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On the flexural rigidity of a micropolar elastic circular cylindrical tube
hf. 1. Engn.9 Sci. Vol. 17. pp. 10154021 @I Pmgmwn Press Ltd.. 1979. Prinled in Great Britain
ON THE FLEXURAL RIGIDITY OF A MICROPOLAR ELASTIC CIRCULAR CYLINDRICAL TUBE? G. V. KRISHNA REDDYS and N. K. VENKATASUBRAMANIANO Departmentof Mathematics,PSG Collegeof Technology,Coimbatore-641004, India Ababnct-Macroscopic stiffness of structural elements are found to be increased when micropolar behaviouris attributedto them. In this paperthe effect of micropolarbehaviouron the flexuralrigidityof a circularcylindricaltube is considered.
1. INTRODUCTION THE PASTtwo
decades have witnessed a surge of activity aimed at extending the range of applicability of the classical theory of continuous media to account for more aspects of structure and atomic interactions of the media. This has lead to the revival and extensions of the Cosserat Continuum theory of physical bodies and formulation of new theories of microstructural media. References to these theories are found in the works of Eringen[l, 21 and Cowin [3]. One among the important contributions to the world of microcontinuum mechanics is the Nonlinear Theory of Simple Micro-Elastic Solids formulated by Eringen and Suhubi[4]. Eringen [23has shown that many of the existing theories of microstructural continua either have contacts with their theory in special situations or could be obtained as restricted cases. Eringen[4] considered a linear theory of elasticity with couple-stresses as a special case of his non-linear theory and later recapitulated the equations of this linear theory under the name Linear Theory of Micropolar Elasticity [5]. The theory of micropolar elasticity, also referred to as the Cosserat theory of elasticity with rigid directors, is concerned with material media whose constituents are dumbbell molecules and is expected to find application in the treatment of mechanics of granular materials with elongated rigid grains and composite fibrous materials. This theory, being one of the simplest extensions of the classical theory of elasticity, has evoked much interest and solutions of some important problems of elasticity have been obtained under this theory. From these solutions it is found that the effect of micropolar behaviour of the material of a body, could be of significance, particularly when some physical dimensions of the body approach certain characteristic micropolar lengths associated with the material of the body. Solving the problem of stress concentration at a cylindrical cavity in an infinite micropolar elastic solid subjected to a field of simple tension at infinity, Kaloni and Ariman[6] have found that the values of the stress concentration factors calculated under the micropolar theory, are less than the classical values and that the decrease is significant when the radius of the cavity approaches a characteristic micropolar length. Gauthier and Jahsman[7] have found that micropolar elastic effects in solid circular cylinders subjected to axial loads or torsion, increase their extensional or torsional stiffness over their classical elastic counterparts and that this effect is more pronounced for very “slender” cylinders, i.e. cylinders whose radii approach a certain micropolar length. They have found[8] a similar effect in the case of a semicircular ring of rectangular cross-section, bent by transverse radial shear resultants at its ends. The effect of micropolar behaviour, on the flexural rigidity of a solid circular cylinder was considered by Krishna Reddy and Venkatasubramanian[9]. They found the increase in flexural rigidity to be significant in the case of “slender cylinders”. In the case of micropolar circular cylindrical tubes, there appear to be two particular cases, in which the increase in the flexural rigidity is to be analysed. In the present paper an expression for the modulus of flexural rigidity of a micropolar cylindrical tube is found and is used to study these two cases. The expression for the modulus of flexural rigidity is obtained by solving the problem of bending of a micropolar circular cylindrical tube by terminal couples, by using the semi-inverse method of Iesan[ 101. tBased on the Doctoralthesis submittedby the second authorto the Universityof Madras,India. SProfessorand Head. #Lecturer. 1015
1016
G.V.KRiSHNAREDDYandN.K.VE~KATAS~3~~AN~
2.BASIC EQUATIONS The equations of equilibriumin the static theory of linear isotropic homogeneous micropolar
elasticity in the absence of body forces and body couples are [4,5] (~+~~+K)V~*U)-(~+K)VX~~U)+KVX~=O
(1)
(a+fi+y)V(V*(P)-yVx(Vxqb)+K(VxU-2+)=0.
(2)
Surface tractions and surface coupbs are given by
where n denotes the unit normal to the surface. The rn~~ro~~~ ~~s~~ A, g and K:have the dimensions of forcetleng&, while the micropolar constants a, /3, r have the diiensions of force. These constants satisfy the inequalities[4,5]
The micropolar moduhi a, /3, y and K vanish for classical elastic materials.
