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Thermo-mechanical measurement of elasto-plastic transitions during cyclic loading
Theoretical and Applied Fracture Mechanics 56 (2011) 1–6
Contents lists available at SciVerse ScienceDirect
Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Thermo-mechanical measurement of elasto-plastic transitions during cyclic loading R.V. Prakash ⇑, T. Pravin, T. Kathirvel, Krishnan Balasubramaniam Department of Mechanical Engineering, Indian Institute of Technology, Madras, Chennai 600 036, India
a r t i c l e
i n f o
Article history: Available online 24 September 2011 Keywords: Infrared thermography Thermo-elastic effect Yield points Stress–strain curve segments
a b s t r a c t The surface temperature of stainless steel SS304 low cycle fatigue specimens subjected to cyclic loading was studied using infrared thermography technique. The thermal data mapped onto the various stages of cyclic stress–strain curve shows the ability of these measurements to identify the yield points in both the compression and tension loading. Based on the results of this study, it is possible to identify the state of stress for materials such as elastic tension, plastic tension, elastic compression, plastic compression during cyclic loading using infrared thermographic data. The thermo-elastic slope and thermo-plastic slope was observed to be dependent on the prior loading cycles. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Infrared thermography is a non-destructive, non-contact, real time technique. It offers the advantage of remote/inaccessible/hazardous area measurements [1]. Any object emits energy by radiation according to its surface temperature. Human eyes are capable of detecting this energy only when the object is warm enough and the energy that is emitted is in the visible band (0.4–0.8 lm). In fact, the human eyes are able to recognize the temperature during heating of a metal, when it passes from red (720 °C), to yellow, to white (1350 °C). For low temperature values falling in the infrared region, thermography provides us with artificial eyes and gives us a chance to see invisible radiation [2]. An infrared (IR) system basically includes a camera, equipped with a series of changeable optics, and a computer. The core of the camera is the infrared detector, which absorbs the IR energy emitted by the object (whose surface temperature is to be measured) and converts it into electrical voltage or current. Finally this electrical voltage is converted into thermograms that provide information on the transient temperature behavior of a given point or area on the surface of the test sample at any given time in the loading cycle. The thermo-elastic effect suggests the existence of a relationship between the amplitude of stress and temperature change in a material during elastic loading. In the elastic region, when a material is subjected to tensile loading, the material undergoes cooling and when a material is subjected to compressive loading, it undergoes heating. Volume dilatation (volume change) takes place in elastic region during tensile loading of material, i.e., atoms get pulled apart slightly and hence, there is a slight increase in bond length. If the loading is done under adiabatic conditions then ⇑ Corresponding author. E-mail address: [email protected] (R.V. Prakash). 0167-8442/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2011.09.001
there is no time for thermal equilibrium to take place between material and surroundings and hence, there is a gross decrease in temperature in this case of elastic tension by a tensile stress [3]. Thermodynamic equation of state and first law of thermodynamics govern these heat effects. The thermo-elastic equation is expressed as [4]:
DT ¼
aT Dr qC p
ð1Þ
where a (lm/m °C) is coefficient of thermal expansion, T (°C) is the ambient temperature, q (g/cc) is density, Cp (J/g °C) is heat capacity at constant pressure and Dr (N/m2) is the change in stress Numerous applications of infrared technique have been developed: (a) to find the distribution of stresses in specimens by applying the principle of thermo-elasticity [5], (b) to characterize in situ fatigue damage during testing of the CFRP specimens using a passive thermography by measuring the temperature increase of the specimen due to hysteretic heating during fatigue testing [6], (c) to determine the size and the location of a subsurface defect from the phase image of infrared thermography [7], (d) to determine the fatigue limit of material by plotting the static temperature in temperature response of material versus stress and fitting linear fit [8]. These studies on temperature evolution during cyclic loading tests have showed that, the temperature response curve of the material is sigmoidal nature with three different slopes [9,10]; the initial rapid change in temperature due to the change in the hysteresis loop area, the secondary steady temperature response due to the stabilization of material stress–strain response along with equilibrium with the surrounding temperature and final sudden temperature rise just prior to final fracture due to the high plasticity induced at the crack tip. Isoenergy density theory was applied to fatigue loading, estimating the hysteresis loops of an hour-glass cylindrical specimen and computing the
2
R.V. Prakash et al. / Theoretical and Applied Fracture Mechanics 56 (2011) 1–6
non-equilibrium temperature determined from the displacement gradients by considering the simultaneous change of surface and volume energy, without invoking the concept of heat, found to oscillate about the ambient temperature [11–13]. The present work deals with a more detailed study on the variation of surface temperature of test sample corresponding to different segments of stress–strain curve of material undergoing cyclic loading.
