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Modelling of the deformation process under thermo-mechanical fatigue conditions
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International Journal of Fatigue 30 (2008) 324–329
International Journalof Fatigue www.elsevier.com/locate/ijfatigue
Modelling of the deformation process under thermo-mechanical fatigue conditions J. Okrajni *, G. Junak, A. Marek Silesian University of Technology, Department of Mechanics of Materials, ul. Krasin´skiego 8, 40 019 Katowice, Poland Accepted 15 January 2007 Available online 20 March 2007
Abstract A method enabling a description of materials behaviour in the fatigue process has been developed. This paper presents some elements of the method and examples of thermo-mechanical fatigue characteristics determined for selected parameters of fatigue tests. Fatigue examinations of the P91 steel, which is used in polish power industry, were carried out. The main problem addressed in the paper is the description of a deformation process under the conditions of mechanical and thermal interactions. An appropriate model has been developed to this end. A model presentation enables extensive investigations of the influence of various types of relations between the fatigue process parameters on changes of the uniaxial stress–strain characteristics. Characteristics of the thermal cycle in thermo-mechanical fatigue conditions, i.e. courses of stress changes as a function of mechanical strain, determined on the model base have been presented for chosen shift angles in the mechanical strain cycle phase in relation to the thermal one. So far, experimental verification of the stress–strain characteristics’ course for a selected value of the phase shift angle has been made. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Thermal and mechanical loadings; Deformation process; Stress–strain curves; Thermo-mechanical fatigue
1. Introduction In the technology which uses appliances subject to a high and variable in time temperature, thermo-mechanical fatigue is one of common causes of cracking of machine parts and equipment components. Currently, an evaluation of the degree to which such parts are damaged is most often conducted based on creep characteristics; data concerning fatigue properties determined at constant temperatures are used as well. However, when trying to enhance the accuracy of forecasting the behaviour of materials, fatigue induced by the action of a cyclically variable temperature, due to its significance and a different nature compared to fatigue at a constant temperature, should be treated in a different way [1–6]. It is impossible to reflect accurately the interrelationships between damages induced by simul-
*
Corresponding author. Tel./fax: +48 32 603 4413. E-mail address: [email protected] (J. Okrajni).
0142-1123/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2007.01.043
taneous mechanical and thermal loads through isothermal tests. In this connection, a rapid development of fatigue investigations has been observed recently. The investigations take into account the influence of a simultaneous action of a variable in time temperature and mechanical loads. The number of teams involved in such research is growing and the results of the research are more and more frequently applied to evaluate and forecast the residual life of industrial appliances’ components. In this scope research is also being conducted at the Department of Mechanics of Materials, Silesian University of Technology, and first of all, refers to materials applied in the power engineering industry. This study is focused on the problem of modelling the deformation of a specimen subjected to thermo-mechanical fatigue. The approach presented in the paper addresses one of the problems the solution of which is necessary to describe the behaviour and to evaluate the durability of components subjected to the action of a cyclically variable temperature which induces a variable, non-uniform thermal stress field in them.
J. Okrajni et al. / International Journal of Fatigue 30 (2008) 324–329
3. Conditions of material investigations
2. Technical example and selection of material for the research
When the material behaviour is observed in the conditions of a cyclically variable temperature, it is necessary to isolate and define the quantities related to deformations. A deformation measured in laboratory conditions during the specimens examination is relative elongation called total deformation. This strain is a sum of thermal strain of measuring length of a specimen and its relative elongation induced by the action of load. The investigation results presented in the paper were carried out with temperature and deformation control. The range of temperature changes amounted to 450 °C with the minimum temperature of 200 °C. Heating was performed by inductive method, whereas cooling was forced using compressed air blown through an axial hole in the sample. Differences in the temperature values throughout the measurement length of the samples did not exceed 10 °C. The sample was subjected to changeable total strain of a triangular symmetric cycle, 90° out of phase in relation to the temperature cycle.
