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On equivoluminal vibration modes of a multipolar elastic plate
Journal of Sound and Vibration (1972) 20 (2), 169-175
ON EQUIVOLUMINAL MULTIPOLAR
VIBRATION ELASTIC
MODES
OF A
PLATE
R. H. KOEBKEP Engineering Mechanics Department and Ordnance Research Laboratory, Pennsylvania State University, Pennsylvania 16801, U.S.A. AND
Y. WE~TSMAN Engineering Science Department, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel (Received 12 July 1971)
A special vibration mode of a plate composed of an elastic material with an oriented micro-structure is investigated according to a multipolar elastic theory. An equivoluminal mode of a finite plate analogous to the Lame mode is sought. However, in the context of multipolar theory, this mode does not exist because the angle at which transverse waves reflect as themselves is not 45” as it is in classical theory. Equivoluminal vibration modes of an infinite plate with traction-free plate faces are developed.
1. INTRODUCTION
There exists a special, equivoluminal vibration mode of a traction free, square plate which is known as the Lame mode of vibration. This particular vibration mode was discussed recently by Mindlin [l] in connection with the development of a general plate vibration theory. In the
reference cited it is shown that the Lame mode vibration can be synthesized by a superposition of four progressive shear waves which successively are incident upon and reflect from the four sides of the plate. The four waves sweep across the plate in a 90”criss-cross pattern making 45” angles with the plate sides. This is a special case of wave reflection since in general, when the incidence angle is not 45”, there is a coupling of incident and reflected shear waves with a dilatation wave. In this paper, the analogous problem for a plate composed of a nonclassical elastic material is examined. The elastic theory employed in this study allows for the inclusion of an elastic oriented micro-structure which is embedded within a matrix of isotropic elastic material. Materials of this nature have been termed multipolar or Cosserat materials and appropriate theories have been published by several authors [2,3]. An interesting application of Cosserat theory was presented by Askar and Cakmak [5], in which the authors demonstrated that a structural lattice model consisting of flexible and extensible beams is properly described by multipolar elastic theory. One may speculate, therefore, that the vibrations of a honeycomb structural plate may be treated by multipolar theory. Reflection of progressive, harmonic waves from a plane surface of a multipolar continuum was treated in some detail by Suhubi and Eringen [3]. 7 Present address: Department Kenya.
of Mechanical Engineering, University of Nairobi, P.O. Box 30197, Nairobi,
169
R. H. KOEBKE AND Y. WEITSMAN
170
2. THE EQUATIONS OF THE THEORY
The formulation of Cosserat theory adopted for this paper is that of reference [2]. A displacement vector uI (with i = 1,2, or 3) describes the bulk displacement of the medium, also called macro displacement, while an antisymmetric tensor #tli, (with i and j = 1,2, or 3) describes a rigid, infinitesimal rotation of the embedded micro-structure. The displacement equations of motion are (A + P - B)%,l, + (P + P)%,il(al + U3 (#tl~1,14+ ~C~*l,i~) + 2u2 hwl,ii and the constitutive equations are
u
Tpq=
Aa,,
CP41 =
Bb4l.P
-
%@pr
1crwl1.l +
I-LPCWI
=
4.1
+
-
2P#t,,,,* = P%,
2P7hw1 + B(%,P
-
u~,4) = 3P’ d29Lf1~
(1)
pup,q,
UP*,
-
2hPd, a,,
s4d
+ 2a2
hrl,P
+
~3ohPcll.r
+
hP1.q.
