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A problem in micropolar elasticity
Inf. J. Engng Sci.,
1973, Vol. I I, pp. 215-234.
A PROBLEM
Pelgamon
Press.
Printed in Great Britain
IN MICROPOLAR
ELASTICITY
S. SRINIVAW Department of Aeronautical Engineering, Indiin Institute of Science, Bangalore 12, India Abstract- In this paper a three-dimensional analysis for statics and dynamics of a class of simply supported rectangular plates made up of micropolar elastic material is presented. The solution is in the form of series, in which each term is explicitly determined. For free vibrations, the frequencies are obtained by the solution of a closed form characteristic equation.
INTRODUCTION
differential equations of micropolar elasticity have been developed by Eringen[ 11. In this paper analysis of a class of simply supported rectangular plates of micropolar elastic material is presented, making use of these equations. The solution is in the form of hyperbolic-trigonometric series, in which each term is explicitly determined. The series are so chosen that the edge boundary conditions are automatically satisfied.
THE GOVERNING
SOLUTION
The governing differential equations for simple harmonic vibrations (frequency O) of a micropolar elastic medium, in the absence of body forces and couples, in rectangular Cartesian coordinares are given by [l] (using usual tensor notation),
The stresses and couple stresses are given in terms of displacements and microrotations by
It is noted here that if K = 0 the displacements and microrotations get decoupled (vi& equation (1)) and also the stresses depend only on displacements. Now, the edge boundary conditions for a simply supported rectangular plate (uide Fig. 1) can be specified as
(4) tAt present NRC-postdoctoral Virginia, USA.
research associate at NASA-Langley 215
Research Center, Hampton,
S. SRINIVAS
Fig. 1. Coordinate system.
The boundary conditions (4) are automatically satisfied by choosing
cb,=
5 2 cos m7rXcos nlrY 4(Z). rn==l
w-1
u llj (6)
Substi~tion of equations (5) and (6) in the governing differential equations (1) yields for each (m, n) combination, @-(~+p)iW-(~+fi)A!fN
-KL (x+j&W -kN -(i;+ji)MN$r(i;+fi)N* (i+,ii)LN ipL 0 iiM V -(i+/%)LM -(i+ji)LN ++(i+ji)L2 r7N --GM 0 W 0 --ix. kN x-(i%++)jMIZ-(Ei+@MN-((a+p)LM f 0 -ZM -(~+~~Nx-(~+~)N*-(~+~)LN q kL 0 --EN iiM (ii+@)LM (&+p)LN x+(&+j!i)L* 4 i
= (0)
(7) where, JI= (Ir.+ii)(L*-g2)+W, x=~(L2-g4)-2r7+Q*~ ~*=pwzhz/8 and I is a reference elastic module, used in nondimensionalisation. In equation (7) -3 denote non~ension~~ quantities. For the non-trivial solution of equation (7) the determinant of the (6 X 6) coefficient matrix must be zero. This condition yields the following twelve roots for L: It r, , + r, and two double roots rt.r, and k r,, where,
217
A problem in micropolar elasticity
and r3,r4
(+
8_{$-i~+ =
(n*j--2ii)(ji+z)
[
+ii’)
zT(ji+z)
signfor r, and - sign for r4). SIMPLE
HARMONIC
VIBRATIONS
For simple harmonic vibrations, the twelve roots of L are f rl, + r, and two double roots +r3, strr,. Now, the eigenvectors corresponding to these roots are found using equation (7). The method to be followed is described in references 121and [3]. Thus, (V, V, W, 6, q, 5)
=F
{O,O,O, M, N, -rl}Ae"Z+{O,
+{M,
N, r2,0,0,0}BenZ+{M,
O,O, M, N, rt)A’e-f*Z
N,-r,O,O,O)B’e-“Z
-K(g2-rf)M/Jli,O,-riM,-rriN,g2)DtertZ + l- r&f, - r&, g2,- ii (g” - r?) Nh. ii (g2 - rf) M/xi, 0) C;e-‘iz + (ii (g” - rt) N&t, - ii (g’-
rj) M/Jr,,0, riM, riN, g*} DWiz
,
(9)
where&= (j.i++)(rf-g2) +fi2,xi=~(rf-g2) -2E+E’jT In equation (9), A, A’, B, B’, C,, C& C,, Ci, D3, D& D4and 0: are the twelve arbitrary constants. It is interesting to note that for roots +r, and +-r, the displacements and microrotations are uncoupled. These roots actually correspond to modes in which rotations and curl of microrotation vector 4 (& +&, + &J~)vanish and the differential equations ( 1a) and ( 1b) get decoupled. Substitu~on of equation (9) in equations (2) and (3) yields the following expressions for stresses and couple stresses. uxz =E4 i
i
[{h(ri-g2)
- (2@+E)M2}(BenZ+B’e-nz)
WI=1 n=i
+&
[ (- (2ji + I() MzrJ (&P~ - Cie+‘@) + (- (3% + 17)MNlc(g” - rf) /+d
x ( D&i2 + Die-‘@) ]] sin m?rX sin nrrY
( 104
218
S. SRINIVAS
$ii(
(J.?m
m=1?a=1 +is4
ri-g’)
-
(2ii+K)N2)(Ben”4B’e-~z)
[(-(2~+K)N2r~}(C~riZ-C~e~P~z)+{(2~+~)MNii(g2-r~)/Jli)
X (D&fz+Dfe-r~z)]] sin m?zXsin n7rY
i [{if rz--g2)f cr,, 45 rn=ln=1
(2CT++)r,2}(Benz+B’e-~)
[ { (2jk + k)g*tj} (C@iz - CIe+@)]] sin m?rX sin nnY
+ J4
+*z4
(lob)
*
Cl&)
[{(2~+rT)MNr,}(Cie~~-C~e-~Z)+~IC[~(N2-M2)tg2-r~)l~t
-a!f2(g2-
rf) /& - g2])
(Z&Pz + l&e-‘@)]] cos mn%y cos nlrY
wa
crux=8j,j, [(-Krl}(Aer’Z-A’e-r~z)+~(2~-tK)MN}(Be”Z+B’e~r~) + I: f{(2~+Ic)~~r~)(Clef*z-CC;e-~~z) f=3,4
+~~[~(N2-M2)(g2-rf)IJlt+~~2(g2-rl)/~~+g2]}
]]
X (DieriZ+ D'e-QZ)
Q=GI”tl + &
[wK4
,
cos m7rX cos naY
(W
er~z+A’e-~~Z)+{(2ji+E)Mr2}(Be”Z-l?’e-r~)
EIMI@fg2+rf) +r7{g2+ri’lg2-r:)/xi}lf(C,e’rz+C;e-’iZ)
+ {Nr,Z [fi(g”-rF)/+i - I]}(Dier@-D~e-‘~z)]] cos mnXsin n7rY =l’=i,=i @z.E
(lof)
[{-~~)(Aer1Z+A’e-“Z)+((2~+K)MT9}(Be~-B’e-nz)
+ Z E(M[p(gZ+rf) +ii{r~-~~g”-t~)/xr}]}(C,er~+C~e-+~) i=3.4
-t{NriKI1+(p+R)(gZ
- rf)f&l) (&erg--Die-V)]l
ax m~Xsin MY
(1%)
A problem in micropolar elasticity
219
erlZ+A’evrtZ) + { (2F+ E)Nr,} (Beti - We--)
+,z4 [{N[Ci(g2+rf)+~{g2+K(g2--rl)lxf}l}(C~’”+C;e-’iZ) +{MrfrT[l-p((g2-
~zll
rf)/Jlr]} (Dg?fz-D;e-rfz)]]
=+inil [WNAerlZ+A’e-r*Z) + {(2p+ i = +fg4 Ew[iw+r:)+~~~:
- i?(g” -
+{-MrfK[l+(ji+K)(g2-rf)/~j]}(Djer*z
it1[(5 (g”-
wrz = Il.8 i,
ri) +
+
[G(g2-rf)
(Cp’” + C;e-‘fz) D;e-r*z)]]
sinm?rX cos n7rY(1 Oi)
(j+$A42}(Aer’Z+A’e-r*Z)
+
(CjerfZ+CIe-r*Z)
x [{ (~+j+WVZ(g2-rrf)/xf}
(Cfe'*z+Cle-rfz)
(D&*z-DIe-r@)]]
htYmil ii1[{fi(g2- r4) -
cos m7rXcos n7rY
*
+
x
i
(lib)
(jij+P)rf}(Aer’Z+A’e-r~Z)
+ ,z, [ { (6 + 7) g3rj} (&erfz - Dfe-r*z)]] cos m?rX cos n7rY
5 mm= hi? rn==l
(lla)
(@+.j;)N2}(Aer1z+A’e-r1Z)
*=3.4
+ {- (B+y)N2rj}
