- Email: [email protected]
The effects of dynamical treatment of thermal stresses on a circular region due to continuous point heat source
ht.
I. Engng Sci., 1974, Vol. 12, pp. 261-271.
Pergamon
Press.
Printed in Great Britain
THE EFFECTS OF DYNAMICAL TREATMENT OF THERMAL STRESSES ON A CIRCULAR REGION DUE TO CONTINUOUS POINT HEAT SOURCE YOITIRO TAKEUTI and NAOTAKE NODA Department of Mechanical Engineering, Shizuoka University, Hamamatsu, Japan Abstract-The investigation referring to the effects of inertia term on the plane transient thermal stress problems are treated for a solid circular cylinder under a continuous point heat source. 1. INTRODUCTION
quasi-static approach to transient thermoelastic problems of a point heat source in a finite region was previously established by one of the present authors [ I]. More recently, Hsu independently solved the same problems [2,3]. However, these investigations rest on the assumption that the inertia terms may be neglected in the governing field equations. Then, in the above analysis, time only enters as a parameter from the transient temperature distribution. In modern technology, as the problems of thermal shock are expected to become significant, so that inertia terms have to be taken into account. The first attempt to examine inertia effects in a transient thermoelastic problem is apparently due to Danilovskaya[4], and after this the similar result has been shown by Mura independently[5]. Recently many papers on dynamic thermoelastic problems have been published for infinite regions, but only one analysis for a finite region has been published so far [6]. Therefore, this paper was examined for the problem of the dynamical thermal stress distribution in a circular region under a continuous point heat source acting on its center. The analysis is developed by the Laplace transform. THE CONVENTIONAL
2. ANALYSIS
(i) Axisymmetrical
fundamental
equation for heat conduction
a27/ar2+ r-‘.
aT/ar
=
l/K.
&T/at
(1)
where T = temperature
change,
k = thermal diffusivity, t = time, r = radial component of polar coordinate.
Initial and boundary
conditions
for T hold: (T)*+ = 0
(r = 0)
lim2rr$=-iQ(t) r-+0 (arlar + hTjrso = 0 261
(2) (3) (4)
262
YOITIRO TAKEUTI and NAOTAKE NODA
where h is the surface heat-transfer coefficient, Q(t) is the strength of a point heat source and a is the radius of the circular cylinder. To obtain the solution of equation (l), we introduce the Laplace transform ~*(r,s) of the function T( r,t ) defined by m F*(r,s) = F(r,t) e-” dt. (5) I0 Performing
the Laplace transform
on equations
d=T* drT+;
ldr* ---7 dr
[z+
(l), (3) and (4), we obtain:
s
*=0
(6)
’
k
hP],_
=0,
and limZnr-dd$=-iQ*. r+O where we have made use of the initial condition The solution T* of equation (6) is r*(r,s) =
A(s
of equation (2).
+ B(s)Ko(rd)
(9
where d is the m, and I,, K. are the modified Bessel functions. and (S), we can determine the constants of A and B.
From equations (7)
B(s) = Q” 2&
Q * hKo(ad A(s)=
-!%
) -
dK,(
ad )
’
ht(ad)+dl,(ad)
Thus solution of r* can now be written: T*(r,s)
-
Q* zTk
h{Io(ad
)Ko(rd
I-
IMad
)L(rd
)} + dU,(ad
)Ko(rd
) + Wad
)lo(rd
>I
(1oj
hlo( ad ) + dl,( ad )
Pe~orming the inverse Laplace transform theorem, we obtain 7(r,t) =
*’ Q(t I0
on equation
-
tMr,tJ
(10)
dt,
and using the convolution
(11)
where
(12)
Dynamical treatment of thermal stresses
263
and 6” is the n th positive root of the equation h.ro(&z)- 5Jl(&) = 0 (ii) The dynamical thermoelastic LA,
(13)
displacement equation of motion can be written as: 1 au, u,
-_7=9_+-_r+ar r ar
1 ve
a%, 1 at
ffl
a7
(14)
ar
where u, = ve = crl = y= A,Y = p= (Ye=
radial displacement component, velocity of propagation of the longitudinal wave = d(h f 2p)ly, material constant = /3/(A + 2~), density, Lame’s constants, (Yt(3A+ 2/J)), coefficient of linear thermal expansion.
