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Description of the stress relaxation process under plane stress conditions
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ELSEVIER
0308-0161(94)00162-6
J. Pres. Ves. & Piping 65 (1996) 97-1(K) 0 1995 Elsevier Science Limited in Northern Ireland. All rights reserved 030%0161/96/$09.50
Description of the stress relaxation process under plane stress conditions W. Osipiuk Technical
University,
al. Wiejska 45, 15-351
Bialystok,
Poland
&
K. Rusinko Technical
University,
ul. Bandery
12, 290646 Lviv. Ukraine
(Received 10 October 1994;accepted 26 October 1994)
This work presents the application of Batdorf-Budiansky theory to the description of the stressrelaxation processunder conditions of concurrent uniaxial tension and torsion. The use of the plastic resistancefunction which takes into account strain hardening and strengtheningas an effect of internal stressconcentration was proposed.The theoretical qualitative description was comparedwith the resultsof the experiment.
1 INTRODUCTION
2 DESCRIPTION
The experimental research of relaxation is more difficult than research of creep. Investigation of the so-called pure relaxation, where the strain is constant is impossible in practice. Because of that, forecasting of relaxation curves on the basis of creep curves and adequate creep hypothesis is necessary. The list of publications concerning this problem is quite long, but they refer mainly to uniaxial stress conditions. Until now relaxation under complex stress has been insufficiently studied. The proposal of using the slip theory for the description of relaxation under complex stress seems to be well founded. The aim of the present work is to present the above description of relaxation utilizing Batdorf-Budiansky’s slip theory.’ The use of a plastic resistance function taking into account anisotropic strain-hardening and strengthening as an effect of internal stress concentration is proposed. The relaxation curve is reflected by the creep process at diminishing stress in time. A plane stress realized in thin-walled cylindrical test pieces and subject to the action of tensile force and torsional moment is considered.
In the theoretical analysis the results of the experiment run by the author of the work’ were used. The experiment was carried out on thin-walled tubular specimens made of austenitic steel. The specimens were put under concurrent tension and torsion at a temperature of 92313. In the loading system of the test machine hydraulic dynamometers were used. In this way it was possible to realize an investigation of the so-called pure relaxation in practice (E, = const, yXZ= const). The tensile stress drop Au, is a function of a value of an initial tangential stress r,,. The experimental results which are presented in Fig. 1 indicate that r,, has considerable influence on Aa:. The above experiment has been described in an aspect of a slip theory using numerical methods. 3 THEORETICAL
OF THE EXPERIMENT
DESCRIPTION
Batdorf-Budiansky’s theory assumes that plastic strain is an effect of an infinite number of slips taking place in all possible planes and directions representing any point of a body. The two 97
W. Osipiuk,
98
K. Rusinko
Budiansky, but the plastic resistance function is introduced in the form:
AU, [MPalI
srl, = So(1 + r1RI/ + Jd (3) where S, ‘is the initial plastic resistance (for (pn,= J,,, = 0); r, is a material constant; J,,/ is the parameter of internal stress concentration expressed by the form:
60
201 0
I 50
I 100
I 150
1 200
I 250
zxz[MPo]
Fig. 1. Dependence of the tensile stress drop Au; on the initial value of the tangential stress 5,; after passage of different times.
vectors y1 and 1 defining the plane and the direction of the slips is called the slip system. The orientation of the slip system is defined by three Euler’s angles (Y,~,o in the half-sphere of unit radius, Fig. 2. The integral stress is expressed by the form:
where $2 is the half-sphere region where the slips occur; dfi = cos p da dp; ol, w2 is the slip limits on the plane’s tangent in the region R to the half-sphere; (pn/is the slip density function; ni, 1, are the directional cosines of the axis II and 1 relative to the system of the coordinates x,y,z (see Ref. 3); r,,, is the component of the shear strain in the system n,l defined by the form: Zn,= (hi = (2) a;jl;nj
x,Y,z).
In the present paper the function (pn,is not used directly in the way proposed by Batdorf and
Fig. 2. The slip plane coordinates.