mzz=
rf)/xf}]}
(6 + 7) M2rj} (Dter*z - D;esrjz)]] cos m7rXcos n7rY
muu= w ji,
(1Oh)
k)Nr,} (Be”Z-B’e-M)
-
+*z4 [{- (~+~)MN~(g2-rf)l~f) + {-
sin m7rXcos n7rY
(llc)
[{-MN(~+~)}(Ae”Z+A’e-r’Z)
n=1
[{K(g2-r~)(N2~-M~)Ixf)(C,e’tz+Cle-’fZ)
f=3.4
+
{jtf~rf(jj+j?)}(D&r@-D~e-r*z)]]
sin mlrXsin nlrY
(lid)
S. SRINIVAS
220
i [{--IMN(BS_y))(Aer’Z3_A’e-“Z) m u.z = #wg ??a=1n=1
i [(Mr,(~+~))(Aer”B--R’e-r~Z) mxz = Wg??I=1 ?l=l +
2
f(-~~~~(g2-r~)Nf~,)(C~r*Z-C~e-r~Z)
1=3,4
(lli) Now, if the lateral sufface z = 0 is loaded by h~o~c
surface stresses of ~p~tudes and surface couples of amplitudes pTt, pyr pzt and the iateral surface z = h is loaded by qz*, qyb,qzb,P,,, pyb,pa&,the lateral surface boundary conditions are, qzt, qut,
qzt
221
A problemin micropolarelasticity
The loadings are expressed in double Fourier series, such that substitution of equations (10) and (11) in (12) and equating term by term one obtains, for each (m, n) combination, an equation,
{Q,, QW,Qy, &,, pyl’ P@,Qsb, Qvb,Q,, Pq, PW, Pz+,1, (13) where Q’s and P’s are Fourier load coefficients. F is a (6 X 12) matrix, with coefficients given by (the Tirst subscript denotes the row number and second subscript the column number) FIPI(Z), FE3(Z) = - GVe’rlZ; FI.(Z),F,,,(Z) F I..~(Z)~ Fuw FwdZ-It
(2) = M(ji(g2+rt)
=&(2ji+r7)Mr&“”
+K[rf-K(gZ--rf)/~r])e”~z
F~mtdZ) = +.Nr,lc(I + (Fi+K)(g2--rif)IJll)e’rfZ
)c;,,(Z), F,,(Z)
= iiMe’+‘Z;F,,(Z),
F2A(Z) =-t (26+ K)NrpefM
F~.w,(Z), F~,,ww(Z) = N(LL(g2+r12)+K[T:-_(g2-r:)/Xr])e’r~Z F~,w(Z)~ &W)(Z) F,,(Z),
= TMrS(l+
FSd(Z) = (h($-g2)
(ET+2) (g2-rf)f+~)e”fz + (2$+ ic)rg)ezW
Fww(Z), Fuw,J(Z) = + (2b+ W%errfz F;,I (Z), F.M(Z> =+Mr,(P+y)eTrRz
FeBl(Z), FBs(Z) = (ii(g2--rf) F wwfZ)r
Ft3m12~
- (~+y)r:)efnZ
(Z) = + (p+ r)gV&‘iz
(14)
(All Ff &‘snot defined by equation ( 14) are zero). In the above equations wherever 2 or i sign occurs, the top sign corresponds to the first quantity on the left hand side and the bottom sign to the second quantity. For example, from the last equation FB,,&Z)
UESVd.11No.2-B
= + (~+~)g%,e*~P; FBSec12,(Z) = - (@+y)g2rie-‘iz.
222
S. SRINIVAS
Further, in the above equations
Solution of equation (13) for given value of excitation frequency parameter a*, yields the constants. Now, by summation of series (5), (6), (10) and (1 l), to desired accuracy, the displacements and stresses can be obtained. FREE
VIBRATIONS
Consider the case when the lateral surfaces are free of surface stresses and couples. Now, the right hand side of equation (13) becomes zero. For non-trivial solution the determinant of the coefficient matrix must be zero:
IF (1)(0)i= * 0
F
(1%
This is the characteristic equation, defining the natural frequency parameter R2. For any given (m, n) this transcendental equation yields infinite number of natural frequencies each corresponding to a different thickness mode. It is also observed that this equation can yield velocities of wave propagation in invite plates. FLEXURE
If the plate is static a2 = 0 and the twelve roots of L are rfcx1, double roots -+s, and triple roots + g, where
(16) The eigenvectors corresponding to these roots are &JO, 0,M, N, sl)A’e-81z
(V, V, W,,!j,q, 4) = {O,O,O, M,N,-s,)AesyZ+
+ (s2M, s2N, 8, - 2 (9 - 522) NIx2, E(8 - $1 MIx~,O)B~~~ +{~k2-.G)Nl+2,-~k2-
+
(-
s"z)MIJI,,0,--2M,-s2N,g2}Ce8~
s2M, - s,N, g”, - 17(8” - sf) N/x2, r7(g” - ~22) M/x2, 0) B’e-8a
+ 42 (s” - 4) Nail, - r7(g” - 4) Ml~~, 0, s2M, s2N, g2fC’e-S4Z
+
-MN
l,O,O,~,~,~
+ {O,O, l,O,
-NZ
-N
O,O}D,t@+ {O,O, l,O, 0, O}DjeTBZ
A problem in micropolar elasticity
223
NM + o,o,o,-- g, -g ,o 2(2ji+K+h)&F N-M + o,o,o,- g, - g
-M -N + -, g ’ g
2(2$+$_++)B’e-*z
l,O,O,O (2ji+rC+U)WZe-sz,
(17)
where ~?8= (MD,+
ND,-ggDs)/(61;i+3i+2i)
9’ = (MD:+
N~~+gD~)/{6~+3~+~);
1,5s=(j.i+i?) (s”z-g”) and xz = +(s$-8)
and
-2;.
In equation (17), A, A’, B, B’, C, C’, D1, &, Dz, D;, D3 and 0; are the twelve arbitrary constants. Substitution of equation (17) in equations (2) and (3) yields the following expressions for stresses and couple stresses.
+ {-M}
( D,ePz+ D;ewgZ) (-- 2k - MZ (2j.i+ i -t %)Z/g} &%Yz
+ {- 2i + M2 (2p 4 ii + 2?1)Z/g} $t?‘e-nz]sin mrX sin ntrY
a,,=8(2~+~)
i i nwi n=l
@a)
[{-N2s2}(BeP"Z-B'e-dZ)
+ (- N ) (D+F + D;evnz) + {- 2i - NZ (2@ + ii + 5) Z/g} &Ye + {- 2i + N2 (2ji + r7+ %) Z/g} We-nz] sin mrX sin nlrY
(18b)
S. SRINIVAS
224
a,,
=
8
[{ji,~,)(Ae~~~-_4'e--~~~)
i
i
rn=l
n=1
-I- (k[ji(N2--
+{(2~++)MNs2)(Be""Z--Be-S"Z)
M2)(g2-~~)~JIZ-M2(g2-s~)/JIZ-gL]}(Ce8~4C’e--saZ)
-I-{ (2p + ii) N/2} (DIeBz+ D;e-gZ) + { (2@ + r?)M/2} ( D,eSz+ D6evffZ) 1-{MN(2~+rc)(2~+Ic+2j;)/g)~z~z + (- MN (2ji + k) (26 + ii + 2i) /g)5FZe-Bz] cos rmX cm nrrY
--A’eaSIZ) f { (2p+ii)MNs,}
(18t
(BeS”Z-B’e-S”Z)
~(~[p(N2-M2)(g2-~~)IJIZ$.I?N2(g2-~~)lJIZ+~]}(CeszZ$.C’e-szZ) + ((2p + ii) N/2} (D,esz+ Die-&) + { (2jiL ~)~/2~(~~~z+~~e-gz) + {MN(2~+~)(2jZ+~+2ji)/g}~ZeBz +- (-- MN (2,~ + K) (2ji + ii + 2i) jg} B’Ze-Bz] cm m?zX cos my
+{j&+!$}
=8 Ql?.?z
i m=1
(D,eBz - Dle-gz) t {~}(D2@z-Die-az)
5 [{-iTN}(Ae81z+A'e-8~z)
n=1
+ {M[p(g-t-Sg)
+t7[s2,-- Z (g” - sjq)/xJ ]} (Se*+- B’e-S*Z)
~{Ns,lc[l+(iu+IZ)(g2-~22)/$]}(Ce~-C’e-~)
(18i
A problem in micropolarelasticity
+{Mji}(D,~Z+D~e-Dz) + +(2$+2+2X)
(2p+K)gZl
I$ [(p+K)(2fi+K+*ii) I
S?eBz+ + t
m=1
+
(1%)
2 [{-KM}(Ae81Z+A’e-81Z)
n=1
(Die”” - D~e-az~) + { pg
+{N(fi+K)}(D3@Z+D~e-BZ) + (2p+i+2Q
+
[
+“+}(D2e”” -
:[2f?(2p+a+T)
(2/z+K)gZ]}swZ+
I
;
Die-&) +p(2p+ii+2X)
[2l?(2/rL+r7+Q
+ b (2p + E + 2x) - (2,G+ K + 2x) (2p + K) gZ]
m=t
S9’e-Bz cosm?~XsinmY I
I
I-y I
UZY= 8 i
-*K(*p+-t+X)
[(p+K)(2p+r7+2X)
-2k(2p+K+ii)-(2fi+K+fi)(2fi+E)gZl
~lI.8 = 8 i
225
I
1
LB’e-uz sin m?rX cos nmY
(18h)
5 [{KM}(Ae81Z+A’e-81Z) n=1
(D,esZ
+
- Die-*)
+ { N,k} ( D,egz + D;eeaz) +
-2~‘(2p+ii+h)-(2F.+~+22X)(2p+~)gZ)
m,=
M’
2i
m=1Rl
{ii(g2-s:)
I
Se-Bz
+ (~+~)M2}(Ae81Z+A’e-8~Z)
+ {- (~~++MMNK(g2-~S22)/~2}(Be8~+B’e-8~)
+ {- (a + 7) M2s2} (Ce*” - C’e-‘2z)
1
sinm?~XcosnrY
(18i)
S. SRINIVAS
226
+
(P+% i‘
M2N 2g
-i- -(@+j+)(ZjIi+tZ-t-X)?
