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Plane-disturbance propagation in elastic nonhomogeneous media. General solution
Int. J. Engng
Sci.,
1973, Vol. 11, pp. 1197-1220.
Pergamon
Press.
PLANE-DISTURBANCE NONHOMOGENEOUS
Technion-
Printed
in Great
Britain
PROPAGATION IN ELASTIC MEDIA. GENERAL SOLUTION
G. ROSENHOUSE Israel Institute of Technology, Department of Mechanics, Haifa, Israel (Communicated by 1. N. SNEDDON)
Abstract-This paper deals with the problem of uniaxial propagation of disturbances in continuous axiallynonhomogeneous media. The general mathematical solution obtained is applicable to physical problems dealing with media with longitudinally-variable parameters. Solution of the resulting differential equations is based on symmetry of their space and time variables. This symmetry leads to transformation of boundary problems into their initial and terminal counterparts. By working out a special version of Fourier’s transform, a law for uniaxial propagation in nonhomogeneous media is derived. It is shown that D’Alembert’s well-known solution for propagation in a homogeneous thin bar or string is a particular case of this law. The significance of such a general theory lies in the fact that most dynamic phenomena in nature occur in nonhomogeneous media, and the solution presented here may contribute to their physical evaluation. As an example, we solve the propagation of a pulse in a heated thin bar. I. INTRODUCTION
A ONE dimensional elastic model is used in deriving the equation of motion for plane disturbance propagation in nonhomogeneous media as an example of a problem involving variable parameters, neglecting energy losses. Assuming that there is no time variation, and with thermal effects neglected, these variable parameters are the elastic modulus E* = E$ . E*(Z) and the density P = PO . p(z). (See Fig. 1). The problem is of practical importance in various applications involving stress and displacement propagation in materials. The solutions obtained in the present paper were applied to cases discussed in [ 11. These include a mechanical model of the inner ear in the form of a nonhomogeneous bar as suggested by Ollendorff[2], variation of the ‘bar velocity’ in a nonuniform temperature field[3], plane propagation in soil (with vertical variation of the parameters) and in the atmosphere, and finally, radial wave propagation in space. A literature survey [4- 131 indicates specific nonhomogeneities obtainable by separation of variables. The solutions have the form of a series with unknown coefficients, solved with the aid of boundary conditions. In this paper, propagation is treated as a boundary-value problem for which a general mathematical solution is obtained. Examples of propagation in media with the nonhomogeneities:
E* = E,*,
E” = E,f(a2kz2)2
are solved for specified boundary conditions. 1197
IJESVol.lINo.Il-E
(1)
1198
G. ROSENHOUSE
Direction of propagation. and variation of parameters
Fig. 1. Representation of element of elastic medium. (a) Unit area cylinder cut from medium along Z-axis. (b) Deformed shape of cylinder element. (c) Variation of parameters along the z-axis. (d) Orientation of medium.
2. TRANSFORMATION
OF THE
EQUATION
OF MOTION
The equations of motion for a free dilatational plane displacement finite medium with finite period are given by the momentum equation
aW* (aw/az) I = az
a[p( aw/ar)1 at
and the conditions for plane displacement
9
aw ax
r)
(2)
propagation
u=v=o -=-=
w = w(Z,
in an elastic
aw ay
o
I
where x, y, Z = the coordinates of the reference system (See Fig. 11, E* = A+ 2~ = elastic parameter, A, p = LamC’s constants, u = displacement in x-direction, II = displacement in y-direction, w = displacement in Z-direction, P = density, t = time.
(3)
Plane-disturbance
propagation in elastic nonhomogeneous
to use non-dimensional
It is convenient
equations
media
1199
by using the following trans-
formation of variables q-z- 2m T r.--2vz L where T= 4L = 7= z=
‘length’ of wave along time-axis (period), extension of the medium in Z-direction, (2rt/T) = non-dimensional time, (21rz/L) = non-dimensional length in Z-direction
and the single equation of motion will have the form:
I
E*(z)2g =
g
[
a2.
p(z)$,
w = W(L 7)
(4)
where
C2 = 5 *Ct = non-dimensional Ci = F
velocity at z = 0
= bar velocity at z = 0. 0
The solution of equation (4) is to obtain an ordinary differential one. Having variable coefficients carried out. The transformation begins by as follows:
our aim and in order to arrive at this aim we shall try equation instead of equation (4) which is a partial in equation (4) a special transformation should be multiplying equation (4) by sin (no) and integrating
(5)
77 77 1 Iw
We next integrate by parts the right-hand integral in equation (4) twice %sin
(no) --wcos
(127) -n2 0
= t(z) -n2 I
w sin (nr) dr,
sin (no) dr
D
w=
W(Z,T).
Because of the variable coefficients of equation (4), a new variable is introduced:
(6)
G. ROSENHOUSE
1200
which yields the stress as a function of time and space
Eon fl(Zr 7) = y(z, 7) . -
’
E*(z)
E*(z) =1+/J
(8)
where E,,E(z) = Young’s modulus.
By developing y(z, 7) into the Fourier series y(z, 7) = + i Y,(n, z) sin (n7) ?I=1
(9)
where E* (z)$
sin (m) d7:
we eliminate 7. If the initial and terminal conditions c(z) = 0, and equation (6) is given by:
(10)
of the problem are zero, then
(11) Differentiating
equation ( 1 1) with respect to z, and multiplying by e*(z) we have
dZY, -- E*(Z) . ddd P(z)2 dz P(Z) dz2
E*(Z) --
dYn -
_a2
dz
x
E*
(z)z
n
sin (n?) d7
(12)
or
E*(Z) +(z) dyn a”,Yn = 0. E*(Z) d2Yn--.. dz+ p(z) dz2 p(z)” dz The last equation is an ordinary second-order
(13)
linear differential equation of the form
=0
A,(z) "y"+B,(z)~+c.(z)y. dz2
(14)
(Really only C, is a function of n and not A, and II,). Y, = Clbl(n, z) +C2b2(n, z),
C1, C,-constants.
(15)
Formally, whatever the functions 6, (z) and b,(z) will be, we can introduce equation (15)by (16) Y,=F,(z)[C,coscw,+C,sin&l, cvn=ck(z) where F,(z)
= m
Plane-disturbance
propagation in elastic nonhomogeneous
This substitution, which still describes the following mathematical treatment.
3. SIMPLIFIED
1201
media
nothing new, fits better than equation (15)
SOLUTION FOR CASE WITH TWO BOUNDARY INCLUDING SOME FORMAL LIMITATIONS
CONDITIONS
As an example of equation (16), we choose the following two arbitrary boundary conditions (17) to which we apply the Fourier sine transform n
fn = Y,(O) =
I
f(r) sin (nr) dr,
0
g
n
s
-aYY&(O)_ = g(r) a.2
0
sin(m)
dr,
subject to the following arbitrary limitations z=O+sina;,=O+~=O, ~=o-+cosa;,=1, (19)
The constants C1 and C, are determined as follows fn = F,(O)C, = Y,(O),
(20)
cl=&
(21)
R
dYtt(O)= Y&C, aF (0) dz
+F.(O)d* G = Rn
(22) (23)
1202
G. ROSENHOUSE
and their substitution in equation ( 16) yields (24) Now we transform Y,(z) into the (z, T) domain and obtain the solution Yk 7) = f
m(z) sin (n7) 2 n=1,2...
(25) =_.-1 2 2 lr .A,,,
#[[sin
-[dn,&,&,
[,(r+?)]+sin
{‘OS [n(T-:)]+cos
[+$]]h
[‘(T+:)]]]’
Observing the last solution, one must pay attention to the fact that (Yis a function of z rather than an independent variable (see equation 16)). Moreover, the Fourier expansion in equation (25) is not influenced by the type of the function 01.In the case that a is a linear function of z, and %(O) = 0, the wave equation with constant coefficients is obtained for propagation of disturbances in homogeneous media or in media with some special nonhomogeneities. The first series in equation (25) will be changed by using the transform: f(7)
= $
i f, sin (m). n=1,2...
(26)
We shall refer to the incomplete series h(T) =+
i
h sin
(117)
(27)
n=1.2...
then we have Afk(7)=fk+l (T)
-fk(7)
=
$fk
sin (k~) , (see Fig. 2)
(28)
hence
(2% Summing the terms A& of equation (29) we have (30)
Plane-disturbance
propagation in elastic nonhomogeneous
media
1203
Fig. 2. Elements offfor line spectrum.
The second series of equation (25) will be changed by using the expression g(7) = -$
2
g, sin (~27)
(31)
g, sin (n7)
(32)
n=1,2...
referring to the incomplete series Sk(T) =;5
5 n=1,2...
we have
hh,(T) =gkfl(T) -gk(T) = gk sin(h).