J,,, = B :’ 2 I
Q(t - s) ds
(4)
where B is a material constant: Q(t -s) is the diminishing function of time differences t (later instant which is studied) and s (earlier instant at which a change in stress occurred); t, is the time in which external stress grew. The choice of eqn (4) was justified in the work described in Ref. 4. If we apply stress with constant speed and assume function Q(t -s) = exp[-b(t - s)], (where b is a structural constant), eqn (4) can be expressed with sufficient precision in the form: J,, = Bz,, exp(-bt) where rn, is the maximum value of the stress in the system n,l; t is the elapsed time from the moment when this stress was applied. It is assumed that the slips occur in those half-sphere regions for which the following equality holds: L/ = Sll
(5) and that there are no slips outside these regions and the inequality rn, < S,, is satisfied. Equations (1) to (5) are the principal relationships of the concept presented in this paper. These equations were used to describe relaxation under complex stress. Relaxation is a process of stress reduction as an effect of creep. A description of relaxation needs a description of creep under diminishing stress. That is why first calculations of creep will be presented and then they will be used for an analysis of relaxation. In the authors’ opinion this procedure makes the presented slip theory more clear. A strict analytical calculation of creep deformations in the case of a complex, varying with time, stress is very difficult. Difficulties occur in determining slip region limits as well as in the solution of eqn (1). The above problem can, however, be easily solved by using numerical methods. With. this aim in mind, the half-sphere presented in Fig. 2 is divided into a large number
Stress relaxation
of sufficiently small planes AR, = cos PiAP,Aai, while integrals in eqn (1) are approximated by sums. If we divide the time t of relaxation into short periods p, the number of which is k, eqn (3) in the kth period will have the form: k-l S:,
=
So
[
1 +
r,
p=l 2
A&
+
r&t,
+
J,“,
1 (6)
where Acpn,defines the increment of function (pnl. After using eqns (6), (5), (4), (2) and forms of directional cosines we obtain: r,Acp:, = A: + A; - 1
(7)
where:
99
process
function calculated on the basis of eqn (7) for w = Wk. Limits of slip range Rk are assigned from the equation: A& = 0 (12) The curve given by eqn (12) divides the half-sphere into regions in which, during the kth period, slip occurs if A& > 0, and into regions in which there are no slips if Aqpi,5 0. Values of increments of creep strains in the kth period are conditioned by slips occurring on the ith surface Ani. These values are expressed on the basis of eqns (10) and (11) in the following way: (A&$ = f (sin 2p cos p sin wkA&);
A; = (St - J:,)( cos a cos 2p sin 0
- sin (Ysin p cos o), gLg
- f sin (Y sin 2/3 cos w”);
(I
0 Considering that slip is a result of dislocation movement we shall notice that in the definite point of the half-sphere, at the given instant of time, slip will be proceeding only in one direction. Using the fact that in the whole region of the half-sphere the relation S$,2 zs, is satisfied, we can define the slip direction o = ~~(a, /3) in any point in the half-sphere. The value of gk is calculated from the tangency condition of the plastic resistance curve S,, and the shear stress curve z,,:
as;, a& -=at.d aid
(w = cd)
(8)
Equation (8) implies: k _ (26: - 5;) cos p _ cos a cos 2/3 tg0 - (6; - J&) sin (Y sin (Y sin /3
(13)
1 (AyQi = - (cos (Ycos /3 cos 2/3 sin gk 2
while +g
x Aha;A/3;
(74
(9)
On the basis of eqn (1) the increments of linear and non-dilatational creep strains in the kth period will be expressed by the forms:
X CAP:>;Aai Api (14) Values of uk are determined from eqn (9), while values of Aqi, by eqn (7) taking into account that o = wk. The above calculations are performed for all infinitesimal surfaces AQ, forming the surface of the half-sphere. The results are summed to obtain increments of strains in the kth slip period, that is: i=l
I=1