In a majority of power engineering and chemical industry installations, fatigue of a thermal and mechanical nature works parallel to creep processes, whereas the creep and fatigue phenomena cannot be separated. The only problem to be still solved is which of the above-mentioned processes should be considered as predominant and which research methodology reflects the best operating conditions of an appliance. When trying to select a representative object, where thermo-mechanical fatigue predominates, an example of a power unit component has been used. Such example can be a steam collector in a superheater, subjected to variable temperature and internal pressure. Steam collectors are thick-walled pipes with external diameters most often between 100 and 350 mm. Coils which supply and carry away a medium are connected to the pipes. Experiments conducted in operating conditions indicate the presence of fissures on the internal surface of collectors. Thermal stresses which occur in individual cycles of temperature changes are one of the causes of such cracking. The common phenomenon of cracking of the superheater collectors working in the Polish power engineering industry made the authors undertake the material research. For this purpose, the P91 steel was selected. The longstanding operation of this material has corroborated its usefulness for work at elevated temperatures. A still significant problem, however, is the cracking of the P91 steel, especially under variable temperature conditions. This stage of the research is focused on the issues of a methodological nature. When choosing the object and material for the investigations, particular attention was paid to its operating conditions. On this basis, the assumption of appropriate parameters of material investigations was possible.
a
THERMAL STRAIN
325
MECHANICAL STRAIN
4. Method of investigation results processing Within the thermo-mechanical fatigue process characteristics, one can enumerate characteristics in the form of dependencies between strains, stresses, temperature and time, fatigue process parameters and the number of cycles required until sample failure [1–5]. During fatigue tests, hysteresis loops are recorded in the force/total strain system. On the basis of the total strain values recorded, diagrams describing the changes in mechanical and thermal strain with relation to time are drawn up (Fig. 1a), and changes of mechanical strain in relation to temperature (Fig. 1b), as well as diagrams illustrating the dependencies between stress calculated as a quotient of axial force and initial section area, and mechanical strain (Fig. 2).
b
0.003
0.003 MECHANICAL STRAIN
0.002 0.002
STRAIN
0.001 0 -0.001 -0.002 -0.003 500
550
600 650 TIME, s
700
750
0.001 0 -0.001 -0.002 -0.003 100
200
300 400 500 TEMPERATURE, OC
600
700
Fig. 1. The thermal and mechanical cycles in thermo-mechanical fatigue conditions: (a) thermal and mechanical strain for a TMF cycle for displacement angle in the total strain cycle phase in relation to the thermal one u = 90° and (b) mechanical strain as temperature function, determined for the thermomechanical test cycle, shown in (a).
326
J. Okrajni et al. / International Journal of Fatigue 30 (2008) 324–329
Such assumption may be fulfilled in certain cases only. Most frequently, characteristics of the state known as steady depend on the load history and the problem connected with the accuracy evaluation of their description remains to be solved. At present, its solutions are based on laboratory investigations. Keeping in mind the fact that the complexity of the evaluation of material behaviour in thermo-mechanical fatigue conditions is far greater in comparison with the low-cycle fatigue at stable temperatures, an assumption about material characteristics stability made at the present stage of the research seems to be justified. The consequence of such assumption is the possibility of presenting stress as a function of strain and temperature, taking into account appropriate initial conditions. In the present work, the description of the deformation process with thermo-mechanical fatigue is based on the steady state characteristics with the low-cycle isothermal fatigue. In this connection, investigations were performed at stable temperatures to determine the courses of stabilised hysteresis loops. Mathematical relationships describing the course of one half of the hysteresis loop were developed under increasing strain, the loop having been determined for individual stable temperature values. The beginning of the coordinate system was assumed at the peak of the hysteresis loop for minimal strain. The curves in Fig. 3a illustrate the top sections of the hysteresis loops. Fig. 3b, shows the section between the points of the following coordinates: from (eR, rR) to (eC, rC). The method proposes a mathematical model of characteristics shown in Fig. 2a in the general form:
500 400
STRESS, MPa
300 200 100 0 -100 -200 -300 -0.003
-0.002
-0.001
0
0.001
0.002
0.003
MECHANICAL STRAIN
Fig. 2. Stress changes as a function of mechanical strain for the selected type of cycle of Fig. 1.
