- Email: [email protected]
The experimental and theoretical study of plastic deformation in the fatigue crack tip based on method of digital image correlation
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Structural Integrity Procedia 00 (2018) 000–000 Available online www.sciencedirect.com Available online at at www.sciencedirect.com Structural Integrity Procedia 00 (2018) 000–000
ScienceDirect ScienceDirect
Procedia Structural Integrity 1300 (2018) 1189–1194 Structural Integrity Procedia (2016) 000–000
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ECF22 - Loading and Environmental effects on Structural Integrity ECF22 - Loading and Environmental effects on Structural Integrity
The experimental and theoretical study of plastic deformation in the The experimental and theoretical study of plastic deformation in the crack tip based on method digital image correlation XVfatigue Portuguese Conference on Fracture, PCF 2016, of 10-12 February 2016, Paço de Arcos, Portugal fatigue crack tip based on method of digital image correlation a b b a A. Vshivkovaa*, A. Iziumova , A. Zakharov , V.pressure Shlyannikov , O. Plekhov Thermo-mechanical modeling of a high turbine blade a b b A. Vshivkov *, A. Iziumova , A. Zakharov , V. Shlyannikov , O. Plekhova of an Institute of Continuous Media Mechanics Russian Academy of Sciences Ural Branch, Perm, Russian Federation airplane gas turbine engine Institute of Continuous Media Mechanics RussianAcademy AcademyofofSciences, SciencesKazan, Ural Branch, Russian Federation Kazan Scientific Center of Russian RussianPerm, Federation a a
b b
Kazan Scientific Center of Russian Academy of Sciences, Kazan, Russian Federation
P. Brandãoa, V. Infanteb, A.M. Deusc*
Abstract a Abstract Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa,
Portugal Thisb work is devoted to the experimental study of strain distribution at crack tip using digital image correlation IDMEC, Department ofto Mechanical Engineering, study InstitutoofSuperior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, This work is devoted the experimental strain distribution at crack tip using digital image correlation technique. The real strain fields were compared with an elastic Portugalsolution for verification of a hypothesis about link of c elastic-plastic technique. realand strain fields were compared with anusing elastic forbetween verification ofRovisco a hypothesis about of the strain fields at Instituto crack tip thesolution coupling Young’s modulus the link secant CeFEMA,The Department ofelastic Mechanical Engineering, Superior Técnico, Universidade de Lisboa, Av. Pais, 1,and 1049-001 Lisboa, the elastic-plastic and elastic strain fields at crack tip using the coupling between Young’s modulus and the secant Portugal plasticity modulus. The precracked plane specimens of titanium alloy VT1-0 with a thick of 1 mm were used in plasticity modulus. The The precracked specimens of correlation titanium alloy VT1-0 with a thick of 1 mm was wereused usedfor in experimental program. methodplane of digital image based on system StrainMaster experimental the program. method with of digital correlation basedThe on strain system StrainMaster for measurement plastic The deformation spatial image resolution up to 1 mkm. field was obtainedwas for used different Abstract measurement the different plastic deformation with spatial up tobiaxial 1 mkm.testing The strain fieldBiss wasBI-00-502. obtained for different crack length and biaxial coefficient was resolution obtained using machine Numerical crack lengthofand different biaxial coefficient was obtained using biaxial testing machine Biss BI-00-502. Numerical simulation deformation fields at the fatigue crack tip was done. A qualitative correspondence between the During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, simulation of deformation fields at theresults fatiguehascrack tip was done. A qualitative correspondence between the theoretical, calculation and experimental been shown. especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent theoretical, calculation andis experimental hasfinite beenelement shown.method (FEM) was developed, in order to be able to predict degradation, one of which creep. A modelresults using the