where Y defines a number of infinitesimal surfaces AR;, .on which slips occur (A& > 0). Stresses u; and z,, are introduced into the eqn (7) by parameters A; and A$, which are defined according to eqn (7a). Increments of plastic strain A$ and Ay& are calculated according to eqn (15). Stress drop Au: after the qth number of time periods in the case of the pure relaxation describes the form: Au, = a&
= 0) - g<
==E 2 A&;@,, t,.,)
(16)
k=l
where 12, 1J: are values of directional cosines for o = ok; A& is the increment of the slip density
where E is Young’s modulus. It can be shown that from eqns (7), (7a) and (10) with consideration of (16) that the initial stress z,, influences the relaxation of stress gz by
100
W. Osipiuk,
reaction on A.$. This reaction can be interpreted as an effect of the occurrence of common slip regions in a material during tension and torsion. Figure 1 presents the comparison between the experimental results and the theoretical description according to eqn (16) for the following values: E,,,, = 158 000 MPa; So= 10 MPa; B = 0.052 MPa-‘; b = 0.5 h-l; r, = 9300; uyx = 420 MPa. The half-sphere is divided into r = 1296 fields (ACI = Ap = 5”). The constants B, b, rl, S,, are defined on the basis of a tension curve and classical creep curves. 4 CONCLUSIONS The description of the influence of the shear stress rVzon the relaxation of the tensile stress o,
K. Rusinko
was confirmed experimentally. The presented slip theory makes the physical interpretation of the discussed problem convincing. The good side of the proposed theory is the possibility of its application to description of relaxation under any complex stress. REFERENCES 1. Batdorf, S. B. and Budiansky, B., A Mathematical Theory of Plasticity Based on the Concept of Slip, NACA, T.N. (1949) 1871. 2. Osasyuk, V. V., Thermal Strength of Materials and Structural Components, Naukova Dumka, Kiev, 1967,p. 539, (in Russian). 3. Osipiuk, V., Stress relaxation in aspect of slip theory. Arch. Bud. Masz. 38 (1991), 223-231. 4. Rusinko, K. N., The Plasticity and Initial Creep Theory, Vi&a Skola,L’vov, 1981,p. 148 (in Russian).
ELSEVIER
0308-0161(94)00162-6
J. Pres. Ves. & Piping 65 (1996) 97-1(K) 0 1995 Elsevier Science Limited in Northern Ireland. All rights reserved 030%0161/96/$09.50
Description of the stress relaxation process under plane stress conditions W. Osipiuk Technical
University,
al. Wiejska 45, 15-351
Bialystok,
Poland
&
K. Rusinko Technical
University,
ul. Bandery
12, 290646 Lviv. Ukraine
(Received 10 October 1994;accepted 26 October 1994)
This work presents the application of Batdorf-Budiansky theory to the description of the stressrelaxation processunder conditions of concurrent uniaxial tension and torsion. The use of the plastic resistancefunction which takes into account strain hardening and strengtheningas an effect of internal stressconcentration was proposed.The theoretical qualitative description was comparedwith the resultsof the experiment.
1 INTRODUCTION
2 DESCRIPTION
The experimental research of relaxation is more difficult than research of creep. Investigation of the so-called pure relaxation, where the strain is constant is impossible in practice. Because of that, forecasting of relaxation curves on the basis of creep curves and adequate creep hypothesis is necessary. The list of publications concerning this problem is quite long, but they refer mainly to uniaxial stress conditions. Until now relaxation under complex stress has been insufficiently studied. The proposal of using the slip theory for the description of relaxation under complex stress seems to be well founded. The aim of the present work is to present the above description of relaxation utilizing Batdorf-Budiansky’s slip theory.’ The use of a plastic resistance function taking into account anisotropic strain-hardening and strengthening as an effect of internal stress concentration is proposed. The relaxation curve is reflected by the creep process at diminishing stress in time. A plane stress realized in thin-walled cylindrical test pieces and subject to the action of tensile force and torsional moment is considered.