AxisOr of the cylindrical polar co-ordinate system we use in the subsequent work, is taken along the axis of the &c&r cyhndricai tube. The plane ends of the tube, which are circular annuli of radii a and b (a < b) are taken to lie on the planes z = 0 and z = 1.We assume that body forces and body couples are absent, the curved surfaces of the tube are free of applied tractions and couples and that the load on the tube is distributed over the ends on the planes z = 0 and z = t, the loading on the ends being statically equivalent to couples M] and -Mj, respectively (4,j denote unit vectors atOngthe x and y axes of the carte&an coordinate system associated with the cyiindri~alcoordinate system chosen). Therefore, on the plane z = 0
where P is the cross-section (annuhrs) of the tube. Curved surfaces of the tube bein@free of forces and couples
on the curved surfaces.
On the flexuralrigidityof a micropolarelastic circularcylindricaltube
1017
where
Y = 1/(2A + 2p + K) is the micropolar Poisson’s ratio and bl, cl are constantsto be determined. Expressions for the non-vanishing components of stress and couple-stress are t, = b,[(A + 2~ + K)U,, tti = bd(p
+ K)uo,,
ter = bdpe,l+
+ Ar-‘u&e]
- Kd’z - v-‘(ue
- ~r,lv)l
K$z -(CL •I-K)r-‘(Ue - use)1
CM= bl[Au,,, + (A + 2~ + K)T-‘(U, tzz =
m,
blA[o,, + r-‘(l),
= bl[81611.r
-(Y
m, = bdyll,,
+
+ Ue,e)] •t E[blr
+ 134
Ue,e)]
COS 8 + CI]
Sin 01
- @ + ~4 sin 61
m,e = bl[br-‘llr,e
-(Y + Bv) ~0s 01
me, = h[yr-‘f,e
- (13 + 7~) ~0s 01
WN
where E = (2~ + ~)(3r\ + 2p + K)/(~A + 2p + K) is the micropolar Young’s modulus and comma denotes partial differentiation. Substituting the expressions for u and $ given by (8) and (9) in (l), (2) and (71, the following equations and boundary conditions are obtained. (A+2jA+K)V@‘V)-(/L+K)VX@XV)+KVX$=O
(11)
(a+B+r)V@‘$)-rvX@X$)+K@XV-2211r)=O
(A + 2/.~+ K)TV,,
(12)
+ A (u, + ue.0) = 0
(13)
r[(cL+ Kb8.r - K&l - p_(ue - tb,e) = 0
(14)
r&,, = (B + yv] sin 8
(13)
for r=a, b inOGBa2r. Solution of the plane problem (1 l)-(U) can be obtained from the general solution of the first plane problem of micropolar elasticity in a ring shaped region, obtained by Chiu and Lee [ 111. Omitting the details, we only record the results. v = 2& cos ee, + hs sin 0eel+ Aee+(A,
cos 8 +
AZsin e)e, + (A2 cos 8 -A, sin 8)ee
I) = 2ha sin Be, + A3ez
(16) (17)
where
N2 = K/2(jh + K) a2 = K(2p +
K)/Y(jA
+
K),
C = (P + K)/(A
I.( ) and K,,( ) are modifiedBessel functions of order n.
+ 2P + K)
(18)
1018
G.V.KRISHNA REDDY andN.K.VENKATASUBRAMANIAN
A,, AZ, A, are arbitrary constants and Bi, Bz, Bs, & are given by the equations
=
N2(/3/y+V)[l 1 0 O]?
(19)
Terms of v and $ corresponding to the arbitrary constants A,, AZ, AS represent a rigid body motion and may be omitted. On z = 0 q.,=-[2b,hcos8{hi+r-‘(h~+h~)}+E(b,rcosO+c,)le,
m(,,,= -b, sin 6[2/3& - (y + /3v)]e,+ b, cos fI[(y + Pv) - 2/3r-‘hh]e,+
(20)
When these expressions are used in conditions (6)r and (6)~we get cl = 0 and b, = M/J where J = nE(b4-a4)/4+4hcr(b4-a4)B,/(1
-3~)
-2?rkQ{bZ,(Sb) - aZ,@a))B4+ {bK,(Sb) - UK,(&)}&] 16~/i?B,
-r(b2-u2)
yS2(1- 3c)
-(y +/Iv)]
(21)
5.DISCUSSIONS AND REMARKS
Macrodisplacement solution for the problem of a circular cylindrical tube bent by terminal couples derived above may be put in the form
where u, = -;b,[z2+
v(x2- y2)1+ b,[(h-hs)+(x2-
y2)(h4+ hs)/r2]
u, = - b,vxy + 2b,xy(h4 + IQ/r2 u, = b,xz
and
r2 = x2+ y2.
(22)
Macrodisplacement of points on the line x = 0, y = yo (a c yo d b) is - b1[(1/2)z2+ Dr]i where D, is a constant whose value depends on yo. Therefore, every line of the tube parallel to its axis and lying in the plane x = 0 deforms into a parabola x=-b,[;z2+D,],
y=yo.