Thus, one obtains the characteristics of the strain process in the form of a hysteresis loop – stress as a function of mechanical strain. The course of such characteristics depends on the of total strain range, the temperature range, phase lag between the cycles of these quantities, the type of cycle and the initial parameters’ values. Having taken into account the number of factors decisive for the course of such characteristics, their experimental determination is possible practically only for selected parameters of experimental investigations. The problem which remains to be solved is the determination of fairly general models of material characteristics which, after having been verified for particular cases, could be applied to forecast the strain course with optional relations between the factors crucial for the course of the process.
r0 ¼ f ðe0 ; T Þ;
ð1Þ
Performing the transformation of function (1) and taking into consideration the initial conditions for the hysteresis loop section which illustrates the strain course under increasing strain, the following relationship is obtained:
5. Model presentation of the deformation process One of the methods commonly applied in the strain process analysis of the low-cycle fatigue conditions is an approach referring to the steady state which should be characterised by stability of the characteristics in the form of a hysteresis loop for the strain range selected at random.
a
e0 2 ð0; DeÞ
r ¼ f ½ðe eR Þ; T þ rR
ð2Þ
A similar transformation for the part of the cycle with decreasing deformation yields:
b
εC,σC
600
STRESS
σ, MPa
400 200 0
-200 -400
εR, σR
-600 -0.006 -0.004 -0.002
0
STRAIN
0.002
0.004 0.006
ε
Fig. 3. Characteristics of mathematical models of the deformation process: (a) half of the hysteresis loop for strain in isothermal conditions – part of the cycle with increasing deformation and (b) hysteresis loop for isothermal low-cycle fatigue – example.
J. Okrajni et al. / International Journal of Fatigue 30 (2008) 324–329
In the case of the loop section corresponding to decreasing strain, one obtains:
500
εC, σC
400
rC ¼ rC0 þ u0 ðe; T Þ
300 STRESS σ, MPa
327
ð5Þ 0
Functions u(e,T), u (e,T) have been introduced in the form:
200
1 uðe; T Þ ¼ ff ½ðe eR Þ; T R f ½ðe eR Þ; T g 2 1 0 u ðe; T Þ ¼ ff ½je eC j; T C f ½je eC j; T g 2
100 0 -100 -200
εR,σR
-300 -0.006 -0.004 -0.002
0.002
0.004
0.006
Function f[e 0 ,T] is assumed for a given material on the basis of the low-cycle fatigue investigation results at constant temperatures. The simple form of this function has been assumed:
Fig. 4. Hysteresis loop in the thermo-mechanical fatigue conditions – example.
r ¼ f ½je eC j; T þ rC
f ðe0 ; T Þ ¼ ðA CT n Þ arctanðDe0 Þ
ð3Þ
rR ¼ rR0 þ uðe; T Þ
ð4Þ
MODEL
420
b
620
400
1 000
300
800 600
EXPERIMENT
500
1 200
STRESS, MPa
STRESS, MPa
220
ð8Þ
where A, C, D, n are material constants determined from mathematical description of isothermal low-cycle fatigue hysteresis loops course. The results of the presented method application describe the course of the strain process in the conditions of thermomechanical fatigue of the P91 steel. The paper focuses on a general presentation of the problem. Its detailed consideration would require more extensive studies. Low-cycle fatigue investigations were performed for the steel, based on which appropriate function f[e 0 ,T], were selected (Fig. 5a). Next, using the relations (1)–(5), the strain behaviour of a sample subjected to thermo-mechanical fatigue was described. Calculation results are provided in Fig. 5b which illustrates the verification results for the model presentation on the basis of laboratory investigations (Fig. 5a). Figs. 6 and 7 illustrate the impact of displacement in the mechanical strain cycle phase in relation to the temperature cycle on the course of strain characteristics. A model presentation enables extensive investigations of the influence of various types of relations on changes of the
Eqs. (2) and (3) contain values which are the coordinates of the hysteresis loop peaks. In the case of isothermal fatigue, these coordinates have constant values for stabilised hysteresis loops. The problem of determining the value of (eR,rR) and (eC, rC) (Fig. 4) becomes more complicated in the case of strain with variable temperature. Temporary relations between stress and strain will, therefore, depend on the temperature that occurs as an independent variable as a function of f[e 0 ,T]. But the temperature and strain are also decisive for the values rR and rC. The influence of temperature on the values rR and rC was taken into account in Eqs. (2) and (3) by introducing additional functions u(e,T), u 0 (e,T), that provide for the effect of remembering the initial strain values by the material with relation to the actual temperature and strain. By that means, for the hysteresis loop section corresponding to the increasing strain, the following is obtained:
20
ð7Þ
6. Verification and application examples 0
MECHANICAL STRAIN
a
ð6Þ
200 100 -
400 -100
200
-200
0
0.002 0.004 0.006 0.008
STRAIN
0.01
0.012
-300 -0.003
-0.002
-0.001
0
0.001
0.002
0.003
MECHANICAL STRAIN
Fig. 5. Function f[e 0 ,T] determined on the basis of experimental results (a) and strain-stress cycle determined experimentally and from the model for selected the thermo-mechanical cycle (b).