Initial and boundary
conditions
for u, hold:
(Ur)r=o = (af.frlat),=o = 0 (um)r=o = 0. I Stress-displacement
(15)
relations:
(16)
To obtain the solution of equation (14), we perform the Laplace equation (14) using the initial conditions of equation (15):
transform
on
(17) Utilizing the Lagrange’s method of variation of parameters, (14) is found to be uT(r,s) = {C(s) - Jl*(r,s)}Il(sr/ve)
the solution of equation
+P(s) + rp*(c s)UWrlvd
(18)
where
cp*(r,s)=
a1
cp*(r,s> =
aI
(19) larKI
($r)zdr.
(20)
264
YOITIRO
Substituting
TAKEUTI
and NAOTAKE NODA
equation (10) into equations rL(rdVl(sr/u,)
(19) and (20), we obtain
- (s/u,)
rL(srlv,)lo(rd)dr
rKoWVdsr/ve) -(s/v,) r&(rd)L(srlu,)dr
rKl(sr/veVo(rd)+
(s/v,)
(21)
rL(rd)K,,(srlv,)dr
”
+(s/u,)
r&(rd)Ko(sr/v,)dr
From equations
r
.
(22)
(21) and (22), we find next relations
The displacement
q*“(a,s) = $*(a$) = 0
(23)
d+*(r,s) dr
(24)
u, will now be finite at the center since D(s) = - cp”(O,s).
From the boundary
(25)
condition of equation (15), we may determine
s){(h + 2P@o(3 +243)
paT*(a, s)- cp”(O,
(26)
C(s) =
(A +2&+(F)
- 243
Thus we have u?(r,s)={C(s)-6*(r,s)}I,
($j+{D(s)+p*(r,s)}K,(~).
By evaluating integral in equations uT(r, s) =
(27)
(21) and (22), we obtain
(s’+ pov%la)ll(sr/u,)
(YIQ* -2
s(uf-
ks){hlo(da)+
dl,(da)}{To($-
poI,($}
[
-
s~{K,($(:) +Io(+&)} +wo{K(~)I@ -I($K(z)} (vi-
ks$+da)-Poq$} 1
+& h{Ko(da)ll(rd) k
in which p0 = 2p/(A + 2~).
+ L(da)K,(rd)}+ d{ll(da)Kl(rd) d(ul- ks){hL(da) + dl,(da)}
- K,(da)l,(rd2)
cw
265
Dynamical treatment of thermal stresses
We perform the inverse Laplace transform on equation (28). Using the convolution theorem, we obtain u&t) = OrQ(t - t,)u,(r,tl)dtt I
(29)
where
x
M&Z,,,) COS(fm -
@.@m 1)+ s M&Z,,) cos (m t -61G)+~) ha
>)I
- e,(z,,,)) + 2 M&T,,) sintm-e,(zm)+~
in which rim is the nth positive root of the equation T/Jo(s) - E.LOJl(r)) = 0, and Mi(Zm) and &(Zm) denote functions bet and bei;
(31)
next relation between
M, (Z,,) = (be&G + bei%,)‘” 4 (Z, ) = tan-’ @Xii 2% /berfZ:, )
the Kelvin’s Bessel
(i = 0,1).
In above expressions the following notations are introduced for simplicity.
(32)
266
YOITIRO
TAKEUTI
and NAOTAKE
NODA
(33) Therefore the components of stress can be derived from the general solution of displacement uI in equation (29). Now, we shall discuss a particular example of Q(t) = constant. Setting Q(t) = Q. in equations (11) and (29), and performing the required integration, it is found that (1
e-kc*)
-
(34)
&fo(Zm)M,(Zm)cos x
M’(Zm)
x
M&z,
+
+
(2)
:,z
(&(Zm)
-
wzn)
+
T/4)
m /ha)
){sin (t, - &(Z, )) - sin Mzm ))I
2 M,(z,
XCOS
e
m )+2(2
(t, -
eo(zm 1)-
~0s
eGm )I
[
+$M,(z,)(~~~ (t-
-
e,vd++os
(ww$]] (35)
We define the dimensionless
quantities as follows:
267
Dynamical treatment of thermal stresses
p=f,
K=&,
** =
t,=$
c
t,,,=yh,,
H=ha,
kurr ” = cxtQo(h+ 2~)’
km% crtQo(h+ 2~ )’
Thus final expressions for temperature, sionless quantities are given as:
displacement
(36) U=--$.