© the 2018creep The Authors. Published by Elsevier B.V.data records (FDR) for a specific aircraft, provided by a commercial aviation behaviour of HPT blades. Flight © 2018 Published by Elsevier B.V. © company, 2018The TheAuthors. Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. wereresponsibility used to obtain and mechanical data for three different flight cycles. In order to create the 3D model Peer-review under of thethermal ECF22 organizers. Peer-review responsibility ECF22 needed forunder the FEM analysis,ofa the HPT bladeorganizers. scrap was scanned, and its chemical composition and material properties were Keywords: crack, deformation, digitalwas image obtained.fatigue, The data that was gathered fedcorrelation; into the FEM model and different simulations were run, first with a simplified 3D Keywords: fatigue, crack, deformation, image correlation; rectangular block shape, in order digital 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 can be useful in the goal of predicting turbine blade life, given a set of FDR data. 1.model Introduction
1. Introduction © 2016 The Authors. Published by Elsevier B.V. Fatigue crack propagation in metals is one the important problems of fracture mechanics. During wide range of Peer-review underpropagation responsibilityinofmetals the Scientific Committee of PCF 2016. of fracture mechanics. During wide range of Fatigue crack one important problems crack rates the kinetics of a crack growthiscan bethe described by correlation with a value of stress intensity factor (Paris`s crack rates the kinetics of a crack growth can be described by correlation with a value of stress intensity factor (Paris`s law). This correlation is the result of the approximation of many experimental Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation. data and doesn`t explain the physical law). This correlation is the result of the approximation of many experimental data and doesn`t explain the physical
* Corresponding author. Tel.: +73422378312; * Corresponding Tel.: +73422378312; E-mail address:author. [email protected] E-mail address: [email protected] 2452-3216 © 2018 The Authors. Published by Elsevier B.V. 2452-3216 © 2018 Authors. Published Elsevier B.V. Peer-review underThe responsibility of theby ECF22 organizers. Peer-review underauthor. responsibility the ECF22 organizers. * 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 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. 10.1016/j.prostr.2018.12.246
A. Vshivkov et al. / Procedia Structural Integrity 13 (2018) 1189–1194 Author name / Structural Integrity Procedia 00 (2018) 000–000
1190 2
nature of this process. A number of approaches, Yates (2008), Mokhtarishirazabad (2017), Izumi (2014), Short (1989), has been developed to study the processes of nucleation and propagation of fatigue cracks based on the J-integral, the work of plastic deformation, the size of the zone of a plastic deformation, the amount of dissipated energy. It is well known that irreversible deformation and failure of metals deformation is accompanied by the structural evolution at all scale levels and leads to energy accumulation and dissipation. Investigation of thermodynamics of deformation and failure is a key issue in solid mechanics. 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. The real engineering structure works under complex types of loading. So, it is of considerable interest to study the behavior of materials under mixed loading conditions that combine mode 1 and 2. In sufficiently plastic structural materials, the propagation of the crack begins when the plastic deformation near its tip becomes large (of the order of a 10 percent). The previous study of the authors was mainly focused on an equation describing the evolution of plastic work at the crack tip under uniaxial loading. In this work, following by Raju (1972), the plastic work and, as a consequence, heat dissipation at crack tip were divided into two parts corresponding to reversible (cyclic) and monotonic plastic zones. Analysis of this approximation has shown the independence of heat dissipation in cyclic plastic zone from the crack advance. This dissipation is fully determined by the spatial size of a cyclic plastic zone and characteristic diameter of the yield surface. This approach gives well know correlation between fatigue crack rate and dissipated energy, Izyumova (2014) and Ranganathan (2008). This equation based the hypothesis about link between the elastic solution and the elastic-plastic deformation in the fatigue crack tip using the Young’s modulus and the secant plasticity modulus by Dixon (1965). 1