2. Experimental details 2.1. Experimental setup Cyclic loading experiments were carried out on a 100 kN MTS servo-hydraulic testing machine. Fig. 1 shows the experimental set up. The experiments were carried out under completely reversed loading conditions, in displacement control mode for elastic–plastic loading and under force controlled mode for purely elastic loading. The infrared thermal imaging system used for the temperature measurement is a CEDIP Jade LWIR camera (refer Fig. 1). This
consists of an infrared camera cooled by stirring cooler. The camera contains an Hg–Cd–Te detector which is sensitive to infrared radiation in the wavelength range 8–10 lm. The window size used in the experiments is 60 56 pixels and an integration time of 200 ls. The camera was positioned on a tripod approximately about 1 m from the specimen surface. The temperature images were acquired in the Automatic Gain Controlled (AGC) mode at a various frame rates and the digital data stored in a computer hard disk. Post-processing software was used for analyzing and extracting the timing graph of temperature data. The temperature values are averaged over a rectangular area along the gage length of the specimen. The temperature variation measured during the experiment is defined as the difference between the temperature at the center of the specimen and the temperature of an unloaded reference specimen. The use of a reference specimen, located next to the loaded sample, eliminates uncertainties introduced by room temperature fluctuations during a long duration test. Polished material surfaces have a small absorptivity and a weak emissivity in infrared spectrum; to overcome the emissivity problem, the specimens were coated with matt-finish black board paint. 2.2. Material and specimen Test specimens conforming to ASTM 606 standard [14] with a gage length diameter of 7 mm and a gage length of 27 mm and grip portion diameter of 10 mm were CNC machined as shown in Fig. 2 for as-received stainless steel (Grade 304) material. As a prerequisite of cyclic testing, the surface of the specimens were mirror polished prior to cyclic testing and then painted black, and the specimen was held using hydraulic grips. Axial extensometer with a gage length of 25 mm and a travel of ±5 mm was used to estimate gage section strain. Table 1 presents the chemical composition of stainless steel used in the experiments which suggests that, the material conforms to ASTM.A.276 SS Type 304.
IRT Camera
2.3. Testing conditions Two types of cyclic loading tests were carried out to study the temperature response variation in a cycle corresponding to various segments of stress–strain curve. Initially to study the material temperature response under fully elastic cyclic loading, the specimen was loaded in the force control with load amplitude equal to 10 kN and stress ratio (ratio of minimum to maximum load) of 1. This load corresponds to a value below yield point load of
Fig. 1. Experimental set up.
Fig. 2. Specimen geometry.
Table 1 Material chemical composition.a
a
Name of the element
Carbon
Chromium
Manganese
Nickel
Phosphorous
Sulfur
Silicon
Wt.%
0.0370
18.100
1.3100
8.020
0.0273
0.0245
0.3790
Balance Fe.
R.V. Prakash et al. / Theoretical and Applied Fracture Mechanics 56 (2011) 1–6
3
the material (the yield point of the sample used was estimated to be 13.5 kN). The test was carried out at 5 Hz frequency and the frames were grabbed using thermal images were captured from IR camera at a frame rate of 100 Hz. To study the material temperature response to elastic–plastic fatigue loading, tests were carried out under strain control with strain amplitude of 8000 micro-strains (the yield strain value here was estimated to be 1800 micro-strains). The test was carried out at 0.5 Hz frequency and the thermal images were captured at a frame rate of 50 Hz.
3. Results and discussion 3.1. Temperature response of material to various segments of stress– strain curve under elastic–elastic fatigue loading conditions During elastic–elastic cyclic loading, material is loaded below its yield point load, so plasticity induced is very small (generally negligible). The area of hysteresis loop formed is normally considered to be negligible as shown in Fig. 3. The overall temperature rise observed during this kind of cyclic test is small. Fig. 4 shows the relationship of stress, strain and temperature with respect to
Fig. 5. Temperature variation within single loading cycle due to elastic cyclic loading.
Fig. 3. Hysteresis loop for 5000th cycle of elastic–elastic cyclic loading.
time for initial three cycles of elastic–elastic cyclic loading. Fig. 5 shows that, the temperature response of material in elastic–elastic loading has only two segments corresponding to elastic tension and elastic compression. The first segment suggests a decrease in temperature due to the thermo-elastic effect in tension loading and second segment suggests an increase in the temperature which is due to thermo-elastic effect in compression loading. The temperature increase during unloading is found to be higher than the temperature drop during loading cycle leading to a residual temperature increase at the end of each elastic loading–unloading cycle. Using the mathematical expression for thermo-elastic effect Eq. (1), and constants from Ref. [15], for SS 304; a = 17.3 lm/ m °C, q = 8 g/cc, Cp = 0.5 J/g °C, the drop or rise in temperature due thermo-elastic effect can be calculated which is equal to 0.68 °C. The experimental results for this material suggest a drop in temperature to be about 0.66 °C, which is in good agreement with the analytical predictions. Also it is observed that, the slope of both the loading and unloading segments in temperature response cycle is similar (0.89) and it does not change with the number of cycles of loading. However, in view of the small increase in temperature during every unloading, the net temperature response was found to progressively increase for the specimen with the number of cycles in the elastic loading experiment. 3.2. Temperature response of material to various segments of stress– strain curve under elastic–plastic fatigue loading conditions
Fig. 4. Strain, stress, temperature variation for initial two cycles of elastic–elastic cyclic loading.
In this type of cyclic loading, it was observed that the material load–displacement curve had four different segments
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Fig. 6. Hysteresis loop for 300th cycle of elastic–plastic cyclic loading.
Fig. 8. Typical material responses of SS 304 during a strain controlled cyclic test during early fatigue cycles: (a) axial force, and (b) temperature evolution versus number of cycles curves.