* 0
and thermal stresses in dimen-
(37)
Jl(Pl)rn)
X
2
M;(Z, ) + $M:(Z x jt&(Z,){sin (t, [
) + 2$$iwO(2~ )M,(Z, )cos
Mo(zm ){c0s
eO(Zm)- MZm ) + :
>
e,(z,))+ sineo(zm )I+fj”ICzm) (e4zkJJ)
x { en. ( t, -B,(Zm)+$-sin + ~~~
(
eo(zm >I-
(t, -
x{cos (t_ -e,(zd++o~
~0s
eo(zmN+ fj M1cZm )
(cl(z)-f)}]]
1 1_,-a?
_ =J KpoJl(p~,,,){K~,(l - cos L) - sin tm) ,,,=I ~,(l+ K*~‘,){1-(~)72m+rf)IJ1(Vm)Jd%n)
p;:
x
(1 +K2P;)J;(pn)
ylKp:Io(Kp:p)+F~(Kp:p)
-3,,,=, q,,,(l+
(38)
-P”(K~P:+cLO)JO(PP~)+~~I(PP~)
K(Po-g) K’rlfiJ{l -(~i+Y:)}~l(%)
1
268
YOITIRO
X
TAKEUTI
and
y*q*Jo(pqm
>+
2
H* &.g(Z”) M&z”)+~
+ 2
$
NAOTAKE
7
NODA
JlkJ”) cm
Mo(z”)A4l(z”)
(wzn) - MZ”) + 41
x A&(Z,){sin (t, - &(Z, 1) + sin OO(Zm )I + +MI(Zm 1 [ x[sin(t,-eI(Zm)+$-sin(81(ZmI-$] + Kq,
[
MO(z,){~O~ (t, -
m &o
-lxml=,
=
T(l+
qm(I+ K2qZm){l -(qfL+
(q
+ PO)JI(Pn)
%(KP :I - PJI(KP :I 1
1 iwo(zm)m(z,)
cos
eo(zm I-
Mo(Zm){sin (t, - eo(zm )) + sin eo(zm )) + $+MI(Z,(2
- sin
I (tm-e,(z)+$
x cos
(39)
- ( K2p3.Jo(pnp) + 7 Jdpnp)
2
M~(zm)+$f:(zm)+2~ x
*
p’, l+$ (1-t K2p:)J:(pn) ( 3
KpzJo(Kp2,p)-p POMKdp)}
X
1
rf)IJ1(Vm)Jo(Vm)
1 _ e-p:*o
v)
$ MICZ*)
r~qml,(pqm)+%J~(~qm)]~Ktlm(l --OS t”)-Sin t”)
1-v pp
eo(zm 1)- cm eo(zm)l +
1
h(z)
+ 4
tm - e,(z,)+T
Mo(Z,,,){cos (t, - &dZ,)) - ~0s eo(%)) + %MlGJ
>
-COS
(e,(zm)-$ Nl
where pn is the n th positive root of the equation HJo(p) - PJdP 1 = 0.
(41)
Dynamical treatment of thermal stresses
269
Fig. 1. Distributions of temperature (H = m, K = 1.935 X IO-').
Fig. 2. Distributions of hoop stress (H = m,K = 1.935 x lo-‘, v = 4).
