tnr. J. Engng Sci., 1973, Vol. 11, pp. 5 19-530.
Pergmon Press.
THERMAL STRESSES SQUARE CYLINDER
Printed in Great Britain
AND COUPLE-STRESSES IN WITH A CIRCULAR HOLE
Y. TAKEUTI and N. NODA Department of Mechanical Engineering, Shizuoka University, Hamamatsu, .Iapan Abstract-This article deals with the effect of couple-stresses on thermal stress distribution in a square cylinder with a central circular hole under steady state temperature distribution. The analysis is developed by the point-matching technique. From our calculation, it may be concluded that stress concentration around a circular hole, taking couplestress into account, results in lower values than those accepted heretofore.
1. INTRODUCTION
previous papers [ 1,2], we dealt with the classic thermoelasticity for a multiplyconnected region. As several examples of this treatment, we solved the thermal stress distribution of the polygonal cylinder with a central circular hole under heat generation without consideration of couple-stress theory. In the last few years considerable attentions have been given by many investigators to the couple-stress theory. This theory, initiated by W. Voigt and developed by brothers E. and F. Cosserat, takes into account an additional assumption that on the surface of a deformable body there act, besides surface forces, also couples. In recent paper a general theory of couple-stress was established by Mindlin[3] and after this Nowacki proposed an extension of couplestress theory to the thermoelasticity [4]. Applying these theories, in this paper, we solved the problem of the effect of couple-stress on thermal stress distribution in a polygonal cylinder with a central circular hole under stationary temperature distribution. In the process of this treatment, we derived Michell’s conditions for the multiply-connected region with a consideration of couple-stress. The analysis is developed by use of point-matching technique accompanied with the least square method. A simple description of the technique is that the solutions used satisfy the three governing differential equations of heat conduction, Airy’s stress function and Mindlin’s stress function exactly, while approximating the boundary conditions at discrete points on the boundary. Numerical works are carried out for the square region with a circular hole under a steady temperature distribution with no heat source. This research is a part of our studies on two-dimensional thermoelastic problems [S-10].
IN THE
2. ANALYSIS
For steady-state
heat conduction equation in cylindrical coordinates A7(r, 19)= 0.
The general solution of equation (I) is given for non-dimensional
(1) radius
~(r, 6) = A,* + B,* In r’ _tE {(A ,frfmn+ B,*fn) cos no+ (C,*Pn+D,*r’n) n=1 519
sin no}.
(2)
520
Y. TAKEUTI
and N. NODA
The fundamental equation of Airy’s stress function x (r, 0) under stationary perature distribution with no hear source is
tem-
AAx = 0.
(3)
As is known, the general solution of equation (3) in polar coordinates
are given by
x = A0 -t B,-,In r’ + COP + &,P In r’ + (Air’-’ + Blr’ + Clr’ In r’ -I-L&G) cos 6 + (L,r’-‘+ + i n=2
M,r’ + Nlr’ In r’ + 01rf3) sin 8
{ (&fen + B,r’” + C,r’2-n + D,r’ 2+B)cos nfl
(L,r’-n+ M,r’n+N,r’2-n+
+
0,r’2+n) sin ne}.
(4)
The Mindlin’s couple stress function $J is governed by the next differential equation A{+-
(ll~)~A~~ = 0.
(5)
The general solution of equation (5) is by polar form [ 1 I ] I,!J= R’ + So In r’ -I-U,I, (ar’/l) + P’&, (d/l)
+ i [{R,r’-“+S,r’“+U,f,(ar’/l)+F/,K,(ar’~l)}cosnB R=l +{W,r’-n+X,r’n+YptZn(ar’/f)+Z,K,(ar’/f))sinn6].
