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Thermal stresses in a heat generating elastic-plastic cylinder with free ends
ht. J. En~n~ Sci. Vol. 32. No. 6, pp. 883-X%, 1994
Pergamon
Copyright 0 1994Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225,‘94$7.(W) + O.(Xt
THERMAL STRESSES IN A HEAT GENERATING ELASTIC-PLASTIC CYLINDER WITH FREE ENDS YUSUF ORCAN Department of Engineering Sciences, Middle East Technical University, 06.531 Ankara, Turkey (Communicated
by H. DEMIRAY)
Abstract-The exact solution of the distribution of stress and deformation in an elastic-perfectly plastic cylindrical rod with uniform internal heat generation under generalized plane strain condition is presented. The treatment is based on Tresca’s yield condition and the associated flow rule. The stress image points of the outer and inner plastic regions lie on two different edges of Tresca prism. With increasing heat generation parameter, the image points of the intermediate regions spread out through two side regimes and meet at another edge regime in the fully plastic state.
I.
INTRODUCTION
stresses in a body due to non-uniform temperature distribution play a very important role in many engineering designs and operation. In the wide domain of application two particular examples are the heating of wires conducting direct current and the solid nuclear fuel elements in which internal energy is generated by nuclear fission. The elastic solutions of the problem are studied in detail, e.g. [l-3], and since the results are genera1 they can easily be adapted to any kind of temperature distribution. However, in the design of such elements power generation can be substantially increased if the material is allowed to yield. Furthermore an elastic-plastic analysis is necessary to improve the low cycle fatigue life of the material. The thermoelastic-plastic deformation of a solid cylinder in the presence of a distributed Gaussian heat source was considered by Kammash et al. [4] using Tresca’s yield condition and its associated flow rule. They paid particular attention to a cylindrical nuclear fuel element exhibiting linear strain hardening surrounded by an elastic cladding. Due to the assumption of plane strain fez = 0) initial yield occurs at the axis of the cylinder and the plastic region which is composed of two different zones expand outward. The outer elastic region vanishes and the cylinder becomes fully plastic when the elastic-plastic boundary reaches the surface. Nevertheless, the cylinder could sustain further loading, and since in the fully plastic state the fuel element consists of only two plastic zones Kammash er al. were able to carry the analysis further without much difficulty. Recently Organ and Gamer [5] investigated the elastic-perfectly plastic behaviour of a centrally heated solid cylinder with fixed ends. For all possible ratios of hot core diameter to outer diameter of the cylinder the temperature of the core was increased up to the value for which the cylinder becomes fully plastic. One of the interesting findings was a critical core radius which leads to instantaneous onset of plastic flow in the entire outer elastic region in spite of the existing inhomogeneous stress distribution. In the present investigation the complete solution of the elastic-perfectly plastic cylindrical rod with uniform internal heat generation under the state of generalized plane strain (free ends) is presented. Leading to closed form solution Tresca’s yield condition and its associated flow rule is used. The material parameters E, CT,), v and a! are assumed to be independent of temperature. Unlike the case considered by Kammash et al. [4], in the absence of any lateral pressure and axial force initial yield occurs at the surface where the circumferential stress is equal to the axial stress. The stress state at the surface and at its neighbourhood lies in an edge regime of
Thermal
883
Y. ORCAN
884
D
Fig. 1. Stress trajectories of fixed material points lying on the current interfaces.
Tresca prism with uB = oZ > a,. In this first stage of plastic deformation, for the nondimensional internal heat generation parameter 9“’ = 9’;‘. two different plastic zones emerge simultaneously at the surface and propagate inwards. The yield conditions in these plastic zones, plastic zone I and plastic zone II, read cB - a, = cry - a, = cr,, and (T@- a, = cr,,, respectively. The first stage of plastic deformation is terminated when the stresses at the axis of the cylinder attain the values ue = g, = $a, = --go for #” = 4’. This state of stress at the axis corresponds to another edge regime of Tresca prism and thus two additional plastic zones, plastic zone III and plastic zone IV, emerge at the centre and expand outwards. The yield conditions read a6 - eZ = a, - (T, = a0 and gB - g7 = go respectively. With increasing load parameter, the intermediate elastic layer gets narrower and vanishes entirely for @” = &‘. Figure 1 shows thematically the stress image trajectories of various fixed material points on principal stress space. The image of the lateral surface r = b moves directly to the edge B of Tresca prism and stays there throughout the course of plastic deformation. A material point lying in plastic zone I, e.g. on the border Y= r,, first moves to the side regime BC and with ongoing plastic deformation migrates to the edge B. The image of a fixed material point on the elastic-plastic interface, r = r,, has just reached the side BC for the current load parameter under consideration. The material points in plastic zone II where plastic Ilow is governed by the side regime BC are in fact lying at different locations from point N to B. On the other hand, the image of the axis moves to the edge D, and remains there while those of plastic zone III travel along the face CD to the edge D. The material points of plastic zone IV are spread out along the side from point K to L>. Finally it is noteworthy that the image of the material points on the radius r = rP in the fully plastic state (see Section 5) move directly to a position on edge Cwitha,>Oandrr,=s;
SOLUTION
AND
ONSET
OF
YIELD
The equation of equilibrium
(2.1) and the geometrical
relations El=--,
Ef)
dr
=
-
r
(2.2a.b)
Thermal stresses in an elastic-plastic cylinder
885
hold in the entire rod irrespective of the material behaviour. In the purely elastic body, the generalized Hooke’s law reads EE, = a, - v(ao + (a,) + EaT
(2.3)
Eee = ue - v(mz + a,) + EaT
(2.4)
E.cz = a, - ~(a~ + cre) + EaT.
(2.5)
Consideration of a rod with free ends leads to the generalized E, = co~sf~~r. The compatibility relation
h?) = Q,
f
plane strain condition,
(2.6)
obtained by the elimination of u from the geometrical relation (2.2a,b) and the use of stress function cp, such that a, = rp/r and oe = d4,ldr satisfy the equilibrium equation (2.1), lead to a differential equation which can be easily solved to yield (2.7)
(2.8)
(2.9) where 8:=f Therewith the radial displacement
‘Trdr. I0
is given by (2.10)
For a solid cylinder CZ = 0, and the condition of vanishing radial stress at the outer surface, o,(b) = 0, yields +2Ea
1 - v O(b)
(2.11)
with B(b) = 3 1’ Tr dr. 0 Hence, the elastic solution is given by u, = fi
[e(b) - f9]
+=E[O(b)+O-T] a, = sv u = (,(sp
[2@(b) - T] f EE, + (1 - 2y)B(b)] - T) - ,i),
(2.12)
(2.13) (2.14) (2.15)
Y. ORCAN
886
IJsing the free end condition (2..16)
the axial strain in the elastic range is obtained to be E, = 2&(b)
(2. I?)
and the axial stress can now be expressed as
(2.18)
2. I Temperature distribution In a cylindrical rod with uniform internal energy generation the steady state temperature distribution is given by
q”’ per unit volume per unit time
(2.20)
In a cylindrical rod with uniform internal energy generation and free ends plastic flow sets in at the outer surface due to tensile circumferential and axial stresses. By setting gH(b) = (r;(b) --~~~from (2.13) or (2.14) one obtains
;“y [2@(b) which, upon substitution
- T(b)] = (r,,
(2.21)
of T(b) and B(b) gives the load parameter (2.22)
for the initiation of yielding at the outer surface of the rod.
