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Wave propagation in thin crystal plates
1n1.J. Engng Sci. Vol. 32, No. 4, pp. 715-717, IW4 Copyright 0 1994 Elscvicr Science Ltd Printed in Great Britain. All rights rcmvcd tlOZ)-7225/Y4 $6.00 + 0.00
Pergamon
LE’ITERS IN APPLIED AND ENGINEERING
WAVE PROPAGATION
SCIENCES
IN THIN CRYSTAL PLATES
C. CONSTANDA Department of Mathematj~,
University of Strathclyde, Glasgow, Gi 1XH U.K.
M. E. PBREZ Departamento de Matemgtica Aplicada, Universidad de Cantabria, Santander, Spain AbstrW-A Mindlin-to theory is const~~ted for cubic crystat plates, and the propagation of small amplitude waves in such plates is discussed in terms of the physical parameters of the material.
The propagation of flexural waves in a homogeneous and isotropic Mindlin plate was studied in Ref. [l], where the problem was reduced to a fourth-order equation for the plate deflection, which incorporates rotary inertia and transverse shear deformation terms. Below we aim to use the full system of the corresponding theory for cubic crystal plates to make an asymptotic analysis of such waves, for small values of the plate thickness and for small or large values of the elastic constants. In contrast to Ref. [l], here the constants include no artificial corrections. Crystal plates play an im~rtant role in the manufacture of integrated circuits [2]. In what follows, Greek and Latin subscripts take the values 1, 2 and 1, 2, 3, respectively, the convention of summation over repeated indices is understood, (- * -),i = a(- . ~)/&,, and a superposed dot indicates differentiation with respect to time. Consider a homogeneous cubic crystal material occupying the region R2 x [-ho/2, h,,f2], where h fj = const. The constitutive relations for this type of anisotropy are of the form [3]: tll = WI.1 + &,2 + %31 f22 = &.t
+ YU2.2 + AU3,3,
t33 = hut.
*f
tij = Pt”i,j
(1)
AU2.2 + YU3.3,
+
i=+j,
uj.i),
where rig= tji and Ui are the components of the stress tensor and displacement and y the elastic constants. The equations of motion are: tij.j
+f;:
=
vector, and h, p
(2)
Piii,
where& are the body forces and p is the (constant) density of the material. The internal energy per unit volume is given by: 8 = $tjjUi,p (3) In view of the shape of the body, displacement field is of the form: &I
we make
= ~32fa(Xyr
0,
the Mindlin-type
a3 = 213(x,,
assumption
f).
that
the
(4)
Replacing (4) in (1) and then (1) in (2) and (3), and using the averaging operators:
we find that the equations of motion for the bending plate can be written as: h2W,,,,
+
~1.22
h2KA + ~h12 ~(21t.t
+ (A +
clN2.,21-
+ ~2.11 +
+ ~2.2 + Av3)
+ 4
yv2.221= ~$3,
715
Au,
+ ~3,d + 6 = h2@,,
cl(v2
+ ~3.2)
+ Fz = h2@2,
(5)
716
C. CONSTANDA
and
where h* = hil12, F, = Blfn + 9,t3, and 4 = of area of the middle plane is: E = $G’= @+G:,,
+
4.2)
+
2h.1~2.2
+
M. E. PEREZ
$i,f3+ .&t,, and that the internal energy per unit
~(“1.2
+
u2.J21
+
P[(UI
+
+,I)*
+
(~2
+
~3.2)~1).
(6)
We assume that p> 0, y > 0 and IAl< y, which means that (6) is a positive quadratic form and (5) a hyperbolic system. We consider small amplitude harmonic waves propagating in the x,-direction, of wavelength A (wave number k), frequency w, phase velocity cP, and group velocity cg. Assuming that the body forces and moments, and the forces and moments acting on the faces, are negligible (that is, F; = 0), and seeking the solution of (5) in the form: u,(x, t) = ajei(kxI--wr),
Uj
=
const ,
we find that the aj must satisfy the system: (h*yk* + p - h2p02)a,
+ ipka3 = 0,
(h*pk* + p - h2pm2)a2 = 0, ipka, + (pm2 - pk2)a3 = 0.