(2)
In the tensor notation used in this work, repeated indices call for summation over the range 3, the comma denotes differentiation with respect to x1, and the superposed dot denotes time differentiation. The symbols rpq, ucpql,and ~plqrldenote three kinds of stresses : the symmetric, the relative antisymmetric, and the couple stress, respectively. The counterparts of the usual Lame moduli are X and p, while CL~,CL~, and CL~, and /3 are micro-material moduli. The total mass density is p; the micro-mass distribution is assumed to be such that the micro-moment of inertia per unit volume is isotropic and expressed by the quantity p’d2/3. 3. TRANSVERSE DISPLACEMENT
WAVES IN AN INFINITE MEDIUM
In multipolar theory there may exist wave types not found in the classical theory and some of these waves exhibit dispersion even in an infinite medium [2, 5, 61. For purposes of this work, we introduce only an equivoluminal wave. To this end consider a plane progressive wave in which a transverse macro-displacement is coupled with a micro-rotation in the plane of the macro-displacement. Let a Cartesian frame xi, x2, x3 be oriented so that the displacements due to the wave lie in the plane of xl and x3 and so that the wave normal is inclined at an angle 8’to the x,-axis. The components of displacement and rotation of this wave are u; = A’cos 0’exp [ik’(x, sin 8’- x3 cos 0’- vt)], U; = A’sin B’exp [ik’(x, sin 8’ - x3 cos 8’ - a)], B’ exp [ik’(x, sin 8’- x3 cos 8’- vt)], *;311=
(3) where the symbols A’ and B’ denote amplitudes, v is the phase velocity, and k’ is a wavenumber. In this equation and in the sequel primes will be used on symbols to distinguish various waves. We note that for equivoluminal motion u;,~ = 0 and for plane strain all gradients in the x2-direction vanish as well as u; = &2, = &r = 0. With this specialization, the equations of motion become (P +/3)&i - 28&i,,, = P%> (P + @U;.u (2~52
-
a~
-
~3~~;31,,11-
@;,I,
= pii;,
+ %%&I +
,@;.3
-
u;,,)
=
3p’d2rC;;31p
(4)
Substitution of the displacements, equations (3), into the equations of motion, equations (4), yields two distinct, homogeneous equations in the two amplitudes A’ and B’. These are k’Z[pv2 - (p + /I)] A’ + 2pk’(iB’) = 0, /3k’A’ + [+p’d2 k” v* - (2~2 - a, - as) k’2 - 2/l] (iB’) = 0.
(5)
MODES OF A MULTIPOLAR
171
ELASTIC PLATE
Let the following notations be defined: c; =
3(2a, - @I- CQ) #d2
c2 = &
’
B
p
’
& = !? P
and
2_
68
wR-p’dZ’
The c~, cB, and cT have the dimensions of velocity and WRhas the dimensions of frequency. For a non-trivial solution for A’and B’, the determinant of their coefficients must vanish. This determinant yields a dispersion relation :
1 J([
2
1 v2 =j
(ci+
c$)+$
f
(6)
;
The dispersion relation, equation (6), can be put into another, more convenient form by eliminating k’ from the equation through the use of k’= w/v, where w denotes the angular frequency. The dispersion relation so derived specifies v2 as a function of w2 : t?(w) =
1 {[(CZ4 c$>w2 - c: w;] i d[(ci + c;, w2 - c; wi]’ - 4c; C$W2 - w;) w2). 2(w2 - wi) (7)
The dispersion relations given by either equation (6) or equation (7) indicate two branches depending on the sign chosen with the square root. The so-called transverse acoustic (TA) phase velocity is obtained taking the minus sign while the so-called transverse rotational optic (TRO) phase velocity is obtained by taking the plus sign. Correspondingly, we shall use the symbols v = cTAand v = cTRoto distinguish the phase velocities of these two wave types. The lower cut-off frequency of the TRO wave is wR. The ratio of amplitudes is obtained equivalently either from the first or second of equation (5). The relation obtained from the first of equations (5) is simpler; it is iB’/A’ = -k’(u2 - c$/2($
- c$).
(8)
4. REFLECTION OF A TRANSVERSE WAVE INTO ITSELF ON THE FREE BOUNDARY OF A HALF SPACE A half space in which a Cartesian frame xi, x2, xj is oriented with the xi and x2 axes lying on the plane boundary is depicted in Figure 1. We shall investigate a special case of incidence and reflection of equivoluminal waves, where an incident wave of the type discussed in section 3 is reflected into a wave of the same type and is not coupled with dilatation waves. Let the incident wave be the wave expressed by equations (3). In addition consider a similar transverse wave which is reflected at an angle 8”, and the components of which are given by u; = A” cos 8”exp [ik”(x, sin 19”+ x3 cos 8” - vt)], u; = -A” sin 0”exp [ik”(x, sin 8” + x3 cos 0” - UC)], ut)]. &, , = -B ”exp [ik”(xl sin en+ X,COS~"-
(9)
On a traction free, plane boundary, such as the plane x3 = 0 indicated in Figure 1, the stress components r33 and p3c31, as well as the sum of stress components 73, f 7t3,, must vanish. This leads to the boundary stress conditions to be satisfied for x3 = 0: -0, U3#3 -
YG311,3 =o
and (P+/3~%,3+(tL-/9U3,1-2b%3~=0.