ijef
G 2 ijel , Gs
(1)
Equation (1) was originally proposed as an empirical expression for a central crack in a wide sheet loaded in tension normal to the line of the crack. The present work is directed on checking this assumption by numerical simulation and experimental measurement of the elastic-plastic zone near the fatigue crack tip. Nomenclature ε
G GS γ ε1, ε2 εie σ σys σys0 σm u C λ
strain tensor
shear modulus secant shear modulus maximum shear strain = ε1 – ε2 principal strain in plane of sheet intensity of plastic deformation
stress tensor Mises yield stress initial Mises yield stress Mises stress displacement vector fourth-order stiffness tensor Lamé constant
2. Experimental setup A series of samples made from titanium alloy Graid-2 were tested in the servo-hydraulic biaxial test system Biss BI-00-502, located in Kazan Scientific Center of Russian Academy of Sciences, figure 2. The geometry of the samples is shown in Figure 1. During tests the samples were subjected to cyclic loading of 10 Hz with constant stress amplitude and different biaxial coefficient η=Px/Py (1, 0.7, 0.5, 0). The crack length in the course of the experiment was measured
A. Vshivkov et al. / Procedia Structural Integrity 13 (2018) 1189–1194 Author name / Structural Integrity Procedia 00 (2018) 000–000
1191 3
by optic method. The strain field was measured by method of digital image correlation based on StrainMaster system and Lavision software. Before testing the surface of the sample near the crack tip was polished and covered with a black matte paint. Then white paint was sprayed over the black paint to obtain a high contrast image. A macro lens and an elevated Led lamp were used. To restore the deformation field in the crack tip area, each frame was subjected to additional processing: calibration to level distortions caused by distortion of the lens, compensate the movement of the subject as a hard whole, regulation the illumination with digital filters. The spatial resolution of the strain field near the crack tip is 3e-6 m. There was considered two-dimensional biaxial loading of a half of a sample. The applied load corresponds to the experimental. A shape of the crack was taken from the digital image correlation data. The system of equations for modeling of a strain field near the crack tip has the following form: (2)
σ 0
σ С : εεp ε
1 u uT 2
εp
F σ
(3) (4) (5)
where F=σm-σys, σys=σys0+ σh(εpe). The isotropic hardening function σh(εpe) was obtained from experimental data.
Figure. 1. Geometry of samples (all sizes in millimeters).
Figure. 2. Testing machine Biss BI-00-502, Biaxial test System.
3. Result of strain measurement The deformation field was measured at the fatigue crack tip. Figure 3 shows the characteristic result of measurements for the case of uniaxial loading. There is not symmetry of the field due to the influence of the direction of the crack. Figure 4 shows the shape of the plastic area boundary and shows the characteristic sizes which were used for the comparison of the experimental and numerical data. Table 1 gives a quantitative comparison of the obtained data.
4 1192
Author name / Structural Integrity Procedia 00 (2018) 000–000 A. Vshivkov et al. / Procedia Structural Integrity 13 (2018) 1189–1194
B 1.0
0.5
1.0
A
0.5
0.5
1.0
C
0.5
1.0
D
�
1.5
Fig. 3. Measured field of strain for uniaxial loading
�
Fig. 4. Boundary of plastic area.
Table 1. Character size of plastic zone. AB AC AD BD
Numerical, mm
Measure, mm
1,20E-01 1,35E-02 1,19E-01 2,18E-01
9,13E-02 3,13E-02 6,98E-02 1,27E-01
4. Results of numerical simulation A numerical simulation of plastic deformation near the crack tip for have been carried out. The estimation of the plastic strain field was carried out using the elastic solution and equation (1) for the strain components ε11, ε12, ε22, maximum shear strain γ, intensity of plastic deformation εpe (6). Numerical and theoretical calculations were made for four biaxial coefficients (n = 0, 0.5, 0.7, 1). The characteristic deformation fields are shown in figures 5, 6. To compare the theoretical and calculated results, an error was calculated for different strain levels corresponded to the applied load (Fig. 7).
pe
1/2
2 2 2 3 2 2 2 2 11 22 11 33 33 22 12 13 23 3 2
(a) (b) Fig. 5. Plastic strain (n=0, component ε22): (a) by a numerical simulation; (b) analytic solution by equation (1).