The numerical calculations on the distributions of temperature and hoop stress were carried. The distributions of temperature and hoop stress are shown in Figs. 1 and 2, respectively. Figure 3 illustrates the variation of the hoop stress at the boundary p = 1 after the thermal shock. Throughout the computations, the dimensionless quantity K-’ has been taken as 5.71 x 106. In the heat conduction calculation we adopt the value of N = 03. 3. CONCLUSION
In un-steady thermoelastic problems, when the heating rate is slow, it is obviously reasonable to neglect the inertia terms in the equations of motion. However, in modern
270
YOITIRO
Fig. 3. Variation of hoop
TAKEUTI
and NAOTAKE
NODA
stress at the boundary p = 1 after the thermal shock (H = ~0, K = 1.935 x lo-‘, ” = t, p = 1).
technology, the more strict theoretical analyses are required to apply for the thermal shock problems. In this paper we developed an analysis for the case of dynamical treatment of thermal stress problem in a finite region under a heat source. In solution equation (38) of displacement U, the first term is as same as the solution in the theory neglected inertia term. The additive terms mean vibration due to dynamic effect of thermal shock. For the engineering material, the maximum amplitude of these vibrating terms are so small comparing with the first term. It is noticed that there are large differences in time interval between the abscissa of Fig. 2 and the one of Fig. 3. For instance, to/K = 1 in Fig. 3 is equivalent to the value of to = 1a93 x lo-’ in dimensional form for ordinal steel, so we can not indicate the dynamic effect in Fig. 2. Moreover, it is observed that the value of actual dimensional radius becomes a = 0.967 x 10m6cm for ordinal steel in Sternberg’s result [7] when we take v = l/3, p = 7.85 grlcm”, K = 0.12 cm*/sec, and p = 7.55 x 10” dynes/cm’. From our results, it may be concluded that the inertia term does not give so large influence on the thermal stress distribution in the ordinary industrial materials and for the interval of short period, i.e. nondimensional time to=zO*O1.Owing to exaggerate the effects of inertia term, Sternberg and others’ calculations took so large value of K, such as non-actual materials. Finally, in the rest of this paper, we refer to some remaining problem. In our treatment the temperature distribution in the medium is assumed to obey the heat conduction equation in the absence of thermomechanical coupling. It seems likely that one can obtain the more strict result with consideration of thermomechanical coupling effect. REFERENCES [l] [2] [3] [4]
Y. TAKEUTI, Bull. Jap. Sot. Mech. Engng 10, 423 (1%7). T. R. HSU, Trans. ASME E-36, 113 (1%9). T. R. HSU, Trans. ASME, B-92, 357 (1970). V. I. DANILOVSKAYA, Prik. Mat. Mekh 14,316 (1950). Also B. A. BOLEY and J. H. WEINER, Theory of Thermal Stresses. Wiley (1960). W. NOWACKI, Themoelasticity. Pergamon (1%2). [5] T. MURA, Proc. 2nd. Jap. natn. Gong. appl. Mech. p. 910 (1952). Also, T. MURA, Res. Rep. Fat. Engng Meiji Uniu 8, 63 (1956). [6] R. S. DHALIWAL and K. L. CHOWDHURY, Arch. Mech. Stos. 20, 47 (1%8). [7] E. STERNBERG and J. G. CHAKRAVORTY, Trans. ASME E-26, 503 (1959). (Receiued 22 April 1973)
Dynamical RbumC-Les recherches concernant planes transitoires sont effectuCes ponctuelle continue.
treatment
of thermal
271
stresses
les effets du terme inertiel dans les problemes de contraintes pour un cylindre circulaire solide sous I’effet d’une source
thermiques de chaleur
Zusammenfassung-Die Untersuchungen in Bezug auf die Wirkungen des Trlgheitsgliedes auf das ebene transiente WSrmespannungsproblem werden fiir einen festen kreisfarmigen Zylinder unter einer stetigen Punktwlrmequelle behandelt. Sommari+L’indagine piane viene applicata
degli effetti del fattore inerzia nei problemi delle sollecitazioni termiche transitorie ad un cilindro circolare solid0 sotto una sorgente di calore continua puntiforme.
A~CT~WT- kcnenoBaHo BnllRHAe rneHa YeCKOrO HalTpaXeHHa IfelTpepbtBHOrO Tenna.