(10)
Utilizing a superposition of the motions associated with two assumed waves given by #equations (3) and equations (9), we have, from the first of equations (lo), -A’ sin B’cos 8’exp [ik’(x, sin 0’- vt)) - A” sin 0”cos 0”exp [ik”(x, sin 8” - ut)] = 0,
(11)
172
R. H. KOEBKE AND Y. WEITSMAN
and, from the second of equations (lo), -B’k’cosd’exp
[ik’(x, sin& - ut)] - B” k” cos6”exp [ik”(x, sine” - vt)] = 0.
(11)
These equations are satisfied only if we let k = k’ = k”,
AZ/~‘=-A’,
B=B’=-B”,
0 = 6’= 8”.
and
(12)
From the third of equations (10) and by utilizing the results above, we obtain pAk(cos2 l3- sin2 0) + /lAk - 2/l(iB) = 0, or iB/A = (k/2/3) (p cos 28 + /?).
(13)
>
XI
Figure 1. Incident and reflectedtransversewaveson a plane boundary.
Figure 2. Dispersionrelation: c: > ct > cc. Combining equation (8) and equation (13) yields a relation specifying the angle of incidence and reflection in terms of the frequency w. This relation is cos28 = 1 - U’(W)/C$.
(14)
As there are in general two possible values of v corresponding to particular frequency o, so there are in general two values of 8 at which the postulated special wave reflection prevails. A study [6] of the dispersion relation, equation (6) or equation (7), when account is taken of the restrictions on the range of values that the material moduli may have, shows that there may be three distinct cases. These are sketched in Figures 2, 3 and 4. From these figures one observes that the lower branch specifies u = c T only when w = 0. For w --f co, the lower branch approaches the smaller of the values c, or cB, whereas the upper branch approaches the larger of c, or CB.The value of c, may be smaller or larger than cr., but CBis always larger than CT.
MODESOF A MULTIPOLARELASTICPLATE
I
0
I
173
.w2
@RZ
Figure 3. Dispersion relation : ci > c.’> ct. 2
CYZ C Figure 4. Dispersion relation: ci > c$ > c,‘.
With reference to the dispersion curves and equation (14), one must conclude that 8 = 45” is possible only for w = 0. The squared wave velocity c&o can attain the value 2c$ in all cases except when ci > 2& and ct -C2c$, while & can attain the value 2& whenever the smaller of ci or ci is greater than 2~:. The case when either c&o or c+~ is equal to 2c$ at some finite frequency represents an upper limit for the special angle of incidence sought, which in that case is B = 90”. Moreover, it is clear that for any finite frequency there is at least one special angle of incidence less than 90” at which an incident equivoluminal wave reflects in itself and is not coupled with dilatation waves.
5. EQUIVOLUMINAL
VIBRATIONS OF A PLATE
An equivoluminal, plane strain vibration mode of an infinite plate is now easily synthesized by a superposition of four progressive waves of transverse motion. To this end, consider in addition to the motions specified by equations (3) and equations (9), another pair of similar waves specified by u’; = A cos &J exp [&(--x1 sin 8 + x3 cos 8 - et)], u’; = -A sin 8 exp [ik(-x, sin e
f x3 cos 8 - vt)],
#r3,] = Bexp [ik(-x, sin 8 + x3 cos e -
and
a)],
x3 cos 8 - ut)], ~4”= A sin 0 exp [ik(-xl sin 0 - x3 cos e - ut)], &;, , = -B exp [ik(-x, sin 0 - x3 cos e - or)]. u1iV= A cos Bexp [ik(-x, sin 8 -
12
(15)
174
R. H. KOEBKE
AND Y. WEITSMAN
A superposition, u1 = u; + u; + UTf u:“, etc., of the four progressive waves results in the following vibration mode : u, = Uk3sin ot cos kl x1 cos k3 x3,
uj = Uk, sin wt sin k, x1 sin k3 x3 (17) where U = 4A/k, k, = ksin9, and exp(-iwt)
k3=kcos8, w = vk
was replaced by sin&.