(6)
A. Vshivkov et al. / Procedia Structural Integrity 13 (2018) 1189–1194 Author name / Structural Integrity Procedia 00 (2018) 000–000
1193 5
(a) (b) Fig. 6. Plastic strain (n=1, component εi): (a) by a numerical simulation; (b) analytic solution by equation (1). 30
25
e11 e12 e22 shear intensity
25
20
20 15 15
10
5
e11 e12 e22 shear intensity
10
1
2
3
4
Load, kN
5
6
7
5
1
2
3
(a)
4
Load, kN
5
6
7
5
6
7
(b) 25
22 e11 e12 e22 shear intensity
20 18
e11 e12 e22 shear intensity
20
16 15
14 12
10
10 8 6
1
2
3
4
5
6
7
5
1
2
3
4
Load, kN Load, kN (c) (d) Fig. 7. An error in the plastic deformation (a) – n = 0; (b) – n = 0.5; (c) – n =0.7; (d) – n = 1.
These loading levels correspond to 10-8 – 10-3 m/cycle crack growth rate.
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A. Vshivkov et al. / Procedia Structural Integrity 13 (2018) 1189–1194 Author name / Structural Integrity Procedia 00 (2018) 000–000
5. Conclusion A series of experiments was carry out. For different biaxial coefficients the strain field at the fatigue crack tip was measured. The results of the measurements have a good agreement with numerical simulation. The theoretical calculation of plastic deformation based on the elastic solution and the secant modulus of elasticity is carried out. According to the results of calculations, the error in determining the plastic deformation through the elastic solution does not exceed 30% for each component of the strain tensor, the maximum shear strain, and the intensity of plastic deformations. This allows us to conclude that it is possible to use equation (1) for the theoretical calculation of the deformation field at the fatigue crack tip and the subsequent calculation of energy dissipation. 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 1648-590148]. References Yates, J.R., Zanganeh, M., Tomlinson, R.A., Brown, M.W., DiazGarrido F.A., 2008. Crack paths under mixed mode loading. Engineering Fracture Mechanics 75, 3-4, 319-330. Mokhtarishirazabad, M., Lopez-Crespo, P., Moreno, B., Lopez-Moreno, A., Zanganeh M., 2017. Optical and analytical investigation of overloads in biaxial fatigue cracks. International Journal of Fatigue 100, 2, 583-590. Izumi, Y., Sakagami, T., Yasumura, K., Shiozawa, D., 2014. A new approach for evaluating stress intensity factor based on thermoelastic stress analysis. APCFS/SIF, 47-51. Short, J. S., Hoeppner, D. W., 1989. A Global/local theory of fatigue crack propagation. Engineering Fracture mechanics, 33, 2, 175-184. 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, 1-14. 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. Ranganathan, N., Chalon, F., Meo, S., 2008. Some aspects of the energy based approach to fatigue crack propagation Original research article. International Journal of Fatigue 30, 10–11, 1921-1929. 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.