In view of the smallness of deformations, the curvature R-’ of the parabola (23) is given by d2x/dz2 approximately. Therefore, for small deflections, R-’ = MJ-‘. J is then a measure for the rigidity of the tube against bending. J is known as the modulus of micropolar flexural rigidity. Its classical counterpart J, is given by J, = &(b4
- u4)/4
where EC is the classical Young’s modulus. The ratio RI = J/J, for equal Young’s modulii, can be used to study the change in the flexural rigidity of a cylindrical tube, when micropolar behaviour is attributed to it. fir will be referred to as the bending stiffness ratio of the tube.
On the flexural rigidity of a micropolar elastic circular cylindrical tube
1019
Evaluating Br, Bz, &, II, from the eqns (19) and substituting in (21), fi, can be expressed in the form
(v + B/r)* @a)* + (Sb)‘+ 8N2(1 - v) + <@a, Sb)
1
(24)
where 5(6u, Sb) = [(~3b)~-(&z)4][Z;(&z)K;(c3b) - K;(Su)Z;(Sb)l x [{Z;(Sb) - Zl(Su)X(Sb)*K,(Sb) - (i3u)*K,(~u)} + {K;(Sb) - K;(Su)~(Sb)*Z2(Sb) - (Sa)*Z2(Su)}]-‘.
The bending stiffness ratio R’ for a solid circular cylinder of radius b is given by [9]
(v + B/r)*
8N*( 1 - v) + t(Sb )
1
(25)
where [(Sb) = (Sb)*Z;(Sb)/Z#ib)
(a’ can be recovered from fI, by allowing 6u to tend to zero). fIi (as also a) depends on v- the Poisson’s ratio, N- the coupling number, /3/y- a micropolar ratio and the length ratios Sa and Sb. S-’ is the characteristic length for bending. The ranges of these parameters may be obtained from (5) as OSNSl,
-la/3/rGl,
Osva1/2.
For positive values of 6a and 6b, [@a, Sb) can be shown to be positive. Therefore, fli > 1, indicating an increase in the flexural rigidity of the tube due to micropolar behaviour. From the analytical expression (24) for RI, it can be shown that, for given Su and Sb, 0, increases from 1 as N* increases from 0 and that fli increases as /3/y increases from - 1 to some positive value (which depend on v and N*) and then decreases as /3/y increases beyond this value. R’ was also found to behave in a similar manner191. A-I’was further found to ditfer from 1 significantly for certain ranges of values of N* and p/y and that this difference increases as the physical dimension b approaches the characteristic micropolar length S-‘. & depends on the two physical dimensions a and b and, therefore, there appear to be two special cases of interest. (i) When S(b - a) is very small compared to either &a or Sb-this is the case of a thin walled tube. (ii) When Sa and Sb are both small-this is the case of a “slender*’ tube. (i) Denoting (Sb - Su)/Gu as c and assuming E 4 1 we have the following approximation for al
-[.JLIaYq+ i-l,=*+*N2 v + 1 2(1 +c)(Sa)
WY + VI 8N2(1- v)+4+4(1 +r)(Su)2 I *
(26)
It is easy to see that
Therefore, for the whole ranges of v, p/y and N*, fli can be shown to differ from 1 by a quantity less than 3.5/(1+ c)(Su)*. This quantity is not sign&ant unless Sa is small. When the thickness (b - a) of the walls of the tube approaches the micropolar length S-‘, for E = 0.1, Sa will be nearly 10 and (0i- 1) C 0.035. Thus for thin walled cylindrical tubes the modulus of micropolar flexural rigidity does not differ significantly from the classical modulus of flexural rigidity. (ii) In the case of a slender tube, fi, is found to differ significantly from 1 for certain ranges of the parameters N* and B/y. Values of RI for Sb = 4, 8 and for ranges of N2, /3/y for which the change is comparatively significant are presented in Tables 1 and 2. Table 3 gives the values
G. V. KRISHNA REDDY and N. K. VENKATASUBRAMANIAN
1020
Table 1. Values of 0, for 6b = 4, a/b = 3/8, l/2, S/8 p/y = -0.5 v=o
v=o.3
g/y = 0.0 Y = 0.5
v=o
p/y = 0.5
Y = 0.3
