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The study of the fatigue crack propagation in mixed mode crack growth
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Available online www.sciencedirect.com Available online at at www.sciencedirect.com Structural Integrity Procedia 00 (2017) 000–000
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Procedia Structural (2017) 438–445 Structural IntegrityIntegrity Procedia500 (2016) 000–000
www.elsevier.com/locate/procedia
www.elsevier.com/locate/procedia
2nd International Conference on Structural Integrity, ICSI 2017, 4-7 September 2017, Funchal, Madeira, Portugal
The study Conference of the fatigue crack in mixed mode crackPortugal XV Portuguese on Fracture, PCFpropagation 2016, 10-12 February 2016, Paço de Arcos, growth Thermo-mechanical modeling of a high pressure turbine blade of an A. Vshivkov, Iziumova, O. Plekhov* airplaneA.gas turbine engine Institute of Continuous Media Mechanics UB RAS, 614014 Perm, Russia a b c
P. Brandão , V. Infante , A.M. Deus *
a
Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Abstract Portugal c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, This work is devoted to analysis of thermodynamics properties of the fatigue cracks propagation in metals. A theoretical Portugal b
description of the elastoplastic condition at the fatigue crack tip is proposed on the base of an elastic solution and a secant elastic modulus. An experimental confirmation of the theoretical approach to the heat flux calculation at the fatigue crack tip is carried out. The character of heat dissipation at different stages of crack propagation is studied. The investigation of the fatigue crack Abstract propagation was carried out on flat samples with stress concentrator made from stainless steel AISE 304. The stress concentrator During their operation, aircraft engine are flux subjected increasingly demanding operating conditions, was the side notch. Infrared modern thermography method andcomponents the contact heat sensor to based on the Seebeck effect are used to monitor high pressure turbine (HPT) blades. Such conditions cause these parts toThe undergo types of time-dependent theespecially dissipatedthe thermal energy. The stress intensity factor was constant during the loading. plasticdifferent zone shape under monotonic degradation, ofcalculated which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict uniaxial loadingone was theoretically. the creep behaviour of HPT blades. Flight © 2017 The Authors. Published by Elsevier B.V.data records (FDR) for a specific aircraft, provided by a commercial aviation © 2017 The Authors. Published by Elsevier B.V.and mechanical data for three different flight cycles. In order to create the 3D model company, were used to obtain Peer-review under responsibility ofthermal the Scientific Committee of ICSI 2017. Peer-review under of theaScientific Committee ICSI 2017 and its chemical composition and material properties were needed for theresponsibility FEM analysis, HPT blade scrap of was scanned, obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D Keywords: Fatigue crack, mixed mode loading, dissipated energy, the plastic zone shape. rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data.
1. Introduction
© 2016 The Authors. Published by Elsevier B.V. It is well known that real metals a complex structure, Peer-review under responsibility of thehave Scientific Committee of PCFwhich 2016. is a hierarchy of different scale levels. Under
deformation, the structural evolution is observed at all scale levels and leads to irreversible deformation and failure Keywords: High Pressure Blade; Creep; Finiteand Element Method; 3DInvestigation Model; Simulation. that is accompanied byTurbine energy accumulation dissipation. of thermodynamics of deformation and
* Corresponding author. Tel.: +73422378312; fax: +73422378487. E-mail address: [email protected] 2452-3216 © 2017 The Authors. Published by Elsevier B.V. Peer-review underauthor. responsibility the Scientific Committee of ICSI 2017. * Corresponding Tel.: +351of 218419991. E-mail address: [email protected] 2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016. 2452-3216 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ICSI 2017 10.1016/j.prostr.2017.07.193
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O. Plekhov et al. / Procedia Structural Integrity 5 (2017) 438–445 A. Vshivkov et al. / Structural Integrity Procedia 00 (2017) 000–000
439
failure is a key issue in solid mechanics. The heat generation process depends on both the thermo elastic effect and plastic energy dissipation. The measurement of heat flux near the crack tip allows one to calculate the energy balance under crack propagation and to obtain a new equation for crack propagation. For a long time, infrared thermography is regarded as the most effective method for estimating the power of the heat sources in the process of mechanical testing. The principal solution of the problem of energy dissipation measurement under deformation and failure can be reach by the development of additional system for direct monitor of heat flow. This idea was affectively used for investigation of energy dissipation in hydrodynamics tasks (Pradere C., 2006). Many authors proposed dependencies linking the rate of crack growth and such quantities as the J-integral, the work of plastic deformation, the size of the zone of plastic deformation, the amount of dissipated energy and other (Matvienko Yu.G., 2004; Rosakis P., 2000; Oliferuk W., 2004; Izyumova A., 