2MN 1
i
mu, = Mil
($i?egz-S?‘ed8z)
I
cosmlrXcosn?rY
(19
s:) + (~t-~)N2)(Ae8~z+A’e-s~z)
jl [iw-
+((P+;ii)MNt7(gZ--s2,)f~,}(Be~+B’e-~) i- {- (6 + 7) N2s,) ( Cesaz- C’e-8”) N3 (Dle@z-D~e-gz) C -f- tP+F@ {-
I
I
-t- (is+?)
%z = h8 i
i
(2ji-++a)
[iii(gz-sy)
2MN g
I
I-(P-t-$
F}
( G?esz- @‘e-BZ)
I
( DzegZ- Die-uZ)
cos m7zX cos nnY
(191
- (~+~)f~)(Aes~Z+A’e-SIZ)
rn==l ?I=1
C
{
(P+$@sz}(CeszZ-C’e-8”Z)
-I- (p+r)F 1
I
4 (-- (P+r)
~](Z31egz-D~e-gz)
(D&z - D;eegZ) cos mrX cos nnY I
(191
A problem in micropolar elasticity
+
‘12x2= W
+
@+9)}(D,F-D;e-Qz)
I
y
+If
sinm?rXsinn7rY
(2~+K+X)(N27-M2~)}(%9z-.We~~)]
i
227
i
(19e)
[{Msl(~+~)}(Ae8~Z-A’e‘8~Z)
m=1 n=1
{-@s2(g2 - @N/x2} (Be”-B’epSg)
+ {-M(~~2,+3/~)}(Ce~~+C’e-~~)
+
{MN(~+~)/2~tD~~Z+Dle-aZ)
+ {- M2 (6 + 7) /2} ( D2eBz+ Ddeeaz) + {- 2Np (2p + Z + x) } ( SWz + StZI’eesz)sin m?rX cos MY 1
(19f)
mzx= h8 i i [{Msl(~+~)}(Ae8’Z-A’e-B~z) rn=ln=1 + {-- W2 (L? - 4 N/x2) (Be sG- B’e-*G) + {- M (78: + pg”) } ( CeSfi+ C’ewgti) + {MN (p + 9) /2}( D,esz+ Dle-gz) + {- M2 (p + 7) /2}(D,egZ + Dde-az) + {- 2N7 (2fi + IT+ i) } (,@ipeBz + SJ’e-Qz) sin m?rX cos nrY J
Wg)
i i [ {Ns,(~+~)}(AeslZ-~Ae-SIZ) my2= hiifm=1 ?I=1 + { /317s, (g2- sl) M/x2}(Besti - B’emaa)+ {- N (PsZ,+ yg”>} ( CesrZ+ C’e-aa) + { (fi + 7) N2/2}( D,eOz+ D;e-aZ) + {- (p + T) MN/2} (D2eBz-I- &e-OZ) +{2M~(2p+K+X)}(B9Z+#e-BZ) m,,=
1
cosm?rXsinnrY
(19h)
IS’ i 2 [ {Nsl(~+~)}(AeS1Z--A’e-81Z) m=1n=1 + {W2 (s’ - ~2,)M/x2) (Be sS-B’e-sfi)
+{-N(~~~+~g2)}(Ce~+C’e-~~)
+ { (p + 7) N2/2}( DleaZ+ D;edfl) + {- (p + 7) MN/2} ( D2ePZ+ Die-&) +{2M~(2ji+~+~)}(~%?@+We-~~)
1
cosm?rXsinn~Y.
(W
Now, the lateral surface boundary conditions are given by equation (12). Here also the loadings are expressed in terms of Fourier series and making use of equations (12), (18) and (19), one obtains, for each (m, n) combination,
[
;
I;;
1
{A,A’,B,B’,C,Cf,D1,D;,Dz,D;,D3,D~}=
{Qzt, Qu,, Qzp PzpPut,Pz,,Qzb, Qu,, Qzb,P,, Pub,PzJ,
(20)
228
S. SRINIVAS
where F is a (6 X 12) matrix with coefficients given by
F,,,(Z), F,,2(Z)
= - kNe’s’Z
A42
N2ti+
I
F,,,(Z), FL&? = + (Er.+4g----
2g
-2i(2&+i+j;)
((ji+k)(2c7i+iw?i
g(@i+3K+2i)
+- (2fii-fT+2i)(2ji+i-t)gZ)
e*z I
F,,,(Z),
F&Z)
= -c MA’
1
- -+I 2g g(6$+-3r7+2j;)
-2H.(2ji+rc+j;)
{(*+ii)
rf: (2ji+K+2iI)
(2F++Kf2X)
(2CL+K)gZ}1e~z _I
~1,11tZ)r~1,12(a
=
M
-
’
(6p+3K+2x)
+- (2ji+K+2i)(2ji+k)gZ}
-2i7(2ji+K+i) F,,,(Z)
= F,,2(Z)
{(E;L+K)(2fi+K+2jI)
e3z
= iTMe~s~z
F2,3(Z) = F2,4(Z) = N[li(g2+$)
-t-r7[~~-r7(g~-s~)/x~]]e~~~
F2,5(Z) = F,,s(Z)
= Th4s2r7[I + (E;i+k) (g2-sf)/t&je*szZ
F2,,(Z)
=+-MN
= F,,(Z)
I
Ir+
2g
1 g(6,&3E+2h)
-2i(2p+k+i)
{(/Ii+i)(21*+z+2~)
ekGz
rt (2fi+i+ti)(2ji+ii)gZ) 1
-2i(qi+K+j;) FmG),
F2,12W
=
N
P-
rt (2&+i+2h)
(6p+;R+2Q
(2p+k)gZ}
1
ewz
i(F+wqi+~+2~)
[
-2k(2ji+k+i) F3,3(Z)
,
F,,, (2) = Ilt (26 + K) gzsze’slz
Fs,AZ)r FdZ) F&Z),
k(21*+k+2X)(2@+Z)gZ)
FdZ)
=
_ *
(6/.d-331c+2i)
= t6fi+~+2i)
{2ji+Kk
{2ji+Kk
I
(2jli+r7+2j;)gZ}e”QZ
(2ji-t-ii-t2X)gZ}e’gZ
ekgZ
A problem in micropolar elasticity
229
FS,I(Z) , F5,2(Z) = Itr Nsl (6 + 7) eAslZ F5J@),F~,4fZ) =+yKsz(g2--sg)
;e-
F5,5tZ),F5.6(23 = -N(jG2,+$g2)ertseZ
Fs,l(Z),F6,2(Z) = {&(g2--s:)
- (~-I-~)s’;}~*~~~
F6,5(Z),F6,6tZ) = rt:CS + 7) gZs2e’*pz
F.&Z),
Fs,,o(Z) = It:
@+fjMg emz.
(21)
Solution of equation (20) yields the constants A, A’, . . ., Di. The series (5), (61, (18) and (19) are summed to obtain displacements, micro-rotations, stresses and couple stresses. NUMERICAL
RESULTS
Numerical results are presented for the case of free vibrations. Solution of the transcendental characteristic equation ( 15) can yield an infinite set of frequencies for a given
230
S. SRINIVAS
modal parameter. In elastic plates the first one of the infinite set of frequencies corresponds to a flexural mode and is called the primary flexural frequency. In case of micropolar elastic materials also, there exists a frequency corresponding to which the mode is fIexural and the couple stresses are relatively small compared to stresses. This frequency is designated here as primary flexural frequency. This frequency might also be the lowest of the infinite set of frequencies. In Fig. 2 the primary flexural frequency is plotted against the micropolar elastic moduli 01, p, y for various values of the coupling modulus K, micromoment density factorjand modal parameter [ (~~~~)z + (&~/b)~]. In Fig. 3 the ratio of microrotation &. to rotation (~w~~y) is plotted against CX,& y. In Fig. 4 the ratio of couple stress m xs to stress (r, is plotted against Q, /3, y. In the evaluation of numerical results LX,/3 and y were taken equal, just for the sake of convenience in presentation.