(33)
Integration of g(r) may have the form [
-7
g(r) d7 =
i
A[ r
n=1,2...
=
$ -jj
g(r) dT] = f --T
i
g, r
n=1,!2...
F[COS (127)-COS
sin
(nT)
dr
--T
(-11T)],
n=1.2...
(34)
hence g(T*)
dr* =f
i n=12,,, [cos[,(,-~)]-cos[n(T+~)]~~}.
The results in equations (30) and (35) now replace the corresponding equation (2S), the final result reads:
(35) terms in
31 u.IIAfn(~+%)+Afn(~-:) y(zq T,=.=;.., F,(O) 2 r+$Jn)
-[da,;O),&jA Ir_(an,n) &(T*)
I
dT* t a, = a(& Z>.
In the case that E* = E,, p = PO, the transformed
(36)
wave equation (13) reduces to
1204
G. ROSENHOUSE
ordinary differential equation
d2Y,+a;y, dz2
=
0
(37)
with the general solution Y,= C 1 coS a,2 + C, sin a,2
(38)
hence, observing equation ( 16):
F,(z) = 1 C&= UnL.
(39)
In the case of thin bars we may assume the approximation E,* = E. which is Young’s modulus of elasticity, hence the boundary conditions are the stress and the stress gradient respectively: f(7) = [ E,?$;’
“1 =~(0.7) (40)
g(7) = $[ EoaWL: r’] = ‘og
‘) ,
(see equation ( 17))
hence, applying equation (36), we obtain the stress in the bar
(41) The sum notation is dropped becausefis
5
Af=f,
n=1,2...
i
not a function of n:
A 1 g(T*) dT* = 1 g(r*) dr*.
(42)
ft=l,Z...
hence, the stress is given by
which is in full agreement with D’Alembert’s solution for homogeneous bars with zero initial conditions and given boundary conditions. The development of the solution for line spectrum (finite period) can be extended to continuous spectral analysis (t > 0). Beginning with the same equations of motion (equations (2), (3)), and boundary conditions f(t)
=
[e*(o)*],
g(t)
=$*(o)“*]
(44)
Plane-disturbance
propagation in elastic nonhomogeneous
media
1205
we apply the Fourier sine transform $ “’ mf(l) sin (&) dt, 0 I0
ff = Ye(O) =
,
y, =
i 0
“’
mE(z)*%sin([t)
(45)
dt.
J0
An ordinary differential equation general solution of this equation is
resembling
equation
(14) is obtained
YE= FS(z)[C1 cos ++C2 sin ~~1, * = CY&).
and the
(46)
Applying the boundary conditions (44) we obtain
t
fs‘OS % +
[da,
I
fb;/&]sin%.
(47)
The solution in the (z, t) domain is given by the inverse transform ~(2, t) = (s)“’
= (:)‘”
lrn Y,sin
1
$$[fr
(5t) d5
cos a,+ ~d~~~),~~
sin*,]
sin (&) dt.
(48)
There are in the solution two integrals containing trigonometric functions. We shall try to eliminate them in order to obtain a solution depending only on the boundary conditions and nonhomogeneity effects, as was done in the case of finite period. For the first integral we use the integration [
d~=[$d~=(~~1’2[&sin(&t)
d&
sr
(seeFig.3)
f I
Fig. 3. Element offfor continuous spectrum: (a) General case, (b) Step function.
(49)
G. ROSENHOUSE
1206
hence m dflr-t
(o&)1 d5
_ 2 1’2 m d[-(--) i .&sin[S(j*y)]d5.
(50)
The second integral of equation (48) will be changed by the integration g(t*)
g(t*) dt* =
=
(i)“’ [
It;(cos
dt* d,$ = (3)“’ -
lam g, ~t~~~~~ sin (nt*) dt* dt
[++y)]}
[@)]+cos
d*
(51)
hence g(t*)
dt* d< = (t)“’ -
[
y{cos
[+-y)]+cos
[++y)]}
The results in equations (501, (52) replace the corresponding and the final result for semi-infinite period is:
d&. (52)
terms in equation (48),
01~= cu(S, z)
(53)
which is also a general case for any continuous non-homogeneity and, as shown for line spectrum, reduces to the solution of D’Alembert in homogeneous media. 4. GENERAL SOLUTION BOUNDARY CONDITIONS,
OF PLANE-WAVE PROPAGATION FOR ANY PAIR OF WITH NO FORMAL LIMITATIONS AND ZERO INITIAL CONDITIONS
The case considered here is equation (2), which is the general form of the linear differential wave equation with real coefficients varying as a function of z, i.e. L(w)
=
A(z)
[
.-$+B(z)&-C(z)-$ 1w=o.
The boundary conditions are a combination (required for the two constants):
(54)
of two out of the four possible alternatives
.h =fi(O,j),
fi=fi(~9j)
&=gl(o,t),
g,=g,(~,t)
I
0s
ZSTr
(55)
The solution is based on elimination of the variable (t), using a technique which resembles the Fourier transform (except for non-convergent cases). The result is
Plane-disturbance
propagation in elastic nonhomogeneous
1207
media
analogous to equation ( 14):
c(z) 2.%+,*(z) d[ Ildpz(z)1 .2
+ b2y, (fi, z) = 0
(56)
where a: =
L
2
( 1
n2
TC,
,
The solution in general is as presented domain, the final result reads:
c(z)2=-
E*(2) P(Z)
c2 ’
O
=
E*
0
PO’
in equation (16). Transforming
into the (z, r)
Y(z,T) = n=123,,, i F n a1 2[ /3rl[ Af I(~ +!”n) +Af l(r-~)]+P=[A~r(f+$+A+~)] 2,
+PIS [ Af + +%n) +Af 2(+] +nA
+P~r[Agt(r+$+ A+$1
T+(an'n) [B2~~(7*)+P22g~(~*)+P23fi(7*)+PZ4fi~(~*)ld~*} (57) Ir-(%/n)
& = &(n, T), or pu = pil (n,0) are the coefficients of the two given boundary conditions (others are not taken into account). D’Alembert’s solution (equation (43)) is the general solution (equation (57)) reduced to a very simple case. The solution presented here refers to a line spectrum, i.e. to finite time. In the case of an infinite or semi-infinite medium (semi-infinite time) we have a continuous spectrum (no frequency filtering by boundary limitations).
5. SOLUTION OF COEFFICIENTS-p The first case is the boundary conditions applied in equation (17). Substituting the first condition in equation ( 16) after transformation as in equation ( 18), we have fn= F,(O) . C1 cos a,(O) + F,,(O) . C2 sin a,(O). For the second boundary condition, equation ( 16) is differentiated
(58)
with respect toz:
dy,, --d-[C,cosa,+C2sinol,]+F,[-CIsin~~+C2cos%1~ dF,
(59)
dz
hence
dY,_ dz -
cos (Y,- F,%sin
OL,C, + *sin 1 [ dz
01,+ F,,$cos dz
a,, C2
1
(60)
and (61)
G. ROSENHOUSE
1208
where:
cos
=
5
(dFnlW
~0s
n
sin
5
a,- Fn(dddz)
sin01,.
K,, =
n
(@Jdz) sinan+ F,(dddz)
cos a,
K,,
which is a formal trigonometric substitution. The second boundary condition is substituted in equation (6 l), yielding the second equation for calculating the coefficients
gn= F,(O) [Cl cosL(O) + Cz sinL(O)]:
(62)
L(O) = [(O, TZ).
The functional relation between C1 and Cp is obtained from equation (58): Cl =
F,
fn
-C,
(0) ~0s an(O)
(63)
tan a,(O).
Now we determine Cz. using equations (58), (62), (63) ~~0s~(O)-F~(O)~sina,(0)
.
I[ f
-C, tana, F,(O)c~sa,(O)
1
~sma,(O)+F~(O)~cosa,(O)
1
Cz=g,
(64)
and explicitly:
c, = p2 =
dF (0) -~cosu.(O)+F,(O)~
sin 01,(0)
11
-F,(O)-
sin o,(O)
+ ysinoa(0)
+F,(O)v
cos o,(O)
~cosa,(O)
f,
F,(~)cos~,(~)+~~ tan 01,(0)
(65)
C, is similarly obtained from equation (59):
fn c, = F,(O)sina,
- c, cot CX,(0)
which is substituted in equations (59), (62):
f, - c, cot o,(O) , sin a,(O) F,,(O)
Plane-disturbance
propagation in elastic nonhomogeneous
I209
media
hence
cos 01, (0)
+
i
cot OL,(0)
dFiz(0) cos a,(O) -F,(O)Y sin a, (0)
.