3. FIRST
STAGE
OF
PLASTIC
~EF~R~lAT~ON
3.1 The elastic-plastic behaviour
In the first stage of plastic deformation the rod is composed of an elastic region at the centre 0 CCr < r,, surrounded by the outer plastic region which consists of two zones: plastic zone I for r4 < r 5 b and plastic zone II for r, < r < r,. The stress and deformation fields in the plastic zones I and II are derived as follows. Plastic zone I: r, < r 5 b
The stress state lies in an “edge-regime” condition takes the form ge - 0; = Q-0,
of Tresca’s prism with f18 = g, and the yield CT,- 0; = Go
(3.1)
Thermal
stresses
in an elastic-plastic
cylinder
n87
of the equilibrium equation (2.1) together with the first expression of the yield condition (2.23) gives g+e= crZ= ao(l + In r) + C3
(3.2)
a, = u. In r + Cj.
(3.3)
Due to plastic incompressibility, EI:= 0, the dilatation consists of its elastic and thermal parts only. Using generalized Hooke’s law, Eeii = (1 - 2v)cr, + 3EcxT
(3.4)
and the geometrical relations (2.2a,b), one obtains
$)ru)=y
[a,)(2 + 3 In Y) + 3C3] + 3aT - l,
(3.5)
with the solution (3.6) Knowing the radial displacement field the plastic parts of the strain components are calculated by subtracting the elastic and thermal parts from the total strain components. They read Epe=~(ui,lnr+C,)-~~,,+~[30-T]-~t,+$ lPr = y
(u.
ep=E Z 3
(3.7)
7-6~ In r + C,) + ~~o1[3&2T]-ft;-+
i [(l - v)cro -t (1 - 2v)(uoln r + C,)] - (YT.
r
(34
(3.9)
Plasm zone II: r3 < r I r4 In plastic zone II which develops simultaneously
together with plastic zone I according to the inequality a, < u, < ue the yield condition has the form oe - u, = u,,.
(3.10)
Substituting the yield condition (3.10) in the equation of equilibrium (2.1) and integrating, one obtains a, = ug In r + Cs
(3.11)
u0 = uo(ln r + 1) + Cs.
(3.12)
The flow rule associated with the yield condition (3.10) leads to de: = -de;,
de; = 0.
(3.13)
Hence the axial strain consists of the elastic and thermal parts only, EE, = u, - Y(U* + a,) + EcxT.
Therefore,
(3.14)
the axial stress becomes uZ = ~[a42 In r + 1) + 2Cs] + EE; - EaT.
(3.15)
888
Y. ORCAN
With the stress components known, making use of plastic incompressibility, the geometrical relations (2.2a,b) are inserted into the dilatation and the radial displacement is found to be u =
(1 + v)(l - 2v) E
Now the plastic strain components
G
Ir,,r( In r + C5) f 2( 1 + v)r6 - w,r 4- -
(3.16)
r
can be derived as .
eg=-E,P=-(l-vZ)~+(l+v)a[2e--J+~, ‘F,
=
(3.17)
r
0.
(3.18)
3.2 Conditions and solution procedure Besides the interface radii r, and r,, and the uniform axial strain E?, there are six unknown functions of the load parameter; C, , . . . , C,. The unknown C2 has to vanish and C3 = -cTg In b since stresses and deformation are bounded at the axis and the outer surface is free of any traction, respectively. For the determination of the remaining unknowns the nonredundant conditions are r = r,
cl_ it @, -fir
(3.19)
(Epe)”= 0
(3.20)
ir’d - Urel _~ flo
(3.21)
II --a; CT,
(3.22)
r = r,
&I* -- e1_1
(3.23)
(Ef;)‘l = (Epg)l
(3.34)
suppIemented by the free end condition (2.16). It is noted that condition replaced by the (redundant) condition (cfl)” = 0 at r = r,.
(3.23) could be
C, = 2a. In z + +J-v 6(r,)
C5=C3= C,=C,-Gr4
CT{]* 1 + 2v - 4v2 2
(3.25)
-u,,lnb
(3.26)
- (1 - 2v)ln ;] - r$[vT(r,) + (1 - 2v)@(r,)] - $ c,r:
(3.27)
(3.28)
1 - v + (1 - 2v)ln z~ + EaT(r4). I
Condition (3.21) relates the border radius r, to the load parameter Ea[28(r3)
- T(r?)] - (1 - v)(T,~= 0.
(3.29) by the equation (3.30)
The free end condition (2.16) may be expressed as f”(0, rs) + Z”(r?, r4) + Z’(r4, b) = 0.
(3.31)
Thermal stresses in an elastic-plastic
cylinder
889
The quantities introduced in (3.31) stand for the integrals J ru, dr in the three different zones. They are given in the Appendix. Substituting T(rj) and 0(r,) from (2.19) and (2.20) into (3.30) one obtains the nondimensional border radius & := rJb as a function of the nondimensional heat generation rate qlll,
6; = 8(1 - v)/g’.
(3.32)
It has already been mentioned that plastic flow commences at the surface with & = & = 1 and the corresponding load parameter follows from (3.14) to be q”’ = 8(1 - v) as it is given by (2.22). With increasing load parameter the plastic zones propagate inwards. The solution of the problem in the first stage of plastic deformation is obtained by calculating first the border radius & from (3.32). Then the nonlinear equation (2.16) representing the free end condition is solved for &. The stresses and deformation are given by the equations derived in the preceding sections and the solution of the problem in the first stage is completed. The validity of this solution ceases as soon as secondary plastic flow sets in at the axis of the rod. This occurs for the load parameter 4”’ = &J’which leads to a: - a:’ = a:’ - a:’ = (T()at the axis of the rod.
4. SECOND
STAGE
OF PLASTIC
DEFORMATION
4.1 The elastic-plastic behaviour Beyond the critical load parameter &!’secondary plastic flow sets in at the centre. The image points of the stresses lie in another “edge regime” of Tresca prism in deviatoric stress space (point D in Fig. 1). In this stage of plastic flow, besides the outer plastic zones which have been treated in the previous section, two additional plastic zones, plastic zone III and plastic zone IV expand outwards starting from the axis. The derivation of displacement, plastic strain and stress field is as follows. Plastic zone III: 0 5 r < r,
In this innermost plastic zone due to the “edge regime” condition adopts the form ul? - o, = (To,
with ue = g,. > a, Tresca yield
gr - a* = UC,.
(4.1)
Equation of equilibrium (2.1) and the yield condition (4.1) lead to u_q=u,=C7,
and
u, = C, - uo,
(4.2a,b)
Using Hooke’s law with generalized plane strain, the geometrical relations (2.2a,b) and plastic incompressibility the following differential equation for u can be derived
$ru)=F Integration
(3C, - uo) + 3c~T - E,.