2014). The classical assumption of an almost complete dissipation of the energy of deformation into heat (Farren W.S., 1925) turns out to be correct only in a limited number of cases. Any real engineering construction contains stress concentrators, welded joints and other potential sources of defects. The analysis of the kinetics of damage accumulation, the process of crack nucleation and kinetics of the crack development allows specialists to predict the time of structure failure and to perform in proper time a partial replacement or repair of deteriorated units of complex structures. Moreover, the repair or replacement of the wornout parts on a timely basis is more effective than their complete replacement after mechanical damage. It is therefore very important to know the time during which the defects in the ill-behaved areas are reaching critical values. The previous authors’ investigations were focused on crack growth problems under an opening or mode I mechanism (Vshivkov А., 2016). However, most structures are failed due to mixed mode loading. Many uniaxial loaded materials, structures and components often contain randomly oriented defects and cracks which develop a mixed mode state by rotation of their orientation with respect to the loading axis. This work is devoted to the investigation of the dissipated energy in the process of crack propagation under mixed mode loading. For this purpose, the original contact heat flux sensor was developed to detect energy dissipation value in the process of crack propagation and verify the data of infrared thermography. This device is based on the Seebeck effect and includes two Peltier elements and temperature controlling feedback. This sensor allows us to study in details a dissipated energy evolution in metal samples (AISI304) with uniaxial and multi axial loadings and propose relations between heat dissipation and fatigue crack rate. 2. Experimental setup A series of samples made from stainless steel AISE 304 were tested. The geometry of the samples is shown in Figure 1. The experimental study was carry out in University of the Federal Armed Forces Munich, Institute for Materials Science, Neubiberg, Germany. During tests the samples were subjected to cyclic loading of 20 Hz with constant stress intensity factor and ratio R = -1. The crack length in the course of the experiment was measured by the potential drop method (Nayeb-Hashemi H., 2004; Hartman G.A., 1987). The electrical potential drop method is accepted as being capable of monitoring the fatigue crack propagation in steel structures. The size of a crack in a steel sample is predicted by applying a constant d.c. (direct current) or a.c. (alternating current) to the sample and by measuring an increase in electrical resistance due to the crack. In this case, the potential method is capable of a sensitivity as fine as 0.02 mm for a d.c. 5 A.
Fig. 1. Geometry of samples.
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To analyze the dissipated energy at the crack tip a contact heat flux sensor was designed and constructed. The proposed sensor is based on the Seebeck effect, which is the reverse of the Peltier effect. The Peltier effect is a thermoelectric phenomenon, in which the passage of electric current through conducting medium leads to the generation or absorption of heat at the point of contact (junction) of two dissimilar conductors. The quantity of heat and its sign depend on the type of materials in contact, the direction and the strength of the electric current. The quantity of heat absorbed or dissipated by the element is directly proportional to the current intensity and the time of its passage. (1) P ABIP P – the power of heat flux; I – the direct current; ПАБ – Peltier coefficient.
Fig. 2. Schematic of the device.1 – testing sample; 2 – “measuring” Peltier element; 3 – “cooling” Peltier element; 4 – radiator; 5, 6 – thermocouple; 7 – resistor.
Figure 1 presents a schematic diagram of the heat flux sensor. The following notation is used in figure 1: sample (1), the heat flux sensor (2). A thermal contact between the sample and the sensor is provided due to the introduction of the thermal paste. Structurally, the sensor comprises two Peltier elements ("measuring" (2) and "cooling" (3)), thermocouples (5), (6) and the radiator (4). The measuring Peltier element is connected to a low-resistance resistor of 1.2 Om (7). To measure the heat flow through the "measuring" Peltier element during the experiment the temperature on its free surface should be a constant. The cooling Peltier element caulked with a radiator was connected with the "measuring" Peltier element. This cooling system has feedback and is controlled based on two temperature sensors located between "measuring" and cooling Peltier elements and far from the studied sample in the zone with constant temperature. The signal from the sensor (voltage at the resistor (7)) is measured by the amplifier and registered in the ADC of the microcontroller. The data are transmitted from the microcontroller to the personal computer for further processing. The "cooling" Peltier element is controlled via pulse width modulation. These sensors were calibrated using a device with a controlled heat flux. A wire resistor with the known resistance is glued on a plastic plate with a size equal to that of test samples. The heat isolating system provides the heat flux from the resistance to the sensor only. The heat flow was calculated using the values of the resistor voltage and the electric current across the resistor.
