JOSEPH S. JANICKI, DALI J. PATEL, JOHN T. YOUNG AND RAMESH N. VAISHNAV 
575 
(aF). Then the initial stresses will change by 
amounts aS#, aSz and ASr, and the initial 
stretches by amounts aX^, aX^ and AAr. Again, 
because of orthotropy, no incremental shear- 
ing stretches will develop and the incre- 
mental stretches will completely define the 
incremental deformation. These incremental 
stretches must also satisfy the incompressible 
condition; therefore, 
(Xe -f- A\«) (\, + AX,) (K + AXr) = 
XeKK = 1 (4) 
R 1/ \ , , PAR , aF 
2Lh 27rRLh J 
-l/gAp 
27rRh ^ 
(9b) 
(9c) 
At this point in the development of the the- 
ory, the complex representation of the sinusoi- 
dally varying quantities should be discussed. 
Consider as an example the quantity Ap, rep- 
resented by 
Let e^, ez and er be the incremental strains de- 
fined by 
e# 
AX« 
X. ' X 
AX, _ AX,. 
(5) 
Or, equivalently, if aR, aL and Ah are the 
sinusoidally varying increments of R, L and h, 
respectively, then using equations (2) gives 
e« 
aR 
R ' 
e. 
Al. 
; e,. 
Ah 
(6) 
In terms of the incremental strains, equation 
(4) becomes 
e« + ez -f Ci. = 0 
(7) 
when one assumes that the increments in 
stresses and stretches are sufficiently small so 
that they can be replaced by their corresponding 
differentials. Using equations (6) and (7) to 
express er in terms of eg and e,, gives 
Ah 
h 
aR 
R 
(8) 
Let Fg, Pz and P,. denote the sinusoidally 
varying incremental stresses. By computing 
the differentials of the stresses in equation (1) 
and using equation (8) to eliminate Ah, the fol- 
lowing relations for the incremental stresses 
are obtained : 
Ap + 
2pAR , pRaL 
h 
Lh 
(9a) 
Ap = |Ap|ej'"' + 0) = |Ap|{sin(&)t -f (^) -f 
j cos(a)t + (/))} (10) 
Here | Apj is the maximum amplitude of the in- 
cremental pressure above the mean pressure 
(p) and (f) is the phase angle which depends on 
when time (t) is taken to be zero, co is the fre- 
quency in radians per second and j = V— 1, 
We have arbitrarily chosen t = 0 to be that 
time when the phase angle of the complex quan- 
tity aR is zero. Thus, at t = 0, Ap, aR, aF and 
aL have the following complex representa- 
tions : 
Ap = lAple^''^ 
aR = IaRI 
aF = lAFIej'^ 
aL = lALIe^''' 
(11) 
(12) 
(13) 
(14) 
Since Fg, P,, Pr, eg, e,, and e, are computed 
fromAp, aR, aF and aL, they too will be com- 
plex numbers- It is important to keep in mind 
that the static case (i.e., oj = 0 and all phase 
angles become zero) is simply a special case 
of this dynamic formulation. 
The complex incremental stresses and 
strains are related as 
e^ = CooFg — CfePz — C^rPr 
— — Cz^P^ -)- CzzPz — • CzrPr 
e,. = — CrgPg — CizP?, + CrrPr 
(15) 
(16) 
(17) 