(4.3)
of (4.3) with respect to the radius results in l-2Y
u =E
(3C, - a,,)r + 3a6r - $ l,r + Llr!. r
Here it is noted that the integration bounded at the axis of the rod.
constant
(4.4)
C, should vanish since the displacement
is
Y. ORCAN
X90
The plastic
parts of the strain
components
are then derived
Es = ;‘;; [(l ~~21’)C:,
2v)C’,
(Jo]
+
-- CJ(,] -
a(%
Due to the inequality
7‘)
~~ ;
2T)
u(.w
t:,
(3.5)
$ E,.
(4.6)
~~cwT + E;.
[u,, ~ (1 ~- 2v)C,]
region
as
(T_< u,. < (To,the yield condition
(4.7)
in this outer
part of the inner
plastic
takes the form CJ; ~- (J,,.
(JO
The flow rule associated increment
of plastic
with the yield
condition
(4.X) yields
= ~ I,
dtj’ -=0.
the following
relations
the total strains
are expressed
(4.9,
by
E,, = r;; + E!~ -- E- + 2a7
(4.10)
.Zr = F,” i crT
(4. I I )
where the condition of generalized plant strain is used. Inserting the total strains and (4.11) into the compatibility equation (2.6) and expressing their elastic parts stresses via generalized Hooke’s law, there follows m, ~ v(u~, + a,) + EaT The elimination (4.8). respectively.
The solution
for the
strains d@dt”
Therefrom,
(4.8)
of (r,- and results
= ${r[(I
(7, using
- V)(CT~+ (7;) - vc~, + 2EcuT ~ EC,]}.
equation
in the following
of equilibrium
differential
equation
(2.1) and the yield
from (4.10) in terms of
(4.12) condition
for CJ(,
is
(4.14) where 0,
:zr
e2
:=
M’:_
”
y
,\‘)
(‘+‘W
I I
Tr
TrM
”
dr,
dr,
’ 2( 1 ~ v)
From the yield condition u; =
u(j
-
u,,,
(4.15)
Thermal
stresses in an elastic-plastic
cylinder
891
and the radial stress is obtained from the integration of the equation of equilibrium as a,=-
1 M ( +-
The circumferential
Cyr -(I-M) _ C IO+l+M)
A[(1 + 4(1 - Y)
- 2A4>e, - (1 + 2M)8,
I
( ~0 + EL).
l l-2Y
(4.16)
strain from (4.10) gives the displacement as
+ (1 + v)aor + E.s,r.
The circumferential
(4.17)
plastic strain is then derived from EBB = ET’+ CXT- E; which results in
Note that the expression (4.18) for the plastic strain components does not contain explicitly the uniform axial strain E, as in the case of the cylinder with fixed ends [4]. In fact, by letting E, = 0 the results in the two inner plastic zones, plastic zone III and plastic zone IV, for the centrally heated cylinder with fixed ends [4] are recovered. 4.2 Conditions and solution procedure With 10 integration constants C,, CZ, . . . , Clo, axial strain E,, and the border radii r,, r,, r, and r, the total number of unknowns is 15. It is already mentioned that C, vanishes. The remaining unknowns are determined as follows. From the continuity of radial stress at the interface r = r, and the condition of vanishing radial stress at the free surface r = b one finds (4.19)
18 16
Elastic
0.0
0.2
behaviour
0.4 Border
Fig. 2. Development
0.6
0.8
1.0
Radius
of the plastic zones with increasing deformation.
load parameter
in the two stages of plastic
892
Y. ORCAN
0.0
0.2
0.4
0.6 Radius
0.8
Fig. 3. Onset of yield at the surface. @“’-= q;”= 5.64.
The axial plastic strain component E, = z The circumferential conditions result in
should vanish at Y= r:,. Together
I
1 -- V + (1 - 2v)ln 2 + cuT(r,)
plastic strain is continuous
C,==r$j(t
- v’):+(l
with (4.19) it gives
1
(4.20)
at I’ = r,, and it vanishes at r = r3:,.These
-t v)a[T(r,J)-28/r,l]}
(4.21)
and
+ cw{(l+ v)r$[T(r,)
--- 2B(r,)] - r$ vT(r,) + (1 - 2v)6(r4)1}.
(4.22)
Using the condition that radial plastic strain vanishes at r = r,. one obtains
c, = &v{2Ea[3H(r,)
- 2T(r,)]
-i- Er, + taco>.
(4.23)
With the expression (4.23) for C, the two equations imposing the continuity of radial and circumferential stresses at r = r,, are solved simultaneously for C, and C,,,. The result is 6(l + M) 1 _2v Q(h)+
1 --1 - I’
4(1 + M) 1 _ 2v
Ttri) - ~&PI))
(4.24)
(4.25)
Thermal stresses in an elastic-plastic
cylinder
893
The integration constants C, and C, are obtained in terms of Cy and Cl0 from the continuity of radial and circumferential stresses at the elastic-plastic interface r = r,. They read
c, =
( 1 1+ ;
c$r;(‘-M)
X [(I + h)(l
+
(1- -+r;(“M) +Ea 4(1 -
- 2M)@,(rz) + (1 - $1
Y)
+ 2M)e,(r,)]
+ ;”
(a0 + I!%,)
(4.26)
c*=-f[l’-~jC,‘i’M+jl+~)~,,,r:-“+~
X [8@(Q) - (1 - $1
- zfV)e,(a)]r:.
(4.27)
This completes the determination of the integration constants and besides the free end condition (2.16) there are three other nonredundant conditions left: the circumferential plastic strain vanishes at r = r2, radial stress is continuous and the expressions for stresses in the elastic region should satisfy the yield condition a, - a, = (T() at the elastic-plastic interface r = r.+ These conditions can be written in the following forms:
(l-
v-X)[C,r~(‘-“‘+~EaB,(r,)j
+(1-v+X)IC,,,rl”‘““+~EaB,(r,)]=O
-!%
l-v
[28(r3) - T(rJ]
- 22
- cffl= 0,
flcbIn 3b -
(4.28)
(4.29)
(4.30)
and the free end condition can be expressed by f"'(0, r,) + Ifv(rl, rJ + Ze’(rz, r3) + Z1’(rx, rt) + Z1(rq,b) = 0
(4.31)
where the functional forms of the integrals representing JmT, dr in the five different zones are listed in the Appendix. With the integration constants all having been determined (4.28)-(4.31) constitute a system of four nonlinear equations in the border radii r,, r,, r, and r, with the heat generation rate q”’ as the loading parameter.