(66)
II
The final solution is obtained by substituting the resufts in equations (64), (66) in the fundamental solution equation ( 16):
Yn=Fn[Ptcos~+Pzsina;tl;
/A==P1@,n), Pz=&(0,f2)
(67)
hence Y, = Fn{t%Sn cos % +&EL cos % + &.& sin %i- Pzzg, sin %I; & = fiii (0, n) . Retransforming
to the (z, 7) domain, we have Y(ZtT)
=z i
Y, sin {fiT)
(6%
r=1,2...
Y(Z,T)
=
(68)
i F, n=1,2... T+(a@)
+nA
I
+-
CPzJW> +&g(r*)
3 d7” . 1
This is part of the general solution (equation (.57)), still more general than D’Alembert’s, as.is seen by comparing it with equation (43). The other cases of boundary conditions, solved in a similar manner, are given in convenient form as follows. Case Z-W:
The bounda~
conditions are:
f=f(%r r), g=g(z137),
(70)
z1 and 2%may be zero or P, so we have four possibilities:
(1) fK4 Q-1, S(O> T), (2) f(n, 7), g(O, r), (3) f(O, T), g(r* r), (41 f (-9 7) 1gfv, 7) *
(71)
G. ROSENHOUSE
1210
The coefficients are PI2 and pZ2for g(0, T) ; T) ; PI4 and pZ4for g (m, 7) .
PII and PZl forf(0,
T) ;
PU and PCUforf(m.
The expressions for the @ii’Sare I
Pl2orp,,=-_, Al
P*1or p13= -t;
A3 P21
orP23
=
1,
pz2
(72)
or p24 = -B, A,
.
where: A, =A,. A2 =$
B,+C,, n
and
A3 =$,
n
A =_dMzl) . Tsin%(zl) R B,=
-F,(zl)
da, (2,) dz cos %(zl).
cot%(z2), sin h(zl),
cn D, = Fn(z2) sin h,(zZ). Case V fi =h(O, 71, fi =f2(7T, T),
(73)
and the coefficients are:
‘11 = F,(r)
-sin a,(a) cos (U,(P) sin h(O) -F,(O)
cos h(O) sin%(n)
p13 = F,(a)
sin%(O) cos %(7r) sin G(O) -F,(O)
coscu,(O) sin cu,(T) ’
“l=Fn(r)
sina,
-cos a,(7r) coso1,(0) -F,(O)
P23=F,(rr)
sina,
cos (Y*(O) cosa,(O) --F,(O)
’
(74)
(75)
(76) sina,
coscr,(7r)’
sincr,(O) COSOL,(~~)
(77)
Case VZ g1
=
g,(O,
g2
=
gz(n,
71, 71,
(78)
Plane-disturbance
propagation in elastic nonhomogeneous
media
1211
The basic equation in this case is equation (61), and the coefficients for equation (57) are:
-sinL(O) . [&(z)lF,(z)l
Pl2 = &(7r) cos tn(7i-) sin tn(0) -K,(O)
cos in(O) sin en(r)’
sin5n(0). [JL(z)/F,(z)] ‘I4 = K,(r)
cos [n(m) sin[n(O> -K,(O)
P2”=Kn( 7r) sink(P) =
L(m) [KI(z)l~n(z)I
--a
cos 5,(O) --K,(O) cm
P24
K,(IT)sin J&(p)
cos &(O) sincn(7r) ’
a,
(0)
* [K”
(z)/Fncd
cos tn(0) -K,(O)
sin t,(O) cos en(g)’
1 sin tn(0) cos tn(7r).
(79)
(80)
(81)
(82)
It is possible in all cases to use the coefficients C1 and C2 for equation (16) instead of pit. In the sixth case we obtain Cl
=
PIa&
+Pl&gn(~),
c2
=
PzziL(O)
+P24&(p),
(83)
but the generalized version of D’Alembert’s of the physical interpretation. 6. STRESS
solution is apparently
preferable because
PROPAGATION ALONG A THIN BAR DUE TO A QUASI-STATIC NON-UNIFORM TEMPERATURE FIELD
The case of a heated bar in a non-uniform temperature field is solved with the aid of Bodner-Clifton test[3], (see Fig. 4). The temperature gradient is determined on the basis of thermodynamic considerations [ I], and then the modulus of elasticity may be calculated at every point along the examined bar. The density is considered to be constant, because it remains nearly constant in spite of the change of temperature, in many solid materials. Along the bar presented in Fig. 5, we obtain the temperature gradient e(z, t) =
lOO[ l--8($)],W).
(84)
The modulus of elasticity as a function of 0 (z, t) is given in Fig. 4, using the relation E,-,= P,,C;.
(85)
By minimizing the error defined by the square deviation of the assumed function from the function of the exact modulus of elasticity, we obtain
E = E(z)
(86) with maximal linear deviation within 3 per cent. It is not only the Bodner-Clifton
test
G. ROSENHOUSE
1212
200
100
200
300
400
500
I t-
700
800
900
1000
“F
T,
Fig. 4. Transformation
600
of elastic wave velocity into modulus of elasticity dependence temperature.
L/2
L/2
b
b
on
Y
Fig. 5. Geometrical and thermal data of the heated bar.
that leads us to the relation (86), but we may also apply the general relation E=E,[1-(~)2]2=Mo[b2-22]Z1 which is convenient
from the mathematical
+=;,
Mll=E064
(87)
point of view, and may be used for many
Plane-disturbance
propagation in elastic nonhomogeneous
media
1213
materials. The solution of the stress propagation is solved generally according to equation (57), but using the relation (86) condition (18) is fulfilled and we may apply the solution (36) or (53), which is simpler than the solution presented in equation (57). It was not intuition that led us to the function (87), but instead of taking into account E and p arbitrarily, and solving afterwards the differential equation (13), we choose the inverse method. We tried to find the elementary functions of E and p, the differential equations of which give the solution ( 16) including the limitation (18). Doing the last procedure we used [ 141. For a demonstration of the line and continuous spectral analysis, two cases are solved. The first is the case of finite period and the initial and ‘terminal’ conditions are zero. The second case deals with semi-infinite period and the initial condition is zero. (The ‘terminal’ condition is eliminated).
Case I The geometry of the rod is defined by
The period is defined by
The material mechanical properties are defined by E = E,(l -JI”)” = M,,(b2-~2)2, P = PO= const.
lb1 a T
The equation of motion is obtained by substituting (88,89,90)
3(a2_z2)2y9”]
= ~.g.~. 0
(90) in equation (2) (91)
The initial and ‘terminal’ conditions are zero
$(z, 0)
= 0,
$z,
7r) = 0.
(92)
The two boundary conditions are given at z = 0
(93)
IJESVol.
II No. II-F
1214
The transformed
G. ROSENHOUSE
equation of motion is d2Y (b2-22)2 -“+a”,m dz2
= 0
&2= n._ L2 ,c2 =-.MO n ( Co T ) PO
(94)
The solution of (94) is
(95) ~~=$(~~+l]1’21nI~I,
I$/ < 1.
The solution of stress is according to equation (3 6):
For the case that the conditions at z = 0 are (97) we get
nd(L/bTc)’
I‘-+I)
- ( l/n2) In I-rJ,
n
(98) Relative to the propagation of stress in homogeneous medium, the result here is influenced by two terms of nonhomogeneity In (( 1 + $)I( 1 - $I,)]instead of z and ( 1 - I/?)~/~ instead of unity. The first term has the graphical representation in Fig. 6. First of all, the infinite singularity at I,IJ= kl.0 should be a geometrical limit. The last conclusion could be understood physically by calculating the bar velocity as a function of I/J.We notice that at I,!J= kl, the bar velocity is zero (which is equivalent to the plastic region
Plane-disturbance propagation in elastic nonhomogeneous media
I.0
a6
OG?/
0.2
O-6
I :O
I.4
I.6
1215
,
Fig. 6. Description of the function In ( ( 1+ JI)/ ( 1- $) I.
in ‘strength of materials’) and the wave propagation is stopped at $ = 21. The bar velocity decreases from point to point according to the function (87), so that if we look again at Fig. 6, and if A denotes the front of the wave and B-the end of the wave, we observe that A is faster than B, which means that the wave shrinks (ATI > A2B2) to be at I,!J= 2 1 the S (Dirac)-function (if the energy ata was conserved). The function (1 - I,!?)l/2 gives the decay of the wave within At [ ( I- I,!?)l/2 decreases as a function of +I,!J(]IJ?]< 1) (see Fig. 7)]. (1 - I,!?)~/~appears as a coefficient in the solution of (+(z, 7)) and with its decrease a(~, 7) decreases too. At JI = +I we have u(z, r) = 0. The physical explanation of this result is based on the impedance change (Z = impedance = m. In our case the impedance decreases as a function of I,!J (see Fig. 8). z=
(l-V>_
This decrease causes infinitesimal reflections I,!J= +l is given b y using the relation
(99) along the elements
(T(7, JI) = Z($) awg:, +)
de. The stress at
(100)
and equation (99). Substituting JI = ?I we obtain: (T(7,O) =ow=o
(101)
1216
G. ROSENHOUSE
(l-3Y2
The magnitude
I
of a unit origin
The magnitude of stress generated from the origin at $= 0.4
-
Fig. 7. Description
of the function (1-v)“*. Dotted lines-propagation from the origin to $I= r?r.1.