Figure 5. Superposition of four transverse waves on a plate.
An infinite plate of thickness 2h is depicted in Figure 5. The directions of the four progressive waves simultaneously impinging and reflecting on the plate faces are indicated by arrows. The non-vanishing stresses corresponding to the motion specified by equations (17) are 733 = -T,, = 2pk, k3 U sin cut sin k, x3 cos k3 x3, 731 = -ujl
=
p(kf - k:) Usinotcosk, xl sink3x3, k3Usinwtcosklxlcosk3xj,
/Qr3,,=(2a2-‘?-a3)
k, sink1 x1 plt311
= jQosk,
CL2tr21=
(&,
I*.2C231 = p,
xI p3[3119
-2,
_
a3)Y3c311~
_“,‘,
_
a3~~lC311~
(18)
The condition that the faces of the plate at x3 = +h and x3 = -h be traction free is satisfied by taking with n = 1, 3, 5, . . .. k, = k cos 0 = nh/2h, (19) now vanish The force traction component 733 and the couple traction component ~~~~~~ on the plate faces and on parallel planes at x3 = &h/n. The traction component TV,+ uc3,,
MODES OF A MULTIPOLAR ELASTIC PLATE
175
vanishes on all planes. With the wavenumber k, selected as in equation (19), we obtain, with the aid of equation (14), kl =ksin0=2
_
1
2h 1/(2c$/z?) - 1’
where, as mentioned before, v2 depends on the frequency and is restricted to values less than 2~:. There are no planes x1 = constant or x2 = constant which are completely traction free. Planes x2 = constant are in general subject to the couple tractions p2ti2, and p2ti3, which vanish only if the modulus a, vanishes. One way of explaining the presence of tractions on all planes xi = constant is by noting that the four progressive waves do not impinge on these planes at the special self-reflecting angle 19.Since this special self-reflecting angle is not 45” in the present theory, the conditions for vanishing tractions cannot be met simultaneously at the plate faces and sides. It is clear, therefore, that there are no modes of plate vibration in the present multipolar theory which are directly analogous to the Lame modes of the classical theory. The particular choice of the four component waves and the choice of an odd integer for the parameter n in equation (19) yields a symmetric equivoluminal vibration mode, meaning the motion is symmetric with respect to the middle plane of the plate. An antisymmetric vibration mode can be easily synthesized in a similar manner by appropriate phasing of the four waves and choosingn to be even instead of odd. The motion yielding an antisymmetric, equivoluminal mode of vibration is given by u, = -Uk, sin wt cos kl x, sin k3 x3, uj = Uk, sin wt sin k, xl cos k, x3, q&l1 = Ur$kf
- ‘$k:)sinutcosk,
x, cosk3x3.
(21)
5. ACKNOWLEDGMENT
This work was, in part, under the financial support of the Ordnance Research Laboratory of The Pennsylvania State University under contract with the Naval Ordnance Systems Command. REFERENCES 1. R. D. MINDLIN 1957 Monograph prepared by U.S. Army Signal Engineering Laboratories, Fort Monmouth, New Jersey. An introduction to the mathematical theory of vibration of elastic plates. 2. R. D. MINDLIN 1964 Archive for Rational Mechanics and Analysis 16, 51-78. Micro-structure in linear elasticity. 3. E. G. SUHUBIand A. C. ERINGEN 1964 International Journal of Engineering Science 2, 389-404. Nonlinear theory of micro elastic solids-II. 4. A. ASKAR and K. S. CAKMAK 1968 International Journal of Engineering Science 6, 583-589. A structural model of a micro-polar continuum. 5. V. R. PARFI~~and A. C. ERINGEN1969 Journalof the AcousticalSociety of America 45,1258-1272. Reflection of plane waves from the flat boundary of a micropolar elastic half space. 6. R. H. KOEBKE and Y. WEIT~MAN 1971 Journal of the Acoustical Society of America (in press). Surface wave propagation over an elastic Cosserat half space.