5. NUMERICAL
RESULTS
Taking v = 0.295, the expansion of the plastic zones with the nondimensional heat generation rate 41”in both stages of plastic deformation is displayed in Fig. 2. The distribution of stress and displacement at onset of yield which corresponds to the heat generation rate $’ = 8(1 - Y) = 5.64 is plotted in Fig. 3. Beyond this load parameter two outer plastic zones develop simultaneously and propagate inwards. The stress distribution and deformation governing the thermo-mechanical behaviour of the rod in the first stage of plastic deformation are given by the equations (3.1)-(3.18) supplemented by (3.25)-(3.31). Figure 4 shows the stresses, displacement and plastic strains for the load parameters $‘I = 10 and @ = 12.428. For the purpose of comparison the displacement for 9”’ = 5.62 is also plotted on the same graph.
Y. ORCAN
0.0
2.0
E ._ ;
0.2
0.4
0.6 Radius
0.8
1.0
0.2
0.4
0.6 Radius
0.8
1.0
(b)
1.5
1.0
SI; 0.5 E m
0.0
6 2 -0.5 E $ -1.0 -0 *f-l.5 -2.0 0.0 Fig. 4. (a) Stress\
and (b) displaccrncnt
and plastic
~II-AIIIS Ior q”’
IO and ‘I”’
1247X
By imposing the restriction &I - CT:’ m.try’ ~ tr:’ = u,, at r = 0, (3.13) and (3.14) are solved simultaneously for q”‘, tJ and [.,. The result is $ = 12.428, 5; = 0.67366 and {a = 0.79820. For plastic zones emerge at the Q”’ > &’ due to the “edge regime” with Cr, = (7, two additional centre and expand outwards. The stress state and deformation at an intermediate level in this second stage of plastic deformation is displayed in Fig. 5. The load parameter is q,,, = 16.200, and the corresponding border radii obtained from the solution of nonlinear system of equations (4.28)-(4.31) are 5, = 0.17000, t2 = 0.43767, 5 = 0.58612 and & = 0.74833. With further increase in the load parameter the annular elastic layer gets narrower and finally disappears. The inner and outer plastic regions join and the rod reaches the fully plastic state. By putting r, = r3 = r,, the system of equations (4.28)-(4.30) is solved for the three
Thermal
1.0
stresses
in an elastic-plastic
cylinder
(a)
0.5 ul %i 5 0.0 r E g -0.5 tz i
-1.0
-1.5
-2.0 0.6 Radius
0.0
(b) 2.0 ._E z z .g $ a z 0
1.0
0.0
$ -1.0 E t -0 .u, 0. -2.0 0 -3.0 0.0 Fig. 5. (a) Stresses
0.2
and (b) displacement
0.4
0.6 Radius
and plastic strains for y”’ = 16.200.
0.8
in the second
1 .O
stage of plastic deformation
unknowns at the fully plastic state to yield #‘, = 18.468, 5, = 0.21017 and & = t3 = 5, = 0.54109. Note that the free end condition (4.31) is not involved in this solution. Since at this state the elastic zone completely disappears, the latter two equations, (4.29) and (4.30) are equivalent to the relations obtained for the continuity of circumferential and radial stresses, c~“(YJ = av(r,,) and al”(r,,) = a:‘(r,), at the interface where plastic zones IV and II meet. The interface radius td at the fully plastic state is then calculated using the free end condition (4.31) to be 0.73229. The stress and deformation field in the fully plastic state is given in Fig. 6. An interesting and complementary research which has a significant practical importance is the determination of the residual stress field remaining in the rod when the internal heat source is removed. As long as reversed plastic flow does not take place the residual stresses and
8%
Y. ORCAN
(a) I
I
1
I
1.0 0.5 In 5 $ 0.0 0” k-O.5 2 z -1.0 9 * -1.5 -2.0 -2.5 Radius (b) 3.0 2 2.0 ._ 2 ;;i
1.0 u z _?? L 0.0 ?? o -7.0 2 E-m . 5
z -3.0
-4.0 0.0
0.2‘
0.4
0.6
0.8
1.0
Radius Fig. 6. (a) Stresses and (hf displaccmcnt
and plastic Ttrain in the fully plastic state. (I”’ = I/‘;.
displacement are obtained by subtracting the unrestricted elastic response to the temperature field under consideration from the actual elastic-plastic response. Figure 7 shows the residual stresses and displacement when the rod is unloaded from a state in the first stage of plastic deformation with the load parameter $” = 16(1 - Y) = 11.28. (Note that this load parameter is equal to 2$‘.) The corresponding interface radii are t3 = 0.70711 and sJ = 0.82013. From Fig. 7 it is observed that the residual stress state in the elastic region is uniform and tensile with moderately small magnitude. However, in the outer plastic zone there is a sharp rise in the compressive circumferential and axial stresses. On the lateral surface they attain the value 0 _ 0 -CT,,. This shows that reversed plastic flow will occur if the cylinder is unloaded cr@-u:= after further increase in the load parameter. In a forthcoming paper the residual stresses and
Thermal stresses in an elastic-plastic
cylinder
897
0.50
< 0.25 E $ 0 e 0.00 6 z 0 -0.25 fn % fn z g -0.50
5 .-4 $ -0.75 fY
-
-
I
-1.00 0.0
f
I 0.2
I 0.4
I
I
II
0.6 Radius
0.8
I 1.0
Fig. 7. Residual stresses after complete unloading from 4”’ = 11.28.
deformation will be investigated by the consideration of reversed plastic flow which takes place in the zones with plastic predeformation [6]. Closely related to this study is the recent paper by Mack [7] who derived the residual stress state in a hollow cylinder by taking into account the secondary plastic flow when the cylinder is brought to standstill after supercritical rotation. Acknowledg~e~r-The
author wishes to thank Professor Dr U. Gamer for his helpful suggestions.
REFERENCES [I] R. A. VALENTIN and J. J. CAREY, Nucl. Engng. Design 12,277 (1970). [2] E. CITAKOGLU, METLi J. Pure Appl. Sci. 9, 167 (1976). [3] E. CITAKOGLU and N. EREN, METUJ. Pure App. Sci. 12,91 (1979). [4] T. B. KAMMASH, S. A. MURCH and P. M. NAGHDI, J. Mech. Phys. So/ids 8, I (1960). [S] Y. ORCAN and U. GAMER, Actu Mech. 90,61 (1991). [6] Y. ORCAN, Residual stresses and secondary plastic flow in a heat generating elastic-plastic ends. in preparation. [7] W. MACK, 2. angew. Math. Me&. 72,6S (1992).
cylinder with free
APPENDIX Theintegral
f(s. r) = .f: ‘cc dr where s and I denote two arbitrary border radii of the region under consideration take the following forms: (a) for the elastic region I”(S, t) = - 16(FJy)h
[26’(? -sq
+s3 - I‘+]+$c,
+ E&)(12-S?)
(Al)
(b) for plastic zone I
I’(s, I) = T
[
I’ In t - s7 Ins + i(rz - s’)
(A2)
1
(c) for plastic zone II Jr&q” l”(s, t) = VL~& In t - sz In s) + t &(I~ - s’) - r 1 ES 32:6-B
(A3)
808 (d) for plastic
Y. ORCAN zone 111
I 1 > P’(s. I) = -(C, - c,J(t- ~ s-) = 2( ,~J j$2E”[M(r,) 2
‘T(r,)l
+ Et; + 2W,}(f-’
S‘)
(A4)
(e) for plastic zone IV
(As) where.