Fig. 7. Typical temperature evolution during continuous cyclic loading test. Both elastic–elastic and elastic–plastic cyclic loading exhibit similar behavior.
corresponding to elastic-tension, plastic-tension, elastic compression and plastic compression. The area of hysteresis loop formed is also significant as shown in Fig. 6. The load–displacement response was acquired by the MTS system. The low-cycle fatigue tests was carried out under strain control with a triangular waveform with strain amplitude of ±8000 micro-strains. Fig. 7 shows the temperature evolution during the continuous cyclic test under above loading conditions. The temperature profile of material (both the maximum as well as the average temperature in a cycle plotted against time) has sigmoidal nature with three different slopes. In Fig. 7, the initial rapid temperature rise (region a) is due to change in hysteresis loop area at first few fatigue cycles [16]. Then temperature response becomes stable (region b) due to the stable stress–strain response of the material and temperature equilibrium with the surrounding. Finally temperature rises suddenly (region c) due to the high plasticity induced at the crack tip just prior to failure. The evolution of temperature with load for each loading cycle during the initial two cycles corresponding to region a in Fig. 7, are presented in more detail in Fig. 8a and b. When the load was initially applied on the specimen, the strain was in the elastic range, and the load or strain–temperature relation can be described by the thermoelastic effect, i.e., the temperature changes were directly proportional to the stress. At the initial stress-free
stage, the temperature difference was 0 °C, but the temperature dropped by 0.50 °C when the load increased from no-load (A) to the yield-load level (B), due to the thermoelastic effect. Beyond the yield load level, up to the maximum load level (C), the temperature increases, due to heat dissipation because of irreversible plastic deformation. When the load elastically decreased from the maximum level (C), 22–0 kN (D), the temperature continued to increase because of the thermoelastic effect in compression loading. During the plastic compression loading beyond the noload condition (E), the temperature increase continues because of the plastic deformation until the load reached the minimum load level of 20 kN (F). Subsequently, when the load again elastically increases from 20 to the yield value at 7 kN (G), the temperature
Fig. 9. Typical material responses of SS 304 during a strain controlled cyclic load test during 76th and 77th cycles: (a) axial force, and (b) temperature evolution versus number of cycles curves.
R.V. Prakash et al. / Theoretical and Applied Fracture Mechanics 56 (2011) 1–6
Fig. 10. Typical material responses of SS 304 during a strain controlled cyclic load test during 201th and 202th cycles: (a) axial force, and (b) temperature evolution versus number of cycles curves.
decreased again due to the thermoelastic effect. After yield point (G), the temperature increased again because of plastic deformation. This cycle of temperature response continued with mechanical loading response. For subsequent cycles, the strain and temperature responses repeated, and the mean temperature continuously increased. Figs. 9 and 10 shows the temperature variation corresponding to different stress–strain segments at intermediate cycles, i.e. 76th–77th and 201th–202th loading cycles in region ‘b’ of Fig. 7. It can be seen from Fig. 8–10 that, it is possible to identify the various segments of stress–strain curve from the temperature response. In temperature response of material, four different slopes are observed corresponding to four segments of stress–strain curve. It can also be seen that, it is possible to locate the yield point of material both in tension and compression using this material temperature response. As it can be seen from the Fig. 9 that both
5
Fig. 11. Hysteresis strain energy density comparison for elastic–elastic and elastic– plastic cyclic loading.
the temperature response slopes corresponding to elastic-compression and plastic-compression are linear and they meet exactly at the point which matches with the yield point in compression from load versus number of loading cycles curve. Also the tensile yield point can be found out by identifying the point of non-linearity in temperature response of material, where linear drop in temperature due to elastic-tension loading ends. Thus it is possible to find the yield point of material in tension and compression using the infrared thermography technique which is otherwise difficult task in field components. Also it can be observed from the Fig. 9 that, the rate at which the temperature increases due to elasticcompression is higher than that of the plastic-compression. As three out of four segments of stress–strain curve namely plastic tension, elastic compression and plastic compression corresponds to temperature increase and only one segment that is elastic tension corresponds to temperature decrease, the net mean temperature of material keeps on increasing after each loading cycle. After every three segments corresponding to temperature increase
Fig. 12. Change in the temperature profile slopes within single loading cycle with number of cycles and the corresponding best fit curves.
