec t t h a t an exper imen ta l procedure u t i l i z i n g a sim­
p l i f i e d geometry might produce a s a t i s f a c t o r y answer.
2.
Response, in v o lv in g a l a r g e p o r t i o n o f th e s ku ll moving as a r i g i d
body, i n t i m a t e l y inv olv es th e p h y s i o l o g i c a l su p p o rt f o r th e head,
i . e . th e s p in a l column along with th e muscular and f l e s h y t i s s u e s
o f t h e neck.
The a n a l y t i c a l modeling o f the neck may c o n s t i t u t e as
for mi dab le a t a s k as t h a t involved with modeling t h e head i t s e l f .
A pp ro p ri a te s i m p l i f y i n g assumptions or an adequate model of the
s u p p o rt s t r u c t u r e mu'st be inc lu de d in th e a n a l y t i c a l p o r t i o n of t h e
investigation.
R e p r o d u c i b i l i t y o f experimental r e s u l t s w ill a l s o
be a d v e r s e l y a f f e c t e d due to i n c o n s i s t e n c i e s in t h e s t a t e of
r e l a x a t i o n of t h e neck.
3.
The damping and i n e r t i a l e f f e c t s o f t h e b ra in must be included in
t h e a n a l y s i s , Orl th e d e t e r m i n a t i o n made t h a t such e f f e c t s are
negligible.
4.
For t h e RBLD mode, th e experimental i n s t r u m e n t a t i o n o f the sk ull
must produce d e f i n i t i v e in f or m at io n re g a rd in g t h e r e s u l t a n t mode
59
shape.
An imp or tan t and perhaps v i t a l measurement f o r th e i n - v i v o
s k u l l i s t h e e x t e n t o f th e a re a o f lo c al d e fo rm a tio n.
Again, t h i s
r e q u i r e s , s o p h i s t i c a t e d ex perimental c a p a b i l i t i e s .
Summary
A f i n i t e element model o f t h e i n - v i v o
sku ll i n d i c a t e s a funda­
mental f r e e - b o d y bending resonance a t a pprox im at el y 3700 Hz.
A modified
system with an end-mounted boundary c o n d i t i o n produces a r i g i d body with
lo c a l deformatio n mode shape a t ap pro xi ma te ly 450 Hz.
This mode a ppea rs
t o be c o n s i s t e n t with t h e r e s u l t s o f s i m i l a r s t u d i e s and sugg est s a r e ­
c o n s i d e r a t i o n o f pre v io u s i n t e r p r e t a t i o n s .
N e i t h e r o f th e two mode shapes o f f e r s p a r t i c u l a r encouragement to
th e i n v e s t i g a t o r i n t e r e s t e d in u t i l i z a t i o n o f t h e s e mode shapes f o r
determination of rheological p ro p e rtie s .
The former becomes u n a t t r a c ­
t i v e p r i m a r i l y because o f t h e high f r e q u e n c i e s i n v o lv e d , and the l a t e r
because o f th e e n t r a n c e o f t h e neck as a modeling problem.
The need t o
a r r i v e a t m a t e r i a l p r o p e r t i e s f o r th e i n - v i v o human s k u l l remains.
D esp ite th e above mentioned shor tc omi ngs, th e proce dures d is c u s s e d may
s t i l l be th e b e s t means a v a i l a b l e t o determine r h e o l o g i c a l p r o p e r t i e s o f
t h e human s k u l l .
REFERENCES
1.
Anderson, R. A ., Fundamentals o f V i b r a t i o n s . New York: The
Macmillan Company, 1967.
2.
Baker, W. E ., "Axfsymmetric Modes o f V ib ra ti o n o f a Thin S p h e r i­
cal S h e l l " , The J ourn al o f th e A co ust ic al S o c i e t y o f
America, Vol. 33, 1961, pp. 1749-1758.
3.
B e n e d ic t, J . V . , e t . a l . , "An A n a l y t i c a l I n v e s t i g a t i o n o f the
C a v a ta ti o n Hypothesis o f Brain Damage", Jou rnal o f Basic
E n g i n e e r i n g , Vol. 92, 1970, pp. 597-603.
4.
Cook, R. D., Concepts and A p p l i c a t i o n s o f F i n i t e Element A nal ys is
New York: Wiley, 1974.
5.