This system has non-zero solutions for all the aj if and only if: h2pk2 + ,u - h2pm2 = 0.
h2y,uk4 - h2(y + ,u)pm*k* + (h2pco2 - &pm* = 0,
(7)
The discriminant of the first equation (7) is h4(y + ~FL)~p~co~ - 4h2yp(h2pw2
- ,u)pw* = h4(y - ,~)~p*w~ + 4h2y,u2pm2 > 0.
This means that k* is always real, therefore, all propagating waves are undamped. Since we prefer to conduct the discussion in terms of wavelength rather than frequency, rewrite (7) as: h2p2w4 - [p + h2(y + ,u)k*]po.t* + h2ypk4 = 0,
h*pm* -
we
h2pk2 = 0.
~1 -
When A is small compared to h (in other words, k is large compared to h-l), the plate can be regarded as an infinite body in all directions for the purpose of wave analysis, and we recover the 3-D results. In the isotropic case (y = h + 2~) these results can be found, for example, in Ref. [3]. From now on we assume that all equalities are asymptotic with respect to the appropriate small parameter. Since, as we have pointed out above, large values of k are uninteresting, we also assume that k is sufficiently small so that no secular terms occur to the order of magnitude considered. If h is small, then a wave of wave number k can propagate with any of the frequencies: w
I
=
w2 = hk2c,,
I9
m3=h-‘++hk2c2
,*
Hence, &)I = ($
(cg),=
+y
h)c,,
yhc, <(c&,
(&
= d + +hkc:,
(c,)R
=
(cp)2 = hkc,,
(42
=
2(cp)2,
hkc:
<
(cJ3r
where cI = (y/p)“* and c, = (~lp)“~. The first and third waves travel much faster than the second one, and are normally dispersive. The second wave is anomalously dispersive. As A+ 03 (or, equivalently, k+ 0): w,+
h-‘c,,
w,+O,
w3+ h-‘,
(cp),+T
(42-+
0,
If y is small, then: o, = wR= h-‘(1 + h2k2)“*ct,
w2 = hk*(l
+ h2k2)-“2c,.
(CpL-,
w.
Letters in Applied and Engineering Sciences
717
Therefore,
&)I = c, (l-i- &J (CJ, = c, (1+ &)-,,
(cp)z = c, (1 + j&)-+*7
< @Jr>
kg)2 = (l~h!&
M’
(42.
The first wave travels faster than the second one, and is normally dispersive. The second wave is anomalously dispersive. As A-, QJ: w, 4 h-‘c,,
%-+ 0,
($)I + ‘=Q* (cp)z+ 0.
If y is large, then: &=.,(k+&),
co2 =
kc,,
w2
=
co3= h -‘c,( 1 + h*k’)“*,
so that (c~),=ci(l+~), P kg)1 =c, ( l-- 2h2yk2 f < (CJl,
@,)2
=
co
ct,
(c&J3
-c,(l
-
-k&-J?
(Cb)3=C,(I+~)-‘n<(Cp)3.
The first wave is the fastest, followed by the third and second ones. The first and third waves have normal dispersion, the second wave is non-dispersive. As A-+ 00: w3+ h-‘c,,
%-*O,
e+-+m,
(c&+=7
(cp)z = co
(c,)e
Ob*
If p is small, then u1 = kc,,
cv2= kc,,
w3 = h-%,(1 + h*k*)‘“,
from which (cp), = %
(c&z = c, 7
(c&J3=c, (1 +&J
kg), = c/r
(c&z = G,
(cJ3=c,
(1 +&)-‘%cp)3.
The first two waves correspond to the longitudinal and transverse waves in the isotropic case, and are non-dispersive. The third wave is normally dispersive. When or.is large, the discussion is similar to that for y small, with the obvious modifications. Acknowledgement-
The work of M.E.P has been partially supported by the D.G.K.Y.T.
REFERENCES fl] R. D, MINDLIN, Influence of rotary inertia and shear on flexurai motions of isotropic elastic plates. J. Appf. Me&. l&31-38 (1951). [2] J. W. MAYER and S. S. LAU, Electronic paternal Science: For integrated Circuits in Si and GaAs. Macmillan, New York (1990). [3] M. E. GURTIN, T!ze Linear Theory uf Elusticity. ~a~d6uch der Physi&, Bd. Viu/.?, pp. l-Z%. Springer-Veriag, Berlin (1972). (Received 23 February 1593; accepted 20 April 1993)
Pergamon
LE’ITERS IN APPLIED AND ENGINEERING
WAVE PROPAGATION
SCIENCES
IN THIN CRYSTAL PLATES
C. CONSTANDA Department of Mathematj~,
University of Strathclyde, Glasgow, Gi 1XH U.K.