328
J. Okrajni et al. / International Journal of Fatigue 30 (2008) 324–329
a
b
0.003
400
0.002
300
0.001
STRESS, MPa
MECHANICAL STRAIN
500
0
-0.001
200 100 -
-0.002
-100
-0.003 0
200
400
600
800
-200 -0.003 -0.002 -0.001
TEMPERATURE, OC
0
0.001
0.002
0.003
MECHANICAL STRAIN
Fig. 6. Characteristics of thermal and mechanical fatigue cycles: (a) temperature changes as a function of mechanical strain determined for displacement angles in phase of mechanical strain cycle in relation to the thermal one u = 180° and (b) stress changes as a function of mechanical strain for the type of cycle displayed in Fig. 6a.
a
0.005
b
400
0.004
300
0.002
200 STRESS, MPa
MECHANICAL STRAIN
0.003
0.001 0 -0.001 -0.002
100 -100
-0.003
-200
-0.004 -0.005 200
400
600
800
1000
-300 -0.006 -0.004 -0.002
TEMPERATURE, OC
0
0.002 0.004 0.006
MECHANICAL STRAIN
Fig. 7. Characteristics of thermal and mechanical fatigue cycles: (a) temperature changes as a function of mechanical strain determined for displacement angles in the phase of mechanical strain cycle in relation to the thermal one u = 90° and (b) stress changes as a function of mechanical strain for the type of cycle displayed in Fig. 7a.
strain process characteristics between the fatigue process parameters. The following figures provide characteristics of various types of thermal and mechanical cycles. Figs. 6a and 7a illustrate the thermal and mechanical cycle characteristics’ in the system: mechanical strain–temperature. Figs. 6b and 7b illustrate the courses of the dependencies between the mechanical strain and stress for the selected thermal and mechanical cycles. The course marked as in Fig. 7b illustrates the characteristics corresponding to the rhomboidal anticlockwise cycle. Fig. 6b corresponds to the out of phase cycles u = 180°. The two courses differ with the phase displacement angle u. So far, experimental verification of the strain characteristics’ course for a selected value of the phase displacement angle has been made. Correctness of the model may be also evaluated based on qualitative criteria which illustrate the resemblance between the diagrams developed and the characteristics determined experimentally by other authors [6].
In further part of the investigations, the mathematical description will be subjected to detailed verification for different materials as well as thermal and mechanical cycles. Acknowledgements The studies have been performed as a part of Research Project No. 3 T08A 02027 financed by KBN (Polish Research Scientific Committee). The authors kindly acknowledge Polish Research Scientific Committee for the support in accomplishment of the research programme. References [1] Okrajni J. Niskocyklowa trwałos´c´ stali z_ arowytrzymałych w warunkach oddziaływan´ mechanicznych i cieplnych [Low-cycle fatigue of creep-resistant steels under mechanical and thermal loading]. ZN Politechniki S´la˛skiej, Hutnictwo z. 32; 1988 [in Polish].
J. Okrajni et al. / International Journal of Fatigue 30 (2008) 324–329 [2] Coffin LF. Fatigue at high temperatures in fatigue at elevated temperatures. ASTM STP 520 1973:5–34. [3] Weron´ski A. Zme˛czenie cieplne metali [Thermal fatigue of metals]. Warszawa: WNT; 1983 [in Polish]. [4] Christ H-J, Mughrabi H, Kraft S, Petry F, Zauter R, Eckert K. The use of plastic strain control in thermo-mechanical fatigue testing. In: Bressers J, Re´my L, editors. Fatigue under thermal and mechanical loading. Netherlands: Kluwer Academic Publishers; 1996. p. 1–14.
329
[5] Okrajni J, Marek A, Junak G. Zme˛czenie cieplno-mechaniczne stali stosowanych w energetyce [Thermo-mechanical fatigue of steels used in power industry]. In: XXI Sympozjum Mechaniki Eksperymentalnej Ciała Stałego, Jachranka, Poland; 2004. p. 355–60 [in: Polish]. [6] Bressers J, Re´my L, editors. Fatigue under thermal and mechanical loading. Netherlands: Kluwer Academic Publishers; 1996.