C ur re y, J . D., "The R e l a t i o n s h i p Between th e S t i f f n e s s and the
Mineral Content o f Bone", J ourn al o f Biomechanics, Vol. 2,
1969,. pp. 477-479.
6.
Engin, A. E . , and King Lu, Y., "Axisymmetric Response o f a F l u i d f i l l e d S p h e ri c a l Shell in Free V i b r a t i o n s " , Jou rnal of
Biomechanics, Vol. 3, 1970, pp. 11-22.
7.
Engin, A. E . , "The axisymmetric Response o f a F l u i d - f i l l e d Spher­
i c a l Shell t o a Local Radial Impulse--A Model f o r Head In­
jury" , Jo^rnaJ_of_Bjo^
Vol. 2, 1969, pp. 325-341.
8.
Evans, F. G., "Mechanical P r o p e r t i e s and H i s t o l o g i c a l S t r u c t u r e
of.Human C o r t i c a l Bone", ASME Paper Noi 70-WA/BHF-7, 1970.
9.
Flugge, W., V i s c o e l a s t i c i t y . Walt h a n . M as s. : B l a i s d e l l P u b l i s h ­
ing Company, 1967.
10.
F r an ke , E. K., "Response o f th e Human Skull t o Mehcanical Vibra­
t i o n s " , J ourn al o f th e A c o u st ic a l So c ie ty o f America, V ol.
28, 1956, pp. 1277-1284.
■
:
11. . F r o s t , H. M., Done Dynamics in O st eo poro si s arid Q^erhalacia .
S p r i n g f i e l d , 111. Charles C. Thomas, 1966.
12.
G a l l a g h e r , R. H., F i n i t e Element Anal ysi s Fundamentals. Engle­
wood C l i f f s , N . J . : P r e n t i c e - H a l l , In c , 1975.
61
13.
Garner, E. R ., and B l a c k k e t t e r , D. 0 . , . " In - v iv o D et ermination o f
Macroscopic B io lo g ic a l M at e ria l P r o p e r t i e s " , Jou rnal o f .
Engineering M a t e r i a l s and Technology, Vol. 97, 1975,
pp. 350-356.
14.
Gordon, E. L . , e t . a l . , "Analysis o f a M u l t i - l a y e r e d Spheric al
Head Impact Model", J ourn al o f Engineering f o r I n d u s t r y ,
Vol. 96, 1974, p p . 534-540.
15.
G u rd ji a n , E. G., e t . a l . , "S tu d ie s on Mechanical Impedance of
th e Human S k u l l : P r e l i n i n a r y R ep ort " , Jou rnal o f Biomechan­
i c s , Vol. 3, 1970, pp. 239-247.
16.
Hardy, C. H., and M ar ca l, R. V . , . " E l a s t i c A na lys is o f a S k u l l ",
Jou rnal o f Applied Mechanics, Vol. 40, 1973, pp. 838-842.
17.
H i c k l i n g , R ., and Wenner, M. L . , "Mathematical Model o f a Head
S u bje c te d t o an Axisymmetrical Impact", J ourn al o f Bio­
m e ch an ic s, Vol. 6, 1973, pp. 115-132.
18.
Hubbard, R. P . , " Fle xur e o f Layered Cranial Bone", Jou rnal of
Biomechanics, Vol. 4, 1971, pp. 251-263.
19.
J u r i s t , J . M., "Measurement o f Ulnar Resonant Frequency", Ortho­
pedic Research L a b o ra to ry , I n t e r n a l Report No. 36, Univer­
s i t y o f Wisconsin Medical C e n t e r , Madison Wisconsin.
20.
Kenner, V.H., and Goldsmith, W., "Dynamic Loading o f a Fl u i d f i l l e d S p h e ri c a l S h e l l " , I n t e r n a t i o n a l J ourn al o f Mechani­
cal S c i e n c e , Vol. 14, 1972, pp. 557-568.
21.
McElhaney, J . H., e t . a l . , "Mechanical P r o p e r t i e s o f Cranial
Bone", Jou rnal o f Biomechanics, Vol. 3, 1970, pp. 495-511.
22.
McElhaney, J . . H . , e t . a l . , " E f f e c t o f Embalming on t h e Mechani­
cal P r o p e r t i e s o f Beef Bone", Journal o f Applied Ph y s io lo g y ,
V ol. 19, 1964, pp. 1234-1236.
23.
McLean, F. C ., and U r i s t , M. R ., Bone. Chicago: The U n i v e r s i t y
o f Chicago P r e s s , 1964.