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The sensitivity of the heat flux sensor was verified during the fatigue crack propagation with the constant amplitude of loading. In this experiment the heat flux sensor was located at a distance from the crack tip. The results of such measurements qualitatively correspond to the case of the direct location of the sensor over the top of the fatigue crack tip. This effect is a consequence of the high thermal conductivity of the material. Measurement results are presented in figure 3
Fig. 3. Direct measurement of the heat flux at the crack tip and measurement at the distance.
There are three sections on the heat flux curve under the mixed loading. Short initial increasing part corresponds to starting of crack propagation (part 1). The second part with constant heat flux corresponds to the regime of short crack propagation (part 2). The last part of the plot (part 3) is characterized by sharp increasing of heat dissipation. During this part we observe the long crack propagation process. The last part is finished by specimen failure. The evolution of the temperature field was recorded by infrared camera FLIR SC 5000. The spectral range of the camera is 3-5 µm. The maximum frame size is 320×256 pixels; the spatial resolution is 10-4 meters. The temperature sensitivity is 25 mK at 300 K. Calibration of the camera was made based on the standard calibration table. It was used FLIR SC5000 MW G1 F/3.0 close-up lens (distortion is less than 0.5%) to investigate the plastic zone in details.
Fig. 4. Schematic of the measured equipment.
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3. Results of fatigue experiments During the experiments, a series of samples was tested where the crack length and the heat flux were recorded. Figures 5 show the characteristic time dependences of heat flux during the experiment. The heat flux during the experiment can be divided into two stages. In the first stage (up to point *), cyclic loading in the elastic mode with a constant amplitude of the applied load is carried out to achieve the required value of the stress intensity factor. In the second stage (after the point *), the load is controlled to maintain a constant value of the stress intensity factor, while the rate of the fatigue crack propagation also remains constant (Fig. 7). This stage is the subject of investigation.
*
Fig. 5. The characteristic heat flux during the experiments with
Fig. 6. The characteristic crack length during the experiments with
constant stress intensity factor.
constant stress intensity factor.
According to the classical concepts the heat flux from the top of the crack should remain constant with constant stress intensity factor. However, in the experiment a monotonic decrease in the heat flux is observed. 4. Energy dissipation at crack tip under cyclic loading Following the work Raju (1972), we can propose a relation between elastic and real deformation at crack tip: 1
E 2 ijef ijel , (2) Es where 𝐸𝐸– the Young’s modulus, 𝐸𝐸𝑠𝑠 - secant plasticity modulus. Equation (2) was originally proposed by Dixon (1965) as results of photo elastic experiments data treatment based on the Ramberg-Osgood relationship (1943): n
A . G 0 We can write a following estimation for octahedral stress and couple it with an elastic solution:
1
oct where B
GA e 0 0
n
2(1 ) 2 B n 1 1 3 el oct , n 1 1 B
, oct , e - elastic limit. e
(3)
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The energy of plastic deformation in representative volume located near crack tip can be estimated as follow: 3 3 An oct d oct e n n1 . Up 2 2 n 1 The energy increment for crack under monotonic loading can be written as d 3 (4) dU p An e n n dl, 2 dl where 𝑙𝑙- crack length.