International Journal of Fatigue 30 (2008) 324–329
International Journalof Fatigue www.elsevier.com/locate/ijfatigue
Modelling of the deformation process under thermo-mechanical fatigue conditions J. Okrajni *, G. Junak, A. Marek Silesian University of Technology, Department of Mechanics of Materials, ul. Krasin´skiego 8, 40 019 Katowice, Poland Accepted 15 January 2007 Available online 20 March 2007
Abstract A method enabling a description of materials behaviour in the fatigue process has been developed. This paper presents some elements of the method and examples of thermo-mechanical fatigue characteristics determined for selected parameters of fatigue tests. Fatigue examinations of the P91 steel, which is used in polish power industry, were carried out. The main problem addressed in the paper is the description of a deformation process under the conditions of mechanical and thermal interactions. An appropriate model has been developed to this end. A model presentation enables extensive investigations of the influence of various types of relations between the fatigue process parameters on changes of the uniaxial stress–strain characteristics. Characteristics of the thermal cycle in thermo-mechanical fatigue conditions, i.e. courses of stress changes as a function of mechanical strain, determined on the model base have been presented for chosen shift angles in the mechanical strain cycle phase in relation to the thermal one. So far, experimental verification of the stress–strain characteristics’ course for a selected value of the phase shift angle has been made. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Thermal and mechanical loadings; Deformation process; Stress–strain curves; Thermo-mechanical fatigue
1. Introduction In the technology which uses appliances subject to a high and variable in time temperature, thermo-mechanical fatigue is one of common causes of cracking of machine parts and equipment components. Currently, an evaluation of the degree to which such parts are damaged is most often conducted based on creep characteristics; data concerning fatigue properties determined at constant temperatures are used as well. However, when trying to enhance the accuracy of forecasting the behaviour of materials, fatigue induced by the action of a cyclically variable temperature, due to its significance and a different nature compared to fatigue at a constant temperature, should be treated in a different way [1–6]. It is impossible to reflect accurately the interrelationships between damages induced by simul-
*
Corresponding author. Tel./fax: +48 32 603 4413. E-mail address: [email protected] (J. Okrajni).
0142-1123/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2007.01.043
taneous mechanical and thermal loads through isothermal tests. In this connection, a rapid development of fatigue investigations has been observed recently. The investigations take into account the influence of a simultaneous action of a variable in time temperature and mechanical loads. The number of teams involved in such research is growing and the results of the research are more and more frequently applied to evaluate and forecast the residual life of industrial appliances’ components. In this scope research is also being conducted at the Department of Mechanics of Materials, Silesian University of Technology, and first of all, refers to materials applied in the power engineering industry. This study is focused on the problem of modelling the deformation of a specimen subjected to thermo-mechanical fatigue. The approach presented in the paper addresses one of the problems the solution of which is necessary to describe the behaviour and to evaluate the durability of components subjected to the action of a cyclically variable temperature which induces a variable, non-uniform thermal stress field in them.
J. Okrajni et al. / International Journal of Fatigue 30 (2008) 324–329
3. Conditions of material investigations
2. Technical example and selection of material for the research
When the material behaviour is observed in the conditions of a cyclically variable temperature, it is necessary to isolate and define the quantities related to deformations. A deformation measured in laboratory conditions during the specimens examination is relative elongation called total deformation. This strain is a sum of thermal strain of measuring length of a specimen and its relative elongation induced by the action of load. The investigation results presented in the paper were carried out with temperature and deformation control. The range of temperature changes amounted to 450 °C with the minimum temperature of 200 °C. Heating was performed by inductive method, whereas cooling was forced using compressed air blown through an axial hole in the sample. Differences in the temperature values throughout the measurement length of the samples did not exceed 10 °C. The sample was subjected to changeable total strain of a triangular symmetric cycle, 90° out of phase in relation to the temperature cycle.