(6)
The conjugate equations are ~(t,b-12A+)/&‘=-212r’-*. a((1 -v)Ax+a,E~)/d8 r’-‘. ~{~-fzA~)/a~= 2P. a((1 -~)Ax+~tE~}/ar’. The fundamental
stress-strain y11 =
kl
=
IAId.
fJJ391 =
relations in plane strain with couple-stress
-y12 =
(UZ?l
-
y21
=
~,t2),,/2r
c~tv= 2~~2k~~~(I+ ~1, 1+u
- E
Yii=
(h,-tU2,,)/2, k32 =
’ /A k3rk Is3
2
=
xt22-$Ja2r
/43
=
$,I
7
CL23 =
VZl
=
1/1m-X,21r
=
tu2r,
+(~+v)cQ&
The components of stress and couple-stress functions x and I,!Jby the equations Cl1
~392
theory?
u212
(8)
19)
-u,,2),2/2
pzs = 2E~2k~~lt I+ ~1
12 CK(j-lKTkk8~j--E
722 =
(7)
I
f+12 =
(10)
(i&k=
may be expressed
-xm--IJIa2,
a22
1,2).
(11)
in terms of stress
=
XI11 +vh,
f12)
(13)
$52.
In case of a multiply-connected body, the rotation oS and the displacement uI and u2 are to be single-valued. Therefore we have to introduce necessary conditions for these physical conditions. Here we consider S be a connected region bounded by tA comma in the subscript represents
the partial differentiation with respect to the spational coordinate,
e.g. u,,, =
au,/ax,.
Thermal stresses and couple-stresses
521
in square cylinder with a circular hole
N + 1 nonintersecting contours L,, L,, . . ., L, of which L,, contains all the others. Let us begin by deducing the condition for single-valuedness of the rotation, ~L,dws=~L,d(t(UZ,1-Ulr2)}=PL,d(y12-1(l.2)=0. 1
I
1
Then we consider at first f,,
d(u,d =
1
(jL, {( U1,2),1dxI + 1
(U,,2),2dX2} = 0
(i= 1,2,...,N).
Using equation (8) to introduce the strain into the integrand, we have $,, d(u,,,) z
= $,, {Yll,zdx, + (2y,,,,-~Y,,,,) 1
dXJ = 0.
Expressing the strain in terms of the stress function by means of equations (lo), ( I I), ( 12) and (I 3) and rearranging, we obtain the condition for the single-valuedness of the rotation +${(I-u)AX+Eatr}ds The condition for the single-valuedness f,.
t d”l=
1
(14,)
=O.
of the displacement
u1 can be written as
I,.{(ul,A dx, + (u,,,) d-4 = 0. 1
Integration by parts yields
The condition for the single-valuedness of the derivative satisfied. Thus, after introducing the strain, we have x1 {(Umdxl+ $,,du,= - rp,,
=-
u1,2 has already been
U1,ndXz) +Xz(U1,,& + u,mW)
f Li [Cw~~r,+wu,d &+
= 0.
{xly,,,,+x,(2y,,,*--y,,,,)}dx,l
Expressing the strain in terms of the stress function, we finally obtain the following condition for the single valuedness of ul,
f(
x1 $-xz$
Li
((1 -Y)Ax+c&r}dS >
= 0.
(14,)
Similar reasoning leads to the third condition for the single-valuedness x,$+x,$ These
three equations
>
of up,
{(l-~)A~+~tE~}d~=O.
(14,), (14,), ( 143) become the additional
condition
of the
Y. TAKEUTi
522
and N. NODA
stress function in plane strain problem. It is easily seen that the later two equations agree with the classical conditions. For the present problem, it is convenient to express these conditions in polar form, +-${(l-~)Ax+&~T}
I(
a
a
277
cosB.z-r.sint?-G
>
0
I
(1%)
rde=O
Ax.rdB+G Eat
2n
li
o cos8-$--r.sint?*$
r.rdfI=O 1 iI%)
21r
2?r
sinB*$-t-r.cosO*$
1
I( 0
Ax.rdt?+z
J^i0
sinH*$+r.cosH*$
r.rde=O. > (1%)
Expressions
*rr
of thermal stresses and couple-stresses
1ax ~__+l!%__!*+L!2 r ar
r2a0
rarae
r2aO’
I - a2x+l!!3_1?!i_‘iE!! CTrB--‘Yaraer2ae r at- r2ae*' -
in terms of polar form,
a2x I a2+ q@=-$---~-’ r arae
per=---
1 a2x rarae
1 a4 r ae
+lax+a’JI r2a@
’
(16)
at-z‘
As an application of unknown problem, consider the effects of couple-stresses of thermal stress problem in a square prism with a central circular hole under steady temperature distribution with no heat source as shown in Fig. I. Now, assume To is the temperature on the inner boundary of the prism and the temperature on the outer is zero, Considering the symmetry of the body, equation (2)
i----bL
Fig. 1. Square prism with a central circular hole.