(Z,+dZ,.$,
-1.0
-06
-02
06
02
of a stress function
+d+,)
IO
Q
Fig. 8. The impedance function Z = (1 - I/?) m
and as was shown before, this result was predicted by the solution (98). Hence, 1bl s m, the limit of the wave propagation along the z axis is -1 bJ s z s )4.
if
Case 11
The geometry of the rod is defined by -L
s
z
s L.
(102)
The period is defined by 0 =S t.
(103)
Plane-disturbance
The material mechanical motion is
propagation in elastic nonhomogeneous
properties
media
1217
are defined by equation (90). The equation of
a [(~z_zz)z~]= -$.$; az
w=w(z,t).
(104)
The initial and ‘terminal’ conditions are
$,O) = b&z, The boundary
t) =
@&$(z,t) = 0.
(105)
conditions are according to definition (17):
iq(O, t)
=f(O,
t),
f)
[b2-L212$
=_m f).
(106)
According to Fig. 9, the boundary conditions should be defined in a region that include the space and the time, and everything that happens to the bar is limited in this region. Applying the continuum approach, the displacement is unique, hence the functions describing the boundary conditions should be equal at the meeting points. Meeting
points
Fig. 9. Finite rod, semi-infinite time. (Zero initial and terminal conditions).
In our case, the initial and terminal conditions should coincide at the points (0, 0), (0, L), Jii~ (t, 0), {$J (r, L), hence, by applying equations (105, 106), we have the limiting condition: (107) The solution of the stress propagation in case II is obtained by solving the equation
(b2-22)2d2Y, dz2
+
a,"Y<
=
0,
ai
2$-~, 0
(108)
1218
G. ROSENHOUSE
and finding the constants of integration due to the boundary conditions (106). Finahy. the stress is
flo9 t”) dt* 1 where
7. PHYSICAL
SIGNIFICANCE
OF GENERAL
SOLUTION
The above analysis shows that the shape and propagation of a disturbance in a nonhomogeneous medium depends on the nonhomogeneity. The resuhing solution contains two mutually-opposed travelling waves as in homogeneous media, but the time function of the disturbance is non-linear, so that the latter solution depends on the nonhomogeneity of the medium and the velocity of propagation depends on the location of the disturbance (see equation 57). Thus, since propagation in homogeneous media has a constant velocity and the shape of the disturbance does not change (cf. the Lorentz transformation in the special theory of relativity) motion in nonhomogeneous media involves accelerations and changes in shape,
ffO,tl=
initial shape
locity
b -
A ~o~inuous
reflection
f(0.t
I= initial
shape
Fig. 10. Shape of disturbance at time t in homogeneous infinite medium (a) and in nonhomogeneous infinite medium(b), with initial shapef(0, t) the same in both cases.
Plane-disturbance
propagation in elastic nonhomogeneous
media
1219
Unlike D’Alembert’s solution, we obtain two unequal waves of arbitrary shape, each moving with its variable velocity along the z-axis, with distortion and attenuation, one in the positive sense and the other in the negative sense. (See Fig. 10). Acknowledgemenrs-The author wishes to thank Professor F. Ollendorlf for the many constructive cussions which enabled bringing this paper to its final form, and to Professor J. J. Golecki (see[l]).
dis-
REFERENCES [I] G. ROSENHOUSE, D.Sc. Thesis, Technion-Israel Institute of technology (1971) (Under the supervision of Prof. J. J. Golecki). [2] F. OLLENDORFF, Z. angew. Marh. Phvs. 19.520 ( 1968). [3] S. R. BODNER and R. F. CLIFTON, An Experimentul Investigation of Elastic-Plastic Pulse Propagation in Aluminum Rods ut Elevated Temperature. Brown Univ., Div. of Engineering, Report Nonr562(20)/44. [4] LJ. S. LINDHOLM and K. D. DOSHl,./. uppl. Mech. 32, 135 (1965). [5] T. Y. TSUI,J. uppl. Mech. 35.824 (1968). [6] P. H. FRANClS,J. appt. Mech. 34,226 (1967). [7] R. C. PAYTON, Q. J. Mech. appl. Moth. XIX, 83 (1961). [8] A. N. DATA, Lnd. J. theoret. Phys. 4,43 (1956). [9] S. P.SUR,fnd.J. theoret.Phys.9,61 (1961). [lo] P. H. FRANCISJ. appl. Mech. 33,702 (1966). [I I] P. M. MORSE, Vibration LtndSound, 2nd Ed., pp. 107,265. McGraw-Hill (1948). (121 E. L. REISS,J.appLMech. 36,803(1969). [I31 J. S. WH1TTIER.J. appl. Mech. 32.947 (1965). [14] E. KAMKE, Differential gleichungen Liisungs Methoden und Liisungen. Leipzig (I. Gewilhnliche Differential Gleichungen (1959). [IS] K. ANGERMAYER, Structural Aluminum Design, 1st Ed. Reynolds Metals (1967). (Received
12 January 1973)
RbsumC-
Cet article traite du probleme de la propagation uniaxiale de perturbations dans des Tilieux continus axialement non homogenes. La solution mathtmatique generale obtenue est applicable a des problemes physiques concernant des milieux presentant des parametres variables lonitudinalement. La solution des equations differentielles resultantes est basee sur la symetrie de leurs variables de I’espace et du temps. Cette symetrie conduit a la transformation de problemes de limites en leurs contre-parties initiales et finales. En elaborant une expression particuliere de la transformee en est deduite. II est montre que la solution bien connue de d’Alembert pour la propagation dans un barreau mince homogtne ou dans une corde represente un cas particulier de cette loi. La signification dune telle theorie gtnerale reside dans le fait que la plupart des phenomtnes dynamiques dans la nature ont lieu dans des milieux non homogenes, et la solution present&e ici peut contribuer B leur evaluation physique. Comme exemple. nous resolvons la propagation dune impulsion dans un barreau mince chauffe. Zusammenfassung-
Diese Arbeit beschlftigt sich mit dem Problem einachsiger Fortpflanzung von Storungen in kontinuierlichen achsial-nichthomogenen Stoffen. Die erhaltene allgemeine mathematische Losung kann auf physikalische Probleme angewandt werden, die Stoffe mit longitudinal-variablen Parametern behandeln. Die Losung der resultierenden Differentialgleichungen basiert auf der Symmetrie ihrer Raum- und Zeitvariablen. Diese Symmetrie fuhrt zur Umwandlung von Grenzproblemen in ihre Anfangs- und Endgegenstiicke. Durch Ausarbeit einer speziellen Version von Fourier’s Transform wird ein Gesetz fur einachsige Fortpflanzung in nichthomogenen Stoffen abgeleitet. Es wird gezeigt, dass D’Alembert’s wohlbekannte Liisung fur Fortpflanzung in einer homogenen diinnen Stange oder Schnur ein besonderer Fall dieses Gesetzes ist. Die Bedeutung einer solchen allgemeinen Theorie liegt in der Tatsache, dass die meisten dynamischen Phenomane in der Natur sich in nichthomogenen Stoffen ereignen, und die hier gezeigte Losung mag zu ihrer physikalischen Bewertung beitragen. Als Beispiel l&en wir die Fortpflanzung eines Pulses in einer erhitzten diinnen Stange. Sommvio-Quest’articolo tratta il problema della propagazione uniassiale di disturbi in mezzi continui assialmente non omogenei. La soluzione matematica generale ottenuta e applicabile a problemi fisici in mezzi con parametri variabili Iongitudinalmente.
1220
G. ROSENHOUSE
La soluzione delle equazioni differenziali risultanti t basata sulla simmetria delle loro variabili nello spazio e net tempo. Questa simmetria conduce alla trasformazione di problemi di confini nei loro corrispondenti problemi iniziali e terminali. Usando una versione speciale della trasformazione di Fourier, viene derivata una legge per la propagazione uniassiale in mezzi non omogenei. Viene dimostrato the la ben nota soluzione di D’Alembert, per la propagazione in una barretta o corda sottile omogenea, e un case particolare di questa legge. II significato di una tale teoria generale giace nel fatto the quasi tutti i fenomeni dinamici occorrono in natura in mezzi non omogenei, e la soluzione qui presentata pub contribuire alla loro valutazione fisica. Come esempio viene risolto il problema della propagazione di un impulso in una barretta sottile riscaldata. A~CT~KT“pn
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Sci.,
1973, Vol. 11, pp. 1197-1220.