1
1
rp f e 2 el Kf e2 (here K - stress intensity factor, rp– estimation for plastic zone Using definition oct e 3 r e r
size, r – polar coordinate, fe – function of polar coordinate determining the relation of octahedral stress versus ) we can rewrite equation (4) as follow: 3 d d (5) dU p An e n n dl. 2 d dl It was shown earlier that plastic zone at crack tip could be divided into two parts: plastically loaded zone (dissipation zone) and elastically unloaded. The geometry of dissipation area would be determined by relation: r df d 1 f e C p sin e f e cos . (6) r d dl 2 r p f e r It was shown by Raju (1972) that for l rp the equation (6) gives two straight lines 79.9 determined
the areas of plastic uploading and elastic unloading. Figure 7a presents the plastic zone shape under monotonic uniaxial loading. Zone A corresponds to the plastic loading caused by crack advance, zone B - the elastic unloading. a
b 1.0
1.0
B
0.5
0.5
A 1.0
1.0
0.5
B
0.5
0.5
0.5
1.0
A
1.0 0.5
0.5
B
B
1.0
1.0
1.5
Figure 7. Structure of plastic zone at crack tip under monotonic loading (a) – uniaxial loading, (b) – multiaxial loading with biaxial coefficient equal to 0.3
For description of crack behavior under mixed mode loading we have to change the function fe in equations (5) and (6). Taking into account elastic solution of the first and second fracture modes at crack tip: K I , II I , II f ( I , II )1 , xx 2r
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I , II yy I , II xy
K I , II 2r K I , II 2r
7
f ( I , II ) 2 , f ( I , II )3 ,
we can write the 𝑓𝑓𝑒𝑒 for multiaxial loading as follow:
(7) f e af I1 bf II1 2 af I 2 bf II 2 2 af I 1 bf II1 af I 1 bf II1 3af I 3 bf II 3 2 . Substitution of equation (7) into equation (6) gives us an estimation of the evolution of plastic zone caused by application of shear stress. The structure of plastic zone at crack tip under monotonic loading is presented in figure 7b. As a result, we can analyze the plastic deformation at crack tip under multiaxial loading based on equation (5). The biaxial coefficient can be taken into account using equation (7). For cyclic loading we have to consider energy dissipation in cyclic plastic zone at crack tip: cyc mon U tot p U p U p . To explain the experimental fact reported in previous paragraph we have to calculate the total energy increment dU cyc p 0 and plastic energy at U tot p . For loading condition considered in our experiment it can be shown that dl
crack tip U tot can be written as follow: p
2 U tot p W1 A W2
dl , dN
(8)
where A – stress amplitude. Based on the equation (8) we can conclude that for small crack rate the plastic work and, as a consequence, energy dissipation at crack tip is proportional to the applied stress amplitude and can decrease during experiments with constant stress intensity factor. The calculation of analytical relations for W1, W2 can be carried out similarly to the monotonic loading. The biaxial coefficient changes the function fe but keeps the constant the structure of the equation (8). It allows us to predict the existence of peculiarities of energy dissipation at crack tip reported in Iziumova (2016) for multiaxial loading. 5. Conclusion In this work the experimental and theoretical study of dissipated energy were carried out during fatigue crack propagation. Based on a contact heat flux sensor, an experimental technique for the application of the method of infrared thermography for measuring the energy dissipation during fatigue test has been developed. The device allows us to measure heat dissipation under uniaxial and multiaxial loadings. Decrease in the power of the dissipated energy was shown during the fatigue crack propagation with a constant stress intensity factor. In order to explain this effect, the theoretical analysis of the energy dissipation zone formation at the crack tip was carried out. Using the coupling between Young’s modulus and the secant plasticity modulus, the analysis of the plastic zone shape and dissipated energy value at the crack tip under uniaxial and multiaxial loadings has been made. A good agreement between theoretical and experimental results illustrates the possibility of the application of the method to description of the energy dissipation under multiaxial loading. Acknowledgements This work was supported by the grant of the President of Russian Federation for support of young Russian scientists and leading scientific schools [MK-1236.2017.1] and the Russian Foundation for Basic research [grant number 16-31-00130]. The authors would like to thank Prof. Jürgen Bär for experimental support of the work.