ScienceDirect ScienceDirect
Structural Integrity Procedia 00 (2018) 000–000 Available online www.sciencedirect.com Available online at at www.sciencedirect.com Structural Integrity Procedia 00 (2018) 000–000
ScienceDirect ScienceDirect
Procedia Structural Integrity 1300 (2018) 1189–1194 Structural Integrity Procedia (2016) 000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
www.elsevier.com/locate/procedia
ECF22 - Loading and Environmental effects on Structural Integrity ECF22 - Loading and Environmental effects on Structural Integrity
The experimental and theoretical study of plastic deformation in the The experimental and theoretical study of plastic deformation in the crack tip based on method digital image correlation XVfatigue Portuguese Conference on Fracture, PCF 2016, of 10-12 February 2016, Paço de Arcos, Portugal fatigue crack tip based on method of digital image correlation a b b a A. Vshivkovaa*, A. Iziumova , A. Zakharov , V.pressure Shlyannikov , O. Plekhov Thermo-mechanical modeling of a high turbine blade a b b A. Vshivkov *, A. Iziumova , A. Zakharov , V. Shlyannikov , O. Plekhova of an Institute of Continuous Media Mechanics Russian Academy of Sciences Ural Branch, Perm, Russian Federation airplane gas turbine engine Institute of Continuous Media Mechanics RussianAcademy AcademyofofSciences, SciencesKazan, Ural Branch, Russian Federation Kazan Scientific Center of Russian RussianPerm, Federation a a
b b
Kazan Scientific Center of Russian Academy of Sciences, Kazan, Russian Federation
P. Brandãoa, V. Infanteb, A.M. Deusc*
Abstract a Abstract Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa,
Portugal Thisb work is devoted to the experimental study of strain distribution at crack tip using digital image correlation IDMEC, Department ofto Mechanical Engineering, study InstitutoofSuperior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, This work is devoted the experimental strain distribution at crack tip using digital image correlation technique. The real strain fields were compared with an elastic Portugalsolution for verification of a hypothesis about link of c elastic-plastic technique. realand strain fields were compared with anusing elastic forbetween verification ofRovisco a hypothesis about of the strain fields at Instituto crack tip thesolution coupling Young’s modulus the link secant CeFEMA,The Department ofelastic Mechanical Engineering, Superior Técnico, Universidade de Lisboa, Av. Pais, 1,and 1049-001 Lisboa, the elastic-plastic and elastic strain fields at crack tip using the coupling between Young’s modulus and the secant Portugal plasticity modulus. The precracked plane specimens of titanium alloy VT1-0 with a thick of 1 mm were used in plasticity modulus. The The precracked specimens of correlation titanium alloy VT1-0 with a thick of 1 mm was wereused usedfor in experimental program. methodplane of digital image based on system StrainMaster experimental the program. method with of digital correlation basedThe on strain system StrainMaster for measurement plastic The deformation spatial image resolution up to 1 mkm. field was obtainedwas for used different Abstract measurement the different plastic deformation with spatial up tobiaxial 1 mkm.testing The strain fieldBiss wasBI-00-502. obtained for different crack length and biaxial coefficient was resolution obtained using machine Numerical crack lengthofand different biaxial coefficient was obtained using biaxial testing machine Biss BI-00-502. Numerical simulation deformation fields at the fatigue crack tip was done. A qualitative correspondence between the During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, simulation of deformation fields at theresults fatiguehascrack tip was done. A qualitative correspondence between the theoretical, calculation and experimental been shown. especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent theoretical, calculation andis experimental hasfinite beenelement shown.method (FEM) was developed, in order to be able to predict degradation, one of which creep. A modelresults using the
© the 2018creep The Authors. Published by Elsevier B.V.data records (FDR) for a specific aircraft, provided by a commercial aviation behaviour of HPT blades. Flight © 2018 Published by Elsevier B.V. © company, 2018The TheAuthors. Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. wereresponsibility used to obtain and mechanical data for three different flight cycles. In order to create the 3D model Peer-review under of thethermal ECF22 organizers. Peer-review responsibility ECF22 needed forunder the FEM analysis,ofa the HPT bladeorganizers. scrap was scanned, and its chemical composition and material properties were Keywords: crack, deformation, digitalwas image obtained.fatigue, The data that was gathered fedcorrelation; into the FEM model and different simulations were run, first with a simplified 3D Keywords: fatigue, crack, deformation, image correlation; rectangular block shape, in order digital 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 can be useful in the goal of predicting turbine blade life, given a set of FDR data. 1.model Introduction
1. Introduction © 2016 The Authors. Published by Elsevier B.V. Fatigue crack propagation in metals is one the important problems of fracture mechanics. During wide range of Peer-review underpropagation responsibilityinofmetals the Scientific Committee of PCF 2016. of fracture mechanics. During wide range of Fatigue crack one important problems crack rates the kinetics of a crack growthiscan bethe described by correlation with a value of stress intensity factor (Paris`s crack rates the kinetics of a crack growth can be described by correlation with a value of stress intensity factor (Paris`s law). This correlation is the result of the approximation of many experimental Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation. data and doesn`t explain the physical law). This correlation is the result of the approximation of many experimental data and doesn`t explain the physical