Pergamon
Press.
PLANE-DISTURBANCE NONHOMOGENEOUS
Technion-
Printed
in Great
Britain
PROPAGATION IN ELASTIC MEDIA. GENERAL SOLUTION
G. ROSENHOUSE Israel Institute of Technology, Department of Mechanics, Haifa, Israel (Communicated by 1. N. SNEDDON)
Abstract-This paper deals with the problem of uniaxial propagation of disturbances in continuous axiallynonhomogeneous media. The general mathematical solution obtained is applicable to physical problems dealing with media with longitudinally-variable parameters. Solution of the resulting differential equations is based on symmetry of their space and time variables. This symmetry leads to transformation of boundary problems into their initial and terminal counterparts. By working out a special version of Fourier’s transform, a law for uniaxial propagation in nonhomogeneous media is derived. It is shown that D’Alembert’s well-known solution for propagation in a homogeneous thin bar or string is a particular case of this law. The significance of such a general theory lies in the fact that most dynamic phenomena in nature occur in nonhomogeneous media, and the solution presented here may contribute to their physical evaluation. As an example, we solve the propagation of a pulse in a heated thin bar. I. INTRODUCTION
A ONE dimensional elastic model is used in deriving the equation of motion for plane disturbance propagation in nonhomogeneous media as an example of a problem involving variable parameters, neglecting energy losses. Assuming that there is no time variation, and with thermal effects neglected, these variable parameters are the elastic modulus E* = E$ . E*(Z) and the density P = PO . p(z). (See Fig. 1). The problem is of practical importance in various applications involving stress and displacement propagation in materials. The solutions obtained in the present paper were applied to cases discussed in [ 11. These include a mechanical model of the inner ear in the form of a nonhomogeneous bar as suggested by Ollendorff[2], variation of the ‘bar velocity’ in a nonuniform temperature field[3], plane propagation in soil (with vertical variation of the parameters) and in the atmosphere, and finally, radial wave propagation in space. A literature survey [4- 131 indicates specific nonhomogeneities obtainable by separation of variables. The solutions have the form of a series with unknown coefficients, solved with the aid of boundary conditions. In this paper, propagation is treated as a boundary-value problem for which a general mathematical solution is obtained. Examples of propagation in media with the nonhomogeneities:
E* = E,*,
E” = E,f(a2kz2)2
are solved for specified boundary conditions. 1197
IJESVol.lINo.Il-E
(1)
1198
G. ROSENHOUSE
Direction of propagation. and variation of parameters
Fig. 1. Representation of element of elastic medium. (a) Unit area cylinder cut from medium along Z-axis. (b) Deformed shape of cylinder element. (c) Variation of parameters along the z-axis. (d) Orientation of medium.
2. TRANSFORMATION
OF THE
EQUATION
OF MOTION
The equations of motion for a free dilatational plane displacement finite medium with finite period are given by the momentum equation
aW* (aw/az) I = az
a[p( aw/ar)1 at
and the conditions for plane displacement
9
aw ax
r)
(2)
propagation
u=v=o -=-=
w = w(Z,
in an elastic
aw ay
o
I
where x, y, Z = the coordinates of the reference system (See Fig. 11, E* = A+ 2~ = elastic parameter, A, p = LamC’s constants, u = displacement in x-direction, II = displacement in y-direction, w = displacement in Z-direction, P = density, t = time.
(3)
Plane-disturbance
propagation in elastic nonhomogeneous
to use non-dimensional
It is convenient
equations
media
1199
by using the following trans-
formation of variables q-z- 2m T r.--2vz L where T= 4L = 7= z=
‘length’ of wave along time-axis (period), extension of the medium in Z-direction, (2rt/T) = non-dimensional time, (21rz/L) = non-dimensional length in Z-direction
and the single equation of motion will have the form:
I
E*(z)2g =
g
[
a2.
p(z)$,
w = W(L 7)
(4)
where
C2 = 5 *Ct = non-dimensional Ci = F
velocity at z = 0
= bar velocity at z = 0. 0
The solution of equation (4) is to obtain an ordinary differential one. Having variable coefficients carried out. The transformation begins by as follows:
our aim and in order to arrive at this aim we shall try equation instead of equation (4) which is a partial in equation (4) a special transformation should be multiplying equation (4) by sin (no) and integrating
(5)
77 77 1 Iw
We next integrate by parts the right-hand integral in equation (4) twice %sin
(no) --wcos
(127) -n2 0
= t(z) -n2 I
w sin (nr) dr,
sin (no) dr
D
w=
W(Z,T).
Because of the variable coefficients of equation (4), a new variable is introduced:
(6)
G. ROSENHOUSE
1200
which yields the stress as a function of time and space
Eon fl(Zr 7) = y(z, 7) . -
’
E*(z)
E*(z) =1+/J
(8)
where E,,E(z) = Young’s modulus.
By developing y(z, 7) into the Fourier series y(z, 7) = + i Y,(n, z) sin (n7) ?I=1
(9)
where E* (z)$
sin (m) d7:
we eliminate 7. If the initial and terminal conditions c(z) = 0, and equation (6) is given by:
(10)
of the problem are zero, then
(11) Differentiating
equation ( 1 1) with respect to z, and multiplying by e*(z) we have
dZY, -- E*(Z) . ddd P(z)2 dz P(Z) dz2
E*(Z) --
dYn -
_a2
dz
x
E*
(z)z
n
sin (n?) d7
(12)
or
E*(Z) +(z) dyn a”,Yn = 0. E*(Z) d2Yn--.. dz+ p(z) dz2 p(z)” dz The last equation is an ordinary second-order
(13)
linear differential equation of the form
=0
A,(z) "y"+B,(z)~+c.(z)y. dz2
(14)
(Really only C, is a function of n and not A, and II,). Y, = Clbl(n, z) +C2b2(n, z),
C1, C,-constants.
(15)
Formally, whatever the functions 6, (z) and b,(z) will be, we can introduce equation (15)by (16) Y,=F,(z)[C,coscw,+C,sin&l, cvn=ck(z) where F,(z)
= m
Plane-disturbance
propagation in elastic nonhomogeneous
This substitution, which still describes the following mathematical treatment.
3. SIMPLIFIED
1201
media
nothing new, fits better than equation (15)
SOLUTION FOR CASE WITH TWO BOUNDARY INCLUDING SOME FORMAL LIMITATIONS
CONDITIONS
As an example of equation (16), we choose the following two arbitrary boundary conditions (17) to which we apply the Fourier sine transform n
fn = Y,(O) =
I
f(r) sin (nr) dr,
0
g
n
s
-aYY&(O)_ = g(r) a.2
0
sin(m)
dr,
subject to the following arbitrary limitations z=O+sina;,=O+~=O, ~=o-+cosa;,=1, (19)
The constants C1 and C, are determined as follows fn = F,(O)C, = Y,(O),
(20)
cl=&
(21)
R
dYtt(O)= Y&C, aF (0) dz
+F.(O)d* G = Rn
(22) (23)
1202
G. ROSENHOUSE
and their substitution in equation ( 16) yields (24) Now we transform Y,(z) into the (z, T) domain and obtain the solution Yk 7) = f
m(z) sin (n7) 2 n=1,2...
(25) =_.-1 2 2 lr .A,,,
#[[sin
-[dn,&,&,
[,(r+?)]+sin
{‘OS [n(T-:)]+cos
[+$]]h
[‘(T+:)]]]’
Observing the last solution, one must pay attention to the fact that (Yis a function of z rather than an independent variable (see equation 16)). Moreover, the Fourier expansion in equation (25) is not influenced by the type of the function 01.In the case that a is a linear function of z, and %(O) = 0, the wave equation with constant coefficients is obtained for propagation of disturbances in homogeneous media or in media with some special nonhomogeneities. The first series in equation (25) will be changed by using the transform: f(7)
= $
i f, sin (m). n=1,2...
(26)
We shall refer to the incomplete series h(T) =+
i
h sin
(117)
(27)
n=1.2...
then we have Afk(7)=fk+l (T)
-fk(7)
=
$fk
sin (k~) , (see Fig. 2)
(28)
hence
(2% Summing the terms A& of equation (29) we have (30)
Plane-disturbance
propagation in elastic nonhomogeneous
media
1203
Fig. 2. Elements offfor line spectrum.
The second series of equation (25) will be changed by using the expression g(7) = -$
2
g, sin (~27)
(31)
g, sin (n7)
(32)
n=1,2...
referring to the incomplete series Sk(T) =;5
5 n=1,2...
we have
hh,(T) =gkfl(T) -gk(T) = gk sin(h).
