Method of characteristics for analysis of elastic wave equations in Cartesian coordinate

Buchanan, G. R.

Huang, J.

Phung, C.

English

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DYNAMIC ELASTICITY BY THE THEORY 
OF CHARACTERISTICS 
by 
GEORGE R. BUCHANAN 
CHI-HUONG PHUNG 
and CASE ~ 1 ~ 6  
JU-CHIN HUANG C O P Y  
- I Prepared For 
- NATIONAL AERONAUTICS AND SPACE ADM INlSTRATlON 
MANNED SPACECRAFT CENTER 
C I 
- ,  HOUSTON, TEXAS 
;, * + , i , " " - ' - - ; J , , : - ?  <, I)( * , Y , '  ,:d: nw :?,mb$= $ - n-n-.$.. -,~-,,-:i;.~, ..,, 
GRANT NUMBER NGR 43-003-007 
SEPTEMBER, 1970 
TENNESSEE TECHNOLOGICAL UNIVERSITY 
DEPARTMENT OF ENGINEERING SCIENCE 
COOKEVILLE, TENNESSEE 38501 
https://ntrs.nasa.gov/search.jsp?R=19710029229 2020-03-11T21:17:15+00:00Z
DYNAMIC ELASTICITY BY THE THEORY OF BWIACTERISTICS 
George R. Buchanan 
Chi-Houng Phung 
and 
Ju-Chin Huang 
Prepared For 
NATIONAL AERONAWICS AND SPACE ADMINISTRATION 
MANNED SPACECRAET CENTER 
HOUSTON, TEXAS 
GRANT NUMBER NGR 43-003-007 
September, 1970 
'IENNESSEE TECHNOLOGICAL UNIVERSITY 
DEPARTMENT OF ENGREEKlNG SCIENCE 
C00KEVILJ.E , TENNESSEE 38501 
ABSTRACT 
. . . . . . . . . . . . . . . . . . . . . . . . . . .  ~ODUC!rION J- 
THE DYNAMTC ELASTICITY PROBLEM . . . . . . . . . . . . . . . . . .  1. 
ANALYSIS USING THE THEORY OF CHARA-STICS . . . . . . . . . . .  3 
CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . .  9 
. . . . . . . . . . . . . . . . . . . . . . . . . . . .  EFERENCES 10 
ILLUSTRATIONS 
Figure 1 . The Direction Cosines in a Spherical Coordinate Sys tm 11 
A characteuistic analysis i s  presented f o r  the equations of e l ~ t i e i w  
in Cartesian coordinates. The characteristic slope equations are derived, 
and it is  verified tha t  two types of waves exis t .  The compatibi.li"cy equa- 
tions are developed in relat ion t o  the direction cosines of a spheric& 
coordinate system. A brief  discussion of the nethod of analysis i s  irreluded, 
INTRODUCTION 
The application of the theory of chaxwteristics and subseguezlt n m r i -  
cal solution of the ciharacrterlestic equations i s  increasing in p o p a m i w  as a 
technique fo r  s~lving.wave propagation problems. The sophisti~~3.ted develop- 
ment of modem digital computers is responsible f o r  the incxsase in =sea& 
effor t  t o  develop and extend fhe technique t o  mre complicated p r o b l m ,  
This report deals with the development of a characteristic andys i s  for the 
three dimensional dynamic e las t i c i ty  problem. 
THE DYNAMIC ELASTICITY PROBLEM 
The equations of motion fo r  a l inear ,  elastic, isotropic and horngenesus 
medium in Cart.esian coordinates are, 
where 
X,  y ,  and z = Cartesian coordinates 
o , o and o. the norsndl stresses in the yz, xz, and 
X Y  z 
xy planes, respectively 
' , ,r , and T = the shear stresses i n  the xy, yz, and 
xy YZ xz 
xz planes, respectively 
p = density 
V V , and V = velocities i n  the x-, y-, and z-dixections, ~specjCiveLy 
x y  Y z 
t = the t h  d h n s i o n .  
The s t ress-d isplacmnt  relations can be written as 
and 
where 
and 
u , u and u = displacemnts in the x-, y-, and z-directions mspectively, 
x y Y  z 
In the theory of cha~*acteu?istics, it is convenient t o  treat first-order 
pa r t i a l  differential. equations; therefore, Eqs, 4 through 9 w i l l  be diffemn- 