~JIR RHepuAH Ha npO6neMbI B CJIy’fae TBepAOrO Kpyi-OBOrO uRJIHHspa npH
IIJIOCKO~O nepexoAHoro X~MHTOYe’lHOrO ACTOSHHKB
&kTBHA
I. Engng Sci., 1974, Vol. 12, pp. 261-271.
Pergamon
Press.
Printed in Great Britain
THE EFFECTS OF DYNAMICAL TREATMENT OF THERMAL STRESSES ON A CIRCULAR REGION DUE TO CONTINUOUS POINT HEAT SOURCE YOITIRO TAKEUTI and NAOTAKE NODA Department of Mechanical Engineering, Shizuoka University, Hamamatsu, Japan Abstract-The investigation referring to the effects of inertia term on the plane transient thermal stress problems are treated for a solid circular cylinder under a continuous point heat source. 1. INTRODUCTION
quasi-static approach to transient thermoelastic problems of a point heat source in a finite region was previously established by one of the present authors [ I]. More recently, Hsu independently solved the same problems [2,3]. However, these investigations rest on the assumption that the inertia terms may be neglected in the governing field equations. Then, in the above analysis, time only enters as a parameter from the transient temperature distribution. In modern technology, as the problems of thermal shock are expected to become significant, so that inertia terms have to be taken into account. The first attempt to examine inertia effects in a transient thermoelastic problem is apparently due to Danilovskaya[4], and after this the similar result has been shown by Mura independently[5]. Recently many papers on dynamic thermoelastic problems have been published for infinite regions, but only one analysis for a finite region has been published so far [6]. Therefore, this paper was examined for the problem of the dynamical thermal stress distribution in a circular region under a continuous point heat source acting on its center. The analysis is developed by the Laplace transform. THE CONVENTIONAL
2. ANALYSIS
(i) Axisymmetrical
fundamental
equation for heat conduction
a27/ar2+ r-‘.
aT/ar
=
l/K.
&T/at
(1)
where T = temperature
change,
k = thermal diffusivity, t = time, r = radial component of polar coordinate.
Initial and boundary
conditions
for T hold: (T)*+ = 0
(r = 0)
lim2rr$=-iQ(t) r-+0 (arlar + hTjrso = 0 261
(2) (3) (4)
262
YOITIRO TAKEUTI and NAOTAKE NODA
where h is the surface heat-transfer coefficient, Q(t) is the strength of a point heat source and a is the radius of the circular cylinder. To obtain the solution of equation (l), we introduce the Laplace transform ~*(r,s) of the function T( r,t ) defined by m F*(r,s) = F(r,t) e-” dt. (5) I0 Performing
the Laplace transform
on equations
d=T* drT+;
ldr* ---7 dr
[z+
(l), (3) and (4), we obtain:
s
*=0
(6)
’
k
hP],_
=0,
and limZnr-dd$=-iQ*. r+O where we have made use of the initial condition The solution T* of equation (6) is r*(r,s) =
A(s
of equation (2).
+ B(s)Ko(rd)
(9
where d is the m, and I,, K. are the modified Bessel functions. and (S), we can determine the constants of A and B.
From equations (7)
B(s) = Q” 2&
Q * hKo(ad A(s)=
-!%
) -
dK,(
ad )
’
ht(ad)+dl,(ad)
Thus solution of r* can now be written: T*(r,s)
-
Q* zTk
h{Io(ad
)Ko(rd
I-
IMad
)L(rd
)} + dU,(ad
)Ko(rd
) + Wad
)lo(rd
>I
(1oj
hlo( ad ) + dl,( ad )
Pe~orming the inverse Laplace transform theorem, we obtain 7(r,t) =
*’ Q(t I0
on equation
-
tMr,tJ
(10)
dt,
and using the convolution
(11)
where
(12)
Dynamical treatment of thermal stresses
263
and 6” is the n th positive root of the equation h.ro(&z)- 5Jl(&) = 0 (ii) The dynamical thermoelastic LA,
(13)
displacement equation of motion can be written as: 1 au, u,
-_7=9_+-_r+ar r ar
1 ve
a%, 1 at
ffl
a7
(14)
ar
where u, = ve = crl = y= A,Y = p= (Ye=
radial displacement component, velocity of propagation of the longitudinal wave = d(h f 2p)ly, material constant = /3/(A + 2~), density, Lame’s constants, (Yt(3A+ 2/J)), coefficient of linear thermal expansion.