It is seen from Fig. 2 that the primary flexural frequency increases with the coupling modulus K. For K = 0 the ordinary elastic stresses and displacements are not coupled to couple stresses and micro-rotations and the primary flexural frequency plotted in Fig. 2 for K = 0 is the same as that for elastic material[3]. It is seen from Figs. 3 and 4 that the relative magnitudes of micro-rotations and couple stresses increase with increasing K. For the frequency under conside~tion the micro-ro~tions and couple stresses are zero when K = 0. Influence of a, /!I, y. It is seen from Fig. 2 that the primary flexural frequency
increases in general with
O.llO
K/G
0.05
0408
0.104 0.01
a
0.100
0.096
0001 0 0.092 00X
0.01
0‘1
a/Gh*-B/Gh’=y/Gh*, F&L 2. Primary
flexural frequency parametir different K’Sandj’s -j/P
log scale
n vs micropolar elastic constants LY,8, y, for = O-01----j/F
= 0.1.
231
A problem in micropolar elasticity I.5 \
1
(g+(g \
\
\
5-
= 0.02
(y+(yhB N.
\
\
\
‘\\
4\
‘\
\
\
\
\
3-
K/G
\
\
\
a.
0.05
n”
\
Qi
“2 0.01
I-
ym
c?00l
I
I 0.1 ^
0.01
I 001
opoo,
^
a/Gh‘=P/Gh’=y/Gh:
,O 01
log scale
Fig. 3. Ratio of micro-rotation c$, to rotation aw/ayvsMicropolar elastic constants OL,8, y, for = O.Ol;----j/P = 0.1. different K’S and j’s. --j/P
for some cases there could be a maxima or minima. It is interesting to note that even if (Y,p and y are zero, the displacements and microrotations are not uncoupled, although couple stresses are then zero. In such a case from equation ( 1b) OL,/3, y, although
cbr=
*K_u~j~2
~-~ [
I
2KTpjw2[if%]
#%I=
+Z = 2K --Kpjw2
[E-g]
and using these relations the micro-rotation terms in equation (la) can be eliminated. The equation (la) then reduces to the form of that of elastic material (A’+ p’) h,lk
+
p’uk,lZ
+
po2uk
=
0
with A’and 11’given by h’=h+
Pjw2K 2K -
pjoZ
a& P’ = P+ (;-p~~;. K-
Thus the value of the primary flexural frequency does not become that of elastic material, even when (Y,/3and y are zero. From Figs. 3 and 4 it is seen that the relative magnitudes of micro-rotations &, &, corresponding to the frequency under consideraion, decreases with increasing OL, p, y, whereas relative magnitudes of couple stresses increase with OL, /3,y. Influence ofj
The primary flexural frequency decreases with an increase of j, the micromoment
S. SRINIVAS
232
s 0
:: 8
0.001
001
:
! b”
K/G=0.05
ES
and 0.01
0.01 ore very cbse
=/Gh*=~/Gh~:y~G~‘,
Cog scale
Fig. 4. Ratio of couple stress m,, to stress u, VNmicropolar elasticconstanta, & Y,fordifferent =@01;----j/ha ~0.1. K’S and j’s. -j/h2
density factor. This is due to an increase in the net inertia of the system with increase in j. For higher values ofj, at lower a, /3, y and K, the frequency could be lower than the co~es~ndi~ frequency for elastic material. The relative m~tudes of both microrotations and couple stresses increase with j. However, the influence of j itself decreases with increase in (Y,/3, y. The influence of j is high at higher values of modal parameter. IF$u~x~ of modal parameter The effect of micropolar elasticity is not highly ~uenced by the modal parameter [ (rn~/u)~ + (nh/b)2]. However, the micropolar effect on frequency seems to be higher for lower modal parameters. For example the ratio of X(K= O*OS;F=0.01; a, 8, y = O*OOl) to X (elastic) is 1*012 and 1407 for modal parameters O-02 and 0. I8 respectively. The ratio of A (2 = 0.05, J,;i*Ol; 3 = /? = 7 = O-1)to h (elastic) is I.7 1 and 149 for modal parameters 042 and ‘
Micro-rotations
and couple stresses
The relative magnitudes of micro-rotations C&and (pywith respect to rotations W/C@ and &v/ax (&., & and w are nearly constant across thickness) are complex functions of micropolar elastic constants 01,p, y, ~,j and modal parameter. The ratios ~~~(~~~~y~ and #~~(~w~~x)are in the ne~~urho~ of 1 only for a smail range of micropolar elastic constants and no generalisation regarding this can be made. The magnitude of microrotation C&is negligible. #& is less than lo-‘* in all cases considered here. The couple stresses mz2, m, and mrrsare also found to be negligible. Asymmet~
in stresses
The asymmetry in shear stresses was significant only-at high values of micropolar elastic constants K, a, 8, y. For example for ri:= 0.05; cii,@,Ij;= 0.1, modal parameter =
A problem in micropolar elasticity
0.02, the asymmetry in crrr, that is (l~~~-o,,I/l~~~+a~~l) lower values of K, OL,/I and y the asymmetry is less. CONCLUDING
233
is as high as 11%. For
REMARKS
In this paper an exact three-dimensional analysis for statics and dynamics of a class of simply supported rectangular plates of micropolar elastic materials is presented, starting from Eringen’s equations [ 11. The solution is in the form of series, in which each term is defined explicitly. For free vibrations, the natural frequencies can be obtained by solution of a closed form characteristic equation. Numerical results are presented for the case of free vibrations. The effect of the micropolar elastic constants on frequency, relative magnitudes of micro-rotations and couple stresses are studied. From the numerical results it is seen that the influence of micropolar elastic property on the primary flexural frequency is not very high. For example for K/G = O-05, a/G/t2 = /3/Gh2 = y/Gh2 = 0.1, the increase in the value of frequency from that of corresponding elastic material’s is just 17 per cent for modal parameter of O-02. However, if higher values of micropolar elastic constants are realised in some materials the inIIuence could be very high. Further the presence of micropolar elastic property itself introduces some additional frequencies, to the infinite spectrum of frequencies, corresponding to which couple stresses are predominant. research was sponsored by Directorate of Research Laboratories, Ministry of Defence, Government of India, as a Grant-in-Aid Scheme and was carried out with Prof. A. K. Rao as Chief Investigator. The numerical investigation was carried out while the author was a research fellow at the Department of Aeronautical Engineering, Delft University of Technology, Delft, The Netherlands.
Acknowledgments-This
NOTATION
a,b,h length, width and thickness of plate (A42+ W) U2 nrh/b integers used in trigonometric
h4,Ng mrhla,
m,n mxx,myy,mzz mxu,mux9 mx, couple stresses mzx,mu,,m,,
I
expansions
surface stresses in x, y, z directions surface couples about x, y, z axes 4 v, w displacements x9 y,z xla, ylb, zlh 8 an elastic property used in non-dimensionalisation material properties ff,89Y,K,~,CL,.i {KP,9 {a, P, rIl@-h2) 4x9 4u, 42
Pm Pm Pt
{IT, i, p: P
utx, UYUYuzr ux,, uux,uxx urr9 uuz9 uzu
jlh” (6 A, PI/g density stresses
S. SRINIVAS
micro-rotations frequency of simple harmonic vibration
REFERENCES [l] A, C. ERINGEN, J. Murh. Mech. 15,909 (1966). [2] S. SRINIVAS and A. K. RAO, J. Franklin Institufe 291,469 (1971). [3] S. SRINIVAS, C. V. JOGA RAO and A. K. RAO, J. Sound Vihafion 12,187 (1970). (Received 2 1 January 197 1) R4sun1&-Dans cet article, on presente une analyse tridimensionnelle de la statique et de la dynamique dune classe de plaques rectangulaires a support simple, cons&&es dun mat&iau Clastique micropolaire. La solution se pr&ente sous forme dun serie dans laquelle chaque terme est explicitement d&mine. Pour les vibrations libres, on obtient les frequences par la solution d’une equation caractCristique a formee. ZusammenfessungKlasse von einfach gemacht sind. Die freie Schwingungen
In dieser Arbeit wird eine dreidimensionale Analyse Wr die Statik und Dynamik einer gestiltzten rechteckigen Platten vorgelegt, die aus mikropolarem elastischen Material Losung hat die Form von Reihen, in denen jedes Glied ausdrilcklich bestimmt ist. Filr werden die Frequenzen durch Liisung einer Kenngleichung geschlossener Form erhalten.
Semmonie-L’A. svolge un’analisi tridimensionale nei riguardi della statica e della dinamica di una classe di piastre rettangolari semplicemente sopportate e costituite da materiale elastic0 micropolare. La soluzione assume la forma di serie, nelle quali ciascun termine 5 esplicitamente determinato. Nel case delle vibrazioni libere si ottengono le frequenze con la soluzione di una equazione caratteristica di forma chiusa. A6cqmurrHOBO
~nasca
l-@RCTWICH np0txo
T~XhfepHbIti tUWIE3
CTBTH'IEKOTO
H J@iH-KOrO
non.xepRABBRHL4x, npaMoyroJIbHbIx rmaclml
COCTOSlHHik OtI~AWIeH-
sis hnxxpouonxp~oro Marepriana. Bcnyxae CEU~~O)UI~IXB~~~~S~~~&
PemeInre EiMWT BEfq PSWOB, B KOTOPbIX KiiXlJbIii ‘IJICSf IlBHO 3&I@HBM. IIOJIY'WOTCK 'IKCTOTbI~IUUiHeM Xap;ucTepHcTAnCCKOI-0 YpasIieHKtI3lIK&hITOfl l$tOOpMbI.