(33)
Integration of g(r) may have the form [
-7
g(r) d7 =
i
A[ r
n=1,2...
=
$ -jj
g(r) dT] = f --T
i
g, r
n=1,!2...
F[COS (127)-COS
sin
(nT)
dr
--T
(-11T)],
n=1.2...
(34)
hence g(T*)
dr* =f
i n=12,,, [cos[,(,-~)]-cos[n(T+~)]~~}.
The results in equations (30) and (35) now replace the corresponding equation (2S), the final result reads:
(35) terms in
31 u.IIAfn(~+%)+Afn(~-:) y(zq T,=.=;.., F,(O) 2 r+$Jn)
-[da,;O),&jA Ir_(an,n) &(T*)
I
dT* t a, = a(& Z>.
In the case that E* = E,, p = PO, the transformed
(36)
wave equation (13) reduces to
1204
G. ROSENHOUSE
ordinary differential equation
d2Y,+a;y, dz2
=
0
(37)
with the general solution Y,= C 1 coS a,2 + C, sin a,2
(38)
hence, observing equation ( 16):
F,(z) = 1 C&= UnL.
(39)
In the case of thin bars we may assume the approximation E,* = E. which is Young’s modulus of elasticity, hence the boundary conditions are the stress and the stress gradient respectively: f(7) = [ E,?$;’
“1 =~(0.7) (40)
g(7) = $[ EoaWL: r’] = ‘og
‘) ,
(see equation ( 17))
hence, applying equation (36), we obtain the stress in the bar
(41) The sum notation is dropped becausefis
5
Af=f,
n=1,2...
i
not a function of n:
A 1 g(T*) dT* = 1 g(r*) dr*.
(42)
ft=l,Z...
hence, the stress is given by
which is in full agreement with D’Alembert’s solution for homogeneous bars with zero initial conditions and given boundary conditions. The development of the solution for line spectrum (finite period) can be extended to continuous spectral analysis (t > 0). Beginning with the same equations of motion (equations (2), (3)), and boundary conditions f(t)
=
[e*(o)*],
g(t)
=$*(o)“*]
(44)
Plane-disturbance
propagation in elastic nonhomogeneous
media
1205
we apply the Fourier sine transform $ “’ mf(l) sin (&) dt, 0 I0
ff = Ye(O) =
,
y, =
i 0
“’
mE(z)*%sin([t)
(45)
dt.
J0
An ordinary differential equation general solution of this equation is
resembling
equation
(14) is obtained
YE= FS(z)[C1 cos ++C2 sin ~~1, * = CY&).
and the
(46)
Applying the boundary conditions (44) we obtain
t
fs‘OS % +
[da,
I
fb;/&]sin%.
(47)
The solution in the (z, t) domain is given by the inverse transform ~(2, t) = (s)“’
= (:)‘”
lrn Y,sin
1
$$[fr
(5t) d5
cos a,+ ~d~~~),~~
sin*,]
sin (&) dt.
(48)
There are in the solution two integrals containing trigonometric functions. We shall try to eliminate them in order to obtain a solution depending only on the boundary conditions and nonhomogeneity effects, as was done in the case of finite period. For the first integral we use the integration [
d~=[$d~=(~~1’2[&sin(&t)
d&
sr
(seeFig.3)
f I
Fig. 3. Element offfor continuous spectrum: (a) General case, (b) Step function.
(49)
G. ROSENHOUSE
1206
hence m dflr-t
(o&)1 d5
_ 2 1’2 m d[-(--) i .&sin[S(j*y)]d5.
(50)
The second integral of equation (48) will be changed by the integration g(t*)
g(t*) dt* =
=
(i)“’ [
It;(cos
dt* d,$ = (3)“’ -
lam g, ~t~~~~~ sin (nt*) dt* dt
[++y)]}
[@)]+cos
d*
(51)
hence g(t*)
dt* d< = (t)“’ -
[
y{cos
[+-y)]+cos
[++y)]}
The results in equations (501, (52) replace the corresponding and the final result for semi-infinite period is:
d&. (52)
terms in equation (48),
01~= cu(S, z)
(53)
which is also a general case for any continuous non-homogeneity and, as shown for line spectrum, reduces to the solution of D’Alembert in homogeneous media. 4. GENERAL SOLUTION BOUNDARY CONDITIONS,
OF PLANE-WAVE PROPAGATION FOR ANY PAIR OF WITH NO FORMAL LIMITATIONS AND ZERO INITIAL CONDITIONS
The case considered here is equation (2), which is the general form of the linear differential wave equation with real coefficients varying as a function of z, i.e. L(w)
=
A(z)
[
.-$+B(z)&-C(z)-$ 1w=o.
The boundary conditions are a combination (required for the two constants):
(54)
of two out of the four possible alternatives
.h =fi(O,j),
fi=fi(~9j)
&=gl(o,t),
g,=g,(~,t)
I
0s
ZSTr
(55)
The solution is based on elimination of the variable (t), using a technique which resembles the Fourier transform (except for non-convergent cases). The result is
Plane-disturbance
propagation in elastic nonhomogeneous
1207
media
analogous to equation ( 14):
c(z) 2.%+,*(z) d[ Ildpz(z)1 .2
+ b2y, (fi, z) = 0
(56)
where a: =
L
2
( 1
n2
TC,
,
The solution in general is as presented domain, the final result reads:
c(z)2=-
E*(2) P(Z)
c2 ’
O
=
E*
0
PO’
in equation (16). Transforming
into the (z, r)
Y(z,T) = n=123,,, i F n a1 2[ /3rl[ Af I(~ +!”n) +Af l(r-~)]+P=[A~r(f+$+A+~)] 2,
+PIS [ Af + +%n) +Af 2(+] +nA
+P~r[Agt(r+$+ A+$1
T+(an'n) [B2~~(7*)+P22g~(~*)+P23fi(7*)+PZ4fi~(~*)ld~*} (57) Ir-(%/n)
& = &(n, T), or pu = pil (n,0) are the coefficients of the two given boundary conditions (others are not taken into account). D’Alembert’s solution (equation (43)) is the general solution (equation (57)) reduced to a very simple case. The solution presented here refers to a line spectrum, i.e. to finite time. In the case of an infinite or semi-infinite medium (semi-infinite time) we have a continuous spectrum (no frequency filtering by boundary limitations).
5. SOLUTION OF COEFFICIENTS-p The first case is the boundary conditions applied in equation (17). Substituting the first condition in equation ( 16) after transformation as in equation ( 18), we have fn= F,(O) . C1 cos a,(O) + F,,(O) . C2 sin a,(O). For the second boundary condition, equation ( 16) is differentiated
(58)
with respect toz:
dy,, --d-[C,cosa,+C2sinol,]+F,[-CIsin~~+C2cos%1~ dF,
(59)
dz
hence
dY,_ dz -
cos (Y,- F,%sin
OL,C, + *sin 1 [ dz
01,+ F,,$cos dz
a,, C2
1
(60)
and (61)
G. ROSENHOUSE
1208
where:
cos
=
5
(dFnlW
~0s
n
sin
5
a,- Fn(dddz)
sin01,.
K,, =
n
(@Jdz) sinan+ F,(dddz)
cos a,
K,,
which is a formal trigonometric substitution. The second boundary condition is substituted in equation (6 l), yielding the second equation for calculating the coefficients
gn= F,(O) [Cl cosL(O) + Cz sinL(O)]:
(62)
L(O) = [(O, TZ).
The functional relation between C1 and Cp is obtained from equation (58): Cl =
F,
fn
-C,
(0) ~0s an(O)
(63)
tan a,(O).
Now we determine Cz. using equations (58), (62), (63) ~~0s~(O)-F~(O)~sina,(0)
.
I[ f
-C, tana, F,(O)c~sa,(O)
1
~sma,(O)+F~(O)~cosa,(O)
1
Cz=g,
(64)
and explicitly:
c, = p2 =
dF (0) -~cosu.(O)+F,(O)~
sin 01,(0)
11
-F,(O)-
sin o,(O)
+ ysinoa(0)
+F,(O)v
cos o,(O)
~cosa,(O)
f,
F,(~)cos~,(~)+~~ tan 01,(0)
(65)
C, is similarly obtained from equation (59):
fn c, = F,(O)sina,
- c, cot CX,(0)
which is substituted in equations (59), (62):
f, - c, cot o,(O) , sin a,(O) F,,(O)
Plane-disturbance
propagation in elastic nonhomogeneous
I209
media
hence
cos 01, (0)
+
i
cot OL,(0)
dFiz(0) cos a,(O) -F,(O)Y sin a, (0)
.