In a majority of power engineering and chemical industry installations, fatigue of a thermal and mechanical nature works parallel to creep processes, whereas the creep and fatigue phenomena cannot be separated. The only problem to be still solved is which of the above-mentioned processes should be considered as predominant and which research methodology reflects the best operating conditions of an appliance. When trying to select a representative object, where thermo-mechanical fatigue predominates, an example of a power unit component has been used. Such example can be a steam collector in a superheater, subjected to variable temperature and internal pressure. Steam collectors are thick-walled pipes with external diameters most often between 100 and 350 mm. Coils which supply and carry away a medium are connected to the pipes. Experiments conducted in operating conditions indicate the presence of fissures on the internal surface of collectors. Thermal stresses which occur in individual cycles of temperature changes are one of the causes of such cracking. The common phenomenon of cracking of the superheater collectors working in the Polish power engineering industry made the authors undertake the material research. For this purpose, the P91 steel was selected. The longstanding operation of this material has corroborated its usefulness for work at elevated temperatures. A still significant problem, however, is the cracking of the P91 steel, especially under variable temperature conditions. This stage of the research is focused on the issues of a methodological nature. When choosing the object and material for the investigations, particular attention was paid to its operating conditions. On this basis, the assumption of appropriate parameters of material investigations was possible.
a
THERMAL STRAIN
325
MECHANICAL STRAIN
4. Method of investigation results processing Within the thermo-mechanical fatigue process characteristics, one can enumerate characteristics in the form of dependencies between strains, stresses, temperature and time, fatigue process parameters and the number of cycles required until sample failure [1–5]. During fatigue tests, hysteresis loops are recorded in the force/total strain system. On the basis of the total strain values recorded, diagrams describing the changes in mechanical and thermal strain with relation to time are drawn up (Fig. 1a), and changes of mechanical strain in relation to temperature (Fig. 1b), as well as diagrams illustrating the dependencies between stress calculated as a quotient of axial force and initial section area, and mechanical strain (Fig. 2).
b
0.003
0.003 MECHANICAL STRAIN
0.002 0.002
STRAIN
0.001 0 -0.001 -0.002 -0.003 500
550
600 650 TIME, s
700
750
0.001 0 -0.001 -0.002 -0.003 100
200
300 400 500 TEMPERATURE, OC
600
700
Fig. 1. The thermal and mechanical cycles in thermo-mechanical fatigue conditions: (a) thermal and mechanical strain for a TMF cycle for displacement angle in the total strain cycle phase in relation to the thermal one u = 90° and (b) mechanical strain as temperature function, determined for the thermomechanical test cycle, shown in (a).
326
J. Okrajni et al. / International Journal of Fatigue 30 (2008) 324–329
Such assumption may be fulfilled in certain cases only. Most frequently, characteristics of the state known as steady depend on the load history and the problem connected with the accuracy evaluation of their description remains to be solved. At present, its solutions are based on laboratory investigations. Keeping in mind the fact that the complexity of the evaluation of material behaviour in thermo-mechanical fatigue conditions is far greater in comparison with the low-cycle fatigue at stable temperatures, an assumption about material characteristics stability made at the present stage of the research seems to be justified. The consequence of such assumption is the possibility of presenting stress as a function of strain and temperature, taking into account appropriate initial conditions. In the present work, the description of the deformation process with thermo-mechanical fatigue is based on the steady state characteristics with the low-cycle isothermal fatigue. In this connection, investigations were performed at stable temperatures to determine the courses of stabilised hysteresis loops. Mathematical relationships describing the course of one half of the hysteresis loop were developed under increasing strain, the loop having been determined for individual stable temperature values. The beginning of the coordinate system was assumed at the peak of the hysteresis loop for minimal strain. The curves in Fig. 3a illustrate the top sections of the hysteresis loops. Fig. 3b, shows the section between the points of the following coordinates: from (eR, rR) to (eC, rC). The method proposes a mathematical model of characteristics shown in Fig. 2a in the general form:
500 400
STRESS, MPa
300 200 100 0 -100 -200 -300 -0.003
-0.002
-0.001
0
0.001
0.002
0.003
MECHANICAL STRAIN
Fig. 2. Stress changes as a function of mechanical strain for the selected type of cycle of Fig. 1.