Pergamon
Copyright 0 1994Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225,‘94$7.(W) + O.(Xt
THERMAL STRESSES IN A HEAT GENERATING ELASTIC-PLASTIC CYLINDER WITH FREE ENDS YUSUF ORCAN Department of Engineering Sciences, Middle East Technical University, 06.531 Ankara, Turkey (Communicated
by H. DEMIRAY)
Abstract-The exact solution of the distribution of stress and deformation in an elastic-perfectly plastic cylindrical rod with uniform internal heat generation under generalized plane strain condition is presented. The treatment is based on Tresca’s yield condition and the associated flow rule. The stress image points of the outer and inner plastic regions lie on two different edges of Tresca prism. With increasing heat generation parameter, the image points of the intermediate regions spread out through two side regimes and meet at another edge regime in the fully plastic state.
I.
INTRODUCTION
stresses in a body due to non-uniform temperature distribution play a very important role in many engineering designs and operation. In the wide domain of application two particular examples are the heating of wires conducting direct current and the solid nuclear fuel elements in which internal energy is generated by nuclear fission. The elastic solutions of the problem are studied in detail, e.g. [l-3], and since the results are genera1 they can easily be adapted to any kind of temperature distribution. However, in the design of such elements power generation can be substantially increased if the material is allowed to yield. Furthermore an elastic-plastic analysis is necessary to improve the low cycle fatigue life of the material. The thermoelastic-plastic deformation of a solid cylinder in the presence of a distributed Gaussian heat source was considered by Kammash et al. [4] using Tresca’s yield condition and its associated flow rule. They paid particular attention to a cylindrical nuclear fuel element exhibiting linear strain hardening surrounded by an elastic cladding. Due to the assumption of plane strain fez = 0) initial yield occurs at the axis of the cylinder and the plastic region which is composed of two different zones expand outward. The outer elastic region vanishes and the cylinder becomes fully plastic when the elastic-plastic boundary reaches the surface. Nevertheless, the cylinder could sustain further loading, and since in the fully plastic state the fuel element consists of only two plastic zones Kammash er al. were able to carry the analysis further without much difficulty. Recently Organ and Gamer [5] investigated the elastic-perfectly plastic behaviour of a centrally heated solid cylinder with fixed ends. For all possible ratios of hot core diameter to outer diameter of the cylinder the temperature of the core was increased up to the value for which the cylinder becomes fully plastic. One of the interesting findings was a critical core radius which leads to instantaneous onset of plastic flow in the entire outer elastic region in spite of the existing inhomogeneous stress distribution. In the present investigation the complete solution of the elastic-perfectly plastic cylindrical rod with uniform internal heat generation under the state of generalized plane strain (free ends) is presented. Leading to closed form solution Tresca’s yield condition and its associated flow rule is used. The material parameters E, CT,), v and a! are assumed to be independent of temperature. Unlike the case considered by Kammash et al. [4], in the absence of any lateral pressure and axial force initial yield occurs at the surface where the circumferential stress is equal to the axial stress. The stress state at the surface and at its neighbourhood lies in an edge regime of
Thermal
883
Y. ORCAN
884
D
Fig. 1. Stress trajectories of fixed material points lying on the current interfaces.
Tresca prism with uB = oZ > a,. In this first stage of plastic deformation, for the nondimensional internal heat generation parameter 9“’ = 9’;‘. two different plastic zones emerge simultaneously at the surface and propagate inwards. The yield conditions in these plastic zones, plastic zone I and plastic zone II, read cB - a, = cry - a, = cr,, and (T@- a, = cr,,, respectively. The first stage of plastic deformation is terminated when the stresses at the axis of the cylinder attain the values ue = g, = $a, = --go for #” = 4’. This state of stress at the axis corresponds to another edge regime of Tresca prism and thus two additional plastic zones, plastic zone III and plastic zone IV, emerge at the centre and expand outwards. The yield conditions read a6 - eZ = a, - (T, = a0 and gB - g7 = go respectively. With increasing load parameter, the intermediate elastic layer gets narrower and vanishes entirely for @” = &‘. Figure 1 shows thematically the stress image trajectories of various fixed material points on principal stress space. The image of the lateral surface r = b moves directly to the edge B of Tresca prism and stays there throughout the course of plastic deformation. A material point lying in plastic zone I, e.g. on the border Y= r,, first moves to the side regime BC and with ongoing plastic deformation migrates to the edge B. The image of a fixed material point on the elastic-plastic interface, r = r,, has just reached the side BC for the current load parameter under consideration. The material points in plastic zone II where plastic Ilow is governed by the side regime BC are in fact lying at different locations from point N to B. On the other hand, the image of the axis moves to the edge D, and remains there while those of plastic zone III travel along the face CD to the edge D. The material points of plastic zone IV are spread out along the side from point K to L>. Finally it is noteworthy that the image of the material points on the radius r = rP in the fully plastic state (see Section 5) move directly to a position on edge Cwitha,>Oandrr,=s;
SOLUTION
AND
ONSET
OF
YIELD
The equation of equilibrium
(2.1) and the geometrical
relations El=--,
Ef)
dr
=
-
r
(2.2a.b)
Thermal stresses in an elastic-plastic cylinder
885
hold in the entire rod irrespective of the material behaviour. In the purely elastic body, the generalized Hooke’s law reads EE, = a, - v(ao + (a,) + EaT
(2.3)
Eee = ue - v(mz + a,) + EaT
(2.4)
E.cz = a, - ~(a~ + cre) + EaT.
(2.5)
Consideration of a rod with free ends leads to the generalized E, = co~sf~~r. The compatibility relation
h?) = Q,
f
plane strain condition,
(2.6)
obtained by the elimination of u from the geometrical relation (2.2a,b) and the use of stress function cp, such that a, = rp/r and oe = d4,ldr satisfy the equilibrium equation (2.1), lead to a differential equation which can be easily solved to yield (2.7)
(2.8)
(2.9) where 8:=f Therewith the radial displacement
‘Trdr. I0
is given by (2.10)
For a solid cylinder CZ = 0, and the condition of vanishing radial stress at the outer surface, o,(b) = 0, yields +2Ea
1 - v O(b)
(2.11)
with B(b) = 3 1’ Tr dr. 0 Hence, the elastic solution is given by u, = fi
[e(b) - f9]
+=E[O(b)+O-T] a, = sv u = (,(sp
[2@(b) - T] f EE, + (1 - 2y)B(b)] - T) - ,i),
(2.12)
(2.13) (2.14) (2.15)
Y. ORCAN
886
IJsing the free end condition (2..16)
the axial strain in the elastic range is obtained to be E, = 2&(b)
(2. I?)
and the axial stress can now be expressed as
(2.18)
2. I Temperature distribution In a cylindrical rod with uniform internal energy generation the steady state temperature distribution is given by
q”’ per unit volume per unit time
(2.20)
In a cylindrical rod with uniform internal energy generation and free ends plastic flow sets in at the outer surface due to tensile circumferential and axial stresses. By setting gH(b) = (r;(b) --~~~from (2.13) or (2.14) one obtains
;“y [2@(b) which, upon substitution
- T(b)] = (r,,
(2.21)
of T(b) and B(b) gives the load parameter (2.22)
for the initiation of yielding at the outer surface of the rod.