ON EQUIVOLUMINAL MULTIPOLAR
VIBRATION ELASTIC
MODES
OF A
PLATE
R. H. KOEBKEP Engineering Mechanics Department and Ordnance Research Laboratory, Pennsylvania State University, Pennsylvania 16801, U.S.A. AND
Y. WE~TSMAN Engineering Science Department, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel (Received 12 July 1971)
A special vibration mode of a plate composed of an elastic material with an oriented micro-structure is investigated according to a multipolar elastic theory. An equivoluminal mode of a finite plate analogous to the Lame mode is sought. However, in the context of multipolar theory, this mode does not exist because the angle at which transverse waves reflect as themselves is not 45” as it is in classical theory. Equivoluminal vibration modes of an infinite plate with traction-free plate faces are developed.
1. INTRODUCTION
There exists a special, equivoluminal vibration mode of a traction free, square plate which is known as the Lame mode of vibration. This particular vibration mode was discussed recently by Mindlin [l] in connection with the development of a general plate vibration theory. In the
reference cited it is shown that the Lame mode vibration can be synthesized by a superposition of four progressive shear waves which successively are incident upon and reflect from the four sides of the plate. The four waves sweep across the plate in a 90”criss-cross pattern making 45” angles with the plate sides. This is a special case of wave reflection since in general, when the incidence angle is not 45”, there is a coupling of incident and reflected shear waves with a dilatation wave. In this paper, the analogous problem for a plate composed of a nonclassical elastic material is examined. The elastic theory employed in this study allows for the inclusion of an elastic oriented micro-structure which is embedded within a matrix of isotropic elastic material. Materials of this nature have been termed multipolar or Cosserat materials and appropriate theories have been published by several authors [2,3]. An interesting application of Cosserat theory was presented by Askar and Cakmak [5], in which the authors demonstrated that a structural lattice model consisting of flexible and extensible beams is properly described by multipolar elastic theory. One may speculate, therefore, that the vibrations of a honeycomb structural plate may be treated by multipolar theory. Reflection of progressive, harmonic waves from a plane surface of a multipolar continuum was treated in some detail by Suhubi and Eringen [3]. 7 Present address: Department Kenya.
of Mechanical Engineering, University of Nairobi, P.O. Box 30197, Nairobi,
169
R. H. KOEBKE AND Y. WEITSMAN
170
2. THE EQUATIONS OF THE THEORY
The formulation of Cosserat theory adopted for this paper is that of reference [2]. A displacement vector uI (with i = 1,2, or 3) describes the bulk displacement of the medium, also called macro displacement, while an antisymmetric tensor #tli, (with i and j = 1,2, or 3) describes a rigid, infinitesimal rotation of the embedded micro-structure. The displacement equations of motion are (A + P - B)%,l, + (P + P)%,il(al + U3 (#tl~1,14+ ~C~*l,i~) + 2u2 hwl,ii and the constitutive equations are
u
Tpq=
Aa,,
CP41 =
Bb4l.P
-
%@pr
1crwl1.l +
I-LPCWI
=
4.1
+
-
2P#t,,,,* = P%,
2P7hw1 + B(%,P
-
u~,4) = 3P’ d29Lf1~
(1)
pup,q,
UP*,
-
2hPd, a,,
s4d
+ 2a2
hrl,P
+
~3ohPcll.r
+
hP1.q.