* Corresponding author. Tel.: +73422378312; * Corresponding Tel.: +73422378312; E-mail address:author. [email protected] E-mail address: [email protected] 2452-3216 © 2018 The Authors. Published by Elsevier B.V. 2452-3216 © 2018 Authors. Published Elsevier B.V. Peer-review underThe responsibility of theby ECF22 organizers. Peer-review underauthor. responsibility the ECF22 organizers. * 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 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. 10.1016/j.prostr.2018.12.246
A. Vshivkov et al. / Procedia Structural Integrity 13 (2018) 1189–1194 Author name / Structural Integrity Procedia 00 (2018) 000–000
1190 2
nature of this process. A number of approaches, Yates (2008), Mokhtarishirazabad (2017), Izumi (2014), Short (1989), has been developed to study the processes of nucleation and propagation of fatigue cracks based on the J-integral, the work of plastic deformation, the size of the zone of a plastic deformation, the amount of dissipated energy. It is well known that irreversible deformation and failure of metals deformation is accompanied by the structural evolution at all scale levels and leads to energy accumulation and dissipation. Investigation of thermodynamics of deformation and failure is a key issue in solid mechanics. 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. The real engineering structure works under complex types of loading. So, it is of considerable interest to study the behavior of materials under mixed loading conditions that combine mode 1 and 2. In sufficiently plastic structural materials, the propagation of the crack begins when the plastic deformation near its tip becomes large (of the order of a 10 percent). The previous study of the authors was mainly focused on an equation describing the evolution of plastic work at the crack tip under uniaxial loading. In this work, following by Raju (1972), the plastic work and, as a consequence, heat dissipation at crack tip were divided into two parts corresponding to reversible (cyclic) and monotonic plastic zones. Analysis of this approximation has shown the independence of heat dissipation in cyclic plastic zone from the crack advance. This dissipation is fully determined by the spatial size of a cyclic plastic zone and characteristic diameter of the yield surface. This approach gives well know correlation between fatigue crack rate and dissipated energy, Izyumova (2014) and Ranganathan (2008). This equation based the hypothesis about link between the elastic solution and the elastic-plastic deformation in the fatigue crack tip using the Young’s modulus and the secant plasticity modulus by Dixon (1965). 1
ijef
G 2 ijel , Gs
(1)
Equation (1) was originally proposed as an empirical expression for a central crack in a wide sheet loaded in tension normal to the line of the crack. The present work is directed on checking this assumption by numerical simulation and experimental measurement of the elastic-plastic zone near the fatigue crack tip. Nomenclature ε
G GS γ ε1, ε2 εie σ σys σys0 σm u C λ
strain tensor
shear modulus secant shear modulus maximum shear strain = ε1 – ε2 principal strain in plane of sheet intensity of plastic deformation
stress tensor Mises yield stress initial Mises yield stress Mises stress displacement vector fourth-order stiffness tensor Lamé constant
2. Experimental setup A series of samples made from titanium alloy Graid-2 were tested in the servo-hydraulic biaxial test system Biss BI-00-502, located in Kazan Scientific Center of Russian Academy of Sciences, figure 2. The geometry of the samples is shown in Figure 1. During tests the samples were subjected to cyclic loading of 10 Hz with constant stress amplitude and different biaxial coefficient η=Px/Py (1, 0.7, 0.5, 0). The crack length in the course of the experiment was measured
A. Vshivkov et al. / Procedia Structural Integrity 13 (2018) 1189–1194 Author name / Structural Integrity Procedia 00 (2018) 000–000
1191 3
by optic method. The strain field was measured by method of digital image correlation based on StrainMaster system and Lavision software. Before testing the surface of the sample near the crack tip was polished and covered with a black matte paint. Then white paint was sprayed over the black paint to obtain a high contrast image. A macro lens and an elevated Led lamp were used. To restore the deformation field in the crack tip area, each frame was subjected to additional processing: calibration to level distortions caused by distortion of the lens, compensate the movement of the subject as a hard whole, regulation the illumination with digital filters. The spatial resolution of the strain field near the crack tip is 3e-6 m. There was considered two-dimensional biaxial loading of a half of a sample. The applied load corresponds to the experimental. A shape of the crack was taken from the digital image correlation data. The system of equations for modeling of a strain field near the crack tip has the following form: (2)
σ 0
σ С : εεp ε
1 u uT 2
εp
F σ
(3) (4) (5)
where F=σm-σys, σys=σys0+ σh(εpe). The isotropic hardening function σh(εpe) was obtained from experimental data.