M. E. PBREZ Departamento de Matemgtica Aplicada, Universidad de Cantabria, Santander, Spain AbstrW-A Mindlin-to theory is const~~ted for cubic crystat plates, and the propagation of small amplitude waves in such plates is discussed in terms of the physical parameters of the material.
The propagation of flexural waves in a homogeneous and isotropic Mindlin plate was studied in Ref. [l], where the problem was reduced to a fourth-order equation for the plate deflection, which incorporates rotary inertia and transverse shear deformation terms. Below we aim to use the full system of the corresponding theory for cubic crystal plates to make an asymptotic analysis of such waves, for small values of the plate thickness and for small or large values of the elastic constants. In contrast to Ref. [l], here the constants include no artificial corrections. Crystal plates play an im~rtant role in the manufacture of integrated circuits [2]. In what follows, Greek and Latin subscripts take the values 1, 2 and 1, 2, 3, respectively, the convention of summation over repeated indices is understood, (- * -),i = a(- . ~)/&,, and a superposed dot indicates differentiation with respect to time. Consider a homogeneous cubic crystal material occupying the region R2 x [-ho/2, h,,f2], where h fj = const. The constitutive relations for this type of anisotropy are of the form [3]: tll = WI.1 + &,2 + %31 f22 = &.t
+ YU2.2 + AU3,3,
t33 = hut.
*f
tij = Pt”i,j
(1)
AU2.2 + YU3.3,
+
i=+j,
uj.i),
where rig= tji and Ui are the components of the stress tensor and displacement and y the elastic constants. The equations of motion are: tij.j
+f;:
=
vector, and h, p
(2)
Piii,
where& are the body forces and p is the (constant) density of the material. The internal energy per unit volume is given by: 8 = $tjjUi,p (3) In view of the shape of the body, displacement field is of the form: &I
we make
= ~32fa(Xyr
0,
the Mindlin-type
a3 = 213(x,,
assumption
f).
that
the
(4)
Replacing (4) in (1) and then (1) in (2) and (3), and using the averaging operators:
we find that the equations of motion for the bending plate can be written as: h2W,,,,
+
~1.22
h2KA + ~h12 ~(21t.t
+ (A +
clN2.,21-
+ ~2.11 +
+ ~2.2 + Av3)
+ 4
yv2.221= ~$3,
715
Au,
+ ~3,d + 6 = h2@,,
cl(v2
+ ~3.2)
+ Fz = h2@2,
(5)
716
C. CONSTANDA
and
where h* = hil12, F, = Blfn + 9,t3, and 4 = of area of the middle plane is: E = $G’= @+G:,,
+
4.2)
+
2h.1~2.2
+
M. E. PEREZ
$i,f3+ .&t,, and that the internal energy per unit
~(“1.2
+
u2.J21
+
P[(UI
+
+,I)*
+
(~2
+
~3.2)~1).
(6)
We assume that p> 0, y > 0 and IAl< y, which means that (6) is a positive quadratic form and (5) a hyperbolic system. We consider small amplitude harmonic waves propagating in the x,-direction, of wavelength A (wave number k), frequency w, phase velocity cP, and group velocity cg. Assuming that the body forces and moments, and the forces and moments acting on the faces, are negligible (that is, F; = 0), and seeking the solution of (5) in the form: u,(x, t) = ajei(kxI--wr),
Uj
=
const ,
we find that the aj must satisfy the system: (h*yk* + p - h2p02)a,
+ ipka3 = 0,
(h*pk* + p - h2pm2)a2 = 0, ipka, + (pm2 - pk2)a3 = 0.