(2)
In the tensor notation used in this work, repeated indices call for summation over the range 3, the comma denotes differentiation with respect to x1, and the superposed dot denotes time differentiation. The symbols rpq, ucpql,and ~plqrldenote three kinds of stresses : the symmetric, the relative antisymmetric, and the couple stress, respectively. The counterparts of the usual Lame moduli are X and p, while CL~,CL~, and CL~, and /3 are micro-material moduli. The total mass density is p; the micro-mass distribution is assumed to be such that the micro-moment of inertia per unit volume is isotropic and expressed by the quantity p’d2/3. 3. TRANSVERSE DISPLACEMENT
WAVES IN AN INFINITE MEDIUM
In multipolar theory there may exist wave types not found in the classical theory and some of these waves exhibit dispersion even in an infinite medium [2, 5, 61. For purposes of this work, we introduce only an equivoluminal wave. To this end consider a plane progressive wave in which a transverse macro-displacement is coupled with a micro-rotation in the plane of the macro-displacement. Let a Cartesian frame xi, x2, x3 be oriented so that the displacements due to the wave lie in the plane of xl and x3 and so that the wave normal is inclined at an angle 8’to the x,-axis. The components of displacement and rotation of this wave are u; = A’cos 0’exp [ik’(x, sin 8’- x3 cos 0’- vt)], U; = A’sin B’exp [ik’(x, sin 8’ - x3 cos 8’ - a)], B’ exp [ik’(x, sin 8’- x3 cos 8’- vt)], *;311=
(3) where the symbols A’ and B’ denote amplitudes, v is the phase velocity, and k’ is a wavenumber. In this equation and in the sequel primes will be used on symbols to distinguish various waves. We note that for equivoluminal motion u;,~ = 0 and for plane strain all gradients in the x2-direction vanish as well as u; = &2, = &r = 0. With this specialization, the equations of motion become (P +/3)&i - 28&i,,, = P%> (P + @U;.u (2~52
-
a~
-
~3~~;31,,11-
@;,I,
= pii;,
+ %%&I +
,@;.3
-
u;,,)
=
3p’d2rC;;31p
(4)
Substitution of the displacements, equations (3), into the equations of motion, equations (4), yields two distinct, homogeneous equations in the two amplitudes A’ and B’. These are k’Z[pv2 - (p + /I)] A’ + 2pk’(iB’) = 0, /3k’A’ + [+p’d2 k” v* - (2~2 - a, - as) k’2 - 2/l] (iB’) = 0.
(5)
MODES OF A MULTIPOLAR
171
ELASTIC PLATE
Let the following notations be defined: c; =
3(2a, - @I- CQ) #d2
c2 = &
’
B
p
’
& = !? P
and
2_
68
wR-p’dZ’
The c~, cB, and cT have the dimensions of velocity and WRhas the dimensions of frequency. For a non-trivial solution for A’and B’, the determinant of their coefficients must vanish. This determinant yields a dispersion relation :
1 J([
2
1 v2 =j
(ci+
c$)+$
f
(6)
;
The dispersion relation, equation (6), can be put into another, more convenient form by eliminating k’ from the equation through the use of k’= w/v, where w denotes the angular frequency. The dispersion relation so derived specifies v2 as a function of w2 : t?(w) =
1 {[(CZ4 c$>w2 - c: w;] i d[(ci + c;, w2 - c; wi]’ - 4c; C$W2 - w;) w2). 2(w2 - wi) (7)
The dispersion relations given by either equation (6) or equation (7) indicate two branches depending on the sign chosen with the square root. The so-called transverse acoustic (TA) phase velocity is obtained taking the minus sign while the so-called transverse rotational optic (TRO) phase velocity is obtained by taking the plus sign. Correspondingly, we shall use the symbols v = cTAand v = cTRoto distinguish the phase velocities of these two wave types. The lower cut-off frequency of the TRO wave is wR. The ratio of amplitudes is obtained equivalently either from the first or second of equation (5). The relation obtained from the first of equations (5) is simpler; it is iB’/A’ = -k’(u2 - c$/2($
- c$).
(8)
4. REFLECTION OF A TRANSVERSE WAVE INTO ITSELF ON THE FREE BOUNDARY OF A HALF SPACE A half space in which a Cartesian frame xi, x2, xj is oriented with the xi and x2 axes lying on the plane boundary is depicted in Figure 1. We shall investigate a special case of incidence and reflection of equivoluminal waves, where an incident wave of the type discussed in section 3 is reflected into a wave of the same type and is not coupled with dilatation waves. Let the incident wave be the wave expressed by equations (3). In addition consider a similar transverse wave which is reflected at an angle 8”, and the components of which are given by u; = A” cos 8”exp [ik”(x, sin 19”+ x3 cos 8” - vt)], u; = -A” sin 0”exp [ik”(x, sin 8” + x3 cos 0” - UC)], ut)]. &, , = -B ”exp [ik”(xl sin en+ X,COS~"-
(9)
On a traction free, plane boundary, such as the plane x3 = 0 indicated in Figure 1, the stress components r33 and p3c31, as well as the sum of stress components 73, f 7t3,, must vanish. This leads to the boundary stress conditions to be satisfied for x3 = 0: -0, U3#3 -
YG311,3 =o
and (P+/3~%,3+(tL-/9U3,1-2b%3~=0.