(66)
II
The final solution is obtained by substituting the resufts in equations (64), (66) in the fundamental solution equation ( 16):
Yn=Fn[Ptcos~+Pzsina;tl;
/A==P1@,n), Pz=&(0,f2)
(67)
hence Y, = Fn{t%Sn cos % +&EL cos % + &.& sin %i- Pzzg, sin %I; & = fiii (0, n) . Retransforming
to the (z, 7) domain, we have Y(ZtT)
=z i
Y, sin {fiT)
(6%
r=1,2...
Y(Z,T)
=
(68)
i F, n=1,2... T+(a@)
+nA
I
+-
CPzJW> +&g(r*)
3 d7” . 1
This is part of the general solution (equation (.57)), still more general than D’Alembert’s, as.is seen by comparing it with equation (43). The other cases of boundary conditions, solved in a similar manner, are given in convenient form as follows. Case Z-W:
The bounda~
conditions are:
f=f(%r r), g=g(z137),
(70)
z1 and 2%may be zero or P, so we have four possibilities:
(1) fK4 Q-1, S(O> T), (2) f(n, 7), g(O, r), (3) f(O, T), g(r* r), (41 f (-9 7) 1gfv, 7) *
(71)
G. ROSENHOUSE
1210
The coefficients are PI2 and pZ2for g(0, T) ; T) ; PI4 and pZ4for g (m, 7) .
PII and PZl forf(0,
T) ;
PU and PCUforf(m.
The expressions for the @ii’Sare I
Pl2orp,,=-_, Al
P*1or p13= -t;
A3 P21
orP23
=
1,
pz2
(72)
or p24 = -B, A,
.
where: A, =A,. A2 =$
B,+C,, n
and
A3 =$,
n
A =_dMzl) . Tsin%(zl) R B,=
-F,(zl)
da, (2,) dz cos %(zl).
cot%(z2), sin h(zl),
cn D, = Fn(z2) sin h,(zZ). Case V fi =h(O, 71, fi =f2(7T, T),
(73)
and the coefficients are:
‘11 = F,(r)
-sin a,(a) cos (U,(P) sin h(O) -F,(O)
cos h(O) sin%(n)
p13 = F,(a)
sin%(O) cos %(7r) sin G(O) -F,(O)
coscu,(O) sin cu,(T) ’
“l=Fn(r)
sina,
-cos a,(7r) coso1,(0) -F,(O)
P23=F,(rr)
sina,
cos (Y*(O) cosa,(O) --F,(O)
’
(74)
(75)
(76) sina,
coscr,(7r)’
sincr,(O) COSOL,(~~)
(77)
Case VZ g1
=
g,(O,
g2
=
gz(n,
71, 71,
(78)
Plane-disturbance
propagation in elastic nonhomogeneous
media
1211
The basic equation in this case is equation (61), and the coefficients for equation (57) are:
-sinL(O) . [&(z)lF,(z)l
Pl2 = &(7r) cos tn(7i-) sin tn(0) -K,(O)
cos in(O) sin en(r)’
sin5n(0). [JL(z)/F,(z)] ‘I4 = K,(r)
cos [n(m) sin[n(O> -K,(O)
P2”=Kn( 7r) sink(P) =
L(m) [KI(z)l~n(z)I
--a
cos 5,(O) --K,(O) cm
P24
K,(IT)sin J&(p)
cos &(O) sincn(7r) ’
a,
(0)
* [K”
(z)/Fncd
cos tn(0) -K,(O)
sin t,(O) cos en(g)’
1 sin tn(0) cos tn(7r).
(79)
(80)
(81)
(82)
It is possible in all cases to use the coefficients C1 and C2 for equation (16) instead of pit. In the sixth case we obtain Cl
=
PIa&
+Pl&gn(~),
c2
=
PzziL(O)
+P24&(p),
(83)
but the generalized version of D’Alembert’s of the physical interpretation. 6. STRESS
solution is apparently
preferable because
PROPAGATION ALONG A THIN BAR DUE TO A QUASI-STATIC NON-UNIFORM TEMPERATURE FIELD
The case of a heated bar in a non-uniform temperature field is solved with the aid of Bodner-Clifton test[3], (see Fig. 4). The temperature gradient is determined on the basis of thermodynamic considerations [ I], and then the modulus of elasticity may be calculated at every point along the examined bar. The density is considered to be constant, because it remains nearly constant in spite of the change of temperature, in many solid materials. Along the bar presented in Fig. 5, we obtain the temperature gradient e(z, t) =
lOO[ l--8($)],W).
(84)
The modulus of elasticity as a function of 0 (z, t) is given in Fig. 4, using the relation E,-,= P,,C;.
(85)
By minimizing the error defined by the square deviation of the assumed function from the function of the exact modulus of elasticity, we obtain
E = E(z)
(86) with maximal linear deviation within 3 per cent. It is not only the Bodner-Clifton
test
G. ROSENHOUSE
1212
200
100
200
300
400
500
I t-
700
800
900
1000
“F
T,
Fig. 4. Transformation
600
of elastic wave velocity into modulus of elasticity dependence temperature.
L/2
L/2
b
b
on
Y
Fig. 5. Geometrical and thermal data of the heated bar.
that leads us to the relation (86), but we may also apply the general relation E=E,[1-(~)2]2=Mo[b2-22]Z1 which is convenient
from the mathematical
+=;,
Mll=E064
(87)
point of view, and may be used for many
Plane-disturbance
propagation in elastic nonhomogeneous
media
1213
materials. The solution of the stress propagation is solved generally according to equation (57), but using the relation (86) condition (18) is fulfilled and we may apply the solution (36) or (53), which is simpler than the solution presented in equation (57). It was not intuition that led us to the function (87), but instead of taking into account E and p arbitrarily, and solving afterwards the differential equation (13), we choose the inverse method. We tried to find the elementary functions of E and p, the differential equations of which give the solution ( 16) including the limitation (18). Doing the last procedure we used [ 141. For a demonstration of the line and continuous spectral analysis, two cases are solved. The first is the case of finite period and the initial and ‘terminal’ conditions are zero. The second case deals with semi-infinite period and the initial condition is zero. (The ‘terminal’ condition is eliminated).
Case I The geometry of the rod is defined by
The period is defined by
The material mechanical properties are defined by E = E,(l -JI”)” = M,,(b2-~2)2, P = PO= const.
lb1 a T
The equation of motion is obtained by substituting (88,89,90)
3(a2_z2)2y9”]
= ~.g.~. 0
(90) in equation (2) (91)
The initial and ‘terminal’ conditions are zero
$(z, 0)
= 0,
$z,
7r) = 0.
(92)
The two boundary conditions are given at z = 0
(93)
IJESVol.
II No. II-F
1214
The transformed
G. ROSENHOUSE
equation of motion is d2Y (b2-22)2 -“+a”,m dz2
= 0
&2= n._ L2 ,c2 =-.MO n ( Co T ) PO
(94)
The solution of (94) is
(95) ~~=$(~~+l]1’21nI~I,
I$/ < 1.
The solution of stress is according to equation (3 6):
For the case that the conditions at z = 0 are (97) we get
nd(L/bTc)’
I‘-+I)
- ( l/n2) In I-rJ,
n
(98) Relative to the propagation of stress in homogeneous medium, the result here is influenced by two terms of nonhomogeneity In (( 1 + $)I( 1 - $I,)]instead of z and ( 1 - I/?)~/~ instead of unity. The first term has the graphical representation in Fig. 6. First of all, the infinite singularity at I,IJ= kl.0 should be a geometrical limit. The last conclusion could be understood physically by calculating the bar velocity as a function of I/J.We notice that at I,!J= kl, the bar velocity is zero (which is equivalent to the plastic region
Plane-disturbance propagation in elastic nonhomogeneous media
I.0
a6
OG?/
0.2
O-6
I :O
I.4
I.6
1215
,
Fig. 6. Description of the function In ( ( 1+ JI)/ ( 1- $) I.
in ‘strength of materials’) and the wave propagation is stopped at $ = 21. The bar velocity decreases from point to point according to the function (87), so that if we look again at Fig. 6, and if A denotes the front of the wave and B-the end of the wave, we observe that A is faster than B, which means that the wave shrinks (ATI > A2B2) to be at I,!J= 2 1 the S (Dirac)-function (if the energy ata was conserved). The function (1 - I,!?)l/2 gives the decay of the wave within At [ ( I- I,!?)l/2 decreases as a function of +I,!J(]IJ?]< 1) (see Fig. 7)]. (1 - I,!?)~/~appears as a coefficient in the solution of (+(z, 7)) and with its decrease a(~, 7) decreases too. At JI = +I we have u(z, r) = 0. The physical explanation of this result is based on the impedance change (Z = impedance = m. In our case the impedance decreases as a function of I,!J (see Fig. 8). z=
(l-V>_
This decrease causes infinitesimal reflections I,!J= +l is given b y using the relation
(99) along the elements
(T(7, JI) = Z($) awg:, +)
de. The stress at
(100)
and equation (99). Substituting JI = ?I we obtain: (T(7,O) =ow=o
(101)
1216
G. ROSENHOUSE
(l-3Y2
The magnitude
I
of a unit origin
The magnitude of stress generated from the origin at $= 0.4
-
Fig. 7. Description
of the function (1-v)“*. Dotted lines-propagation from the origin to $I= r?r.1.