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References Pradere C., Joanicot M., Batsale J-C., Toutain J., Gourdon C., 2006. Processing of temperature field in chemical microreactors with infrared thermography. QIRT Journal 3, 117-135. Matvienko Yu.G., Morozov E M., 2004. Calculation of the energy J-integral for bodies with notches and cracks. International Journal of Fracture 125, 249-261. Rosakis P., Rosakis A.J., Ravichandran G., Hodowany J., 2000. A thermodynamic internal variable model for the partitional of plastic work into heat and stored energy in metals. J. Mech. Phys. Solids 48, 581-607. Oliferuk W., Maj M., Raniecki B., 2004. Experimental Analysis of Energy Storage Rate Components during Tensile Deformation of Polycrystals. Materials Science and Engineering 374, 77-81. Izyumova A., Plekhov O., 2014. Calculation of the energy J-integral in plastic zone ahead of a crack tip by infrared scanning. FFEMS 37, 1330– 1337. Farren W.S., Taylor G.I., 1925. The heat developed during plastic extension of metals. Proc. Royal. Soc. of London. Ser. A 107, 422-451. Vshivkov А., Iziumova A., Bar U., Plekhov O., 2016. Experimental study of heat dissipation at the crack tip during fatigue crack propagation. Frattura ed Integrità Strutturale 35, 131-137. Nayeb-Hashemi H., Swet D., Vaziri A., 2004. New electrical potential method for measuring crack growth in nonconductive materials. Measurement 36, 121–129. Hartman G.A., Johnson D.A., 1987. D-C Electric-Potential Method Applied to Thermal/Mechanical Fatigue Crack Growth. Experimental Mechanics 27, 106-112. Raju K.N., 1972. An energy balance criterion for crack growth under fatigue loading from considerations of energy of plastic deformation. International Journal of Fracture Mechanics 8, 1-14. Raju K.N., 1969. On the calculation of plastic energy dissipation rate during stable crack growth. International Journal of Fracture Mechanics 5, 101-112. Dixon J.R., 1965. Stress and strain distributions around cracks in sheet materials having various work-hardening characteristics, Ministry of Technology, National Engineering Laboratory, Materials Group: East Kilbride, Glasgow, Scotland, 224-244. Iziumova A., Vshivkov A., Plekhov O., 2016. The study of the effect of energy dissipation at fatigue crack tip on its propagation rate. 13th Quantitative Infrared Thermography Conference. Gdańsk, Pologne, paper 101.
ScienceDirect
Available online www.sciencedirect.com Available online at at www.sciencedirect.com Structural Integrity Procedia 00 (2017) 000–000
ScienceDirect ScienceDirect
Procedia Structural (2017) 438–445 Structural IntegrityIntegrity Procedia500 (2016) 000–000
www.elsevier.com/locate/procedia
www.elsevier.com/locate/procedia
2nd International Conference on Structural Integrity, ICSI 2017, 4-7 September 2017, Funchal, Madeira, Portugal
The study Conference of the fatigue crack in mixed mode crackPortugal XV Portuguese on Fracture, PCFpropagation 2016, 10-12 February 2016, Paço de Arcos, growth Thermo-mechanical modeling of a high pressure turbine blade of an A. Vshivkov, Iziumova, O. Plekhov* airplaneA.gas turbine engine Institute of Continuous Media Mechanics UB RAS, 614014 Perm, Russia a b c
P. Brandão , V. Infante , A.M. Deus *
a
Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Abstract Portugal c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, This work is devoted to analysis of thermodynamics properties of the fatigue cracks propagation in metals. A theoretical Portugal b
description of the elastoplastic condition at the fatigue crack tip is proposed on the base of an elastic solution and a secant elastic modulus. An experimental confirmation of the theoretical approach to the heat flux calculation at the fatigue crack tip is carried out. The character of heat dissipation at different stages of crack propagation is studied. The investigation of the fatigue crack Abstract propagation was carried out on flat samples with stress concentrator made from stainless steel AISE 304. The stress concentrator During their operation, aircraft engine are flux subjected increasingly demanding operating conditions, was the side notch. Infrared modern thermography method andcomponents the contact heat sensor to based on the Seebeck effect are used to monitor high pressure turbine (HPT) blades. Such conditions cause these parts toThe undergo types of time-dependent theespecially dissipatedthe thermal energy. The stress intensity factor was constant during the loading. plasticdifferent zone shape under monotonic degradation, ofcalculated which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict uniaxial loadingone was theoretically. the creep behaviour of HPT blades. Flight © 2017 The Authors. Published by Elsevier B.V.data records (FDR) for a specific aircraft, provided by a commercial aviation © 2017 The Authors. Published by Elsevier B.V.and mechanical data for three different flight cycles. In order to create the 3D model company, were used to obtain Peer-review under responsibility ofthermal the Scientific Committee of ICSI 2017. Peer-review under of theaScientific Committee ICSI 2017 and its chemical composition and material properties were needed for theresponsibility FEM analysis, HPT blade scrap of was scanned, obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D Keywords: Fatigue crack, mixed mode loading, dissipated energy, the plastic zone shape. rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data.