t i a t ed  with respect t o  the t o  yield 
and 
Equations 1 through 3 and 11 through 1 6  make up a s e t  of nine l h e w  
first-order pa r t i a l  differential  equations which govern the three d h n -  
sional dynamic elasticity problem, 
ANALYSIS USING THE THEORY OF CHARACTERISTICS 
The analysis used hexein i s  similar t o  tha t  used by Sauerteh (1) 
and Madden (2)  and w i l l  be brief ly outlined, The nine governing equatioi~s 
can be represented i n  a convenient form using the index notation as foLLsws, 
where B o  represents the dependent variables; xk represents the independelat 3 
variables and the aijk are  constants. The characteristic (slope) equa- 
t ions fo r  Eq. (17) car! be developed by changing the independent vmi&l-es 
from xk t o  some arbi trary coordinate system, say n , n g  , and n That 
Ass* tha t  values of all dependent variables and t he i r  derivatives 
with respect t o  n 2  and rt3 are specified on a swface n l  = constant, these 
transformed par t i a l  differential  equations would be expected t o  yield the 
derivatives wfth respect t o  n , ,  i f  they exis t .  These derivatives axe 
- 
a B o  a n .  J a B j  ann 2 -  - -a*. --
ai jk a n ,  ax l ~ k  an, axk 
where n - 2 , 3 ,  
To determine the characteristic equations it is desirable t o  deteua- 
mine the conditions under which the derivatives normal t o  n I  do not ex i s t ,  
tha t  is, the normdl derivatives t o  the surface, n l  = a constant, iure dis- 
continuouso These discontinuity surfaces which have been mentioned a-c 
the beginning of t h i s  chapter are also called characteristic surfaces. 
The requirement f o r  discontinuity in the derivatives with respect to ri, 
is  then the vanishing of the determinant of the coefficients of the 
derivatives with respect t o  n l  in Eqs. (191, o r  
The nine governing equations (1) through (3 )  and (11) through (161, 
det 
a f t e r  tmnsformation , may be w i t t e n  in the form of Eq . ( 20 ) as follows, 
r,, 0  0  n ,  n ,  0  -PTY 0  0  
X Y z t 
0  n ,  0 n ,  0  n ,  0  -PO 0  
Y X z )t 
0  0  n ,  0  n ,  n~ 0  0  -Prl ,  
z x Y t 
rl , 0  o O 0  0  -Ah3 -ACny - A h y  
"t X Y z 
0  n ,  0  o o O -ACny -ABn, -ACn, 
t X Y z 
0  0  q, o O 0  - A h ,  - A h y  
t X Y z 
a n  .) 
.L 
a i j k  
k 
O O O O O n ,  A A 
t 
0  -pz -py 
= 0  
where the comma indicates parkial differentiation with respect t o  the 
variable following the comma. Also, f o r  convenience the subscript I 
has been of i t ted  fo r  the variable n. Expanding Eq. ( 2 1 )  gives the 
following 
Equations ( 2 3 )  and (24) show tha t  t h i s  three-spatial d h n s i a n d  
problem involves two kinds of waves, n m l y ,  a longitudinal and a 
shear wave. Letting cL and cs represent the longitudinal and shear 
wave velocities,  respectively, it follows tha t  
The extra factor  n , in Eq. (25)  indicates the part icle  path which 
t 
is  a characteristic surface with zero velocity. By expressing Eqs . C 2 3 ) 
and (24) as 
where 
c = c  o r c  L s 
and introdu.chg a new parameter J1, the characteristic slope eqyation.~ 
- - 
aF dy , - - 
d@ a n ,  Y - -2nry 
dz - aF 
- _ - -  
a* a n ,  - - 2 b Z  
2 
Elirmfnating the parameter 3, gives 
dz n ) ~  
- -c2 - - s c  ,z 
d t  
,t 2 2 n ,  + n ,  + T I ,  2 
X Y Z 