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temperature starts dropping for the fourth segment, that point of maximum temperature corresponds to the minimum stress level in that cycle. By comparing Figs. 8–10, it can be seen that, for initial two cycles the temperature increase per cycle is nearly 2 °C, for 76th and 77th cycle the temperature increase per cycle is around 1.25 °C and for 201th and 202th it is only 1.1 °C. This change in temperature evolution per cycle could be due to the thermal equilibrium reached by the specimen with surrounding after initial few loading cycles. Hysteresis strain energy density (HSED)was calculated for elastic–elastic and elastic–plastic cyclic loading based on the area enclosed by cyclic stress–strain curve and plotted against number of cycles as shown in Fig. 11. In case of elastic–elastic cyclic loading, HSED (ep = 190.5 micro-strains for 5000th cycle) was in the range of 0.03–0.05 J/mm3 for 1–5e4 cycles, the specimen did not fail. For elastic–plastic cyclic loading, HSED (ep = 9215 micro-strains for 300th cycle) was in the range of 7–10 J/mm3 for 1–750 cycles, the specimen failed at 750th cycle. In case of elastic–plastic fatigue loading, the slope of temperature profile corresponding to various stress–strain segments changes with number of loading cycles. Fig. 12 shows the change in slopes with respect to the number of cycles for elastic-compression, plastic compression and elastic-tension segments of stress– strain curve. The slope for elastic-tension segments is negative due to the thermo-elastic effect but for comparison purpose only magnitude of the slope is considered. The slope of elastic-tension segment of temperature profile increases with the number of cycles due to the fact that, the temperature difference between specimen and surrounding increases with number of cycles and hence during temperature drop due to elastic-tension heat transfer occurs at faster rate. Similarly in case of elastic-compression and plastic compression, it becomes difficult for the specimen to maintain the same rate at which the temperature increases in each cycle and hence the slope of the temperature profile decreases with the number of cycles. 4. Conclusions The experimental results showed that, in case of elastic–elastic cyclic loading within each cycle, the material temperature response has two slopes corresponding to elastic-tension and elasticcompression which is due to thermoelastic effect and in case of elastic–plastic cyclic loading within each loading cycle, the material temperature response curve has four slopes corresponding to four segments in stress–strain curve. The yield point of material in tension and compression can be found out using the infrared thermography technique. In case of compressive yield point, it is the point where the intersection of two distinct slopes in the temperature curve that corresponds to elastic-compression and plastic-compression segments of stress–strain curve takes place. In case of tensile yield point, it is the point where linearity of
temperature drop (in temperature response of material due to elastic tension) ends. In case of elastic loading, the slopes corresponding to elastic-tension and elastic-compression were found to be the same and does do not change with the number of cycles. There is a net increase in temperature in each elastic loading– unloading cycle that cumulatively adds-up as the number of cycles increases. In elastic–plastic fatigue loading, the slopes corresponding to four segments of stress–strain curve changes with number of loading cycles. It is observed that, the slope of the material temperature response curve corresponding to elastic-compression segment of stress–strain curve is higher than that of the plastic-compression segment. Also it is observed that, the highest temperature in each cycle is reached when the stress reaches its minimum value in cycle. References [1] X.P.V. Maldague, Theory and practice of infrared technology for nondestructive testing, Wiley Series in Microwave and Optical Engineering, John Wiley and Sons Publication, 2001. [2] M. Carosena, M.C. Giovanni, L. Giorleo, The use of infrared thermography for materials characterization, Journal of Materials Processing Technology 155– 156 (2004) 1132–1137. [3] K.N. Pandey, S. Chand, Deformation based temperature rise: a review, International Journal of Pressure Vessels and Piping 80 (2003) 673–687. [4] Norbert G.H. Meyendorf, Henrik Rosner, Victoria Kramb, Shamachary Sathish, Thermo-acoustic fatigue characterization, Ultrasonics 40 (2002) 427–434. [5] J.M. Dulieu-Barton, P. Stanley, Development and applications of thermoelastic stress analysis, Journal of Strain Analysis 33 (2) (1998) 93–104. [6] R. Steinberger, T.I.V. Leitao, E. Ladstatter, G. Pinter, W. Billinger, R.W. Lang, Infrared thermographic techniques for non-destructive damage characterization of carbon fibre reinforced polymers during tensile fatigue testing, International Journal of Fatigue 28 (2006) 1340–1347. [7] M. Krishnapillai, R. Jones, I.H. Marshall, M. Bannister, N. Rajic, Thermography as a tool for damage assessment, Composite Structures 67 (2005) 149–155. [8] G. Meneghetti, Analysis of the fatigue strength of a stainless steel based on the energy dissipation, International Journal of Fatigue 29 (2006) 81–94. [9] P.K. Liaw, H. Wang, L. Jiang, B. Yang, J.Y. Huang, R.C. Kuo, J.G. Huang, Thermographic detection of fatigue damage of pressure vessel steels at 1000 Hz and 20 Hz, Scripta Materialia 42 (2000) 389–395. [10] B. Yang, P.K. Liaw, M. Morrison, C.T. Liu, R.A. Buchanan, J.Y. Huang, R.C. Kuo, J.G. Huang, D.E. Fielden, Temperature evolution during fatigue damage, Intermetallics 13 (2005) 419–428. [11] G.C. Sih, D.Y. Jeong, Hysteresis loops predicted by isoenergy density theory for polycrystals. Part I: Fundamentals of non-equilibrium thermal-mechanical coupling effects, Theoretical and Applied Fracture Mechanics 41 (1-3) (2004) 233–266. [12] G.C. Sih, D.Y. Jeong, Hysteresis loops predicted by isoenergy density theory for polycrystals. Part II: Cyclic heating and cooling effects predicted from nonequilibrium theory for 6061-T6 aluminum, SAE 4340 steel and Ti-8Al-1Mo-1V titanium cylindrical bars., Theoretical and Applied Fracture Mechanics 41 (1– 3) (2004) 267–289. [13] G.C. Sih, Mechanics of Fracture Initiation and Propagation, Kluwer Academic Publisher, Boston, 1991. [14] ASTM Standards: E 606 Practice for Strain-Controlled Fatigue Testing, Annual Book of ASTM Standards, vol 03.01. [15] www.matweb.com. [16] L. Jiang, H. Wang, P.K. Liaw, C.R. Brooks, D.L. Klarstrom, Temperature evolution during low-cycle fatigue of ULTIMET alloy: experiment and modeling, Mechanics of Materials 36 (2004) 73–84.