1. Introduction
© 2016 The Authors. Published by Elsevier B.V. It is well known that real metals a complex structure, Peer-review under responsibility of thehave Scientific Committee of PCFwhich 2016. is a hierarchy of different scale levels. Under
deformation, the structural evolution is observed at all scale levels and leads to irreversible deformation and failure Keywords: High Pressure Blade; Creep; Finiteand Element Method; 3DInvestigation Model; Simulation. that is accompanied byTurbine energy accumulation dissipation. of thermodynamics of deformation and
* Corresponding author. Tel.: +73422378312; fax: +73422378487. E-mail address: [email protected] 2452-3216 © 2017 The Authors. Published by Elsevier B.V. Peer-review underauthor. responsibility the Scientific Committee of ICSI 2017. * Corresponding Tel.: +351of 218419991. E-mail address: [email protected] 2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016. 2452-3216 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ICSI 2017 10.1016/j.prostr.2017.07.193
2
O. Plekhov et al. / Procedia Structural Integrity 5 (2017) 438–445 A. Vshivkov et al. / Structural Integrity Procedia 00 (2017) 000–000
439
failure is a key issue in solid mechanics. The heat generation process depends on both the thermo elastic effect and plastic energy dissipation. The measurement of heat flux near the crack tip allows one to calculate the energy balance under crack propagation and to obtain a new equation for crack propagation. For a long time, infrared thermography is regarded as the most effective method for estimating the power of the heat sources in the process of mechanical testing. The principal solution of the problem of energy dissipation measurement under deformation and failure can be reach by the development of additional system for direct monitor of heat flow. This idea was affectively used for investigation of energy dissipation in hydrodynamics tasks (Pradere C., 2006). Many authors proposed dependencies linking the rate of crack growth and such quantities as the J-integral, the work of plastic deformation, the size of the zone of plastic deformation, the amount of dissipated energy and other (Matvienko Yu.G., 2004; Rosakis P., 2000; Oliferuk W., 2004; Izyumova A., 2014). The classical assumption of an almost complete dissipation of the energy of deformation into heat (Farren W.S., 1925) turns out to be correct only in a limited number of cases. Any real engineering construction contains stress concentrators, welded joints and other potential sources of defects. The analysis of the kinetics of damage accumulation, the process of crack nucleation and kinetics of the crack development allows specialists to predict the time of structure failure and to perform in proper time a partial replacement or repair of deteriorated units of complex structures. Moreover, the repair or replacement of the wornout parts on a timely basis is more effective than their complete replacement after mechanical damage. It is therefore very important to know the time during which the defects in the ill-behaved areas are reaching critical values. The previous authors’ investigations were focused on crack growth problems under an opening or mode I mechanism (Vshivkov А., 2016). However, most structures are failed due to mixed mode loading. Many uniaxial loaded materials, structures and components often contain randomly oriented defects and cracks which develop a mixed mode state by rotation of their orientation with respect to the loading axis. This work is devoted to the investigation of the dissipated energy in the process of crack propagation under mixed mode loading. For this purpose, the original contact heat flux sensor was developed to detect energy dissipation value in the process of crack propagation and verify the data of infrared thermography. This device is based on the Seebeck effect and includes two Peltier elements and temperature controlling feedback. This sensor allows us to study in details a dissipated energy evolution in metal samples (AISI304) with uniaxial and multi axial loadings and propose relations between heat dissipation and fatigue crack rate. 2. Experimental setup A series of samples made from stainless steel AISE 304 were tested. The geometry of the samples is shown in Figure 1. The experimental study was carry out in University of the Federal Armed Forces Munich, Institute for Materials Science, Neubiberg, Germany. During tests the samples were subjected to cyclic loading of 20 Hz with constant stress intensity factor and ratio R = -1. The crack length in the course of the experiment was measured by the potential drop method (Nayeb-Hashemi H., 2004; Hartman G.A., 1987). The electrical potential drop method is accepted as being capable of monitoring the fatigue crack propagation in steel structures. The size of a crack in a steel sample is predicted by applying a constant d.c. (direct current) or a.c. (alternating current) to the sample and by measuring an increase in electrical resistance due to the crack. In this case, the potential method is capable of a sensitivity as fine as 0.02 mm for a d.c. 5 A.
Fig. 1. Geometry of samples.
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To analyze the dissipated energy at the crack tip a contact heat flux sensor was designed and constructed. The proposed sensor is based on the Seebeck effect, which is the reverse of the Peltier effect. The Peltier effect is a thermoelectric phenomenon, in which the passage of electric current through conducting medium leads to the generation or absorption of heat at the point of contact (junction) of two dissimilar conductors. The quantity of heat and its sign depend on the type of materials in contact, the direction and the strength of the electric current. The quantity of heat absorbed or dissipated by the element is directly proportional to the current intensity and the time of its passage. (1) P ABIP P – the power of heat flux; I – the direct current; ПАБ – Peltier coefficient.