This system has non-zero solutions for all the aj if and only if: h2pk2 + ,u - h2pm2 = 0.
h2y,uk4 - h2(y + ,u)pm*k* + (h2pco2 - &pm* = 0,
(7)
The discriminant of the first equation (7) is h4(y + ~FL)~p~co~ - 4h2yp(h2pw2
- ,u)pw* = h4(y - ,~)~p*w~ + 4h2y,u2pm2 > 0.
This means that k* is always real, therefore, all propagating waves are undamped. Since we prefer to conduct the discussion in terms of wavelength rather than frequency, rewrite (7) as: h2p2w4 - [p + h2(y + ,u)k*]po.t* + h2ypk4 = 0,
h*pm* -
we
h2pk2 = 0.
~1 -
When A is small compared to h (in other words, k is large compared to h-l), the plate can be regarded as an infinite body in all directions for the purpose of wave analysis, and we recover the 3-D results. In the isotropic case (y = h + 2~) these results can be found, for example, in Ref. [3]. From now on we assume that all equalities are asymptotic with respect to the appropriate small parameter. Since, as we have pointed out above, large values of k are uninteresting, we also assume that k is sufficiently small so that no secular terms occur to the order of magnitude considered. If h is small, then a wave of wave number k can propagate with any of the frequencies: w
I
=
w2 = hk2c,,
I9
m3=h-‘++hk2c2
,*
Hence, &)I = ($
(cg),=
+y
h)c,,
yhc, <(c&,
(&
= d + +hkc:,
(c,)R
=
(cp)2 = hkc,,
(42
=
2(cp)2,
hkc:
<
(cJ3r
where cI = (y/p)“* and c, = (~lp)“~. The first and third waves travel much faster than the second one, and are normally dispersive. The second wave is anomalously dispersive. As A+ 03 (or, equivalently, k+ 0): w,+
h-‘c,,
w,+O,
w3+ h-‘,
(cp),+T
(42-+
0,
If y is small, then: o, = wR= h-‘(1 + h2k2)“*ct,
w2 = hk*(l
+ h2k2)-“2c,.
(CpL-,
w.
Letters in Applied and Engineering Sciences
717
Therefore,
&)I = c, (l-i- &J (CJ, = c, (1+ &)-,,
(cp)z = c, (1 + j&)-+*7
< @Jr>
kg)2 = (l~h!&
M’
(42.
The first wave travels faster than the second one, and is normally dispersive. The second wave is anomalously dispersive. As A-, QJ: w, 4 h-‘c,,
%-+ 0,
($)I + ‘=Q* (cp)z+ 0.
If y is large, then: &=.,(k+&),
co2 =
kc,,
w2
=
co3= h -‘c,( 1 + h*k’)“*,
so that (c~),=ci(l+~), P kg)1 =c, ( l-- 2h2yk2 f < (CJl,
@,)2
=
co
ct,
(c&J3
-c,(l
-
-k&-J?
(Cb)3=C,(I+~)-‘n<(Cp)3.
The first wave is the fastest, followed by the third and second ones. The first and third waves have normal dispersion, the second wave is non-dispersive. As A-+ 00: w3+ h-‘c,,
%-*O,
e+-+m,
(c&+=7
(cp)z = co
(c,)e
Ob*
If p is small, then u1 = kc,,
cv2= kc,,
w3 = h-%,(1 + h*k*)‘“,
from which (cp), = %
(c&z = c, 7
(c&J3=c, (1 +&J
kg), = c/r
(c&z = G,
(cJ3=c,
(1 +&)-‘%cp)3.
The first two waves correspond to the longitudinal and transverse waves in the isotropic case, and are non-dispersive. The third wave is normally dispersive. When or.is large, the discussion is similar to that for y small, with the obvious modifications. Acknowledgement-
The work of M.E.P has been partially supported by the D.G.K.Y.T.
REFERENCES fl] R. D, MINDLIN, Influence of rotary inertia and shear on flexurai motions of isotropic elastic plates. J. Appf. Me&. l&31-38 (1951). [2] J. W. MAYER and S. S. LAU, Electronic paternal Science: For integrated Circuits in Si and GaAs. Macmillan, New York (1990). [3] M. E. GURTIN, T!ze Linear Theory uf Elusticity. ~a~d6uch der Physi&, Bd. Viu/.?, pp. l-Z%. Springer-Veriag, Berlin (1972). (Received 23 February 1593; accepted 20 April 1993)