3. FIRST
STAGE
OF
PLASTIC
~EF~R~lAT~ON
3.1 The elastic-plastic behaviour
In the first stage of plastic deformation the rod is composed of an elastic region at the centre 0 CCr < r,, surrounded by the outer plastic region which consists of two zones: plastic zone I for r4 < r 5 b and plastic zone II for r, < r < r,. The stress and deformation fields in the plastic zones I and II are derived as follows. Plastic zone I: r, < r 5 b
The stress state lies in an “edge-regime” condition takes the form ge - 0; = Q-0,
of Tresca’s prism with f18 = g, and the yield CT,- 0; = Go
(3.1)
Thermal
stresses
in an elastic-plastic
cylinder
n87
of the equilibrium equation (2.1) together with the first expression of the yield condition (2.23) gives g+e= crZ= ao(l + In r) + C3
(3.2)
a, = u. In r + Cj.
(3.3)
Due to plastic incompressibility, EI:= 0, the dilatation consists of its elastic and thermal parts only. Using generalized Hooke’s law, Eeii = (1 - 2v)cr, + 3EcxT
(3.4)
and the geometrical relations (2.2a,b), one obtains
$)ru)=y
[a,)(2 + 3 In Y) + 3C3] + 3aT - l,
(3.5)
with the solution (3.6) Knowing the radial displacement field the plastic parts of the strain components are calculated by subtracting the elastic and thermal parts from the total strain components. They read Epe=~(ui,lnr+C,)-~~,,+~[30-T]-~t,+$ lPr = y
(u.
ep=E Z 3
(3.7)
7-6~ In r + C,) + ~~o1[3&2T]-ft;-+
i [(l - v)cro -t (1 - 2v)(uoln r + C,)] - (YT.
r
(34
(3.9)
Plasm zone II: r3 < r I r4 In plastic zone II which develops simultaneously
together with plastic zone I according to the inequality a, < u, < ue the yield condition has the form oe - u, = u,,.
(3.10)
Substituting the yield condition (3.10) in the equation of equilibrium (2.1) and integrating, one obtains a, = ug In r + Cs
(3.11)
u0 = uo(ln r + 1) + Cs.
(3.12)
The flow rule associated with the yield condition (3.10) leads to de: = -de;,
de; = 0.
(3.13)
Hence the axial strain consists of the elastic and thermal parts only, EE, = u, - Y(U* + a,) + EcxT.
Therefore,
(3.14)
the axial stress becomes uZ = ~[a42 In r + 1) + 2Cs] + EE; - EaT.
(3.15)
888
Y. ORCAN
With the stress components known, making use of plastic incompressibility, the geometrical relations (2.2a,b) are inserted into the dilatation and the radial displacement is found to be u =
(1 + v)(l - 2v) E
Now the plastic strain components
G
Ir,,r( In r + C5) f 2( 1 + v)r6 - w,r 4- -
(3.16)
r
can be derived as .
eg=-E,P=-(l-vZ)~+(l+v)a[2e--J+~, ‘F,
=
(3.17)
r
0.
(3.18)
3.2 Conditions and solution procedure Besides the interface radii r, and r,, and the uniform axial strain E?, there are six unknown functions of the load parameter; C, , . . . , C,. The unknown C2 has to vanish and C3 = -cTg In b since stresses and deformation are bounded at the axis and the outer surface is free of any traction, respectively. For the determination of the remaining unknowns the nonredundant conditions are r = r,
cl_ it @, -fir
(3.19)
(Epe)”= 0
(3.20)
ir’d - Urel _~ flo
(3.21)
II --a; CT,
(3.22)
r = r,
&I* -- e1_1
(3.23)
(Ef;)‘l = (Epg)l
(3.34)
suppIemented by the free end condition (2.16). It is noted that condition replaced by the (redundant) condition (cfl)” = 0 at r = r,.
(3.23) could be
C, = 2a. In z + +J-v 6(r,)
C5=C3= C,=C,-Gr4
CT{]* 1 + 2v - 4v2 2
(3.25)
-u,,lnb
(3.26)
- (1 - 2v)ln ;] - r$[vT(r,) + (1 - 2v)@(r,)] - $ c,r:
(3.27)
(3.28)
1 - v + (1 - 2v)ln z~ + EaT(r4). I
Condition (3.21) relates the border radius r, to the load parameter Ea[28(r3)
- T(r?)] - (1 - v)(T,~= 0.
(3.29) by the equation (3.30)
The free end condition (2.16) may be expressed as f”(0, rs) + Z”(r?, r4) + Z’(r4, b) = 0.
(3.31)
Thermal stresses in an elastic-plastic
cylinder
889
The quantities introduced in (3.31) stand for the integrals J ru, dr in the three different zones. They are given in the Appendix. Substituting T(rj) and 0(r,) from (2.19) and (2.20) into (3.30) one obtains the nondimensional border radius & := rJb as a function of the nondimensional heat generation rate qlll,
6; = 8(1 - v)/g’.
(3.32)
It has already been mentioned that plastic flow commences at the surface with & = & = 1 and the corresponding load parameter follows from (3.14) to be q”’ = 8(1 - v) as it is given by (2.22). With increasing load parameter the plastic zones propagate inwards. The solution of the problem in the first stage of plastic deformation is obtained by calculating first the border radius & from (3.32). Then the nonlinear equation (2.16) representing the free end condition is solved for &. The stresses and deformation are given by the equations derived in the preceding sections and the solution of the problem in the first stage is completed. The validity of this solution ceases as soon as secondary plastic flow sets in at the axis of the rod. This occurs for the load parameter 4”’ = &J’which leads to a: - a:’ = a:’ - a:’ = (T()at the axis of the rod.
4. SECOND
STAGE
OF PLASTIC
DEFORMATION
4.1 The elastic-plastic behaviour Beyond the critical load parameter &!’secondary plastic flow sets in at the centre. The image points of the stresses lie in another “edge regime” of Tresca prism in deviatoric stress space (point D in Fig. 1). In this stage of plastic flow, besides the outer plastic zones which have been treated in the previous section, two additional plastic zones, plastic zone III and plastic zone IV expand outwards starting from the axis. The derivation of displacement, plastic strain and stress field is as follows. Plastic zone III: 0 5 r < r,
In this innermost plastic zone due to the “edge regime” condition adopts the form ul? - o, = (To,
with ue = g,. > a, Tresca yield
gr - a* = UC,.
(4.1)
Equation of equilibrium (2.1) and the yield condition (4.1) lead to u_q=u,=C7,
and
u, = C, - uo,
(4.2a,b)
Using Hooke’s law with generalized plane strain, the geometrical relations (2.2a,b) and plastic incompressibility the following differential equation for u can be derived
$ru)=F Integration
(3C, - uo) + 3c~T - E,.