Contents lists available at SciVerse ScienceDirect
Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Thermo-mechanical measurement of elasto-plastic transitions during cyclic loading R.V. Prakash ⇑, T. Pravin, T. Kathirvel, Krishnan Balasubramaniam Department of Mechanical Engineering, Indian Institute of Technology, Madras, Chennai 600 036, India
a r t i c l e
i n f o
Article history: Available online 24 September 2011 Keywords: Infrared thermography Thermo-elastic effect Yield points Stress–strain curve segments
a b s t r a c t The surface temperature of stainless steel SS304 low cycle fatigue specimens subjected to cyclic loading was studied using infrared thermography technique. The thermal data mapped onto the various stages of cyclic stress–strain curve shows the ability of these measurements to identify the yield points in both the compression and tension loading. Based on the results of this study, it is possible to identify the state of stress for materials such as elastic tension, plastic tension, elastic compression, plastic compression during cyclic loading using infrared thermographic data. The thermo-elastic slope and thermo-plastic slope was observed to be dependent on the prior loading cycles. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Infrared thermography is a non-destructive, non-contact, real time technique. It offers the advantage of remote/inaccessible/hazardous area measurements [1]. Any object emits energy by radiation according to its surface temperature. Human eyes are capable of detecting this energy only when the object is warm enough and the energy that is emitted is in the visible band (0.4–0.8 lm). In fact, the human eyes are able to recognize the temperature during heating of a metal, when it passes from red (720 °C), to yellow, to white (1350 °C). For low temperature values falling in the infrared region, thermography provides us with artificial eyes and gives us a chance to see invisible radiation [2]. An infrared (IR) system basically includes a camera, equipped with a series of changeable optics, and a computer. The core of the camera is the infrared detector, which absorbs the IR energy emitted by the object (whose surface temperature is to be measured) and converts it into electrical voltage or current. Finally this electrical voltage is converted into thermograms that provide information on the transient temperature behavior of a given point or area on the surface of the test sample at any given time in the loading cycle. The thermo-elastic effect suggests the existence of a relationship between the amplitude of stress and temperature change in a material during elastic loading. In the elastic region, when a material is subjected to tensile loading, the material undergoes cooling and when a material is subjected to compressive loading, it undergoes heating. Volume dilatation (volume change) takes place in elastic region during tensile loading of material, i.e., atoms get pulled apart slightly and hence, there is a slight increase in bond length. If the loading is done under adiabatic conditions then ⇑ Corresponding author. E-mail address: [email protected] (R.V. Prakash). 0167-8442/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2011.09.001
there is no time for thermal equilibrium to take place between material and surroundings and hence, there is a gross decrease in temperature in this case of elastic tension by a tensile stress [3]. Thermodynamic equation of state and first law of thermodynamics govern these heat effects. The thermo-elastic equation is expressed as [4]:
DT ¼
aT Dr qC p
ð1Þ
where a (lm/m °C) is coefficient of thermal expansion, T (°C) is the ambient temperature, q (g/cc) is density, Cp (J/g °C) is heat capacity at constant pressure and Dr (N/m2) is the change in stress Numerous applications of infrared technique have been developed: (a) to find the distribution of stresses in specimens by applying the principle of thermo-elasticity [5], (b) to characterize in situ fatigue damage during testing of the CFRP specimens using a passive thermography by measuring the temperature increase of the specimen due to hysteretic heating during fatigue testing [6], (c) to determine the size and the location of a subsurface defect from the phase image of infrared thermography [7], (d) to determine the fatigue limit of material by plotting the static temperature in temperature response of material versus stress and fitting linear fit [8]. These studies on temperature evolution during cyclic loading tests have showed that, the temperature response curve of the material is sigmoidal nature with three different slopes [9,10]; the initial rapid change in temperature due to the change in the hysteresis loop area, the secondary steady temperature response due to the stabilization of material stress–strain response along with equilibrium with the surrounding temperature and final sudden temperature rise just prior to final fracture due to the high plasticity induced at the crack tip. Isoenergy density theory was applied to fatigue loading, estimating the hysteresis loops of an hour-glass cylindrical specimen and computing the
2
R.V. Prakash et al. / Theoretical and Applied Fracture Mechanics 56 (2011) 1–6
non-equilibrium temperature determined from the displacement gradients by considering the simultaneous change of surface and volume energy, without invoking the concept of heat, found to oscillate about the ambient temperature [11–13]. The present work deals with a more detailed study on the variation of surface temperature of test sample corresponding to different segments of stress–strain curve of material undergoing cyclic loading.