In the theoretical analysis the results of the experiment run by the author of the work’ were used. The experiment was carried out on thin-walled tubular specimens made of austenitic steel. The specimens were put under concurrent tension and torsion at a temperature of 92313. In the loading system of the test machine hydraulic dynamometers were used. In this way it was possible to realize an investigation of the so-called pure relaxation in practice (E, = const, yXZ= const). The tensile stress drop Au, is a function of a value of an initial tangential stress r,,. The experimental results which are presented in Fig. 1 indicate that r,, has considerable influence on Aa:. The above experiment has been described in an aspect of a slip theory using numerical methods. 3 THEORETICAL
OF THE EXPERIMENT
DESCRIPTION
Batdorf-Budiansky’s theory assumes that plastic strain is an effect of an infinite number of slips taking place in all possible planes and directions representing any point of a body. The two 97
W. Osipiuk,
98
K. Rusinko
Budiansky, but the plastic resistance function is introduced in the form:
AU, [MPalI
srl, = So(1 + r1RI/ + Jd (3) where S, ‘is the initial plastic resistance (for (pn,= J,,, = 0); r, is a material constant; J,,/ is the parameter of internal stress concentration expressed by the form:
60
201 0
I 50
I 100
I 150
1 200
I 250
zxz[MPo]
Fig. 1. Dependence of the tensile stress drop Au; on the initial value of the tangential stress 5,; after passage of different times.
vectors y1 and 1 defining the plane and the direction of the slips is called the slip system. The orientation of the slip system is defined by three Euler’s angles (Y,~,o in the half-sphere of unit radius, Fig. 2. The integral stress is expressed by the form:
where $2 is the half-sphere region where the slips occur; dfi = cos p da dp; ol, w2 is the slip limits on the plane’s tangent in the region R to the half-sphere; (pn/is the slip density function; ni, 1, are the directional cosines of the axis II and 1 relative to the system of the coordinates x,y,z (see Ref. 3); r,,, is the component of the shear strain in the system n,l defined by the form: Zn,= (hi = (2) a;jl;nj
x,Y,z).
In the present paper the function (pn,is not used directly in the way proposed by Batdorf and
Fig. 2. The slip plane coordinates.
J,,, = B :’ 2 I
Q(t - s) ds
(4)
where B is a material constant: Q(t -s) is the diminishing function of time differences t (later instant which is studied) and s (earlier instant at which a change in stress occurred); t, is the time in which external stress grew. The choice of eqn (4) was justified in the work described in Ref. 4. If we apply stress with constant speed and assume function Q(t -s) = exp[-b(t - s)], (where b is a structural constant), eqn (4) can be expressed with sufficient precision in the form: J,, = Bz,, exp(-bt) where rn, is the maximum value of the stress in the system n,l; t is the elapsed time from the moment when this stress was applied. It is assumed that the slips occur in those half-sphere regions for which the following equality holds: L/ = Sll
(5) and that there are no slips outside these regions and the inequality rn, < S,, is satisfied. Equations (1) to (5) are the principal relationships of the concept presented in this paper. These equations were used to describe relaxation under complex stress. Relaxation is a process of stress reduction as an effect of creep. A description of relaxation needs a description of creep under diminishing stress. That is why first calculations of creep will be presented and then they will be used for an analysis of relaxation. In the authors’ opinion this procedure makes the presented slip theory more clear. A strict analytical calculation of creep deformations in the case of a complex, varying with time, stress is very difficult. Difficulties occur in determining slip region limits as well as in the solution of eqn (1). The above problem can, however, be easily solved by using numerical methods. With. this aim in mind, the half-sphere presented in Fig. 2 is divided into a large number
Stress relaxation
of sufficiently small planes AR, = cos PiAP,Aai, while integrals in eqn (1) are approximated by sums. If we divide the time t of relaxation into short periods p, the number of which is k, eqn (3) in the kth period will have the form: k-l S:,