•24.
N i c k e l ! , R. E . , and Mar cal , R. V., "In-vacuo Modal Dynamic Re­
sponse o f t h e Human S k u l l 11, Jou rnal o f Engineering f o r
I n d u s t r y , Vo l. 96, 1974, pp. 490-494.
62
25.
P r z e m i e n i e c k i , J . S . , Theory o f Matrix S t r u c t u r a l A n a l y s i s . New
York: McGraw-Hill, 1968.
26.
Raven, F. H., Automatic Control E n g i n e e r in g . New York: McGraWH i l l , 1968.
27.
S e d l i n , E. D., "Rheological Model f o r C o r t i c a l Bone", Acta Orth.
Sc a n d ., S u p p l . 8 3 , 1965.
28.
Tennyson, R. C ., e t . a l . , "Dynamic V i s c o e l a s t i c Response o f Bone",
Experimental Mechanics, V ol. 12, 1972, pp. 502-507.
29.
Timoshenko, S . , V i b r a t i o n Problems in E n g i n e e r i n g . 4th E d . , New
York: Wiley, 1974.
30.
Vaughan, J . , Physiology o f Bone. Oxford: Clarendon P r e s s , 1970.
31.
Wood, J . L . , "Dynamic Response o f Human Cranial Bone", Journal
o f Biomechanics, VoT.. 4, 1971, pp. 1-12.
32.
Z i e n k i e w i c z , 0. C ., The F i n i t e Element Method in Engineering
Sc ie nce . 2nd E d. , London: McGraw-Hill, 1971.
APPENDIX
APPENDIX I
SHELL ELEMENT
The f i n i t e element d e r i v a t i o n f o r th e t r i a n g u l a r s h e l l element
c l o s e l y fo ll ows t h a t d e s c r i b e d by Zienkiewicz [3 2 ] .
D ev iati ons from
Z i e n k i e w i c z 1s development inv olv e th e use o f a r i g h t hand c a r t e s i a n
c o o r d i n a t e system, and th e independent development o f the h y d r o s t a t i c
and d e v i a t o r i c s t i f f n e s s .
This i s done so t h a t t h e Correspondence
P r i n c i p l e can be a p p l i e d d i r e c t l y [ 9 ] f o r s o l u t i o n o f th e v i s c o e l a s t i c
problem.
The i n d i v i d u a l t r e a t m e n t o f t h e h y d r o s t a t i c and d e v i a t o r i c
s t i f f n e s s allow s f o r a l g e b r a i c s i m p l i f i c a t i o n s which i n c r e a s e computa­
tional efficiency.
The s h e l l element i s s u b j e c t e d t o both i n - p l a n e and bending l o a d s .
The small deformatio n assumption i s u t i l i z e d so t h a t t h e two deformation.,
modes (bending and i n - p l a n e ) a r e ta ken as mutually in de pen de nt.
The
two elements a r e then combined by l i n e a r s u p e r p o s i t i o n to form the
s h e l l element.
The In - P la ne Element
. .
The dis p la c e m e n ts (two i n - p l a n e deforma tio ns and one r o t a t i o n )
and t h e c o rr e sp ondi ng e x t e r n a l loads f o r a t y p i c a l node a r e shown in
Figur es ( I - I ) and ( 1 - 2 ) .
65
Z
FIGURE I - I .
In -p lan e
deforma tio ns a t node
j for a typical e l e ­
ment shown in local
coordinates.
FIGURE 1 - 2 .
In - pla ne
loads a t node j f o r a
t y p i c a l element shown
in lo c a l c o o r d i n a t e s .
The i n - p l a n e de for ma tio ns (u and v) a r e allowed to vary l i n e a r l y
with r e s p e c t to c o o r d i n a t e l o c a t i o n such t h a t
and
u(x,y)
=
Oi1 + Cx2X + a 3y
v(x,y)
=
n4 + Ct5X + <x6y
The r e s u l t i s s i x unknown a c o e f f i c i e n t s
(I-I)
and
s i x d e g re ss o f freedom
(two a t each node) from which t h e s e can be determined.
The dis pl a ce m e nts in the x c o o r d i n a t e d i r e c t i o n a t th e nodes
can be w r i t t e n
66
«1 + <*2*1 + "3^1
ui
UJ
=
"I + « 2 * j + " s / j
uk
=
*1 + *2*k + "S^k
(1-2)
where s u b s c r i p t s i , j , and k i d e n t i f y t h e r e s p e c t i v e node.