Y = 0.5
v=o
v = 0.3
Y = 0.5
a/b = 318 N* = 0.5 0.75 1.0
1.195 1.290 1.384
1.130 1.195 1.260
1.110 1.164 1.219
1.219 1.329 1.438
1.177 1.266 1.353
1.168 1.251 1.333
1.195 1.290 1.384
1.190 1.280 1.369
1.197 1.292 1.384
a/b = l/2 N2 = 0.5 0.75 1.0
1.177 1.263 1.348
1.119 1.178 1.237
1.100 1.150 1.200
1.200 1.300 1.400
1.161 I.242
1.322
1.152 1.227 1.302
1.177 1.263 1.348
1.170 1.251 I.332
1.175 1.260 1.342
a/b = 5/S N’ = 0.5 0.75 1.0
1.157 1.235 1.311
1.107 1.160 1.213
1.090 1.135 1.180
1.180 1.270 1.360
1.145 1.217 1.289
1.136 1.203 1.270
1.157 1.235 1.311
1.149 1.222 1.293
1.153 1.227 I.u)o
Table 2. Values of G, for 66 = 8, a/b = 3/8, l/2, S/8 /3/y = -0.5 v=o
/3/y = 0.0
Y = 0.3
Y = 0.5 1.027 1.041
a/b = 3/8 N* = 0.5 0.75 1.0
1.052 1.077 1.102
I .033
a/b = l/2 N2 = 0.5 0.75 1.0
1.047 1.070 1.093
I .030
a/b = 518 N* = 0.5 0.75 1.0
1.041 1.062 1.082
1.049 1.066 1.045 1.060
1.027 1.040 1.054
1.055 1.025 I .037 1.050 1.022 1.034 1.045
v=o
B/y = 0.5
Y =0.3
Y = 0.5
lJ=o
Y = 0.3
Y = 0.5
1.055 1.082 1.110
1.045 1.068 1.090
1.044 1.065
1.087
1.052 1.077 1.102
1.053 1.078 1.104
1.056 1.084 1.111
1.050
1.041 I.061 1.082
1.040 1.059 l.u79
1.047 1.070 I .093
1.047 1.070 1.093
1.074 1.099
1.037 1.055 1.073
1.035 1.053 1.070
1.041 1.062 1.082
1.041 1.061 1.081
1.043 1.064 1.085
1.075 1.100
1.045 1.067 1.090
I .050
Table 3. Values of II’ for Sb = 4.8 /3/y = 0.0
B/y = -0.5
v=o
Y = 0.3
Y =
0.5
&b=4
N2 = 0.5 0.75 1.0
1.226 1.335 1.441
1.149 1.223 1.297
1.125 1.188 1.250
Sb=8
N2=0.5 0.75 1.0
1.060 1.089 1.118
1.038 I .056 1.075
1.031 1.047
1.063
v=o
Y = 0.3
1.250 1.375 1.5al
1.063 1.094 1.125
/9/y=
0.5
0.5
v=o
Y = 0.3
Y = 0.5
1.203 1.304 1.405
1.194 1.290 1.384
1.226 1.335 1.441
1.223 1.329 1.431
1.236 1.347 1.455
1.052 1.077 1.103
I .050 1.076 1.100
1.060 1.089 1.118
1.062 1.092 1.122
1.066 1.099 1.131
Y =
of R’ (the bending stiffness ratio for a solid circular cylinder) for the same ranges of N2 and MY* Comparison of the values of 0, and fl given in Tables 1,2 and 3 show that: (i) Due to micropolar behaviour, the increase in bending stillness for a solid circular cylinder of a given diameter is more than that for a circular cylindrical tube of the same outer diameter and for a given outer diameter, fir decreases as (a/b) increases. (ii) For a given value of u/b, f% decreases as 6b increases. It may also be concluded that the increase in the flexural rigidity of a circular cylindrical tube due to micropolar behaviour will not be sign&ant unless the tube is “slender” Acknm&dgmu~~s-The
Madras and Dr. R. investigation.
authors wish to express theirgratitudeto Prof. G. R. Damodaran.Vicechancellor, Universityof
Subbey~ttn,
Principal, PSG College of Technology, for the encouragementgiven during this REFERENCES
[ll A. C. ERINGEN, Fracture (Edited by H. Liebowitz), Vol. II, p. 621. Acadeznic Press, New York (1968). [2] A. C. ERINGEN,Mechanics of Generalised Continua (Editedby E. Kroner),p. 18. Springer-Verb, Berlin(1968).
On the flexural @dity of a micropolar elastic circular cylindrical tube
1021
[3] S. C. COWIN, Int. X Solids Structures 6,389 (1970). [4] A. C. ERINGEN and E. S. SUHUBI, Int. 1. Engng Sci. 2, 189,389 (1964). [5] A. C. ERINGEN, J. Math. Mech. 15,909(1966). [6] P. N. KALONI and T. ARIMAN, Z. Angew. Moth. Phys. 12, 136 (1%7). [7] R. D. GAUTHIER and W. E. JAHSMAN, J. Appf. Mech. Trans. ASME, Series E 97,369 (1975). 181 R. D. GAUTHIER and W. E. JAHSMAN, J. Appl. Mech. Trans. ASME, Series E 98,502 (1976). [9] G. V. KRISHNA REDDY and N. K. VENKATASUBRAMANIAN, 1. Appl. Mech. Tmns. ASME 100,429 (1978). [IO] D. IESAN, Anal. Stiint. Univ. lasi. Mathematics 17,483 (1971). [ill. B. M. CHIU and J. D. LEE, Int. J. Engng Sci. 11,997 (1973).