(10)
Utilizing a superposition of the motions associated with two assumed waves given by #equations (3) and equations (9), we have, from the first of equations (lo), -A’ sin B’cos 8’exp [ik’(x, sin 0’- vt)) - A” sin 0”cos 0”exp [ik”(x, sin 8” - ut)] = 0,
(11)
172
R. H. KOEBKE AND Y. WEITSMAN
and, from the second of equations (lo), -B’k’cosd’exp
[ik’(x, sin& - ut)] - B” k” cos6”exp [ik”(x, sine” - vt)] = 0.
(11)
These equations are satisfied only if we let k = k’ = k”,
AZ/~‘=-A’,
B=B’=-B”,
0 = 6’= 8”.
and
(12)
From the third of equations (10) and by utilizing the results above, we obtain pAk(cos2 l3- sin2 0) + /lAk - 2/l(iB) = 0, or iB/A = (k/2/3) (p cos 28 + /?).
(13)
>
XI
Figure 1. Incident and reflectedtransversewaveson a plane boundary.
Figure 2. Dispersionrelation: c: > ct > cc. Combining equation (8) and equation (13) yields a relation specifying the angle of incidence and reflection in terms of the frequency w. This relation is cos28 = 1 - U’(W)/C$.
(14)
As there are in general two possible values of v corresponding to particular frequency o, so there are in general two values of 8 at which the postulated special wave reflection prevails. A study [6] of the dispersion relation, equation (6) or equation (7), when account is taken of the restrictions on the range of values that the material moduli may have, shows that there may be three distinct cases. These are sketched in Figures 2, 3 and 4. From these figures one observes that the lower branch specifies u = c T only when w = 0. For w --f co, the lower branch approaches the smaller of the values c, or cB, whereas the upper branch approaches the larger of c, or CB.The value of c, may be smaller or larger than cr., but CBis always larger than CT.
MODESOF A MULTIPOLARELASTICPLATE
I
0
I
173
.w2
@RZ
Figure 3. Dispersion relation : ci > c.’> ct. 2
CYZ C Figure 4. Dispersion relation: ci > c$ > c,‘.
With reference to the dispersion curves and equation (14), one must conclude that 8 = 45” is possible only for w = 0. The squared wave velocity c&o can attain the value 2c$ in all cases except when ci > 2& and ct -C2c$, while & can attain the value 2& whenever the smaller of ci or ci is greater than 2~:. The case when either c&o or c+~ is equal to 2c$ at some finite frequency represents an upper limit for the special angle of incidence sought, which in that case is B = 90”. Moreover, it is clear that for any finite frequency there is at least one special angle of incidence less than 90” at which an incident equivoluminal wave reflects in itself and is not coupled with dilatation waves.
5. EQUIVOLUMINAL
VIBRATIONS OF A PLATE
An equivoluminal, plane strain vibration mode of an infinite plate is now easily synthesized by a superposition of four progressive waves of transverse motion. To this end, consider in addition to the motions specified by equations (3) and equations (9), another pair of similar waves specified by u’; = A cos &J exp [&(--x1 sin 8 + x3 cos 8 - et)], u’; = -A sin 8 exp [ik(-x, sin e
f x3 cos 8 - vt)],
#r3,] = Bexp [ik(-x, sin 8 + x3 cos e -
and
a)],
x3 cos 8 - ut)], ~4”= A sin 0 exp [ik(-xl sin 0 - x3 cos e - ut)], &;, , = -B exp [ik(-x, sin 0 - x3 cos e - or)]. u1iV= A cos Bexp [ik(-x, sin 8 -
12
(15)
174
R. H. KOEBKE
AND Y. WEITSMAN
A superposition, u1 = u; + u; + UTf u:“, etc., of the four progressive waves results in the following vibration mode : u, = Uk3sin ot cos kl x1 cos k3 x3,
uj = Uk, sin wt sin k, x1 sin k3 x3 (17) where U = 4A/k, k, = ksin9, and exp(-iwt)
k3=kcos8, w = vk
was replaced by sin&.