1973, Vol. I I, pp. 215-234.
A PROBLEM
Pelgamon
Press.
Printed in Great Britain
IN MICROPOLAR
ELASTICITY
S. SRINIVAW Department of Aeronautical Engineering, Indiin Institute of Science, Bangalore 12, India Abstract- In this paper a three-dimensional analysis for statics and dynamics of a class of simply supported rectangular plates made up of micropolar elastic material is presented. The solution is in the form of series, in which each term is explicitly determined. For free vibrations, the frequencies are obtained by the solution of a closed form characteristic equation.
INTRODUCTION
differential equations of micropolar elasticity have been developed by Eringen[ 11. In this paper analysis of a class of simply supported rectangular plates of micropolar elastic material is presented, making use of these equations. The solution is in the form of hyperbolic-trigonometric series, in which each term is explicitly determined. The series are so chosen that the edge boundary conditions are automatically satisfied.
THE GOVERNING
SOLUTION
The governing differential equations for simple harmonic vibrations (frequency O) of a micropolar elastic medium, in the absence of body forces and couples, in rectangular Cartesian coordinares are given by [l] (using usual tensor notation),
The stresses and couple stresses are given in terms of displacements and microrotations by
It is noted here that if K = 0 the displacements and microrotations get decoupled (vi& equation (1)) and also the stresses depend only on displacements. Now, the edge boundary conditions for a simply supported rectangular plate (uide Fig. 1) can be specified as
(4) tAt present NRC-postdoctoral Virginia, USA.
research associate at NASA-Langley 215
Research Center, Hampton,
S. SRINIVAS
Fig. 1. Coordinate system.
The boundary conditions (4) are automatically satisfied by choosing
cb,=
5 2 cos m7rXcos nlrY 4(Z). rn==l
w-1
u llj (6)
Substi~tion of equations (5) and (6) in the governing differential equations (1) yields for each (m, n) combination, @-(~+p)iW-(~+fi)A!fN
-KL (x+j&W -kN -(i;+ji)MN$r(i;+fi)N* (i+,ii)LN ipL 0 iiM V -(i+/%)LM -(i+ji)LN ++(i+ji)L2 r7N --GM 0 W 0 --ix. kN x-(i%++)jMIZ-(Ei+@MN-((a+p)LM f 0 -ZM -(~+~~Nx-(~+~)N*-(~+~)LN q kL 0 --EN iiM (ii+@)LM (&+p)LN x+(&+j!i)L* 4 i
= (0)
(7) where, JI= (Ir.+ii)(L*-g2)+W, x=~(L2-g4)-2r7+Q*~ ~*=pwzhz/8 and I is a reference elastic module, used in nondimensionalisation. In equation (7) -3 denote non~ension~~ quantities. For the non-trivial solution of equation (7) the determinant of the (6 X 6) coefficient matrix must be zero. This condition yields the following twelve roots for L: It r, , + r, and two double roots rt.r, and k r,, where,
217
A problem in micropolar elasticity
and r3,r4
(+
8_{$-i~+ =
(n*j--2ii)(ji+z)
[
+ii’)
zT(ji+z)
signfor r, and - sign for r4). SIMPLE
HARMONIC
VIBRATIONS
For simple harmonic vibrations, the twelve roots of L are f rl, + r, and two double roots +r3, strr,. Now, the eigenvectors corresponding to these roots are found using equation (7). The method to be followed is described in references 121and [3]. Thus, (V, V, W, 6, q, 5)
=F
{O,O,O, M, N, -rl}Ae"Z+{O,
+{M,
N, r2,0,0,0}BenZ+{M,
O,O, M, N, rt)A’e-f*Z
N,-r,O,O,O)B’e-“Z
-K(g2-rf)M/Jli,O,-riM,-rriN,g2)DtertZ + l- r&f, - r&, g2,- ii (g” - r?) Nh. ii (g2 - rf) M/xi, 0) C;e-‘iz + (ii (g” - rt) N&t, - ii (g’-
rj) M/Jr,,0, riM, riN, g*} DWiz
,
(9)
where&= (j.i++)(rf-g2) +fi2,xi=~(rf-g2) -2E+E’jT In equation (9), A, A’, B, B’, C,, C& C,, Ci, D3, D& D4and 0: are the twelve arbitrary constants. It is interesting to note that for roots +r, and +-r, the displacements and microrotations are uncoupled. These roots actually correspond to modes in which rotations and curl of microrotation vector 4 (& +&, + &J~)vanish and the differential equations ( 1a) and ( 1b) get decoupled. Substitu~on of equation (9) in equations (2) and (3) yields the following expressions for stresses and couple stresses. uxz =E4 i
i
[{h(ri-g2)
- (2@+E)M2}(BenZ+B’e-nz)
WI=1 n=i
+&
[ (- (2ji + I() MzrJ (&P~ - Cie+‘@) + (- (3% + 17)MNlc(g” - rf) /+d
x ( D&i2 + Die-‘@) ]] sin m?rX sin nrrY
( 104
218
S. SRINIVAS
$ii(
(J.?m
m=1?a=1 +is4
ri-g’)
-
(2ii+K)N2)(Ben”4B’e-~z)
[(-(2~+K)N2r~}(C~riZ-C~e~P~z)+{(2~+~)MNii(g2-r~)/Jli)
X (D&fz+Dfe-r~z)]] sin m?zXsin n7rY
i [{if rz--g2)f cr,, 45 rn=ln=1
(2CT++)r,2}(Benz+B’e-~)
[ { (2jk + k)g*tj} (C@iz - CIe+@)]] sin m?rX sin nnY
+ J4
+*z4
(lob)
*
Cl&)
[{(2~+rT)MNr,}(Cie~~-C~e-~Z)+~IC[~(N2-M2)tg2-r~)l~t
-a!f2(g2-
rf) /& - g2])
(Z&Pz + l&e-‘@)]] cos mn%y cos nlrY
wa
crux=8j,j, [(-Krl}(Aer’Z-A’e-r~z)+~(2~-tK)MN}(Be”Z+B’e~r~) + I: f{(2~+Ic)~~r~)(Clef*z-CC;e-~~z) f=3,4
+~~[~(N2-M2)(g2-rf)IJlt+~~2(g2-rl)/~~+g2]}
]]
X (DieriZ+ D'e-QZ)
Q=GI”tl + &
[wK4
,
cos m7rX cos naY
(W
er~z+A’e-~~Z)+{(2ji+E)Mr2}(Be”Z-l?’e-r~)
EIMI@fg2+rf) +r7{g2+ri’lg2-r:)/xi}lf(C,e’rz+C;e-’iZ)
+ {Nr,Z [fi(g”-rF)/+i - I]}(Dier@-D~e-‘~z)]] cos mnXsin n7rY =l’=i,=i @z.E
(lof)
[{-~~)(Aer1Z+A’e-“Z)+((2~+K)MT9}(Be~-B’e-nz)
+ Z E(M[p(gZ+rf) +ii{r~-~~g”-t~)/xr}]}(C,er~+C~e-+~) i=3.4
-t{NriKI1+(p+R)(gZ
- rf)f&l) (&erg--Die-V)]l
ax m~Xsin MY
(1%)
A problem in micropolar elasticity
219
erlZ+A’evrtZ) + { (2F+ E)Nr,} (Beti - We--)
+,z4 [{N[Ci(g2+rf)+~{g2+K(g2--rl)lxf}l}(C~’”+C;e-’iZ) +{MrfrT[l-p((g2-
~zll
rf)/Jlr]} (Dg?fz-D;e-rfz)]]
=+inil [WNAerlZ+A’e-r*Z) + {(2p+ i = +fg4 Ew[iw+r:)+~~~:
- i?(g” -
+{-MrfK[l+(ji+K)(g2-rf)/~j]}(Djer*z
it1[(5 (g”-
wrz = Il.8 i,
ri) +
+
[G(g2-rf)
(Cp’” + C;e-‘fz) D;e-r*z)]]
sinm?rX cos n7rY(1 Oi)
(j+$A42}(Aer’Z+A’e-r*Z)
+
(CjerfZ+CIe-r*Z)
x [{ (~+j+WVZ(g2-rrf)/xf}
(Cfe'*z+Cle-rfz)
(D&*z-DIe-r@)]]
htYmil ii1[{fi(g2- r4) -
cos m7rXcos n7rY
*
+
x
i
(lib)
(jij+P)rf}(Aer’Z+A’e-r~Z)
+ ,z, [ { (6 + 7) g3rj} (&erfz - Dfe-r*z)]] cos m?rX cos n7rY
5 mm= hi? rn==l
(lla)
(@+.j;)N2}(Aer1z+A’e-r1Z)
*=3.4
+ {- (B+y)N2rj}
mzz=
rf)/xf}]}
(6 + 7) M2rj} (Dter*z - D;esrjz)]] cos m7rXcos n7rY