(Received 10 January 1979)
ON THE FLEXURAL RIGIDITY OF A MICROPOLAR ELASTIC CIRCULAR CYLINDRICAL TUBE? G. V. KRISHNA REDDYS and N. K. VENKATASUBRAMANIANO Departmentof Mathematics,PSG Collegeof Technology,Coimbatore-641004, India Ababnct-Macroscopic stiffness of structural elements are found to be increased when micropolar behaviouris attributedto them. In this paperthe effect of micropolarbehaviouron the flexuralrigidityof a circularcylindricaltube is considered.
1. INTRODUCTION THE PASTtwo
decades have witnessed a surge of activity aimed at extending the range of applicability of the classical theory of continuous media to account for more aspects of structure and atomic interactions of the media. This has lead to the revival and extensions of the Cosserat Continuum theory of physical bodies and formulation of new theories of microstructural media. References to these theories are found in the works of Eringen[l, 21 and Cowin [3]. One among the important contributions to the world of microcontinuum mechanics is the Nonlinear Theory of Simple Micro-Elastic Solids formulated by Eringen and Suhubi[4]. Eringen [23has shown that many of the existing theories of microstructural continua either have contacts with their theory in special situations or could be obtained as restricted cases. Eringen[4] considered a linear theory of elasticity with couple-stresses as a special case of his non-linear theory and later recapitulated the equations of this linear theory under the name Linear Theory of Micropolar Elasticity [5]. The theory of micropolar elasticity, also referred to as the Cosserat theory of elasticity with rigid directors, is concerned with material media whose constituents are dumbbell molecules and is expected to find application in the treatment of mechanics of granular materials with elongated rigid grains and composite fibrous materials. This theory, being one of the simplest extensions of the classical theory of elasticity, has evoked much interest and solutions of some important problems of elasticity have been obtained under this theory. From these solutions it is found that the effect of micropolar behaviour of the material of a body, could be of significance, particularly when some physical dimensions of the body approach certain characteristic micropolar lengths associated with the material of the body. Solving the problem of stress concentration at a cylindrical cavity in an infinite micropolar elastic solid subjected to a field of simple tension at infinity, Kaloni and Ariman[6] have found that the values of the stress concentration factors calculated under the micropolar theory, are less than the classical values and that the decrease is significant when the radius of the cavity approaches a characteristic micropolar length. Gauthier and Jahsman[7] have found that micropolar elastic effects in solid circular cylinders subjected to axial loads or torsion, increase their extensional or torsional stiffness over their classical elastic counterparts and that this effect is more pronounced for very “slender” cylinders, i.e. cylinders whose radii approach a certain micropolar length. They have found[8] a similar effect in the case of a semicircular ring of rectangular cross-section, bent by transverse radial shear resultants at its ends. The effect of micropolar behaviour, on the flexural rigidity of a solid circular cylinder was considered by Krishna Reddy and Venkatasubramanian[9]. They found the increase in flexural rigidity to be significant in the case of “slender cylinders”. In the case of micropolar circular cylindrical tubes, there appear to be two particular cases, in which the increase in the flexural rigidity is to be analysed. In the present paper an expression for the modulus of flexural rigidity of a micropolar cylindrical tube is found and is used to study these two cases. The expression for the modulus of flexural rigidity is obtained by solving the problem of bending of a micropolar circular cylindrical tube by terminal couples, by using the semi-inverse method of Iesan[ 101. tBased on the Doctoralthesis submittedby the second authorto the Universityof Madras,India. SProfessorand Head. #Lecturer. 1015
1016
G.V.KRiSHNAREDDYandN.K.VE~KATAS~3~~AN~
2.BASIC EQUATIONS The equations of equilibriumin the static theory of linear isotropic homogeneous micropolar
elasticity in the absence of body forces and body couples are [4,5] (~+~~+K)V~*U)-(~+K)VX~~U)+KVX~=O
(1)
(a+fi+y)V(V*(P)-yVx(Vxqb)+K(VxU-2+)=0.
(2)
Surface tractions and surface coupbs are given by
where n denotes the unit normal to the surface. The rn~~ro~~~ ~~s~~ A, g and K:have the dimensions of forcetleng&, while the micropolar constants a, /3, r have the diiensions of force. These constants satisfy the inequalities[4,5]
The micropolar moduhi a, /3, y and K vanish for classical elastic materials.
AxisOr of the cylindrical polar co-ordinate system we use in the subsequent work, is taken along the axis of the &c&r cyhndricai tube. The plane ends of the tube, which are circular annuli of radii a and b (a < b) are taken to lie on the planes z = 0 and z = 1.We assume that body forces and body couples are absent, the curved surfaces of the tube are free of applied tractions and couples and that the load on the tube is distributed over the ends on the planes z = 0 and z = t, the loading on the ends being statically equivalent to couples M] and -Mj, respectively (4,j denote unit vectors atOngthe x and y axes of the carte&an coordinate system associated with the cyiindri~alcoordinate system chosen). Therefore, on the plane z = 0
where P is the cross-section (annuhrs) of the tube. Curved surfaces of the tube bein@free of forces and couples
on the curved surfaces.