Initial and boundary
conditions
for u, hold:
(Ur)r=o = (af.frlat),=o = 0 (um)r=o = 0. I Stress-displacement
(15)
relations:
(16)
To obtain the solution of equation (14), we perform the Laplace equation (14) using the initial conditions of equation (15):
transform
on
(17) Utilizing the Lagrange’s method of variation of parameters, (14) is found to be uT(r,s) = {C(s) - Jl*(r,s)}Il(sr/ve)
the solution of equation
+P(s) + rp*(c s)UWrlvd
(18)
where
cp*(r,s)=
a1
cp*(r,s> =
aI
(19) larKI
($r)zdr.
(20)
264
YOITIRO
Substituting
TAKEUTI
and NAOTAKE NODA
equation (10) into equations rL(rdVl(sr/u,)
(19) and (20), we obtain
- (s/u,)
rL(srlv,)lo(rd)dr
rKoWVdsr/ve) -(s/v,) r&(rd)L(srlu,)dr
rKl(sr/veVo(rd)+
(s/v,)
(21)
rL(rd)K,,(srlv,)dr
”
+(s/u,)
r&(rd)Ko(sr/v,)dr
From equations
r
.
(22)
(21) and (22), we find next relations
The displacement
q*“(a,s) = $*(a$) = 0
(23)
d+*(r,s) dr
(24)
u, will now be finite at the center since D(s) = - cp”(O,s).
From the boundary
(25)
condition of equation (15), we may determine
s){(h + 2P@o(3 +243)
paT*(a, s)- cp”(O,
(26)
C(s) =
(A +2&+(F)
- 243
Thus we have u?(r,s)={C(s)-6*(r,s)}I,
($j+{D(s)+p*(r,s)}K,(~).
By evaluating integral in equations uT(r, s) =
(27)
(21) and (22), we obtain
(s’+ pov%la)ll(sr/u,)
(YIQ* -2
s(uf-
ks){hlo(da)+
dl,(da)}{To($-
poI,($}
[
-
s~{K,($(:) +Io(+&)} +wo{K(~)I@ -I($K(z)} (vi-
ks$+da)-Poq$} 1
+& h{Ko(da)ll(rd) k
in which p0 = 2p/(A + 2~).
+ L(da)K,(rd)}+ d{ll(da)Kl(rd) d(ul- ks){hL(da) + dl,(da)}
- K,(da)l,(rd2)
cw
265
Dynamical treatment of thermal stresses
We perform the inverse Laplace transform on equation (28). Using the convolution theorem, we obtain u&t) = OrQ(t - t,)u,(r,tl)dtt I
(29)
where
x
M&Z,,,) COS(fm -
@.@m 1)+ s M&Z,,) cos (m t -61G)+~) ha
>)I
- e,(z,,,)) + 2 M&T,,) sintm-e,(zm)+~
in which rim is the nth positive root of the equation T/Jo(s) - E.LOJl(r)) = 0, and Mi(Zm) and &(Zm) denote functions bet and bei;
(31)
next relation between
M, (Z,,) = (be&G + bei%,)‘” 4 (Z, ) = tan-’ @Xii 2% /berfZ:, )
the Kelvin’s Bessel
(i = 0,1).
In above expressions the following notations are introduced for simplicity.