muu= w ji,
(1Oh)
k)Nr,} (Be”Z-B’e-M)
-
+*z4 [{- (~+~)MN~(g2-rf)l~f) + {-
sin m7rXcos n7rY
(llc)
[{-MN(~+~)}(Ae”Z+A’e-r’Z)
n=1
[{K(g2-r~)(N2~-M~)Ixf)(C,e’tz+Cle-’fZ)
f=3.4
+
{jtf~rf(jj+j?)}(D&r@-D~e-r*z)]]
sin mlrXsin nlrY
(lid)
S. SRINIVAS
220
i [{--IMN(BS_y))(Aer’Z3_A’e-“Z) m u.z = #wg ??a=1n=1
i [(Mr,(~+~))(Aer”B--R’e-r~Z) mxz = Wg??I=1 ?l=l +
2
f(-~~~~(g2-r~)Nf~,)(C~r*Z-C~e-r~Z)
1=3,4
(lli) Now, if the lateral sufface z = 0 is loaded by h~o~c
surface stresses of ~p~tudes and surface couples of amplitudes pTt, pyr pzt and the iateral surface z = h is loaded by qz*, qyb,qzb,P,,, pyb,pa&,the lateral surface boundary conditions are, qzt, qut,
qzt
221
A problemin micropolarelasticity
The loadings are expressed in double Fourier series, such that substitution of equations (10) and (11) in (12) and equating term by term one obtains, for each (m, n) combination, an equation,
{Q,, QW,Qy, &,, pyl’ P@,Qsb, Qvb,Q,, Pq, PW, Pz+,1, (13) where Q’s and P’s are Fourier load coefficients. F is a (6 X 12) matrix, with coefficients given by (the Tirst subscript denotes the row number and second subscript the column number) FIPI(Z), FE3(Z) = - GVe’rlZ; FI.(Z),F,,,(Z) F I..~(Z)~ Fuw FwdZ-It
(2) = M(ji(g2+rt)
=&(2ji+r7)Mr&“”
+K[rf-K(gZ--rf)/~r])e”~z
F~mtdZ) = +.Nr,lc(I + (Fi+K)(g2--rif)IJll)e’rfZ
)c;,,(Z), F,,(Z)
= iiMe’+‘Z;F,,(Z),
F2A(Z) =-t (26+ K)NrpefM
F~.w,(Z), F~,,ww(Z) = N(LL(g2+r12)+K[T:-_(g2-r:)/Xr])e’r~Z F~,w(Z)~ &W)(Z) F,,(Z),
= TMrS(l+
FSd(Z) = (h($-g2)
(ET+2) (g2-rf)f+~)e”fz + (2$+ ic)rg)ezW
Fww(Z), Fuw,J(Z) = + (2b+ W%errfz F;,I (Z), F.M(Z> =+Mr,(P+y)eTrRz
FeBl(Z), FBs(Z) = (ii(g2--rf) F wwfZ)r
Ft3m12~
- (~+y)r:)efnZ
(Z) = + (p+ r)gV&‘iz
(14)
(All Ff &‘snot defined by equation ( 14) are zero). In the above equations wherever 2 or i sign occurs, the top sign corresponds to the first quantity on the left hand side and the bottom sign to the second quantity. For example, from the last equation FB,,&Z)
UESVd.11No.2-B
= + (~+~)g%,e*~P; FBSec12,(Z) = - (@+y)g2rie-‘iz.
222
S. SRINIVAS
Further, in the above equations
Solution of equation (13) for given value of excitation frequency parameter a*, yields the constants. Now, by summation of series (5), (6), (10) and (1 l), to desired accuracy, the displacements and stresses can be obtained. FREE
VIBRATIONS
Consider the case when the lateral surfaces are free of surface stresses and couples. Now, the right hand side of equation (13) becomes zero. For non-trivial solution the determinant of the coefficient matrix must be zero:
IF (1)(0)i= * 0
F
(1%
This is the characteristic equation, defining the natural frequency parameter R2. For any given (m, n) this transcendental equation yields infinite number of natural frequencies each corresponding to a different thickness mode. It is also observed that this equation can yield velocities of wave propagation in invite plates. FLEXURE
If the plate is static a2 = 0 and the twelve roots of L are rfcx1, double roots -+s, and triple roots + g, where
(16) The eigenvectors corresponding to these roots are &JO, 0,M, N, sl)A’e-81z
(V, V, W,,!j,q, 4) = {O,O,O, M,N,-s,)AesyZ+
+ (s2M, s2N, 8, - 2 (9 - 522) NIx2, E(8 - $1 MIx~,O)B~~~ +{~k2-.G)Nl+2,-~k2-
+
(-
s"z)MIJI,,0,--2M,-s2N,g2}Ce8~
s2M, - s,N, g”, - 17(8” - sf) N/x2, r7(g” - ~22) M/x2, 0) B’e-8a
+ 42 (s” - 4) Nail, - r7(g” - 4) Ml~~, 0, s2M, s2N, g2fC’e-S4Z
+
-MN
l,O,O,~,~,~
+ {O,O, l,O,
-NZ
-N
O,O}D,t@+ {O,O, l,O, 0, O}DjeTBZ
A problem in micropolar elasticity
223
NM + o,o,o,-- g, -g ,o 2(2ji+K+h)&F N-M + o,o,o,- g, - g
-M -N + -, g ’ g
2(2$+$_++)B’e-*z
l,O,O,O (2ji+rC+U)WZe-sz,
(17)
where ~?8= (MD,+
ND,-ggDs)/(61;i+3i+2i)
9’ = (MD:+
N~~+gD~)/{6~+3~+~);
1,5s=(j.i+i?) (s”z-g”) and xz = +(s$-8)
and
-2;.
In equation (17), A, A’, B, B’, C, C’, D1, &, Dz, D;, D3 and 0; are the twelve arbitrary constants. Substitution of equation (17) in equations (2) and (3) yields the following expressions for stresses and couple stresses.
+ {-M}
( D,ePz+ D;ewgZ) (-- 2k - MZ (2j.i+ i -t %)Z/g} &%Yz
+ {- 2i + M2 (2p 4 ii + 2?1)Z/g} $t?‘e-nz]sin mrX sin ntrY
a,,=8(2~+~)
i i nwi n=l
@a)
[{-N2s2}(BeP"Z-B'e-dZ)
+ (- N ) (D+F + D;evnz) + {- 2i - NZ (2@ + ii + 5) Z/g} &Ye + {- 2i + N2 (2ji + r7+ %) Z/g} We-nz] sin mrX sin nlrY
(18b)
S. SRINIVAS
224
a,,
=
8
[{ji,~,)(Ae~~~-_4'e--~~~)
i
i
rn=l
n=1
-I- (k[ji(N2--
+{(2~++)MNs2)(Be""Z--Be-S"Z)
M2)(g2-~~)~JIZ-M2(g2-s~)/JIZ-gL]}(Ce8~4C’e--saZ)
-I-{ (2p + ii) N/2} (DIeBz+ D;e-gZ) + { (2@ + r?)M/2} ( D,eSz+ D6evffZ) 1-{MN(2~+rc)(2~+Ic+2j;)/g)~z~z + (- MN (2ji + k) (26 + ii + 2i) /g)5FZe-Bz] cos rmX cm nrrY
--A’eaSIZ) f { (2p+ii)MNs,}
(18t
(BeS”Z-B’e-S”Z)
~(~[p(N2-M2)(g2-~~)IJIZ$.I?N2(g2-~~)lJIZ+~]}(CeszZ$.C’e-szZ) + ((2p + ii) N/2} (D,esz+ Die-&) + { (2jiL ~)~/2~(~~~z+~~e-gz) + {MN(2~+~)(2jZ+~+2ji)/g}~ZeBz +- (-- MN (2,~ + K) (2ji + ii + 2i) jg} B’Ze-Bz] cm m?zX cos my
+{j&+!$}
=8 Ql?.?z
i m=1
(D,eBz - Dle-gz) t {~}(D2@z-Die-az)
5 [{-iTN}(Ae81z+A'e-8~z)
n=1
+ {M[p(g-t-Sg)
+t7[s2,-- Z (g” - sjq)/xJ ]} (Se*+- B’e-S*Z)
~{Ns,lc[l+(iu+IZ)(g2-~22)/$]}(Ce~-C’e-~)
(18i
A problem in micropolarelasticity
+{Mji}(D,~Z+D~e-Dz) + +(2$+2+2X)
(2p+K)gZl
I$ [(p+K)(2fi+K+*ii) I
S?eBz+ + t
m=1
+
(1%)
2 [{-KM}(Ae81Z+A’e-81Z)
n=1
(Die”” - D~e-az~) + { pg
+{N(fi+K)}(D3@Z+D~e-BZ) + (2p+i+2Q
+
[
+“+}(D2e”” -
:[2f?(2p+a+T)
(2/z+K)gZ]}swZ+
I
;
Die-&) +p(2p+ii+2X)
[2l?(2/rL+r7+Q
+ b (2p + E + 2x) - (2,G+ K + 2x) (2p + K) gZ]
m=t
S9’e-Bz cosm?~XsinmY I
I
I-y I
UZY= 8 i
-*K(*p+-t+X)
[(p+K)(2p+r7+2X)
-2k(2p+K+ii)-(2fi+K+fi)(2fi+E)gZl
~lI.8 = 8 i
225
I
1
LB’e-uz sin m?rX cos nmY
(18h)
5 [{KM}(Ae81Z+A’e-81Z) n=1
(D,esZ
+
- Die-*)
+ { N,k} ( D,egz + D;eeaz) +
-2~‘(2p+ii+h)-(2F.+~+22X)(2p+~)gZ)
m,=
M’
2i
m=1Rl
{ii(g2-s:)
I
Se-Bz
+ (~+~)M2}(Ae81Z+A’e-8~Z)
+ {- (~~++MMNK(g2-~S22)/~2}(Be8~+B’e-8~)
+ {- (a + 7) M2s2} (Ce*” - C’e-‘2z)
1
sinm?~XcosnrY
(18i)
S. SRINIVAS
226
+
(P+% i‘
M2N 2g
-i- -(@+j+)(ZjIi+tZ-t-X)?