On the flexuralrigidityof a micropolarelastic circularcylindricaltube
1017
where
Y = 1/(2A + 2p + K) is the micropolar Poisson’s ratio and bl, cl are constantsto be determined. Expressions for the non-vanishing components of stress and couple-stress are t, = b,[(A + 2~ + K)U,, tti = bd(p
+ K)uo,,
ter = bdpe,l+
+ Ar-‘u&e]
- Kd’z - v-‘(ue
- ~r,lv)l
K$z -(CL •I-K)r-‘(Ue - use)1
CM= bl[Au,,, + (A + 2~ + K)T-‘(U, tzz =
m,
blA[o,, + r-‘(l),
= bl[81611.r
-(Y
m, = bdyll,,
+
+ Ue,e)] •t E[blr
+ 134
Ue,e)]
COS 8 + CI]
Sin 01
- @ + ~4 sin 61
m,e = bl[br-‘llr,e
-(Y + Bv) ~0s 01
me, = h[yr-‘f,e
- (13 + 7~) ~0s 01
WN
where E = (2~ + ~)(3r\ + 2p + K)/(~A + 2p + K) is the micropolar Young’s modulus and comma denotes partial differentiation. Substituting the expressions for u and $ given by (8) and (9) in (l), (2) and (71, the following equations and boundary conditions are obtained. (A+2jA+K)V@‘V)-(/L+K)VX@XV)+KVX$=O
(11)
(a+B+r)V@‘$)-rvX@X$)+K@XV-2211r)=O
(A + 2/.~+ K)TV,,
(12)
+ A (u, + ue.0) = 0
(13)
r[(cL+ Kb8.r - K&l - p_(ue - tb,e) = 0
(14)
r&,, = (B + yv] sin 8
(13)
for r=a, b inOGBa2r. Solution of the plane problem (1 l)-(U) can be obtained from the general solution of the first plane problem of micropolar elasticity in a ring shaped region, obtained by Chiu and Lee [ 111. Omitting the details, we only record the results. v = 2& cos ee, + hs sin 0eel+ Aee+(A,
cos 8 +
AZsin e)e, + (A2 cos 8 -A, sin 8)ee
I) = 2ha sin Be, + A3ez
(16) (17)
where
N2 = K/2(jh + K) a2 = K(2p +
K)/Y(jA
+
K),
C = (P + K)/(A
I.( ) and K,,( ) are modifiedBessel functions of order n.
+ 2P + K)
(18)
1018
G.V.KRISHNA REDDY andN.K.VENKATASUBRAMANIAN
A,, AZ, A, are arbitrary constants and Bi, Bz, Bs, & are given by the equations
=
N2(/3/y+V)[l 1 0 O]?
(19)
Terms of v and $ corresponding to the arbitrary constants A,, AZ, AS represent a rigid body motion and may be omitted. On z = 0 q.,=-[2b,hcos8{hi+r-‘(h~+h~)}+E(b,rcosO+c,)le,
m(,,,= -b, sin 6[2/3& - (y + /3v)]e,+ b, cos fI[(y + Pv) - 2/3r-‘hh]e,+
(20)
When these expressions are used in conditions (6)r and (6)~we get cl = 0 and b, = M/J where J = nE(b4-a4)/4+4hcr(b4-a4)B,/(1
-3~)
-2?rkQ{bZ,(Sb) - aZ,@a))B4+ {bK,(Sb) - UK,(&)}&] 16~/i?B,
-r(b2-u2)
yS2(1- 3c)
-(y +/Iv)]
(21)
5.DISCUSSIONS AND REMARKS
Macrodisplacement solution for the problem of a circular cylindrical tube bent by terminal couples derived above may be put in the form
where u, = -;b,[z2+
v(x2- y2)1+ b,[(h-hs)+(x2-
y2)(h4+ hs)/r2]
u, = - b,vxy + 2b,xy(h4 + IQ/r2 u, = b,xz
and
r2 = x2+ y2.
(22)
Macrodisplacement of points on the line x = 0, y = yo (a c yo d b) is - b1[(1/2)z2+ Dr]i where D, is a constant whose value depends on yo. Therefore, every line of the tube parallel to its axis and lying in the plane x = 0 deforms into a parabola x=-b,[;z2+D,],
y=yo.