Fig. 2. Schematic of the device.1 – testing sample; 2 – “measuring” Peltier element; 3 – “cooling” Peltier element; 4 – radiator; 5, 6 – thermocouple; 7 – resistor.
Figure 1 presents a schematic diagram of the heat flux sensor. The following notation is used in figure 1: sample (1), the heat flux sensor (2). A thermal contact between the sample and the sensor is provided due to the introduction of the thermal paste. Structurally, the sensor comprises two Peltier elements ("measuring" (2) and "cooling" (3)), thermocouples (5), (6) and the radiator (4). The measuring Peltier element is connected to a low-resistance resistor of 1.2 Om (7). To measure the heat flow through the "measuring" Peltier element during the experiment the temperature on its free surface should be a constant. The cooling Peltier element caulked with a radiator was connected with the "measuring" Peltier element. This cooling system has feedback and is controlled based on two temperature sensors located between "measuring" and cooling Peltier elements and far from the studied sample in the zone with constant temperature. The signal from the sensor (voltage at the resistor (7)) is measured by the amplifier and registered in the ADC of the microcontroller. The data are transmitted from the microcontroller to the personal computer for further processing. The "cooling" Peltier element is controlled via pulse width modulation. These sensors were calibrated using a device with a controlled heat flux. A wire resistor with the known resistance is glued on a plastic plate with a size equal to that of test samples. The heat isolating system provides the heat flux from the resistance to the sensor only. The heat flow was calculated using the values of the resistor voltage and the electric current across the resistor.
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The sensitivity of the heat flux sensor was verified during the fatigue crack propagation with the constant amplitude of loading. In this experiment the heat flux sensor was located at a distance from the crack tip. The results of such measurements qualitatively correspond to the case of the direct location of the sensor over the top of the fatigue crack tip. This effect is a consequence of the high thermal conductivity of the material. Measurement results are presented in figure 3
Fig. 3. Direct measurement of the heat flux at the crack tip and measurement at the distance.
There are three sections on the heat flux curve under the mixed loading. Short initial increasing part corresponds to starting of crack propagation (part 1). The second part with constant heat flux corresponds to the regime of short crack propagation (part 2). The last part of the plot (part 3) is characterized by sharp increasing of heat dissipation. During this part we observe the long crack propagation process. The last part is finished by specimen failure. The evolution of the temperature field was recorded by infrared camera FLIR SC 5000. The spectral range of the camera is 3-5 µm. The maximum frame size is 320×256 pixels; the spatial resolution is 10-4 meters. The temperature sensitivity is 25 mK at 300 K. Calibration of the camera was made based on the standard calibration table. It was used FLIR SC5000 MW G1 F/3.0 close-up lens (distortion is less than 0.5%) to investigate the plastic zone in details.
Fig. 4. Schematic of the measured equipment.
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3. Results of fatigue experiments During the experiments, a series of samples was tested where the crack length and the heat flux were recorded. Figures 5 show the characteristic time dependences of heat flux during the experiment. The heat flux during the experiment can be divided into two stages. In the first stage (up to point *), cyclic loading in the elastic mode with a constant amplitude of the applied load is carried out to achieve the required value of the stress intensity factor. In the second stage (after the point *), the load is controlled to maintain a constant value of the stress intensity factor, while the rate of the fatigue crack propagation also remains constant (Fig. 7). This stage is the subject of investigation.
*
Fig. 5. The characteristic heat flux during the experiments with
Fig. 6. The characteristic crack length during the experiments with
constant stress intensity factor.
constant stress intensity factor.
According to the classical concepts the heat flux from the top of the crack should remain constant with constant stress intensity factor. However, in the experiment a monotonic decrease in the heat flux is observed. 4. Energy dissipation at crack tip under cyclic loading Following the work Raju (1972), we can propose a relation between elastic and real deformation at crack tip: 1
E 2 ijef ijel , (2) Es where 𝐸𝐸– the Young’s modulus, 𝐸𝐸𝑠𝑠 - secant plasticity modulus. Equation (2) was originally proposed by Dixon (1965) as results of photo elastic experiments data treatment based on the Ramberg-Osgood relationship (1943): n
A . G 0 We can write a following estimation for octahedral stress and couple it with an elastic solution:
1
oct where B
GA e 0 0
n
2(1 ) 2 B n 1 1 3 el oct , n 1 1 B
, oct , e - elastic limit. e
(3)
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The energy of plastic deformation in representative volume located near crack tip can be estimated as follow: 3 3 An oct d oct e n n1 . Up 2 2 n 1 The energy increment for crack under monotonic loading can be written as d 3 (4) dU p An e n n dl, 2 dl where 𝑙𝑙- crack length.