(4.3)
of (4.3) with respect to the radius results in l-2Y
u =E
(3C, - a,,)r + 3a6r - $ l,r + Llr!. r
Here it is noted that the integration bounded at the axis of the rod.
constant
(4.4)
C, should vanish since the displacement
is
Y. ORCAN
X90
The plastic
parts of the strain
components
are then derived
Es = ;‘;; [(l ~~21’)C:,
2v)C’,
(Jo]
+
-- CJ(,] -
a(%
Due to the inequality
7‘)
~~ ;
2T)
u(.w
t:,
(3.5)
$ E,.
(4.6)
~~cwT + E;.
[u,, ~ (1 ~- 2v)C,]
region
as
(T_< u,. < (To,the yield condition
(4.7)
in this outer
part of the inner
plastic
takes the form CJ; ~- (J,,.
(JO
The flow rule associated increment
of plastic
with the yield
condition
(4.X) yields
= ~ I,
dtj’ -=0.
the following
relations
the total strains
are expressed
(4.9,
by
E,, = r;; + E!~ -- E- + 2a7
(4.10)
.Zr = F,” i crT
(4. I I )
where the condition of generalized plant strain is used. Inserting the total strains and (4.11) into the compatibility equation (2.6) and expressing their elastic parts stresses via generalized Hooke’s law, there follows m, ~ v(u~, + a,) + EaT The elimination (4.8). respectively.
The solution
for the
strains d@dt”
Therefrom,
(4.8)
of (r,- and results
= ${r[(I
(7, using
- V)(CT~+ (7;) - vc~, + 2EcuT ~ EC,]}.
equation
in the following
of equilibrium
differential
equation
(2.1) and the yield
from (4.10) in terms of
(4.12) condition
for CJ(,
is
(4.14) where 0,
:zr
e2
:=
M’:_
”
y
,\‘)
(‘+‘W
I I
Tr
TrM
”
dr,
dr,
’ 2( 1 ~ v)
From the yield condition u; =
u(j
-
u,,,
(4.15)
Thermal
stresses in an elastic-plastic
cylinder
891
and the radial stress is obtained from the integration of the equation of equilibrium as a,=-
1 M ( +-
The circumferential
Cyr -(I-M) _ C IO+l+M)
A[(1 + 4(1 - Y)
- 2A4>e, - (1 + 2M)8,
I
( ~0 + EL).
l l-2Y
(4.16)
strain from (4.10) gives the displacement as
+ (1 + v)aor + E.s,r.
The circumferential
(4.17)
plastic strain is then derived from EBB = ET’+ CXT- E; which results in
Note that the expression (4.18) for the plastic strain components does not contain explicitly the uniform axial strain E, as in the case of the cylinder with fixed ends [4]. In fact, by letting E, = 0 the results in the two inner plastic zones, plastic zone III and plastic zone IV, for the centrally heated cylinder with fixed ends [4] are recovered. 4.2 Conditions and solution procedure With 10 integration constants C,, CZ, . . . , Clo, axial strain E,, and the border radii r,, r,, r, and r, the total number of unknowns is 15. It is already mentioned that C, vanishes. The remaining unknowns are determined as follows. From the continuity of radial stress at the interface r = r, and the condition of vanishing radial stress at the free surface r = b one finds (4.19)
18 16
Elastic
0.0
0.2
behaviour
0.4 Border
Fig. 2. Development
0.6
0.8
1.0
Radius
of the plastic zones with increasing deformation.
load parameter
in the two stages of plastic
892
Y. ORCAN
0.0
0.2
0.4
0.6 Radius
0.8
Fig. 3. Onset of yield at the surface. @“’-= q;”= 5.64.
The axial plastic strain component E, = z The circumferential conditions result in
should vanish at Y= r:,. Together
I
1 -- V + (1 - 2v)ln 2 + cuT(r,)
plastic strain is continuous
C,==r$j(t
- v’):+(l
with (4.19) it gives
1
(4.20)
at I’ = r,, and it vanishes at r = r3:,.These
-t v)a[T(r,J)-28/r,l]}
(4.21)
and
+ cw{(l+ v)r$[T(r,)
--- 2B(r,)] - r$ vT(r,) + (1 - 2v)6(r4)1}.
(4.22)
Using the condition that radial plastic strain vanishes at r = r,. one obtains
c, = &v{2Ea[3H(r,)
- 2T(r,)]
-i- Er, + taco>.
(4.23)
With the expression (4.23) for C, the two equations imposing the continuity of radial and circumferential stresses at r = r,, are solved simultaneously for C, and C,,,. The result is 6(l + M) 1 _2v Q(h)+
1 --1 - I’
4(1 + M) 1 _ 2v
Ttri) - ~&PI))
(4.24)
(4.25)
Thermal stresses in an elastic-plastic
cylinder
893
The integration constants C, and C, are obtained in terms of Cy and Cl0 from the continuity of radial and circumferential stresses at the elastic-plastic interface r = r,. They read
c, =
( 1 1+ ;
c$r;(‘-M)
X [(I + h)(l
+
(1- -+r;(“M) +Ea 4(1 -
- 2M)@,(rz) + (1 - $1
Y)
+ 2M)e,(r,)]
+ ;”
(a0 + I!%,)
(4.26)
c*=-f[l’-~jC,‘i’M+jl+~)~,,,r:-“+~
X [8@(Q) - (1 - $1
- zfV)e,(a)]r:.
(4.27)
This completes the determination of the integration constants and besides the free end condition (2.16) there are three other nonredundant conditions left: the circumferential plastic strain vanishes at r = r2, radial stress is continuous and the expressions for stresses in the elastic region should satisfy the yield condition a, - a, = (T() at the elastic-plastic interface r = r.+ These conditions can be written in the following forms:
(l-
v-X)[C,r~(‘-“‘+~EaB,(r,)j
+(1-v+X)IC,,,rl”‘““+~EaB,(r,)]=O
-!%
l-v
[28(r3) - T(rJ]
- 22
- cffl= 0,
flcbIn 3b -
(4.28)
(4.29)
(4.30)
and the free end condition can be expressed by f"'(0, r,) + Ifv(rl, rJ + Ze’(rz, r3) + Z1’(rx, rt) + Z1(rq,b) = 0
(4.31)
where the functional forms of the integrals representing JmT, dr in the five different zones are listed in the Appendix. With the integration constants all having been determined (4.28)-(4.31) constitute a system of four nonlinear equations in the border radii r,, r,, r, and r, with the heat generation rate q”’ as the loading parameter.
5. NUMERICAL
RESULTS
Taking v = 0.295, the expansion of the plastic zones with the nondimensional heat generation rate 41”in both stages of plastic deformation is displayed in Fig. 2. The distribution of stress and displacement at onset of yield which corresponds to the heat generation rate $’ = 8(1 - Y) = 5.64 is plotted in Fig. 3. Beyond this load parameter two outer plastic zones develop simultaneously and propagate inwards. The stress distribution and deformation governing the thermo-mechanical behaviour of the rod in the first stage of plastic deformation are given by the equations (3.1)-(3.18) supplemented by (3.25)-(3.31). Figure 4 shows the stresses, displacement and plastic strains for the load parameters $‘I = 10 and @ = 12.428. For the purpose of comparison the displacement for 9”’ = 5.62 is also plotted on the same graph.