Equations (331, (341, and (35) can be reduced t o  a simple f o m  by w b g  
the direction cosines between the normal t o  the surface n = c o n s t a t  
and the x-, y-, and z-axes as 
I '  ,X 
cos (n,x) = 
n , 2  + n , 2  + n ,  2 
X Y Z 
COS (rl,z) = n , z  
2 2 T I ,  + n ,  + n ,  2 
X Y Z 
Equations (36) can be written in terms of spherical coordinates 
8 and + according t o  Figure 1 as 
Cos (?l,x) = Sin 8 Cos 4 
Cos (l-,,y) = Sin 8 Sin 4 
cos ( Q , z )  = cos e 
Substituting in to  Eqs. (331, (341, and (35) yields 
dx 
-- 
d t  - f- c Sin 8 Cos 4 
In Eqs. (38) only the positive sign need be considered since the negati~pe 
sign may be ob~ained by changing the reference f o r  0 and 4 .  T h e ~ f o ~  ?, 
the final f o m  of chaac te r i s t i c  slope equations is  
- dy - sin 0 sin m dt - 
In Eqs. (391, a given value f o r  8 and Q, (0+21~) defines one o f t b e  
characrkeristic directions at a point. These characteristics axe t e ~ m d  
"Bichaacteristics" . Considering the entixe range of 8 and 4 C O+2n 1 , 
Eqs. (39) describe a general sphere in space, namely, a "sharacteristic 
sphere". The f d l y  of bichamcteristics are the generatops of the 
sphere. 
The compa-tibility equation comsponding t o  the bichayt3cteristies 
given by Eqs. (39) i s  obtained by combining the t r an s fomd  equations 
in a manner such tha t  the indeterminable derivatives with respect to 
rl do not appear. Multiplying the transformed equations by weighthg 
factors h , h 2 ,  . . .. and h respectively and sumring w i l l  yield 
1 9 
such a relation. Relations between the X's are found by equating 
t o  zero the coefficients with respect t o  n in the transformed equa- 
t ions. The derivatives can be m i t t e n  in the form: 
Only eight of these homogeneous equations are required t o  find the 
following relations 
X2 = tan $I h l  
A ,  = PA], 
A 4  = + Sin e Cos 4 
C 1 
X 5  = f_  Sin 0 Sin $ tan $I 
C 1 
A, = + P Cos e 
C 1 
- Sin 0 h 7 - f -  C (Sin $I + Cos $I tan $ 1 ~ ~  
+ Cos 8 + P Sin e Cos $ X8 = - C 2 
and 
- 
+ Cos 0 tan 4 + P s in  9 s in  4 A 9  - ' 
C 1 
where 
Thus the compatibility equations can be obtained f r o m  the sum of the 
weighted equations (1) through ( 3 )  and (11) through (16 ) as follows 
2 
+ Sin 8 Cos ${u 
- P C ~ V ~ , ~  - -2pvcc2 - c 2 )  ( V  + v 1) x ,t L s Y,Y z,z 
? Sin 0 (Sin 6 + Cos 4 tan 4) {T - p c 2 ( v  + V  ) I  
XY ,t s X,Y Y,X 
These are the gene& compatibility equations fo r  a t b e - s p a t i &  
d h n s i o n d l  dynamic e las t i c i ty  problem. Specific equations can be 
obtained by choosing values of 9 and 6, and specifying c t o  be c or 
L 
CONCLUSIONS 
The characteristic slope equations and corresponding compcz"cibility 
equations have been developed i n  Cartesian coordinates f o r  the t h ~ e  
dimensional d y n h c  e las t i c i ty  problem. It is verified that two types 
of waves are present, nanely, longitudinal and shear waves. 
1. Sauerwein, H a m y .  "A General Nirmerical Method of Characte~ais"cics,'" 
AIAA Second Aerospace Sciences h e t i n g  , New York , January, 3-96 5 , 
AIAA Paper No. 65-25. 
2 .  Madden, Richard. "Hypervelocity Impact Analysis by the Method of 
Characte~ist ics ,"  NASA TR-298, January, 1969.  



Dynamic elasticity by the theory of characteristics

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