(32)
266
YOITIRO
TAKEUTI
and NAOTAKE
NODA
(33) Therefore the components of stress can be derived from the general solution of displacement uI in equation (29). Now, we shall discuss a particular example of Q(t) = constant. Setting Q(t) = Q. in equations (11) and (29), and performing the required integration, it is found that (1
e-kc*)
-
(34)
&fo(Zm)M,(Zm)cos x
M’(Zm)
x
M&z,
+
+
(2)
:,z
(&(Zm)
-
wzn)
+
T/4)
m /ha)
){sin (t, - &(Z, )) - sin Mzm ))I
2 M,(z,
XCOS
e
m )+2(2
(t, -
eo(zm 1)-
~0s
eGm )I
[
+$M,(z,)(~~~ (t-
-
e,vd++os
(ww$]] (35)
We define the dimensionless
quantities as follows:
267
Dynamical treatment of thermal stresses
p=f,
K=&,
** =
t,=$
c
t,,,=yh,,
H=ha,
kurr ” = cxtQo(h+ 2~)’
km% crtQo(h+ 2~ )’
Thus final expressions for temperature, sionless quantities are given as:
displacement
(36) U=--$.
* 0
and thermal stresses in dimen-
(37)
Jl(Pl)rn)
X
2
M;(Z, ) + $M:(Z x jt&(Z,){sin (t, [
) + 2$$iwO(2~ )M,(Z, )cos
Mo(zm ){c0s
eO(Zm)- MZm ) + :
>
e,(z,))+ sineo(zm )I+fj”ICzm) (e4zkJJ)
x { en. ( t, -B,(Zm)+$-sin + ~~~
(
eo(zm >I-
(t, -
x{cos (t_ -e,(zd++o~
~0s
eo(zmN+ fj M1cZm )
(cl(z)-f)}]]
1 1_,-a?
_ =J KpoJl(p~,,,){K~,(l - cos L) - sin tm) ,,,=I ~,(l+ K*~‘,){1-(~)72m+rf)IJ1(Vm)Jd%n)
p;:
x
(1 +K2P;)J;(pn)
ylKp:Io(Kp:p)+F~(Kp:p)
-3,,,=, q,,,(l+
(38)
-P”(K~P:+cLO)JO(PP~)+~~I(PP~)
K(Po-g) K’rlfiJ{l -(~i+Y:)}~l(%)
1
268
YOITIRO
X
TAKEUTI
and
y*q*Jo(pqm
>+
2
H* &.g(Z”) M&z”)+~
+ 2
$
NAOTAKE
7
NODA
JlkJ”) cm
Mo(z”)A4l(z”)
(wzn) - MZ”) + 41
x A&(Z,){sin (t, - &(Z, 1) + sin OO(Zm )I + +MI(Zm 1 [ x[sin(t,-eI(Zm)+$-sin(81(ZmI-$] + Kq,
[
MO(z,){~O~ (t, -
m &o
-lxml=,
=
T(l+
qm(I+ K2qZm){l -(qfL+
(q
+ PO)JI(Pn)
%(KP :I - PJI(KP :I 1
1 iwo(zm)m(z,)
cos
eo(zm I-
Mo(Zm){sin (t, - eo(zm )) + sin eo(zm )) + $+MI(Z,(2
- sin
I (tm-e,(z)+$
x cos
(39)
- ( K2p3.Jo(pnp) + 7 Jdpnp)
2
M~(zm)+$f:(zm)+2~ x
*
p’, l+$ (1-t K2p:)J:(pn) ( 3
KpzJo(Kp2,p)-p POMKdp)}
X
1
rf)IJ1(Vm)Jo(Vm)
1 _ e-p:*o
v)
$ MICZ*)
r~qml,(pqm)+%J~(~qm)]~Ktlm(l --OS t”)-Sin t”)
1-v pp
eo(zm 1)- cm eo(zm)l +
1
h(z)
+ 4
tm - e,(z,)+T
Mo(Z,,,){cos (t, - &dZ,)) - ~0s eo(%)) + %MlGJ
>
-COS
(e,(zm)-$ Nl
where pn is the n th positive root of the equation HJo(p) - PJdP 1 = 0.
(41)
Dynamical treatment of thermal stresses
269
Fig. 1. Distributions of temperature (H = m, K = 1.935 X IO-').
Fig. 2. Distributions of hoop stress (H = m,K = 1.935 x lo-‘, v = 4).