Thus, one obtains the characteristics of the strain process in the form of a hysteresis loop – stress as a function of mechanical strain. The course of such characteristics depends on the of total strain range, the temperature range, phase lag between the cycles of these quantities, the type of cycle and the initial parameters’ values. Having taken into account the number of factors decisive for the course of such characteristics, their experimental determination is possible practically only for selected parameters of experimental investigations. The problem which remains to be solved is the determination of fairly general models of material characteristics which, after having been verified for particular cases, could be applied to forecast the strain course with optional relations between the factors crucial for the course of the process.
r0 ¼ f ðe0 ; T Þ;
ð1Þ
Performing the transformation of function (1) and taking into consideration the initial conditions for the hysteresis loop section which illustrates the strain course under increasing strain, the following relationship is obtained:
5. Model presentation of the deformation process One of the methods commonly applied in the strain process analysis of the low-cycle fatigue conditions is an approach referring to the steady state which should be characterised by stability of the characteristics in the form of a hysteresis loop for the strain range selected at random.
a
e0 2 ð0; DeÞ
r ¼ f ½ðe eR Þ; T þ rR
ð2Þ
A similar transformation for the part of the cycle with decreasing deformation yields:
b
εC,σC
600
STRESS
σ, MPa
400 200 0
-200 -400
εR, σR
-600 -0.006 -0.004 -0.002
0
STRAIN
0.002
0.004 0.006
ε
Fig. 3. Characteristics of mathematical models of the deformation process: (a) half of the hysteresis loop for strain in isothermal conditions – part of the cycle with increasing deformation and (b) hysteresis loop for isothermal low-cycle fatigue – example.
J. Okrajni et al. / International Journal of Fatigue 30 (2008) 324–329
In the case of the loop section corresponding to decreasing strain, one obtains:
500
εC, σC
400
rC ¼ rC0 þ u0 ðe; T Þ
300 STRESS σ, MPa
327
ð5Þ 0
Functions u(e,T), u (e,T) have been introduced in the form:
200
1 uðe; T Þ ¼ ff ½ðe eR Þ; T R f ½ðe eR Þ; T g 2 1 0 u ðe; T Þ ¼ ff ½je eC j; T C f ½je eC j; T g 2
100 0 -100 -200
εR,σR
-300 -0.006 -0.004 -0.002
0.002
0.004
0.006
Function f[e 0 ,T] is assumed for a given material on the basis of the low-cycle fatigue investigation results at constant temperatures. The simple form of this function has been assumed:
Fig. 4. Hysteresis loop in the thermo-mechanical fatigue conditions – example.
r ¼ f ½je eC j; T þ rC
f ðe0 ; T Þ ¼ ðA CT n Þ arctanðDe0 Þ
ð3Þ
rR ¼ rR0 þ uðe; T Þ
ð4Þ
MODEL
420
b
620
400
1 000
300
800 600
EXPERIMENT
500
1 200
STRESS, MPa
STRESS, MPa
220
ð8Þ
where A, C, D, n are material constants determined from mathematical description of isothermal low-cycle fatigue hysteresis loops course. The results of the presented method application describe the course of the strain process in the conditions of thermomechanical fatigue of the P91 steel. The paper focuses on a general presentation of the problem. Its detailed consideration would require more extensive studies. Low-cycle fatigue investigations were performed for the steel, based on which appropriate function f[e 0 ,T], were selected (Fig. 5a). Next, using the relations (1)–(5), the strain behaviour of a sample subjected to thermo-mechanical fatigue was described. Calculation results are provided in Fig. 5b which illustrates the verification results for the model presentation on the basis of laboratory investigations (Fig. 5a). Figs. 6 and 7 illustrate the impact of displacement in the mechanical strain cycle phase in relation to the temperature cycle on the course of strain characteristics. A model presentation enables extensive investigations of the influence of various types of relations on changes of the
Eqs. (2) and (3) contain values which are the coordinates of the hysteresis loop peaks. In the case of isothermal fatigue, these coordinates have constant values for stabilised hysteresis loops. The problem of determining the value of (eR,rR) and (eC, rC) (Fig. 4) becomes more complicated in the case of strain with variable temperature. Temporary relations between stress and strain will, therefore, depend on the temperature that occurs as an independent variable as a function of f[e 0 ,T]. But the temperature and strain are also decisive for the values rR and rC. The influence of temperature on the values rR and rC was taken into account in Eqs. (2) and (3) by introducing additional functions u(e,T), u 0 (e,T), that provide for the effect of remembering the initial strain values by the material with relation to the actual temperature and strain. By that means, for the hysteresis loop section corresponding to the increasing strain, the following is obtained:
20
ð7Þ
6. Verification and application examples 0
MECHANICAL STRAIN
a
ð6Þ
200 100 -
400 -100
200
-200
0
0.002 0.004 0.006 0.008
STRAIN
0.01
0.012
-300 -0.003
-0.002
-0.001
0
0.001
0.002
0.003
MECHANICAL STRAIN
Fig. 5. Function f[e 0 ,T] determined on the basis of experimental results (a) and strain-stress cycle determined experimentally and from the model for selected the thermo-mechanical cycle (b).