In view of the smallness of deformations, the curvature R-’ of the parabola (23) is given by d2x/dz2 approximately. Therefore, for small deflections, R-’ = MJ-‘. J is then a measure for the rigidity of the tube against bending. J is known as the modulus of micropolar flexural rigidity. Its classical counterpart J, is given by J, = &(b4
- u4)/4
where EC is the classical Young’s modulus. The ratio RI = J/J, for equal Young’s modulii, can be used to study the change in the flexural rigidity of a cylindrical tube, when micropolar behaviour is attributed to it. fir will be referred to as the bending stiffness ratio of the tube.
On the flexural rigidity of a micropolar elastic circular cylindrical tube
1019
Evaluating Br, Bz, &, II, from the eqns (19) and substituting in (21), fi, can be expressed in the form
(v + B/r)* @a)* + (Sb)‘+ 8N2(1 - v) + <@a, Sb)
1
(24)
where 5(6u, Sb) = [(~3b)~-(&z)4][Z;(&z)K;(c3b) - K;(Su)Z;(Sb)l x [{Z;(Sb) - Zl(Su)X(Sb)*K,(Sb) - (i3u)*K,(~u)} + {K;(Sb) - K;(Su)~(Sb)*Z2(Sb) - (Sa)*Z2(Su)}]-‘.
The bending stiffness ratio R’ for a solid circular cylinder of radius b is given by [9]
(v + B/r)*
8N*( 1 - v) + t(Sb )
1
(25)
where [(Sb) = (Sb)*Z;(Sb)/Z#ib)
(a’ can be recovered from fI, by allowing 6u to tend to zero). fIi (as also a) depends on v- the Poisson’s ratio, N- the coupling number, /3/y- a micropolar ratio and the length ratios Sa and Sb. S-’ is the characteristic length for bending. The ranges of these parameters may be obtained from (5) as OSNSl,
-la/3/rGl,
Osva1/2.
For positive values of 6a and 6b, [@a, Sb) can be shown to be positive. Therefore, fli > 1, indicating an increase in the flexural rigidity of the tube due to micropolar behaviour. From the analytical expression (24) for RI, it can be shown that, for given Su and Sb, 0, increases from 1 as N* increases from 0 and that fli increases as /3/y increases from - 1 to some positive value (which depend on v and N*) and then decreases as /3/y increases beyond this value. R’ was also found to behave in a similar manner191. A-I’was further found to ditfer from 1 significantly for certain ranges of values of N* and p/y and that this difference increases as the physical dimension b approaches the characteristic micropolar length S-‘. & depends on the two physical dimensions a and b and, therefore, there appear to be two special cases of interest. (i) When S(b - a) is very small compared to either &a or Sb-this is the case of a thin walled tube. (ii) When Sa and Sb are both small-this is the case of a “slender*’ tube. (i) Denoting (Sb - Su)/Gu as c and assuming E 4 1 we have the following approximation for al
-[.JLIaYq+ i-l,=*+*N2 v + 1 2(1 +c)(Sa)
WY + VI 8N2(1- v)+4+4(1 +r)(Su)2 I *
(26)
It is easy to see that
Therefore, for the whole ranges of v, p/y and N*, fli can be shown to differ from 1 by a quantity less than 3.5/(1+ c)(Su)*. This quantity is not sign&ant unless Sa is small. When the thickness (b - a) of the walls of the tube approaches the micropolar length S-‘, for E = 0.1, Sa will be nearly 10 and (0i- 1) C 0.035. Thus for thin walled cylindrical tubes the modulus of micropolar flexural rigidity does not differ significantly from the classical modulus of flexural rigidity. (ii) In the case of a slender tube, fi, is found to differ significantly from 1 for certain ranges of the parameters N* and B/y. Values of RI for Sb = 4, 8 and for ranges of N2, /3/y for which the change is comparatively significant are presented in Tables 1 and 2. Table 3 gives the values
G. V. KRISHNA REDDY and N. K. VENKATASUBRAMANIAN
1020
Table 1. Values of 0, for 6b = 4, a/b = 3/8, l/2, S/8 p/y = -0.5 v=o
v=o.3
g/y = 0.0 Y = 0.5
v=o
p/y = 0.5
Y = 0.3
Y = 0.5
v=o