2MN 1
i
mu, = Mil
($i?egz-S?‘ed8z)
I
cosmlrXcosn?rY
(19
s:) + (~t-~)N2)(Ae8~z+A’e-s~z)
jl [iw-
+((P+;ii)MNt7(gZ--s2,)f~,}(Be~+B’e-~) i- {- (6 + 7) N2s,) ( Cesaz- C’e-8”) N3 (Dle@z-D~e-gz) C -f- tP+F@ {-
I
I
-t- (is+?)
%z = h8 i
i
(2ji-++a)
[iii(gz-sy)
2MN g
I
I-(P-t-$
F}
( G?esz- @‘e-BZ)
I
( DzegZ- Die-uZ)
cos m7zX cos nnY
(191
- (~+~)f~)(Aes~Z+A’e-SIZ)
rn==l ?I=1
C
{
(P+$@sz}(CeszZ-C’e-8”Z)
-I- (p+r)F 1
I
4 (-- (P+r)
~](Z31egz-D~e-gz)
(D&z - D;eegZ) cos mrX cos nnY I
(191
A problem in micropolar elasticity
+
‘12x2= W
+
@+9)}(D,F-D;e-Qz)
I
y
+If
sinm?rXsinn7rY
(2~+K+X)(N27-M2~)}(%9z-.We~~)]
i
227
i
(19e)
[{Msl(~+~)}(Ae8~Z-A’e‘8~Z)
m=1 n=1
{-@s2(g2 - @N/x2} (Be”-B’epSg)
+ {-M(~~2,+3/~)}(Ce~~+C’e-~~)
+
{MN(~+~)/2~tD~~Z+Dle-aZ)
+ {- M2 (6 + 7) /2} ( D2eBz+ Ddeeaz) + {- 2Np (2p + Z + x) } ( SWz + StZI’eesz)sin m?rX cos MY 1
(19f)
mzx= h8 i i [{Msl(~+~)}(Ae8’Z-A’e-B~z) rn=ln=1 + {-- W2 (L? - 4 N/x2) (Be sG- B’e-*G) + {- M (78: + pg”) } ( CeSfi+ C’ewgti) + {MN (p + 9) /2}( D,esz+ Dle-gz) + {- M2 (p + 7) /2}(D,egZ + Dde-az) + {- 2N7 (2fi + IT+ i) } (,@ipeBz + SJ’e-Qz) sin m?rX cos nrY J
Wg)
i i [ {Ns,(~+~)}(AeslZ-~Ae-SIZ) my2= hiifm=1 ?I=1 + { /317s, (g2- sl) M/x2}(Besti - B’emaa)+ {- N (PsZ,+ yg”>} ( CesrZ+ C’e-aa) + { (fi + 7) N2/2}( D,eOz+ D;e-aZ) + {- (p + T) MN/2} (D2eBz-I- &e-OZ) +{2M~(2p+K+X)}(B9Z+#e-BZ) m,,=
1
cosm?rXsinnrY
(19h)
IS’ i 2 [ {Nsl(~+~)}(AeS1Z--A’e-81Z) m=1n=1 + {W2 (s’ - ~2,)M/x2) (Be sS-B’e-sfi)
+{-N(~~~+~g2)}(Ce~+C’e-~~)
+ { (p + 7) N2/2}( DleaZ+ D;edfl) + {- (p + 7) MN/2} ( D2ePZ+ Die-&) +{2M~(2ji+~+~)}(~%?@+We-~~)
1
cosm?rXsinn~Y.
(W
Now, the lateral surface boundary conditions are given by equation (12). Here also the loadings are expressed in terms of Fourier series and making use of equations (12), (18) and (19), one obtains, for each (m, n) combination,
[
;
I;;
1
{A,A’,B,B’,C,Cf,D1,D;,Dz,D;,D3,D~}=
{Qzt, Qu,, Qzp PzpPut,Pz,,Qzb, Qu,, Qzb,P,, Pub,PzJ,
(20)
228
S. SRINIVAS
where F is a (6 X 12) matrix with coefficients given by
F,,,(Z), F,,2(Z)
= - kNe’s’Z
A42
N2ti+
I
F,,,(Z), FL&? = + (Er.+4g----
2g
-2i(2&+i+j;)
((ji+k)(2c7i+iw?i
g(@i+3K+2i)
+- (2fii-fT+2i)(2ji+i-t)gZ)
e*z I
F,,,(Z),
F&Z)
= -c MA’
1
- -+I 2g g(6$+-3r7+2j;)
-2H.(2ji+rc+j;)
{(*+ii)
rf: (2ji+K+2iI)
(2F++Kf2X)
(2CL+K)gZ}1e~z _I
~1,11tZ)r~1,12(a
=
M
-
’
(6p+3K+2x)
+- (2ji+K+2i)(2ji+k)gZ}
-2i7(2ji+K+i) F,,,(Z)
= F,,2(Z)
{(E;L+K)(2fi+K+2jI)
e3z
= iTMe~s~z
F2,3(Z) = F2,4(Z) = N[li(g2+$)
-t-r7[~~-r7(g~-s~)/x~]]e~~~
F2,5(Z) = F,,s(Z)
= Th4s2r7[I + (E;i+k) (g2-sf)/t&je*szZ
F2,,(Z)
=+-MN
= F,,(Z)
I
Ir+
2g
1 g(6,&3E+2h)
-2i(2p+k+i)
{(/Ii+i)(21*+z+2~)
ekGz
rt (2fi+i+ti)(2ji+ii)gZ) 1
-2i(qi+K+j;) FmG),
F2,12W
=
N
P-
rt (2&+i+2h)
(6p+;R+2Q
(2p+k)gZ}
1
ewz
i(F+wqi+~+2~)
[
-2k(2ji+k+i) F3,3(Z)
,
F,,, (2) = Ilt (26 + K) gzsze’slz
Fs,AZ)r FdZ) F&Z),
k(21*+k+2X)(2@+Z)gZ)
FdZ)
=
_ *
(6/.d-331c+2i)
= t6fi+~+2i)
{2ji+Kk
{2ji+Kk
I
(2jli+r7+2j;)gZ}e”QZ
(2ji-t-ii-t2X)gZ}e’gZ
ekgZ
A problem in micropolar elasticity
229
FS,I(Z) , F5,2(Z) = Itr Nsl (6 + 7) eAslZ F5J@),F~,4fZ) =+yKsz(g2--sg)
;e-
F5,5tZ),F5.6(23 = -N(jG2,+$g2)ertseZ
Fs,l(Z),F6,2(Z) = {&(g2--s:)
- (~-I-~)s’;}~*~~~
F6,5(Z),F6,6tZ) = rt:CS + 7) gZs2e’*pz
F.&Z),
Fs,,o(Z) = It:
@+fjMg emz.
(21)
Solution of equation (20) yields the constants A, A’, . . ., Di. The series (5), (61, (18) and (19) are summed to obtain displacements, micro-rotations, stresses and couple stresses. NUMERICAL
RESULTS
Numerical results are presented for the case of free vibrations. Solution of the transcendental characteristic equation ( 15) can yield an infinite set of frequencies for a given
230
S. SRINIVAS
modal parameter. In elastic plates the first one of the infinite set of frequencies corresponds to a flexural mode and is called the primary flexural frequency. In case of micropolar elastic materials also, there exists a frequency corresponding to which the mode is fIexural and the couple stresses are relatively small compared to stresses. This frequency is designated here as primary flexural frequency. This frequency might also be the lowest of the infinite set of frequencies. In Fig. 2 the primary flexural frequency is plotted against the micropolar elastic moduli 01, p, y for various values of the coupling modulus K, micromoment density factorjand modal parameter [ (~~~~)z + (&~/b)~]. In Fig. 3 the ratio of microrotation &. to rotation (~w~~y) is plotted against CX,& y. In Fig. 4 the ratio of couple stress m xs to stress (r, is plotted against Q, /3, y. In the evaluation of numerical results LX,/3 and y were taken equal, just for the sake of convenience in presentation.