Equations (1-2) can be w r i t t e n in m a tr ix form as
I
u T
>
=
1
, U j
X 1 .
y i
a i
f
>
j
y j
a2 1
X
a3
<
J
By i n v e r t i n g t h e 3X3 nodal c o o r d i n a t e m a t r i x , the f o ll o w i n g e x pr e ss io n
f o r t h e f i r s t t h r e e a lp ha c o e f f i c i e n t s can be o b ta in e d .
1
4
ai
aj
ak
bi
bj
bk
_c i
cj
C sj
r o
>
S
It
0 1 I
ui
<
uJ
lukj
where 2 a i s t h e d e te r m in a n t o f th e pre v io u s 3X3 m a tr ix and i s num e ri c al ly
equal t o twic e th e a re a p f the t r i a n g u l a r element.
The a , b, and c
c o n s t a n t s a r e elements o f the a d j o i n t o f t h e nodal c o o r d i n a t e m a tr ix .
A s i m i l a r e x p r e s s io n can be w r i t t e n f o r t h e remaining t h r e e alpha co­
e f f i c i e n t s using, th e nodal dis p la ce m e n ts in th e y - c o o r d i n a t e d i r e c t i o n .
67
Plane s t r a i n f o r small de for ma tio ns i s de fin e d
3U
3X
fe' T
X
{e}
< Ew
y
> =
I
IV
<
(1-5)
ay.
3U
L
ay +
3V
33T
S u b s t i t u t i n g f o r u and v from e q u a ti o n ( 1 - 4 ) , the s t r a i n becomes
(e)
( 1- 6 )
*3 + "5
and, again s u b s t i t u t i n g from eq u at i o n (1-4) f o r th e alpha c o e f f i c i e n t s
give t h e fo ll o w i n g e x p r e s s i o n r e l a t i n g t h e s t r a i n f i e l d to nodal d i s ­
pla cements.
I
O
y -
_a
=
O
[Bi
where
•r-5
_Q
[B] {6e }
O
=
Lf_
{£}
0
ci 0
cj 0
ck
ci
bi c j
bj c k
bk
7—
and
[ 6e ] T
=
. Luj
Vj Uj
V1 U k V k J
The s t r e s s - s t r a i n r e l a t i o n s h i p i s
68
[Tv
O
X
E
X
J Ev >
( 1- 8 )
y
C
O
O
l_
[x y j
O
—
(1-v)
O
y
P-
<a , I =
O-1
£
£
LxyJ
o r , in terms o f h y d r o s t a t i c and d e v i a t o r i c s t r e s s and s t r a i n
I IT
Is
PO
Il
S
0
I
I
0
e
I
I
0
<e >
0
0
0
0
(1-9)
where t h e h y d r o s t a t i c s t r e s s (s ) i s de fi n e d
s
~
V ax + °y)
(1-10)
and th e h y d r o s t a t i c s t r a i n (e) i s .
? (e x + ey>
(I- H)
Let t h e bulk modulus k be de fi n e d as
. E
2TT-7J
K
( 1-
12 )
We can then w r i t e
s
=
k
*2e
. or
{s}
K[Dh] {e}
(1-13)
69
where
I
I
0
I
I
0
0
0
0
S i m i l a r l y f o r t h e d e v i a t o r i c s t r e s s - s t r a i n r e l a t i o n s h i p we have
1 - 1
0
S;X
Px I
<S,
y f
2(1 + v)
-I
O
O
xy__
I
0
<
(1-14)
E
I xyj
where
- S
and
etc.
eX - e
The s h e a r modulus (G) i s de fi n e d
G
=
2 ( I E+ v)
and, we can then w r i t e
- V
'
G • Exy
(1-15)
G[Dd ]'{E}
(1-16)
o r , in m a tr ix form
{S}
=
where
1
0
Q-
I—
Il
_I
I
I
-I
0
-I
I
0
0
0
I
To fo r m u la t e th e h y d r o s t a t i c and d e v i a t o r Ic s t i f f n e s s we w r i t e th e
e x p r e s s i o n f o r th e p o t e n t i a l energy (Iig ) o f the element.