Thermal stresses and couple-stresses
in square cylinder with a circular hole
523
becomes
Boundary conditions of temperature
distribution:
where NL is finite integer and represents number of division of angle n/4. In above results and throughout in this paper, we treat the dimensionless values. Boundary conditions of stress distribution:
Because of the symmetrical
arrangement
stress functions become
From equation (7), we obtain next relations
From equations (15,), (15,) and (15,), we obtain next relation a2.&Bn” +4&f
f23)
1 - u) = 0.
Thus, introducing the thermat stresses and the couple stresses equations (20) and (2 1) into equation (16), we have,
+ (4~ - 16n2)r’4n-284n- f16n2+4n-2)r’-4nc4,-t ( 16n2 + 4n)~‘-~@%~~ + (4n - 16$) Y’~‘?x~,
and substituting
(16nZ-4n-22)r’4Rda,
524
-
flee
EatTo
Y. TAKEUTI
and N. NODA
-{4nr”(~)l,,,
(p)-
(16n2+4n)r’-v,,
(p)}&
+{W(;)K,.,
(;rt)+
(16n2+4n)rr-2K,,(~r’)]z~~]~o~4n~
= - r’-2bo + 2c, + (21nr’ + 3) d, + i
[ ( 16n2 + 4n) r’-4n-2u4n 7l=l
‘4n--2b4,,+ (4n - 2) (4n - 1) r’-4nc4n + (4n + 2) (4n + 1) r’ln.dqn
+ ( 16n2-44n)r
- ( 16n2 + 4n) r’-4n-2w4n + ( 16n2 - 4n) r’4n-2x4n +[4nr’-1(~)l,,,(~r’)-(16n2+4n)r’-2Z4.(~r’)}y4~ +[-4nr”(~)K,._,(~r~)-(l6~2+4~)r~~2K4~(~r~)]~4n]~os4~~
ur+3 _
EatTo
c[ N
- ( 16n2 + 4n) r’-4n-2u4n + ( 16n2 - 4n) r’4n-2b4n - ( 16n2 - 4n) r’-4nc4,
n=l
+
(1
6n2 + 4n) r’4nd4n + ( 16n2 + 4n) r’-4n-2w4n + ( 16n2 - 4n) r’4n-2x4n
+{-r’-1(~)14~_,
(Trf)+
(16n2+4n)rfp214~(~r’)}y4~
+(r’-1(~)K4,,(~r’)+(16n2+4n)r’-2K4.($r’)}zt.]sin4nB
(+Br’= EatTo
~[-(l6n2+4n)r’~4”~2a4.+(16nZ-4n)r’4”~2b4~-(l6~2-4n)r’~4nc4. n=l + (16r~~+4n)r’~~d~~+ 2
(16n2+4n)r’-4n-2~4,+
(16n2-4n)r’4n-2x4,
+16n2+4n}r’2/4~(~r’)-(~)r’~114..,(~r’)]y4,
2 +
*
[(
$r'
)
+
16n2 + 4nrre2K 4~(~r’)+(~)re1K4fl-I(~r’)].z4R]sin4nB
= i {4nr’-4n-1w4, 7l=* + 4nr’-114n
(24)
+ 4nrf4*-‘qn
a r’ y4,, + 4nr’-‘K 4~(~rf)z4n}cos4n0 ( 1 )
(25)
Thermal stresses and couple-stresses in square cylinder with a circular hole
525
where small letter’s coefficients mean values of large letter’s coefficients divided by EatTo. Now, we show that temperature and stress distribution must satisfy the boundary conditions. For this purpose the numerical calculation performed to get the unknown coefficients are enormous. Therefore we use the point-matching technique to satisfy the boundary conditions at a selected finite set of outer boundary points of the square region. If we replace niI in equations (17), (20) and (2 1) by I$: approximately,
we have
to solve two kinds of the simultaneous equations for temperature and stress distribution, as the numerical calculation we practiced the cases for b/a = 2,3,4 and for lJ = 0,0*5. For illustrative purpose we show one of the examples of the values of the coefficients for the case a/f = 1, v = 0 and b/a = 3. First, temperature function, becomes += u