v = 0.3
Y = 0.5
a/b = 318 N* = 0.5 0.75 1.0
1.195 1.290 1.384
1.130 1.195 1.260
1.110 1.164 1.219
1.219 1.329 1.438
1.177 1.266 1.353
1.168 1.251 1.333
1.195 1.290 1.384
1.190 1.280 1.369
1.197 1.292 1.384
a/b = l/2 N2 = 0.5 0.75 1.0
1.177 1.263 1.348
1.119 1.178 1.237
1.100 1.150 1.200
1.200 1.300 1.400
1.161 I.242
1.322
1.152 1.227 1.302
1.177 1.263 1.348
1.170 1.251 I.332
1.175 1.260 1.342
a/b = 5/S N’ = 0.5 0.75 1.0
1.157 1.235 1.311
1.107 1.160 1.213
1.090 1.135 1.180
1.180 1.270 1.360
1.145 1.217 1.289
1.136 1.203 1.270
1.157 1.235 1.311
1.149 1.222 1.293
1.153 1.227 I.u)o
Table 2. Values of G, for 66 = 8, a/b = 3/8, l/2, S/8 /3/y = -0.5 v=o
/3/y = 0.0
Y = 0.3
Y = 0.5 1.027 1.041
a/b = 3/8 N* = 0.5 0.75 1.0
1.052 1.077 1.102
I .033
a/b = l/2 N2 = 0.5 0.75 1.0
1.047 1.070 1.093
I .030
a/b = 518 N* = 0.5 0.75 1.0
1.041 1.062 1.082
1.049 1.066 1.045 1.060
1.027 1.040 1.054
1.055 1.025 I .037 1.050 1.022 1.034 1.045
v=o
B/y = 0.5
Y =0.3
Y = 0.5
lJ=o
Y = 0.3
Y = 0.5
1.055 1.082 1.110
1.045 1.068 1.090
1.044 1.065
1.087
1.052 1.077 1.102
1.053 1.078 1.104
1.056 1.084 1.111
1.050
1.041 I.061 1.082
1.040 1.059 l.u79
1.047 1.070 I .093
1.047 1.070 1.093
1.074 1.099
1.037 1.055 1.073
1.035 1.053 1.070
1.041 1.062 1.082
1.041 1.061 1.081
1.043 1.064 1.085
1.075 1.100
1.045 1.067 1.090
I .050
Table 3. Values of II’ for Sb = 4.8 /3/y = 0.0
B/y = -0.5
v=o
Y = 0.3
Y =
0.5
&b=4
N2 = 0.5 0.75 1.0
1.226 1.335 1.441
1.149 1.223 1.297
1.125 1.188 1.250
Sb=8
N2=0.5 0.75 1.0
1.060 1.089 1.118
1.038 I .056 1.075
1.031 1.047
1.063
v=o
Y = 0.3
1.250 1.375 1.5al
1.063 1.094 1.125
/9/y=
0.5
0.5
v=o
Y = 0.3
Y = 0.5
1.203 1.304 1.405
1.194 1.290 1.384
1.226 1.335 1.441
1.223 1.329 1.431
1.236 1.347 1.455
1.052 1.077 1.103
I .050 1.076 1.100
1.060 1.089 1.118
1.062 1.092 1.122
1.066 1.099 1.131
Y =
of R’ (the bending stiffness ratio for a solid circular cylinder) for the same ranges of N2 and MY* Comparison of the values of 0, and fl given in Tables 1,2 and 3 show that: (i) Due to micropolar behaviour, the increase in bending stillness for a solid circular cylinder of a given diameter is more than that for a circular cylindrical tube of the same outer diameter and for a given outer diameter, fir decreases as (a/b) increases. (ii) For a given value of u/b, f% decreases as 6b increases. It may also be concluded that the increase in the flexural rigidity of a circular cylindrical tube due to micropolar behaviour will not be sign&ant unless the tube is “slender” Acknm&dgmu~~s-The
Madras and Dr. R. investigation.
authors wish to express theirgratitudeto Prof. G. R. Damodaran.Vicechancellor, Universityof
Subbey~ttn,
Principal, PSG College of Technology, for the encouragementgiven during this REFERENCES
[ll A. C. ERINGEN, Fracture (Edited by H. Liebowitz), Vol. II, p. 621. Acadeznic Press, New York (1968). [2] A. C. ERINGEN,Mechanics of Generalised Continua (Editedby E. Kroner),p. 18. Springer-Verb, Berlin(1968).
On the flexural @dity of a micropolar elastic circular cylindrical tube
1021
[3] S. C. COWIN, Int. X Solids Structures 6,389 (1970). [4] A. C. ERINGEN and E. S. SUHUBI, Int. 1. Engng Sci. 2, 189,389 (1964). [5] A. C. ERINGEN, J. Math. Mech. 15,909(1966). [6] P. N. KALONI and T. ARIMAN, Z. Angew. Moth. Phys. 12, 136 (1%7). [7] R. D. GAUTHIER and W. E. JAHSMAN, J. Appf. Mech. Trans. ASME, Series E 97,369 (1975). 181 R. D. GAUTHIER and W. E. JAHSMAN, J. Appl. Mech. Trans. ASME, Series E 98,502 (1976). [9] G. V. KRISHNA REDDY and N. K. VENKATASUBRAMANIAN, 1. Appl. Mech. Tmns. ASME 100,429 (1978). [IO] D. IESAN, Anal. Stiint. Univ. lasi. Mathematics 17,483 (1971). [ill. B. M. CHIU and J. D. LEE, Int. J. Engng Sci. 11,997 (1973).
(Received 10 January 1979)