The numerical calculations on the distributions of temperature and hoop stress were carried. The distributions of temperature and hoop stress are shown in Figs. 1 and 2, respectively. Figure 3 illustrates the variation of the hoop stress at the boundary p = 1 after the thermal shock. Throughout the computations, the dimensionless quantity K-’ has been taken as 5.71 x 106. In the heat conduction calculation we adopt the value of N = 03. 3. CONCLUSION
In un-steady thermoelastic problems, when the heating rate is slow, it is obviously reasonable to neglect the inertia terms in the equations of motion. However, in modern
270
YOITIRO
Fig. 3. Variation of hoop
TAKEUTI
and NAOTAKE
NODA
stress at the boundary p = 1 after the thermal shock (H = ~0, K = 1.935 x lo-‘, ” = t, p = 1).
technology, the more strict theoretical analyses are required to apply for the thermal shock problems. In this paper we developed an analysis for the case of dynamical treatment of thermal stress problem in a finite region under a heat source. In solution equation (38) of displacement U, the first term is as same as the solution in the theory neglected inertia term. The additive terms mean vibration due to dynamic effect of thermal shock. For the engineering material, the maximum amplitude of these vibrating terms are so small comparing with the first term. It is noticed that there are large differences in time interval between the abscissa of Fig. 2 and the one of Fig. 3. For instance, to/K = 1 in Fig. 3 is equivalent to the value of to = 1a93 x lo-’ in dimensional form for ordinal steel, so we can not indicate the dynamic effect in Fig. 2. Moreover, it is observed that the value of actual dimensional radius becomes a = 0.967 x 10m6cm for ordinal steel in Sternberg’s result [7] when we take v = l/3, p = 7.85 grlcm”, K = 0.12 cm*/sec, and p = 7.55 x 10” dynes/cm’. From our results, it may be concluded that the inertia term does not give so large influence on the thermal stress distribution in the ordinary industrial materials and for the interval of short period, i.e. nondimensional time to=zO*O1.Owing to exaggerate the effects of inertia term, Sternberg and others’ calculations took so large value of K, such as non-actual materials. Finally, in the rest of this paper, we refer to some remaining problem. In our treatment the temperature distribution in the medium is assumed to obey the heat conduction equation in the absence of thermomechanical coupling. It seems likely that one can obtain the more strict result with consideration of thermomechanical coupling effect. REFERENCES [l] [2] [3] [4]
Y. TAKEUTI, Bull. Jap. Sot. Mech. Engng 10, 423 (1%7). T. R. HSU, Trans. ASME E-36, 113 (1%9). T. R. HSU, Trans. ASME, B-92, 357 (1970). V. I. DANILOVSKAYA, Prik. Mat. Mekh 14,316 (1950). Also B. A. BOLEY and J. H. WEINER, Theory of Thermal Stresses. Wiley (1960). W. NOWACKI, Themoelasticity. Pergamon (1%2). [5] T. MURA, Proc. 2nd. Jap. natn. Gong. appl. Mech. p. 910 (1952). Also, T. MURA, Res. Rep. Fat. Engng Meiji Uniu 8, 63 (1956). [6] R. S. DHALIWAL and K. L. CHOWDHURY, Arch. Mech. Stos. 20, 47 (1%8). [7] E. STERNBERG and J. G. CHAKRAVORTY, Trans. ASME E-26, 503 (1959). (Receiued 22 April 1973)
Dynamical RbumC-Les recherches concernant planes transitoires sont effectuCes ponctuelle continue.
treatment
of thermal
271
stresses
les effets du terme inertiel dans les problemes de contraintes pour un cylindre circulaire solide sous I’effet d’une source
thermiques de chaleur
Zusammenfassung-Die Untersuchungen in Bezug auf die Wirkungen des Trlgheitsgliedes auf das ebene transiente WSrmespannungsproblem werden fiir einen festen kreisfarmigen Zylinder unter einer stetigen Punktwlrmequelle behandelt. Sommari+L’indagine piane viene applicata
degli effetti del fattore inerzia nei problemi delle sollecitazioni termiche transitorie ad un cilindro circolare solid0 sotto una sorgente di calore continua puntiforme.
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