1~000000-OG3515201 Inr’+0*7764841
X 10-3(r’-4-r’4)
cos48
+ 0.2362236 x lO-‘j ( P8 - P) cos 88 + 0.1328208 x 1O-s ( r’-12 - P2) cos 128 +0*7407626x
10-‘3(r’-16-P)
cos
168+0*3444857x
10-‘s(r’-20-r’20)
For stress function, we obtain, a, = 0 b. = 0.5553369 co = - 0.3841085 do = 0.2128800
u4 = 0.8932201 x 1O-3 b, = -0.9199141 X 1O-3 c4 = - O-5827528 x 1O-4 d4 = 0.8496928 x 1O-4
us = -0.1597231 X lo+ b, = - 0.2507609 x lo-’ cg = 0.1714054 x 10-G $ = 0.1339377 x lo-’
;:: = ::: =
- -0.7707155 x lo-” ;:I = 0.2058408 x IO-” - 0.5798479 x lo-” :I:~-o.1497319x 10-12
-0.1060516 x lo-l4 b,, = 0.2016184 x lo-14 - -0.818576 x lo-‘5 2: z-0.1370968 x lo-‘5
w4 = -0.2951575 x 1O-2 x4 = O-1845803 x 1O-2 y4 = - 0.3404472 24 = 0.8351701 x 1O-4
wE= X8= ys = 28 =
W 12 = 0.2934916 x lo-’ x 12= -0.1943240 x 1O-s y12 = 0.3030649 x 104 2,2 = - 0.3704055 x 10-18 W 2. = -
0.1244916 x lo-l2 -0.2310117 x lo-‘3 yzo = O-5379038 x 10” 220= 0.1947652 x 1O-35.
X 20 =
UESVd.
I1No.SD
-0.6199635 O-2995625 x 0.3365318 x -0.1613075
x 10-S
10-S 10-S x 10-l”
U z. =
W 1‘3=
0.9126256 x 0.4919045 x - 0.5439708 -0.1461591
1O-5 10-e x 10 x 10-10
0.6956694 X 10-s xl6 = -0.205 1170 X lo-” yls = 0.2365728 x lo8 x 1O-25 Z ,,j = -0.1654380
~0~208.
Y. TAKEUTI
526
t"
0.4
and N. NODA
t l-
w \
0
450
t
b'
-04
~
a/l=1
------
l=c)
Fig. 2. Stress distribution of urSrrover the octant. I.0
0.5 Lo
e B b
45"
0
-05
-1.0
-
a/1=1
-----_
l=O
Fig. 3. Stress distribution of (+88over the octant for v = 0.
0
450
-05
-IO
-1.5 v=o -
5
a/l--l
------~c:o
Fig. 4. Stress distribution of age over the octant for Y = 0.5.
Thermal stresses and couple-stresses
in square cylinder with a circular hole
02 0.1
$
:
450
0
\ 9, b
-01
Fig. 5. Stress distribution of (T~Vover the
OCtant.
I-”
lz \
:
bm
0
450
-0.1
v=o -
a/l:1
-----.
l=O
Fig. 6. Stress distribution of gBreover the octant.
w” \* 1
0
45”
-0.1
15” v=o v
Fig. 7. Couple-stress
0
a/C =I
distribution of prrl over the.octant.
527
528
Y. TAKELJTI
and N. NODA
a/l=1
Fig. 8. Couple-stress
distribution of pez over the octant.
Fig. 9. Relation between max. uesand b/a.
The variations in thermal stresses and couple-stresses are shown in Figs. 2-8. Figure 9 illustrates the relation between max. (see) and b/a, and Fig. 10 illustrates the relation between max. (wee) and u/l. In the rest of the paper we refer to the some check on the accuracy of the point-matching method. We see that the values of ull, cl2 and psrz computated from the approximate solutions of orrr+ oee, uerf, per and prtz on the outer boundary are nearly equal to zero. From our calculation, it may be concluded that stress concentration around a circular hole, taking couple-stress into account, results in lower values than those accepted heretofore.