2. Experimental details 2.1. Experimental setup Cyclic loading experiments were carried out on a 100 kN MTS servo-hydraulic testing machine. Fig. 1 shows the experimental set up. The experiments were carried out under completely reversed loading conditions, in displacement control mode for elastic–plastic loading and under force controlled mode for purely elastic loading. The infrared thermal imaging system used for the temperature measurement is a CEDIP Jade LWIR camera (refer Fig. 1). This
consists of an infrared camera cooled by stirring cooler. The camera contains an Hg–Cd–Te detector which is sensitive to infrared radiation in the wavelength range 8–10 lm. The window size used in the experiments is 60 56 pixels and an integration time of 200 ls. The camera was positioned on a tripod approximately about 1 m from the specimen surface. The temperature images were acquired in the Automatic Gain Controlled (AGC) mode at a various frame rates and the digital data stored in a computer hard disk. Post-processing software was used for analyzing and extracting the timing graph of temperature data. The temperature values are averaged over a rectangular area along the gage length of the specimen. The temperature variation measured during the experiment is defined as the difference between the temperature at the center of the specimen and the temperature of an unloaded reference specimen. The use of a reference specimen, located next to the loaded sample, eliminates uncertainties introduced by room temperature fluctuations during a long duration test. Polished material surfaces have a small absorptivity and a weak emissivity in infrared spectrum; to overcome the emissivity problem, the specimens were coated with matt-finish black board paint. 2.2. Material and specimen Test specimens conforming to ASTM 606 standard [14] with a gage length diameter of 7 mm and a gage length of 27 mm and grip portion diameter of 10 mm were CNC machined as shown in Fig. 2 for as-received stainless steel (Grade 304) material. As a prerequisite of cyclic testing, the surface of the specimens were mirror polished prior to cyclic testing and then painted black, and the specimen was held using hydraulic grips. Axial extensometer with a gage length of 25 mm and a travel of ±5 mm was used to estimate gage section strain. Table 1 presents the chemical composition of stainless steel used in the experiments which suggests that, the material conforms to ASTM.A.276 SS Type 304.
IRT Camera
2.3. Testing conditions Two types of cyclic loading tests were carried out to study the temperature response variation in a cycle corresponding to various segments of stress–strain curve. Initially to study the material temperature response under fully elastic cyclic loading, the specimen was loaded in the force control with load amplitude equal to 10 kN and stress ratio (ratio of minimum to maximum load) of 1. This load corresponds to a value below yield point load of
Fig. 1. Experimental set up.
Fig. 2. Specimen geometry.
Table 1 Material chemical composition.a
a
Name of the element
Carbon
Chromium
Manganese
Nickel
Phosphorous
Sulfur
Silicon
Wt.%
0.0370
18.100
1.3100
8.020
0.0273
0.0245
0.3790
Balance Fe.
R.V. Prakash et al. / Theoretical and Applied Fracture Mechanics 56 (2011) 1–6
3
the material (the yield point of the sample used was estimated to be 13.5 kN). The test was carried out at 5 Hz frequency and the frames were grabbed using thermal images were captured from IR camera at a frame rate of 100 Hz. To study the material temperature response to elastic–plastic fatigue loading, tests were carried out under strain control with strain amplitude of 8000 micro-strains (the yield strain value here was estimated to be 1800 micro-strains). The test was carried out at 0.5 Hz frequency and the thermal images were captured at a frame rate of 50 Hz.
3. Results and discussion 3.1. Temperature response of material to various segments of stress– strain curve under elastic–elastic fatigue loading conditions During elastic–elastic cyclic loading, material is loaded below its yield point load, so plasticity induced is very small (generally negligible). The area of hysteresis loop formed is normally considered to be negligible as shown in Fig. 3. The overall temperature rise observed during this kind of cyclic test is small. Fig. 4 shows the relationship of stress, strain and temperature with respect to
Fig. 5. Temperature variation within single loading cycle due to elastic cyclic loading.
Fig. 3. Hysteresis loop for 5000th cycle of elastic–elastic cyclic loading.
time for initial three cycles of elastic–elastic cyclic loading. Fig. 5 shows that, the temperature response of material in elastic–elastic loading has only two segments corresponding to elastic tension and elastic compression. The first segment suggests a decrease in temperature due to the thermo-elastic effect in tension loading and second segment suggests an increase in the temperature which is due to thermo-elastic effect in compression loading. The temperature increase during unloading is found to be higher than the temperature drop during loading cycle leading to a residual temperature increase at the end of each elastic loading–unloading cycle. Using the mathematical expression for thermo-elastic effect Eq. (1), and constants from Ref. [15], for SS 304; a = 17.3 lm/ m °C, q = 8 g/cc, Cp = 0.5 J/g °C, the drop or rise in temperature due thermo-elastic effect can be calculated which is equal to 0.68 °C. The experimental results for this material suggest a drop in temperature to be about 0.66 °C, which is in good agreement with the analytical predictions. Also it is observed that, the slope of both the loading and unloading segments in temperature response cycle is similar (0.89) and it does not change with the number of cycles of loading. However, in view of the small increase in temperature during every unloading, the net temperature response was found to progressively increase for the specimen with the number of cycles in the elastic loading experiment. 3.2. Temperature response of material to various segments of stress– strain curve under elastic–plastic fatigue loading conditions
Fig. 4. Strain, stress, temperature variation for initial two cycles of elastic–elastic cyclic loading.
In this type of cyclic loading, it was observed that the material load–displacement curve had four different segments
4
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Fig. 6. Hysteresis loop for 300th cycle of elastic–plastic cyclic loading.
Fig. 8. Typical material responses of SS 304 during a strain controlled cyclic test during early fatigue cycles: (a) axial force, and (b) temperature evolution versus number of cycles curves.