Y. ORCAN
0.0
2.0
E ._ ;
0.2
0.4
0.6 Radius
0.8
1.0
0.2
0.4
0.6 Radius
0.8
1.0
(b)
1.5
1.0
SI; 0.5 E m
0.0
6 2 -0.5 E $ -1.0 -0 *f-l.5 -2.0 0.0 Fig. 4. (a) Stress\
and (b) displaccrncnt
and plastic
~II-AIIIS Ior q”’
IO and ‘I”’
1247X
By imposing the restriction &I - CT:’ m.try’ ~ tr:’ = u,, at r = 0, (3.13) and (3.14) are solved simultaneously for q”‘, tJ and [.,. The result is $ = 12.428, 5; = 0.67366 and {a = 0.79820. For plastic zones emerge at the Q”’ > &’ due to the “edge regime” with Cr, = (7, two additional centre and expand outwards. The stress state and deformation at an intermediate level in this second stage of plastic deformation is displayed in Fig. 5. The load parameter is q,,, = 16.200, and the corresponding border radii obtained from the solution of nonlinear system of equations (4.28)-(4.31) are 5, = 0.17000, t2 = 0.43767, 5 = 0.58612 and & = 0.74833. With further increase in the load parameter the annular elastic layer gets narrower and finally disappears. The inner and outer plastic regions join and the rod reaches the fully plastic state. By putting r, = r3 = r,, the system of equations (4.28)-(4.30) is solved for the three
Thermal
1.0
stresses
in an elastic-plastic
cylinder
(a)
0.5 ul %i 5 0.0 r E g -0.5 tz i
-1.0
-1.5
-2.0 0.6 Radius
0.0
(b) 2.0 ._E z z .g $ a z 0
1.0
0.0
$ -1.0 E t -0 .u, 0. -2.0 0 -3.0 0.0 Fig. 5. (a) Stresses
0.2
and (b) displacement
0.4
0.6 Radius
and plastic strains for y”’ = 16.200.
0.8
in the second
1 .O
stage of plastic deformation
unknowns at the fully plastic state to yield #‘, = 18.468, 5, = 0.21017 and & = t3 = 5, = 0.54109. Note that the free end condition (4.31) is not involved in this solution. Since at this state the elastic zone completely disappears, the latter two equations, (4.29) and (4.30) are equivalent to the relations obtained for the continuity of circumferential and radial stresses, c~“(YJ = av(r,,) and al”(r,,) = a:‘(r,), at the interface where plastic zones IV and II meet. The interface radius td at the fully plastic state is then calculated using the free end condition (4.31) to be 0.73229. The stress and deformation field in the fully plastic state is given in Fig. 6. An interesting and complementary research which has a significant practical importance is the determination of the residual stress field remaining in the rod when the internal heat source is removed. As long as reversed plastic flow does not take place the residual stresses and
8%
Y. ORCAN
(a) I
I
1
I
1.0 0.5 In 5 $ 0.0 0” k-O.5 2 z -1.0 9 * -1.5 -2.0 -2.5 Radius (b) 3.0 2 2.0 ._ 2 ;;i
1.0 u z _?? L 0.0 ?? o -7.0 2 E-m . 5
z -3.0
-4.0 0.0
0.2‘
0.4
0.6
0.8
1.0
Radius Fig. 6. (a) Stresses and (hf displaccmcnt
and plastic Ttrain in the fully plastic state. (I”’ = I/‘;.
displacement are obtained by subtracting the unrestricted elastic response to the temperature field under consideration from the actual elastic-plastic response. Figure 7 shows the residual stresses and displacement when the rod is unloaded from a state in the first stage of plastic deformation with the load parameter $” = 16(1 - Y) = 11.28. (Note that this load parameter is equal to 2$‘.) The corresponding interface radii are t3 = 0.70711 and sJ = 0.82013. From Fig. 7 it is observed that the residual stress state in the elastic region is uniform and tensile with moderately small magnitude. However, in the outer plastic zone there is a sharp rise in the compressive circumferential and axial stresses. On the lateral surface they attain the value 0 _ 0 -CT,,. This shows that reversed plastic flow will occur if the cylinder is unloaded cr@-u:= after further increase in the load parameter. In a forthcoming paper the residual stresses and
Thermal stresses in an elastic-plastic
cylinder
897
0.50
< 0.25 E $ 0 e 0.00 6 z 0 -0.25 fn % fn z g -0.50
5 .-4 $ -0.75 fY
-
-
I
-1.00 0.0
f
I 0.2
I 0.4
I
I
II
0.6 Radius
0.8
I 1.0
Fig. 7. Residual stresses after complete unloading from 4”’ = 11.28.
deformation will be investigated by the consideration of reversed plastic flow which takes place in the zones with plastic predeformation [6]. Closely related to this study is the recent paper by Mack [7] who derived the residual stress state in a hollow cylinder by taking into account the secondary plastic flow when the cylinder is brought to standstill after supercritical rotation. Acknowledg~e~r-The
author wishes to thank Professor Dr U. Gamer for his helpful suggestions.
REFERENCES [I] R. A. VALENTIN and J. J. CAREY, Nucl. Engng. Design 12,277 (1970). [2] E. CITAKOGLU, METLi J. Pure Appl. Sci. 9, 167 (1976). [3] E. CITAKOGLU and N. EREN, METUJ. Pure App. Sci. 12,91 (1979). [4] T. B. KAMMASH, S. A. MURCH and P. M. NAGHDI, J. Mech. Phys. So/ids 8, I (1960). [S] Y. ORCAN and U. GAMER, Actu Mech. 90,61 (1991). [6] Y. ORCAN, Residual stresses and secondary plastic flow in a heat generating elastic-plastic ends. in preparation. [7] W. MACK, 2. angew. Math. Me&. 72,6S (1992).
cylinder with free
APPENDIX Theintegral
f(s. r) = .f: ‘cc dr where s and I denote two arbitrary border radii of the region under consideration take the following forms: (a) for the elastic region I”(S, t) = - 16(FJy)h
[26’(? -sq
+s3 - I‘+]+$c,
+ E&)(12-S?)
(Al)
(b) for plastic zone I
I’(s, I) = T
[
I’ In t - s7 Ins + i(rz - s’)
(A2)
1
(c) for plastic zone II Jr&q” l”(s, t) = VL~& In t - sz In s) + t &(I~ - s’) - r 1 ES 32:6-B
(A3)
808 (d) for plastic
Y. ORCAN zone 111
I 1 > P’(s. I) = -(C, - c,J(t- ~ s-) = 2( ,~J j$2E”[M(r,) 2
‘T(r,)l
+ Et; + 2W,}(f-’
S‘)
(A4)
(e) for plastic zone IV
(As) where.