Figure 5. Superposition of four transverse waves on a plate.
An infinite plate of thickness 2h is depicted in Figure 5. The directions of the four progressive waves simultaneously impinging and reflecting on the plate faces are indicated by arrows. The non-vanishing stresses corresponding to the motion specified by equations (17) are 733 = -T,, = 2pk, k3 U sin cut sin k, x3 cos k3 x3, 731 = -ujl
=
p(kf - k:) Usinotcosk, xl sink3x3, k3Usinwtcosklxlcosk3xj,
/Qr3,,=(2a2-‘?-a3)
k, sink1 x1 plt311
= jQosk,
CL2tr21=
(&,
I*.2C231 = p,
xI p3[3119
-2,
_
a3)Y3c311~
_“,‘,
_
a3~~lC311~
(18)
The condition that the faces of the plate at x3 = +h and x3 = -h be traction free is satisfied by taking with n = 1, 3, 5, . . .. k, = k cos 0 = nh/2h, (19) now vanish The force traction component 733 and the couple traction component ~~~~~~ on the plate faces and on parallel planes at x3 = &h/n. The traction component TV,+ uc3,,
MODES OF A MULTIPOLAR ELASTIC PLATE
175
vanishes on all planes. With the wavenumber k, selected as in equation (19), we obtain, with the aid of equation (14), kl =ksin0=2
_
1
2h 1/(2c$/z?) - 1’
where, as mentioned before, v2 depends on the frequency and is restricted to values less than 2~:. There are no planes x1 = constant or x2 = constant which are completely traction free. Planes x2 = constant are in general subject to the couple tractions p2ti2, and p2ti3, which vanish only if the modulus a, vanishes. One way of explaining the presence of tractions on all planes xi = constant is by noting that the four progressive waves do not impinge on these planes at the special self-reflecting angle 19.Since this special self-reflecting angle is not 45” in the present theory, the conditions for vanishing tractions cannot be met simultaneously at the plate faces and sides. It is clear, therefore, that there are no modes of plate vibration in the present multipolar theory which are directly analogous to the Lame modes of the classical theory. The particular choice of the four component waves and the choice of an odd integer for the parameter n in equation (19) yields a symmetric equivoluminal vibration mode, meaning the motion is symmetric with respect to the middle plane of the plate. An antisymmetric vibration mode can be easily synthesized in a similar manner by appropriate phasing of the four waves and choosingn to be even instead of odd. The motion yielding an antisymmetric, equivoluminal mode of vibration is given by u, = -Uk, sin wt cos kl x, sin k3 x3, uj = Uk, sin wt sin k, xl cos k, x3, q&l1 = Ur$kf
- ‘$k:)sinutcosk,
x, cosk3x3.
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5. ACKNOWLEDGMENT
This work was, in part, under the financial support of the Ordnance Research Laboratory of The Pennsylvania State University under contract with the Naval Ordnance Systems Command. REFERENCES 1. R. D. MINDLIN 1957 Monograph prepared by U.S. Army Signal Engineering Laboratories, Fort Monmouth, New Jersey. An introduction to the mathematical theory of vibration of elastic plates. 2. R. D. MINDLIN 1964 Archive for Rational Mechanics and Analysis 16, 51-78. Micro-structure in linear elasticity. 3. E. G. SUHUBIand A. C. ERINGEN 1964 International Journal of Engineering Science 2, 389-404. Nonlinear theory of micro elastic solids-II. 4. A. ASKAR and K. S. CAKMAK 1968 International Journal of Engineering Science 6, 583-589. A structural model of a micro-polar continuum. 5. V. R. PARFI~~and A. C. ERINGEN1969 Journalof the AcousticalSociety of America 45,1258-1272. Reflection of plane waves from the flat boundary of a micropolar elastic half space. 6. R. H. KOEBKE and Y. WEIT~MAN 1971 Journal of the Acoustical Society of America (in press). Surface wave propagation over an elastic Cosserat half space.