Fig. 7. Typical temperature evolution during continuous cyclic loading test. Both elastic–elastic and elastic–plastic cyclic loading exhibit similar behavior.
corresponding to elastic-tension, plastic-tension, elastic compression and plastic compression. The area of hysteresis loop formed is also significant as shown in Fig. 6. The load–displacement response was acquired by the MTS system. The low-cycle fatigue tests was carried out under strain control with a triangular waveform with strain amplitude of ±8000 micro-strains. Fig. 7 shows the temperature evolution during the continuous cyclic test under above loading conditions. The temperature profile of material (both the maximum as well as the average temperature in a cycle plotted against time) has sigmoidal nature with three different slopes. In Fig. 7, the initial rapid temperature rise (region a) is due to change in hysteresis loop area at first few fatigue cycles [16]. Then temperature response becomes stable (region b) due to the stable stress–strain response of the material and temperature equilibrium with the surrounding. Finally temperature rises suddenly (region c) due to the high plasticity induced at the crack tip just prior to failure. The evolution of temperature with load for each loading cycle during the initial two cycles corresponding to region a in Fig. 7, are presented in more detail in Fig. 8a and b. When the load was initially applied on the specimen, the strain was in the elastic range, and the load or strain–temperature relation can be described by the thermoelastic effect, i.e., the temperature changes were directly proportional to the stress. At the initial stress-free
stage, the temperature difference was 0 °C, but the temperature dropped by 0.50 °C when the load increased from no-load (A) to the yield-load level (B), due to the thermoelastic effect. Beyond the yield load level, up to the maximum load level (C), the temperature increases, due to heat dissipation because of irreversible plastic deformation. When the load elastically decreased from the maximum level (C), 22–0 kN (D), the temperature continued to increase because of the thermoelastic effect in compression loading. During the plastic compression loading beyond the noload condition (E), the temperature increase continues because of the plastic deformation until the load reached the minimum load level of 20 kN (F). Subsequently, when the load again elastically increases from 20 to the yield value at 7 kN (G), the temperature
Fig. 9. Typical material responses of SS 304 during a strain controlled cyclic load test during 76th and 77th cycles: (a) axial force, and (b) temperature evolution versus number of cycles curves.
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Fig. 10. Typical material responses of SS 304 during a strain controlled cyclic load test during 201th and 202th cycles: (a) axial force, and (b) temperature evolution versus number of cycles curves.
decreased again due to the thermoelastic effect. After yield point (G), the temperature increased again because of plastic deformation. This cycle of temperature response continued with mechanical loading response. For subsequent cycles, the strain and temperature responses repeated, and the mean temperature continuously increased. Figs. 9 and 10 shows the temperature variation corresponding to different stress–strain segments at intermediate cycles, i.e. 76th–77th and 201th–202th loading cycles in region ‘b’ of Fig. 7. It can be seen from Fig. 8–10 that, it is possible to identify the various segments of stress–strain curve from the temperature response. In temperature response of material, four different slopes are observed corresponding to four segments of stress–strain curve. It can also be seen that, it is possible to locate the yield point of material both in tension and compression using this material temperature response. As it can be seen from the Fig. 9 that both
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Fig. 11. Hysteresis strain energy density comparison for elastic–elastic and elastic– plastic cyclic loading.
the temperature response slopes corresponding to elastic-compression and plastic-compression are linear and they meet exactly at the point which matches with the yield point in compression from load versus number of loading cycles curve. Also the tensile yield point can be found out by identifying the point of non-linearity in temperature response of material, where linear drop in temperature due to elastic-tension loading ends. Thus it is possible to find the yield point of material in tension and compression using the infrared thermography technique which is otherwise difficult task in field components. Also it can be observed from the Fig. 9 that, the rate at which the temperature increases due to elasticcompression is higher than that of the plastic-compression. As three out of four segments of stress–strain curve namely plastic tension, elastic compression and plastic compression corresponds to temperature increase and only one segment that is elastic tension corresponds to temperature decrease, the net mean temperature of material keeps on increasing after each loading cycle. After every three segments corresponding to temperature increase
Fig. 12. Change in the temperature profile slopes within single loading cycle with number of cycles and the corresponding best fit curves.