1/2 [sj. {e}dv +1/2 LSJ {e}dv -[FJ {6 }
The f i r s t term
(1-17)
i s t h e s t r a i n energy due to the h y d r o s t a t i c s t r e s s , t h e
second term i s th e s t r a i n energy due t o t h e d e v i a t o r i c s t r e s s , and the
f i n a l term i s th e c o n t r i b u t i o n o f t h e work done by t h e e x t e r n a l f o r c e s .
S u b s t i t u t i n g e q u a t i o n s ( 1 - 7 ) , ( 1 - 1 3 ) , and (1-16) i n t o eq uat io n
(1-17) and minimizing p o t e n t i a l energy by t a k i n g th e f i r s t v a r i a t i o n o f
d is p la ce m e n t gives
[B]T[Dh][B]dv + G
,Tr
[ B ] ‘ [Dd][B]dv
{6e > - (F)
(1-18)
In both i n t e g r a l s the in t e g r a n d i s independent o f c o o r d i n a t e l o c a t i o n
w i t h i n t h e element so t h a t eq u at i o n (1-18) can be w r i t t e n
[Kh] + CKd ] 1V
<Fe )
(1-19)
where
[Kh]
=
t A [ B ] ‘ [Dh][B]
( 1- 20 )
[Kd ]
=
t A [ B ] , [Dd][B.]
( 1- 21 )
and
Here t i s th e t h i c k n e s s and A i s t h e a r e a o f the element.
71
Bending Element
For th e bending e le m en t, the dis pla ce m e nt s and e x t e r n a l f o r c e s a re
as shown in Figure (1-3)
and ( 1 - 4 ) .
FIGURE 1 - 3 .
Bending
deforma tio ns a t node
j for a typical e l e ­
ment shown in local
coordinates.
FIGURE 1 - 4 .
Bending
loads a t node j f o r a
t y p i c a l element shown
in lo c al c o o r d i n a t e s .
The di s pl a ce m e nts a t node j can be w r i t t e n
fw. T
j
r
W I
V
I
J
<e
xJ
I
>
ay
3W .
8„
-
V--y J .
L
-
J.
J
( 1-
22)
72
The d is p la ce m e n t f i e l d i s allowed to be f i t by a t h i r d o r d e r p o ly ­
nomial expansion over t h e area o f t h e t r i a n g l e , i . e .
w(x,y)
=
U x y x2 xy y 2 x3 xy2 y 3J W
(1-23)
lP(x,y)J{a}
where {«•} i s t h e v e c t o r o f nine c o e f f i c i e n t s which a r e t o be determined
by th e nine (t h r e e , a t each node) nodal d i s p la c e m e n ts .
Then
(6 } .=
[A]{a}
(1-24)
where th e elements o f [A] a r e determined by d i f f e r e n t i a t i n g equation
(1-23) a cc or din g t o e q u a ti o n (1-22) a t nodal l o c a t i o n s i , j , and k.
S u b s t i t u t i o n o f t h i s r e s u l t back i n t o e quat io n (1-23) gives
w(x,y)
=
|_N(x>y)J{<y
(1-25)
where t h e shape f u n c t i o n N(x,y) becomes
[N(x,y)J
= •
Lp( ^ ) J M ' 1
■
•
(1-26)
The s t r a i n f o r small bending deforma tio ns i s approximated as
V
w
l
S X 2
£ x
2
a
N
I
II
W
„
E y
w
2
9 y
E x
y
,
^ a 2 W
-
■
[ 2
B
x a y J
V.
(1-27)
S u b s t i t u t i n g e q u a t i o n s (1-25) and (1-26) i n t o eq u at i o n (1-27) gives
{e}
=
-z [B '] { < 5 }
(1-28)
where
r .
[Er]
[P5XxJ
[P5yyJ
-I
(1-29)
2 [ P 5xyJ
Here t h e s u b s c r i p t s x and y i n d i c a t e p a r t i a l d i f f e r e n t i a t i o n with r e s p e c t
to th e x and y independent v a r i a b l e s r e s p e c t i v e l y .
Using the h y d r o s t a t i c and d e v i a t o r i c s t r e s s - s t r a i n r e l a t i o n s h i p
we can w r i t e t h e p o t e n t i a l energy o f bending.
iT
[ B - ] ' [ D h] [ B - ] + G [ B ' ] [ D j ] [ B ' ]
KJ
dv{6e } - [ F j { y
Minimizing p o t e n t i a l energy gives
=
([^ ]
+ C K p n y
[ r h]
=
' I
[ B ' ] T[ D . ] [ B ' ] Cffl
A
n
I- - - - - - 1
(F)
=
I
[B"]T[Dd ]{B-] dA
C L
I_ _ _ _ I
where
and
12
(1-31)
(1-30)
74 ■ .