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a
b
0.003
400
0.002
300
0.001
STRESS, MPa
MECHANICAL STRAIN
500
0
-0.001
200 100 -
-0.002
-100
-0.003 0
200
400
600
800
-200 -0.003 -0.002 -0.001
TEMPERATURE, OC
0
0.001
0.002
0.003
MECHANICAL STRAIN
Fig. 6. Characteristics of thermal and mechanical fatigue cycles: (a) temperature changes as a function of mechanical strain determined for displacement angles in phase of mechanical strain cycle in relation to the thermal one u = 180° and (b) stress changes as a function of mechanical strain for the type of cycle displayed in Fig. 6a.
a
0.005
b
400
0.004
300
0.002
200 STRESS, MPa
MECHANICAL STRAIN
0.003
0.001 0 -0.001 -0.002
100 -100
-0.003
-200
-0.004 -0.005 200
400
600
800
1000
-300 -0.006 -0.004 -0.002
TEMPERATURE, OC
0
0.002 0.004 0.006
MECHANICAL STRAIN
Fig. 7. Characteristics of thermal and mechanical fatigue cycles: (a) temperature changes as a function of mechanical strain determined for displacement angles in the phase of mechanical strain cycle in relation to the thermal one u = 90° and (b) stress changes as a function of mechanical strain for the type of cycle displayed in Fig. 7a.
strain process characteristics between the fatigue process parameters. The following figures provide characteristics of various types of thermal and mechanical cycles. Figs. 6a and 7a illustrate the thermal and mechanical cycle characteristics’ in the system: mechanical strain–temperature. Figs. 6b and 7b illustrate the courses of the dependencies between the mechanical strain and stress for the selected thermal and mechanical cycles. The course marked as in Fig. 7b illustrates the characteristics corresponding to the rhomboidal anticlockwise cycle. Fig. 6b corresponds to the out of phase cycles u = 180°. The two courses differ with the phase displacement angle u. So far, experimental verification of the strain characteristics’ course for a selected value of the phase displacement angle has been made. Correctness of the model may be also evaluated based on qualitative criteria which illustrate the resemblance between the diagrams developed and the characteristics determined experimentally by other authors [6].
In further part of the investigations, the mathematical description will be subjected to detailed verification for different materials as well as thermal and mechanical cycles. Acknowledgements The studies have been performed as a part of Research Project No. 3 T08A 02027 financed by KBN (Polish Research Scientific Committee). The authors kindly acknowledge Polish Research Scientific Committee for the support in accomplishment of the research programme. References [1] Okrajni J. Niskocyklowa trwałos´c´ stali z_ arowytrzymałych w warunkach oddziaływan´ mechanicznych i cieplnych [Low-cycle fatigue of creep-resistant steels under mechanical and thermal loading]. ZN Politechniki S´la˛skiej, Hutnictwo z. 32; 1988 [in Polish].
J. Okrajni et al. / International Journal of Fatigue 30 (2008) 324–329 [2] Coffin LF. Fatigue at high temperatures in fatigue at elevated temperatures. ASTM STP 520 1973:5–34. [3] Weron´ski A. Zme˛czenie cieplne metali [Thermal fatigue of metals]. Warszawa: WNT; 1983 [in Polish]. [4] Christ H-J, Mughrabi H, Kraft S, Petry F, Zauter R, Eckert K. The use of plastic strain control in thermo-mechanical fatigue testing. In: Bressers J, Re´my L, editors. Fatigue under thermal and mechanical loading. Netherlands: Kluwer Academic Publishers; 1996. p. 1–14.
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[5] Okrajni J, Marek A, Junak G. Zme˛czenie cieplno-mechaniczne stali stosowanych w energetyce [Thermo-mechanical fatigue of steels used in power industry]. In: XXI Sympozjum Mechaniki Eksperymentalnej Ciała Stałego, Jachranka, Poland; 2004. p. 355–60 [in: Polish]. [6] Bressers J, Re´my L, editors. Fatigue under thermal and mechanical loading. Netherlands: Kluwer Academic Publishers; 1996.