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temperature starts dropping for the fourth segment, that point of maximum temperature corresponds to the minimum stress level in that cycle. By comparing Figs. 8–10, it can be seen that, for initial two cycles the temperature increase per cycle is nearly 2 °C, for 76th and 77th cycle the temperature increase per cycle is around 1.25 °C and for 201th and 202th it is only 1.1 °C. This change in temperature evolution per cycle could be due to the thermal equilibrium reached by the specimen with surrounding after initial few loading cycles. Hysteresis strain energy density (HSED)was calculated for elastic–elastic and elastic–plastic cyclic loading based on the area enclosed by cyclic stress–strain curve and plotted against number of cycles as shown in Fig. 11. In case of elastic–elastic cyclic loading, HSED (ep = 190.5 micro-strains for 5000th cycle) was in the range of 0.03–0.05 J/mm3 for 1–5e4 cycles, the specimen did not fail. For elastic–plastic cyclic loading, HSED (ep = 9215 micro-strains for 300th cycle) was in the range of 7–10 J/mm3 for 1–750 cycles, the specimen failed at 750th cycle. In case of elastic–plastic fatigue loading, the slope of temperature profile corresponding to various stress–strain segments changes with number of loading cycles. Fig. 12 shows the change in slopes with respect to the number of cycles for elastic-compression, plastic compression and elastic-tension segments of stress– strain curve. The slope for elastic-tension segments is negative due to the thermo-elastic effect but for comparison purpose only magnitude of the slope is considered. The slope of elastic-tension segment of temperature profile increases with the number of cycles due to the fact that, the temperature difference between specimen and surrounding increases with number of cycles and hence during temperature drop due to elastic-tension heat transfer occurs at faster rate. Similarly in case of elastic-compression and plastic compression, it becomes difficult for the specimen to maintain the same rate at which the temperature increases in each cycle and hence the slope of the temperature profile decreases with the number of cycles. 4. Conclusions The experimental results showed that, in case of elastic–elastic cyclic loading within each cycle, the material temperature response has two slopes corresponding to elastic-tension and elasticcompression which is due to thermoelastic effect and in case of elastic–plastic cyclic loading within each loading cycle, the material temperature response curve has four slopes corresponding to four segments in stress–strain curve. The yield point of material in tension and compression can be found out using the infrared thermography technique. In case of compressive yield point, it is the point where the intersection of two distinct slopes in the temperature curve that corresponds to elastic-compression and plastic-compression segments of stress–strain curve takes place. In case of tensile yield point, it is the point where linearity of
temperature drop (in temperature response of material due to elastic tension) ends. In case of elastic loading, the slopes corresponding to elastic-tension and elastic-compression were found to be the same and does do not change with the number of cycles. There is a net increase in temperature in each elastic loading– unloading cycle that cumulatively adds-up as the number of cycles increases. In elastic–plastic fatigue loading, the slopes corresponding to four segments of stress–strain curve changes with number of loading cycles. It is observed that, the slope of the material temperature response curve corresponding to elastic-compression segment of stress–strain curve is higher than that of the plastic-compression segment. Also it is observed that, the highest temperature in each cycle is reached when the stress reaches its minimum value in cycle. References [1] X.P.V. Maldague, Theory and practice of infrared technology for nondestructive testing, Wiley Series in Microwave and Optical Engineering, John Wiley and Sons Publication, 2001. [2] M. Carosena, M.C. Giovanni, L. Giorleo, The use of infrared thermography for materials characterization, Journal of Materials Processing Technology 155– 156 (2004) 1132–1137. [3] K.N. Pandey, S. Chand, Deformation based temperature rise: a review, International Journal of Pressure Vessels and Piping 80 (2003) 673–687. [4] Norbert G.H. Meyendorf, Henrik Rosner, Victoria Kramb, Shamachary Sathish, Thermo-acoustic fatigue characterization, Ultrasonics 40 (2002) 427–434. [5] J.M. Dulieu-Barton, P. Stanley, Development and applications of thermoelastic stress analysis, Journal of Strain Analysis 33 (2) (1998) 93–104. [6] R. Steinberger, T.I.V. Leitao, E. Ladstatter, G. Pinter, W. Billinger, R.W. Lang, Infrared thermographic techniques for non-destructive damage characterization of carbon fibre reinforced polymers during tensile fatigue testing, International Journal of Fatigue 28 (2006) 1340–1347. [7] M. Krishnapillai, R. Jones, I.H. Marshall, M. Bannister, N. Rajic, Thermography as a tool for damage assessment, Composite Structures 67 (2005) 149–155. [8] G. Meneghetti, Analysis of the fatigue strength of a stainless steel based on the energy dissipation, International Journal of Fatigue 29 (2006) 81–94. [9] P.K. Liaw, H. Wang, L. Jiang, B. Yang, J.Y. Huang, R.C. Kuo, J.G. Huang, Thermographic detection of fatigue damage of pressure vessel steels at 1000 Hz and 20 Hz, Scripta Materialia 42 (2000) 389–395. [10] B. Yang, P.K. Liaw, M. Morrison, C.T. Liu, R.A. Buchanan, J.Y. Huang, R.C. Kuo, J.G. Huang, D.E. Fielden, Temperature evolution during fatigue damage, Intermetallics 13 (2005) 419–428. [11] G.C. Sih, D.Y. Jeong, Hysteresis loops predicted by isoenergy density theory for polycrystals. Part I: Fundamentals of non-equilibrium thermal-mechanical coupling effects, Theoretical and Applied Fracture Mechanics 41 (1-3) (2004) 233–266. [12] G.C. Sih, D.Y. Jeong, Hysteresis loops predicted by isoenergy density theory for polycrystals. Part II: Cyclic heating and cooling effects predicted from nonequilibrium theory for 6061-T6 aluminum, SAE 4340 steel and Ti-8Al-1Mo-1V titanium cylindrical bars., Theoretical and Applied Fracture Mechanics 41 (1– 3) (2004) 267–289. [13] G.C. Sih, Mechanics of Fracture Initiation and Propagation, Kluwer Academic Publisher, Boston, 1991. [14] ASTM Standards: E 606 Practice for Strain-Controlled Fatigue Testing, Annual Book of ASTM Standards, vol 03.01. [15] www.matweb.com. [16] L. Jiang, H. Wang, P.K. Liaw, C.R. Brooks, D.L. Klarstrom, Temperature evolution during low-cycle fatigue of ULTIMET alloy: experiment and modeling, Mechanics of Materials 36 (2004) 73–84.