It is seen from Fig. 2 that the primary flexural frequency increases with the coupling modulus K. For K = 0 the ordinary elastic stresses and displacements are not coupled to couple stresses and micro-rotations and the primary flexural frequency plotted in Fig. 2 for K = 0 is the same as that for elastic material[3]. It is seen from Figs. 3 and 4 that the relative magnitudes of micro-rotations and couple stresses increase with increasing K. For the frequency under conside~tion the micro-ro~tions and couple stresses are zero when K = 0. Influence of a, /!I, y. It is seen from Fig. 2 that the primary flexural frequency
increases in general with
O.llO
K/G
0.05
0408
0.104 0.01
a
0.100
0.096
0001 0 0.092 00X
0.01
0‘1
a/Gh*-B/Gh’=y/Gh*, F&L 2. Primary
flexural frequency parametir different K’Sandj’s -j/P
log scale
n vs micropolar elastic constants LY,8, y, for = O-01----j/F
= 0.1.
231
A problem in micropolar elasticity I.5 \
1
(g+(g \
\
\
5-
= 0.02
(y+(yhB N.
\
\
\
‘\\
4\
‘\
\
\
\
\
3-
K/G
\
\
\
a.
0.05
n”
\
Qi
“2 0.01
I-
ym
c?00l
I
I 0.1 ^
0.01
I 001
opoo,
^
a/Gh‘=P/Gh’=y/Gh:
,O 01
log scale
Fig. 3. Ratio of micro-rotation c$, to rotation aw/ayvsMicropolar elastic constants OL,8, y, for = O.Ol;----j/P = 0.1. different K’S and j’s. --j/P
for some cases there could be a maxima or minima. It is interesting to note that even if (Y,p and y are zero, the displacements and microrotations are not uncoupled, although couple stresses are then zero. In such a case from equation ( 1b) OL,/3, y, although
cbr=
*K_u~j~2
~-~ [
I
2KTpjw2[if%]
#%I=
+Z = 2K --Kpjw2
[E-g]
and using these relations the micro-rotation terms in equation (la) can be eliminated. The equation (la) then reduces to the form of that of elastic material (A’+ p’) h,lk
+
p’uk,lZ
+
po2uk
=
0
with A’and 11’given by h’=h+
Pjw2K 2K -
pjoZ
a& P’ = P+ (;-p~~;. K-
Thus the value of the primary flexural frequency does not become that of elastic material, even when (Y,/3and y are zero. From Figs. 3 and 4 it is seen that the relative magnitudes of micro-rotations &, &, corresponding to the frequency under consideraion, decreases with increasing OL, p, y, whereas relative magnitudes of couple stresses increase with OL, /3,y. Influence ofj
The primary flexural frequency decreases with an increase of j, the micromoment
S. SRINIVAS
232
s 0
:: 8
0.001
001
:
! b”
K/G=0.05
ES
and 0.01
0.01 ore very cbse
=/Gh*=~/Gh~:y~G~‘,
Cog scale
Fig. 4. Ratio of couple stress m,, to stress u, VNmicropolar elasticconstanta, & Y,fordifferent =@01;----j/ha ~0.1. K’S and j’s. -j/h2
density factor. This is due to an increase in the net inertia of the system with increase in j. For higher values ofj, at lower a, /3, y and K, the frequency could be lower than the co~es~ndi~ frequency for elastic material. The relative m~tudes of both microrotations and couple stresses increase with j. However, the influence of j itself decreases with increase in (Y,/3, y. The influence of j is high at higher values of modal parameter. IF$u~x~ of modal parameter The effect of micropolar elasticity is not highly ~uenced by the modal parameter [ (rn~/u)~ + (nh/b)2]. However, the micropolar effect on frequency seems to be higher for lower modal parameters. For example the ratio of X(K= O*OS;F=0.01; a, 8, y = O*OOl) to X (elastic) is 1*012 and 1407 for modal parameters O-02 and 0. I8 respectively. The ratio of A (2 = 0.05, J,;i*Ol; 3 = /? = 7 = O-1)to h (elastic) is I.7 1 and 149 for modal parameters 042 and ‘
Micro-rotations
and couple stresses
The relative magnitudes of micro-rotations C&and (pywith respect to rotations W/C@ and &v/ax (&., & and w are nearly constant across thickness) are complex functions of micropolar elastic constants 01,p, y, ~,j and modal parameter. The ratios ~~~(~~~~y~ and #~~(~w~~x)are in the ne~~urho~ of 1 only for a smail range of micropolar elastic constants and no generalisation regarding this can be made. The magnitude of microrotation C&is negligible. #& is less than lo-‘* in all cases considered here. The couple stresses mz2, m, and mrrsare also found to be negligible. Asymmet~
in stresses
The asymmetry in shear stresses was significant only-at high values of micropolar elastic constants K, a, 8, y. For example for ri:= 0.05; cii,@,Ij;= 0.1, modal parameter =
A problem in micropolar elasticity
0.02, the asymmetry in crrr, that is (l~~~-o,,I/l~~~+a~~l) lower values of K, OL,/I and y the asymmetry is less. CONCLUDING
233
is as high as 11%. For
REMARKS
In this paper an exact three-dimensional analysis for statics and dynamics of a class of simply supported rectangular plates of micropolar elastic materials is presented, starting from Eringen’s equations [ 11. The solution is in the form of series, in which each term is defined explicitly. For free vibrations, the natural frequencies can be obtained by solution of a closed form characteristic equation. Numerical results are presented for the case of free vibrations. The effect of the micropolar elastic constants on frequency, relative magnitudes of micro-rotations and couple stresses are studied. From the numerical results it is seen that the influence of micropolar elastic property on the primary flexural frequency is not very high. For example for K/G = O-05, a/G/t2 = /3/Gh2 = y/Gh2 = 0.1, the increase in the value of frequency from that of corresponding elastic material’s is just 17 per cent for modal parameter of O-02. However, if higher values of micropolar elastic constants are realised in some materials the inIIuence could be very high. Further the presence of micropolar elastic property itself introduces some additional frequencies, to the infinite spectrum of frequencies, corresponding to which couple stresses are predominant. research was sponsored by Directorate of Research Laboratories, Ministry of Defence, Government of India, as a Grant-in-Aid Scheme and was carried out with Prof. A. K. Rao as Chief Investigator. The numerical investigation was carried out while the author was a research fellow at the Department of Aeronautical Engineering, Delft University of Technology, Delft, The Netherlands.
Acknowledgments-This
NOTATION
a,b,h length, width and thickness of plate (A42+ W) U2 nrh/b integers used in trigonometric
h4,Ng mrhla,
m,n mxx,myy,mzz mxu,mux9 mx, couple stresses mzx,mu,,m,,
I
expansions
surface stresses in x, y, z directions surface couples about x, y, z axes 4 v, w displacements x9 y,z xla, ylb, zlh 8 an elastic property used in non-dimensionalisation material properties ff,89Y,K,~,CL,.i {KP,9 {a, P, rIl@-h2) 4x9 4u, 42
Pm Pm Pt
{IT, i, p: P
utx, UYUYuzr ux,, uux,uxx urr9 uuz9 uzu
jlh” (6 A, PI/g density stresses
S. SRINIVAS
micro-rotations frequency of simple harmonic vibration
REFERENCES [l] A, C. ERINGEN, J. Murh. Mech. 15,909 (1966). [2] S. SRINIVAS and A. K. RAO, J. Franklin Institufe 291,469 (1971). [3] S. SRINIVAS, C. V. JOGA RAO and A. K. RAO, J. Sound Vihafion 12,187 (1970). (Received 2 1 January 197 1) R4sun1&-Dans cet article, on presente une analyse tridimensionnelle de la statique et de la dynamique dune classe de plaques rectangulaires a support simple, cons&&es dun mat&iau Clastique micropolaire. La solution se pr&ente sous forme dun serie dans laquelle chaque terme est explicitement d&mine. Pour les vibrations libres, on obtient les frequences par la solution d’une equation caractCristique a formee. ZusammenfessungKlasse von einfach gemacht sind. Die freie Schwingungen
In dieser Arbeit wird eine dreidimensionale Analyse Wr die Statik und Dynamik einer gestiltzten rechteckigen Platten vorgelegt, die aus mikropolarem elastischen Material Losung hat die Form von Reihen, in denen jedes Glied ausdrilcklich bestimmt ist. Filr werden die Frequenzen durch Liisung einer Kenngleichung geschlossener Form erhalten.
Semmonie-L’A. svolge un’analisi tridimensionale nei riguardi della statica e della dinamica di una classe di piastre rettangolari semplicemente sopportate e costituite da materiale elastic0 micropolare. La soluzione assume la forma di serie, nelle quali ciascun termine 5 esplicitamente determinato. Nel case delle vibrazioni libere si ottengono le frequenze con la soluzione di una equazione caratteristica di forma chiusa. A6cqmurrHOBO
~nasca
l-@RCTWICH np0txo
T~XhfepHbIti tUWIE3
CTBTH'IEKOTO
H J@iH-KOrO
non.xepRABBRHL4x, npaMoyroJIbHbIx rmaclml
COCTOSlHHik OtI~AWIeH-
sis hnxxpouonxp~oro Marepriana. Bcnyxae CEU~~O)UI~IXB~~~~S~~~&
PemeInre EiMWT BEfq PSWOB, B KOTOPbIX KiiXlJbIii ‘IJICSf IlBHO 3&I@HBM. IIOJIY'WOTCK 'IKCTOTbI~IUUiHeM Xap;ucTepHcTAnCCKOI-0 YpasIieHKtI3lIK&hITOfl l$tOOpMbI.