=
So
[
1 +
r,
p=l 2
A&
+
r&t,
+
J,“,
1 (6)
where Acpn,defines the increment of function (pnl. After using eqns (6), (5), (4), (2) and forms of directional cosines we obtain: r,Acp:, = A: + A; - 1
(7)
where:
99
process
function calculated on the basis of eqn (7) for w = Wk. Limits of slip range Rk are assigned from the equation: A& = 0 (12) The curve given by eqn (12) divides the half-sphere into regions in which, during the kth period, slip occurs if A& > 0, and into regions in which there are no slips if Aqpi,5 0. Values of increments of creep strains in the kth period are conditioned by slips occurring on the ith surface Ani. These values are expressed on the basis of eqns (10) and (11) in the following way: (A&$ = f (sin 2p cos p sin wkA&);
A; = (St - J:,)( cos a cos 2p sin 0
- sin (Ysin p cos o), gLg
- f sin (Y sin 2/3 cos w”);
(I
0 Considering that slip is a result of dislocation movement we shall notice that in the definite point of the half-sphere, at the given instant of time, slip will be proceeding only in one direction. Using the fact that in the whole region of the half-sphere the relation S$,2 zs, is satisfied, we can define the slip direction o = ~~(a, /3) in any point in the half-sphere. The value of gk is calculated from the tangency condition of the plastic resistance curve S,, and the shear stress curve z,,:
as;, a& -=at.d aid
(w = cd)
(8)
Equation (8) implies: k _ (26: - 5;) cos p _ cos a cos 2/3 tg0 - (6; - J&) sin (Y sin (Y sin /3
(13)
1 (AyQi = - (cos (Ycos /3 cos 2/3 sin gk 2
while +g
x Aha;A/3;
(74
(9)
On the basis of eqn (1) the increments of linear and non-dilatational creep strains in the kth period will be expressed by the forms:
X CAP:>;Aai Api (14) Values of uk are determined from eqn (9), while values of Aqi, by eqn (7) taking into account that o = wk. The above calculations are performed for all infinitesimal surfaces AQ, forming the surface of the half-sphere. The results are summed to obtain increments of strains in the kth slip period, that is: i=l
I=1
where Y defines a number of infinitesimal surfaces AR;, .on which slips occur (A& > 0). Stresses u; and z,, are introduced into the eqn (7) by parameters A; and A$, which are defined according to eqn (7a). Increments of plastic strain A$ and Ay& are calculated according to eqn (15). Stress drop Au: after the qth number of time periods in the case of the pure relaxation describes the form: Au, = a&
= 0) - g<
==E 2 A&;@,, t,.,)
(16)
k=l
where 12, 1J: are values of directional cosines for o = ok; A& is the increment of the slip density
where E is Young’s modulus. It can be shown that from eqns (7), (7a) and (10) with consideration of (16) that the initial stress z,, influences the relaxation of stress gz by
100
W. Osipiuk,
reaction on A.$. This reaction can be interpreted as an effect of the occurrence of common slip regions in a material during tension and torsion. Figure 1 presents the comparison between the experimental results and the theoretical description according to eqn (16) for the following values: E,,,, = 158 000 MPa; So= 10 MPa; B = 0.052 MPa-‘; b = 0.5 h-l; r, = 9300; uyx = 420 MPa. The half-sphere is divided into r = 1296 fields (ACI = Ap = 5”). The constants B, b, rl, S,, are defined on the basis of a tension curve and classical creep curves. 4 CONCLUSIONS The description of the influence of the shear stress rVzon the relaxation of the tensile stress o,
K. Rusinko
was confirmed experimentally. The presented slip theory makes the physical interpretation of the discussed problem convincing. The good side of the proposed theory is the possibility of its application to description of relaxation under any complex stress. REFERENCES 1. Batdorf, S. B. and Budiansky, B., A Mathematical Theory of Plasticity Based on the Concept of Slip, NACA, T.N. (1949) 1871. 2. Osasyuk, V. V., Thermal Strength of Materials and Structural Components, Naukova Dumka, Kiev, 1967,p. 539, (in Russian). 3. Osipiuk, V., Stress relaxation in aspect of slip theory. Arch. Bud. Masz. 38 (1991), 223-231. 4. Rusinko, K. N., The Plasticity and Initial Creep Theory, Vi&a Skola,L’vov, 1981,p. 148 (in Russian).