(Z,+dZ,.$,
-1.0
-06
-02
06
02
of a stress function
+d+,)
IO
Q
Fig. 8. The impedance function Z = (1 - I/?) m
and as was shown before, this result was predicted by the solution (98). Hence, 1bl s m, the limit of the wave propagation along the z axis is -1 bJ s z s )4.
if
Case 11
The geometry of the rod is defined by -L
s
z
s L.
(102)
The period is defined by 0 =S t.
(103)
Plane-disturbance
The material mechanical motion is
propagation in elastic nonhomogeneous
properties
media
1217
are defined by equation (90). The equation of
a [(~z_zz)z~]= -$.$; az
w=w(z,t).
(104)
The initial and ‘terminal’ conditions are
$,O) = b&z, The boundary
t) =
@&$(z,t) = 0.
(105)
conditions are according to definition (17):
iq(O, t)
=f(O,
t),
f)
[b2-L212$
=_m f).
(106)
According to Fig. 9, the boundary conditions should be defined in a region that include the space and the time, and everything that happens to the bar is limited in this region. Applying the continuum approach, the displacement is unique, hence the functions describing the boundary conditions should be equal at the meeting points. Meeting
points
Fig. 9. Finite rod, semi-infinite time. (Zero initial and terminal conditions).
In our case, the initial and terminal conditions should coincide at the points (0, 0), (0, L), Jii~ (t, 0), {$J (r, L), hence, by applying equations (105, 106), we have the limiting condition: (107) The solution of the stress propagation in case II is obtained by solving the equation
(b2-22)2d2Y, dz2
+
a,"Y<
=
0,
ai
2$-~, 0
(108)
1218
G. ROSENHOUSE
and finding the constants of integration due to the boundary conditions (106). Finahy. the stress is
flo9 t”) dt* 1 where
7. PHYSICAL
SIGNIFICANCE
OF GENERAL
SOLUTION
The above analysis shows that the shape and propagation of a disturbance in a nonhomogeneous medium depends on the nonhomogeneity. The resuhing solution contains two mutually-opposed travelling waves as in homogeneous media, but the time function of the disturbance is non-linear, so that the latter solution depends on the nonhomogeneity of the medium and the velocity of propagation depends on the location of the disturbance (see equation 57). Thus, since propagation in homogeneous media has a constant velocity and the shape of the disturbance does not change (cf. the Lorentz transformation in the special theory of relativity) motion in nonhomogeneous media involves accelerations and changes in shape,
ffO,tl=
initial shape
locity
b -
A ~o~inuous
reflection
f(0.t
I= initial
shape
Fig. 10. Shape of disturbance at time t in homogeneous infinite medium (a) and in nonhomogeneous infinite medium(b), with initial shapef(0, t) the same in both cases.
Plane-disturbance
propagation in elastic nonhomogeneous
media
1219
Unlike D’Alembert’s solution, we obtain two unequal waves of arbitrary shape, each moving with its variable velocity along the z-axis, with distortion and attenuation, one in the positive sense and the other in the negative sense. (See Fig. 10). Acknowledgemenrs-The author wishes to thank Professor F. Ollendorlf for the many constructive cussions which enabled bringing this paper to its final form, and to Professor J. J. Golecki (see[l]).
dis-
REFERENCES [I] G. ROSENHOUSE, D.Sc. Thesis, Technion-Israel Institute of technology (1971) (Under the supervision of Prof. J. J. Golecki). [2] F. OLLENDORFF, Z. angew. Marh. Phvs. 19.520 ( 1968). [3] S. R. BODNER and R. F. CLIFTON, An Experimentul Investigation of Elastic-Plastic Pulse Propagation in Aluminum Rods ut Elevated Temperature. Brown Univ., Div. of Engineering, Report Nonr562(20)/44. [4] LJ. S. LINDHOLM and K. D. DOSHl,./. uppl. Mech. 32, 135 (1965). [5] T. Y. TSUI,J. uppl. Mech. 35.824 (1968). [6] P. H. FRANClS,J. appt. Mech. 34,226 (1967). [7] R. C. PAYTON, Q. J. Mech. appl. Moth. XIX, 83 (1961). [8] A. N. DATA, Lnd. J. theoret. Phys. 4,43 (1956). [9] S. P.SUR,fnd.J. theoret.Phys.9,61 (1961). [lo] P. H. FRANCISJ. appl. Mech. 33,702 (1966). [I I] P. M. MORSE, Vibration LtndSound, 2nd Ed., pp. 107,265. McGraw-Hill (1948). (121 E. L. REISS,J.appLMech. 36,803(1969). [I31 J. S. WH1TTIER.J. appl. Mech. 32.947 (1965). [14] E. KAMKE, Differential gleichungen Liisungs Methoden und Liisungen. Leipzig (I. Gewilhnliche Differential Gleichungen (1959). [IS] K. ANGERMAYER, Structural Aluminum Design, 1st Ed. Reynolds Metals (1967). (Received
12 January 1973)
RbsumC-
Cet article traite du probleme de la propagation uniaxiale de perturbations dans des Tilieux continus axialement non homogenes. La solution mathtmatique generale obtenue est applicable a des problemes physiques concernant des milieux presentant des parametres variables lonitudinalement. La solution des equations differentielles resultantes est basee sur la symetrie de leurs variables de I’espace et du temps. Cette symetrie conduit a la transformation de problemes de limites en leurs contre-parties initiales et finales. En elaborant une expression particuliere de la transformee en est deduite. II est montre que la solution bien connue de d’Alembert pour la propagation dans un barreau mince homogtne ou dans une corde represente un cas particulier de cette loi. La signification dune telle theorie gtnerale reside dans le fait que la plupart des phenomtnes dynamiques dans la nature ont lieu dans des milieux non homogenes, et la solution present&e ici peut contribuer B leur evaluation physique. Comme exemple. nous resolvons la propagation dune impulsion dans un barreau mince chauffe. Zusammenfassung-
Diese Arbeit beschlftigt sich mit dem Problem einachsiger Fortpflanzung von Storungen in kontinuierlichen achsial-nichthomogenen Stoffen. Die erhaltene allgemeine mathematische Losung kann auf physikalische Probleme angewandt werden, die Stoffe mit longitudinal-variablen Parametern behandeln. Die Losung der resultierenden Differentialgleichungen basiert auf der Symmetrie ihrer Raum- und Zeitvariablen. Diese Symmetrie fuhrt zur Umwandlung von Grenzproblemen in ihre Anfangs- und Endgegenstiicke. Durch Ausarbeit einer speziellen Version von Fourier’s Transform wird ein Gesetz fur einachsige Fortpflanzung in nichthomogenen Stoffen abgeleitet. Es wird gezeigt, dass D’Alembert’s wohlbekannte Liisung fur Fortpflanzung in einer homogenen diinnen Stange oder Schnur ein besonderer Fall dieses Gesetzes ist. Die Bedeutung einer solchen allgemeinen Theorie liegt in der Tatsache, dass die meisten dynamischen Phenomane in der Natur sich in nichthomogenen Stoffen ereignen, und die hier gezeigte Losung mag zu ihrer physikalischen Bewertung beitragen. Als Beispiel l&en wir die Fortpflanzung eines Pulses in einer erhitzten diinnen Stange. Sommvio-Quest’articolo tratta il problema della propagazione uniassiale di disturbi in mezzi continui assialmente non omogenei. La soluzione matematica generale ottenuta e applicabile a problemi fisici in mezzi con parametri variabili Iongitudinalmente.
1220
G. ROSENHOUSE
La soluzione delle equazioni differenziali risultanti t basata sulla simmetria delle loro variabili nello spazio e net tempo. Questa simmetria conduce alla trasformazione di problemi di confini nei loro corrispondenti problemi iniziali e terminali. Usando una versione speciale della trasformazione di Fourier, viene derivata una legge per la propagazione uniassiale in mezzi non omogenei. Viene dimostrato the la ben nota soluzione di D’Alembert, per la propagazione in una barretta o corda sottile omogenea, e un case particolare di questa legge. II significato di una tale teoria generale giace nel fatto the quasi tutti i fenomeni dinamici occorrono in natura in mezzi non omogenei, e la soluzione qui presentata pub contribuire alla loro valutazione fisica. Come esempio viene risolto il problema della propagazione di un impulso in una barretta sottile riscaldata. A~CT~KT“pn
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