The in t e g r a n d f o r both t h e h y d r o s t a t i c and d e v ia to r. ic bending, s t i f f ­
n e ss e s c o n t a i n elements dependent upon a r e a c o o r d i n a t e s x and y, and,
t h e r e f o r e , must be inc lud ed in t h e i n t e g r a t i o n over th e a re a o f the
element.
•'
' The c o n s i s t e n t mass m a tr ix f o r a v i s c o e l a s t i c m a t e r i a l i s the same
as t h a t f o r t h e e l a s t i c m a t e r i a l .
The mass m a t r i c e s f o r both th e in^
plane and t h e bending element a r e p r e s e n t e d in Zienkiewicz [ 3 2 ] , and w i l l
not be r e p e a t e d here.
Shell Element
The h y d r o s t a t i c s t i f f n e s s , d e v i a t o r i c s t i f f n e s s , and mass f o r the
plane s t r e s s , and bending elements a r e superimposed to form t h e r e s p e c t i v e
s t i f f n e s s and mass m a t r i c e s f o r t h e s h e l l element.
An a r b i t r a r y f i c t i c i o u s s t i f f n e s s i s su p p li e d to complete th e r e l a ­
t i o n s h i p between t h e a n g u l a r deform at ion 0
[32] p r o v id e s th e form f o r t h i s m a t r i x .
and l o a d in g M^.
Zienkiewicz
When a d j a c e n t elements a r e co-
p l a n e r , t h e i d e n t i t y 0 = 0 a r i s e s , which can produce erro ne ous r e s u l t s .
By i n t r o d u c i n g an a r b i t r a r i l y l a r g e s t i f f n e s s the p o s s i b i l i t y o f t h i s
i
'
i d e n t i t y i s e l i m i n a t e d and only small e r r o r i s i n j e c t e d .
The development f o r t h e s h e l l element has been c a r r i e d out in l o c a l
c o o r d i n a t e s as shown in Figure (I -T ) and ( 1 - 2 ) .
Trans for ma tio n in t o
global c o o r d i n a t e s i s n e c e s s a r y f o r c o n s i d e r a t i o n o f t h e t o t a l pro­
blem [ 3 2 ] .
)
Appendix I I
FLOW DIAGRAM FOR REDUCED MASS & STIFFNESS GENERATION
Initialize
CALL f o r g e n e r - J
■inn n-F a c _
*
sumed modes.
SUBROUTINE
GAMGEN
Generate assumed
modes.
Generate node
identification.
SUBROUTINE
STIFF
Transform from
globa l to loc al
c o o r d i n a t e s . ___
Generate global
nodal c o o r d i n a t e s
f o r nex t element.
/CALL f o r s t i f f ­
ness and mass y
generation.
/= "
/CALL f o r mem^ br a ne mass &
s t i f f n e s s gen. /
Perform c o o r d i n a t e
r e d u c t i o n o f mass
and s t i f f n e s s
matrices.
/ CALL f o r bendM ng mass & s t i f f - ?
ness g e n e r a t i o n /
Generate the
f i c t i c i o u s mass &
s t i f f n e s s matrice
Overlay reduced
m a t r i c e s in
global reduced
mass and s t i f f ­
ness m a t r i c e s .
Superimpose the
bending and mem­
brane m a t r i c e s .
Transform from
lo c al to global
coordinates.
RETURN
Output reduced
mass, h y d r o s t a t i c
s t i f f n e s s , and
deviatoric s t i f f ­
ness m a t r i c e s .
END
)
SUBROUTINE
STIFFP
Generate in - p l a n e
mass, h y d r o s t a t i c
s t i f f n e s s , & devi­
atoric stiffness
f o r th e element.
SUBROUTINE
STIFFS
Generate bending
mass, h y d r o s t a t i c
s t i f f n e s s , & devi­
atoric stiffness
f o r the element.
M O f l f T A t f A CTjI
TC
i
___
3 1762 10013899
JJ3T8
GT62
c o p .2
D A T E
Graramens , Gerald M
Frequency response
a n a ly s is o f th e in -v iv o
human s k u ll
/S n V ^ ssuedt0 '
- H.J
S B k r