1
1
rp f e 2 el Kf e2 (here K - stress intensity factor, rp– estimation for plastic zone Using definition oct e 3 r e r
size, r – polar coordinate, fe – function of polar coordinate determining the relation of octahedral stress versus ) we can rewrite equation (4) as follow: 3 d d (5) dU p An e n n dl. 2 d dl It was shown earlier that plastic zone at crack tip could be divided into two parts: plastically loaded zone (dissipation zone) and elastically unloaded. The geometry of dissipation area would be determined by relation: r df d 1 f e C p sin e f e cos . (6) r d dl 2 r p f e r It was shown by Raju (1972) that for l rp the equation (6) gives two straight lines 79.9 determined
the areas of plastic uploading and elastic unloading. Figure 7a presents the plastic zone shape under monotonic uniaxial loading. Zone A corresponds to the plastic loading caused by crack advance, zone B - the elastic unloading. a
b 1.0
1.0
B
0.5
0.5
A 1.0
1.0
0.5
B
0.5
0.5
0.5
1.0
A
1.0 0.5
0.5
B
B
1.0
1.0
1.5
Figure 7. Structure of plastic zone at crack tip under monotonic loading (a) – uniaxial loading, (b) – multiaxial loading with biaxial coefficient equal to 0.3
For description of crack behavior under mixed mode loading we have to change the function fe in equations (5) and (6). Taking into account elastic solution of the first and second fracture modes at crack tip: K I , II I , II f ( I , II )1 , xx 2r
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I , II yy I , II xy
K I , II 2r K I , II 2r
7
f ( I , II ) 2 , f ( I , II )3 ,
we can write the 𝑓𝑓𝑒𝑒 for multiaxial loading as follow:
(7) f e af I1 bf II1 2 af I 2 bf II 2 2 af I 1 bf II1 af I 1 bf II1 3af I 3 bf II 3 2 . Substitution of equation (7) into equation (6) gives us an estimation of the evolution of plastic zone caused by application of shear stress. The structure of plastic zone at crack tip under monotonic loading is presented in figure 7b. As a result, we can analyze the plastic deformation at crack tip under multiaxial loading based on equation (5). The biaxial coefficient can be taken into account using equation (7). For cyclic loading we have to consider energy dissipation in cyclic plastic zone at crack tip: cyc mon U tot p U p U p . To explain the experimental fact reported in previous paragraph we have to calculate the total energy increment dU cyc p 0 and plastic energy at U tot p . For loading condition considered in our experiment it can be shown that dl
crack tip U tot can be written as follow: p
2 U tot p W1 A W2
dl , dN
(8)
where A – stress amplitude. Based on the equation (8) we can conclude that for small crack rate the plastic work and, as a consequence, energy dissipation at crack tip is proportional to the applied stress amplitude and can decrease during experiments with constant stress intensity factor. The calculation of analytical relations for W1, W2 can be carried out similarly to the monotonic loading. The biaxial coefficient changes the function fe but keeps the constant the structure of the equation (8). It allows us to predict the existence of peculiarities of energy dissipation at crack tip reported in Iziumova (2016) for multiaxial loading. 5. Conclusion In this work the experimental and theoretical study of dissipated energy were carried out during fatigue crack propagation. Based on a contact heat flux sensor, an experimental technique for the application of the method of infrared thermography for measuring the energy dissipation during fatigue test has been developed. The device allows us to measure heat dissipation under uniaxial and multiaxial loadings. Decrease in the power of the dissipated energy was shown during the fatigue crack propagation with a constant stress intensity factor. In order to explain this effect, the theoretical analysis of the energy dissipation zone formation at the crack tip was carried out. Using the coupling between Young’s modulus and the secant plasticity modulus, the analysis of the plastic zone shape and dissipated energy value at the crack tip under uniaxial and multiaxial loadings has been made. A good agreement between theoretical and experimental results illustrates the possibility of the application of the method to description of the energy dissipation under multiaxial loading. Acknowledgements This work was supported by the grant of the President of Russian Federation for support of young Russian scientists and leading scientific schools [MK-1236.2017.1] and the Russian Foundation for Basic research [grant number 16-31-00130]. The authors would like to thank Prof. Jürgen Bär for experimental support of the work.
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