Thermal stresses and couple-stresses
insquarecylinder with
a circular hole
l+O
o/t FJi#.IO. Relation betwem max. aMand a/l, ~~~~~~~~T~~~ yfj
components ofstrain teasor
components ofstrt%s t@nsor ui components of displacement Kii components of curvature &s corn~~~e~t of rutatbn &5 c~~~e~ts ~fcu~p~~-s~r~sstensar E Y~u~~‘s m#dulus f new material constants as to eoupbstress (C= B/G) B elastic: coefficient of Curvature (48 = HJKJ -cOti Eddington’s notation a, c~~c~e~t of 3inear thermaI expansion v Poisson3 ratio G ~~~~~~S of rigidity p material constant as to thermal stress I/3 G=EffJt I - 2~) ) i& Kronecker delta x Airy”s stress function $6 ~~~d~j~‘Sstress f~~ct~~~ 06
I,, Ka
~~~~~~~ Bessel ~~~~t~~~~
polar coordinates time constant temperature radius ofcircular hole ~~~f~~~~~of square prism ~a~l~c~a~operator I,2orx,y r’ nondimension radius (u”’= v/a) ‘i partial differentiation with regard to r”
r, B
f TO a b B i,j
530
[I] [2] [3] [4] [5] [6] [7] [8] [9] [IO] [ 1I]
Y. TAKEUTI
and N. NODA
REFERENCES Y.TAKEUTIandT.SEKIYA,Z.A.M.M.48,237(1968). Y. TAKEUTI and T. SEKIYA, Proceed. Jup. Nar. Gong. App. Mech. 8,119 (1958). R. D. MINDLIN, Arch. ration. Mech. Analysis 11,415 (1962). W. NOWACKI, Bull. Polo. Sci. 12, 129 (1966). Y. TAKEUTI, Z.A.M.M. 45, 177 (1965). Y. TAKEUTI, Inl. J. Engng Sci. 6,539 (1968). Y.TAKEUTIandN.NODA,Z.A.M.M.50,587(1970). Y. TAKEUTI, Nuclear Eng. Des. 14,201 (I 970). Y. TAKEUTI, Bull. Jup. Sm. Mech. Eng. 10,423 ( 1967). Y. TAKEUTI and K. YAMAZATO, Z.A. M. M. (to be published). W. C. BERT and F. J. APPL, AIAA. J. 6,968 (1968). (Received
9 August 1972)
RCsume-Cet article traite des effets du couple de contrainte sur la distribution d’une contrainte thermique dans un cylindre car& avec un trou central circulaire et avec une distribution de temptrature constante. L’analyse est d&elopp&e au moyen de la technique ponctuelle. A partir de nos calculs, on peut conclure que la concentration de contraintesautour d’un trou circulaire, tenant compte du couple de contrainte, rCsulte en des valeurs plus faibles que celles accepttes jusqu’ici. Zusammenfassung-Diese Arbeit beschstigt sich mit der Wirkung von Krititepaarspannungen auf thermische Spannungsverteilung in einem rechteckigen Zylinder mit einem zentralen kreisfiirmigen Loch unter Stetigzustand-Temperaturverteilung. Die Analyse wird nach dem Punktangleichverfahren entwickelt. Aus unserer Berechnung kann geschlossen werden, dass Spannungskonzentration urn ein kreisftirmiges Loch, wobei Krgftepaarspannung beriicksichtigt wird, zu niedrigeren Werten fiihrt als bisher angenommen. Sommario- In quest0 articolo si tratta l’effetto delle sollecitazioni di accoppiamento sulla distribuzione della sollecitazione termica in un cilindro quadrato con un foro centrale sottoposto a distribuzione di temperatura uniforme. L’analisi viene sviluppata mediante il metodo dell’adattamento dei punti. In base ai nostri calcoli si pu concludere the la concentrazione delle soilecitazioni intorno a un foro circolare, prendendo in considerazione la sollecitazione d’accoppiamento, dh valori pili bassi di quelli sinora accettati. A6npau~ - B pa6o-re HanpSiC‘ZHEi5I TeMII‘ZpaTypbI. BOSMO,KHO IIapHbIX
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