Figure. 1. Geometry of samples (all sizes in millimeters).
Figure. 2. Testing machine Biss BI-00-502, Biaxial test System.
3. Result of strain measurement The deformation field was measured at the fatigue crack tip. Figure 3 shows the characteristic result of measurements for the case of uniaxial loading. There is not symmetry of the field due to the influence of the direction of the crack. Figure 4 shows the shape of the plastic area boundary and shows the characteristic sizes which were used for the comparison of the experimental and numerical data. Table 1 gives a quantitative comparison of the obtained data.
4 1192
Author name / Structural Integrity Procedia 00 (2018) 000–000 A. Vshivkov et al. / Procedia Structural Integrity 13 (2018) 1189–1194
B 1.0
0.5
1.0
A
0.5
0.5
1.0
C
0.5
1.0
D
�
1.5
Fig. 3. Measured field of strain for uniaxial loading
�
Fig. 4. Boundary of plastic area.
Table 1. Character size of plastic zone. AB AC AD BD
Numerical, mm
Measure, mm
1,20E-01 1,35E-02 1,19E-01 2,18E-01
9,13E-02 3,13E-02 6,98E-02 1,27E-01
4. Results of numerical simulation A numerical simulation of plastic deformation near the crack tip for have been carried out. The estimation of the plastic strain field was carried out using the elastic solution and equation (1) for the strain components ε11, ε12, ε22, maximum shear strain γ, intensity of plastic deformation εpe (6). Numerical and theoretical calculations were made for four biaxial coefficients (n = 0, 0.5, 0.7, 1). The characteristic deformation fields are shown in figures 5, 6. To compare the theoretical and calculated results, an error was calculated for different strain levels corresponded to the applied load (Fig. 7).
pe
1/2
2 2 2 3 2 2 2 2 11 22 11 33 33 22 12 13 23 3 2
(a) (b) Fig. 5. Plastic strain (n=0, component ε22): (a) by a numerical simulation; (b) analytic solution by equation (1).
(6)
A. Vshivkov et al. / Procedia Structural Integrity 13 (2018) 1189–1194 Author name / Structural Integrity Procedia 00 (2018) 000–000
1193 5
(a) (b) Fig. 6. Plastic strain (n=1, component εi): (a) by a numerical simulation; (b) analytic solution by equation (1). 30
25
e11 e12 e22 shear intensity
25
20
20 15 15
10
5
e11 e12 e22 shear intensity
10
1
2
3
4
Load, kN
5
6
7
5
1
2
3
(a)
4
Load, kN
5
6
7
5
6
7
(b) 25
22 e11 e12 e22 shear intensity
20 18
e11 e12 e22 shear intensity
20
16 15
14 12
10
10 8 6
1
2
3
4
5
6
7
5
1
2
3
4
Load, kN Load, kN (c) (d) Fig. 7. An error in the plastic deformation (a) – n = 0; (b) – n = 0.5; (c) – n =0.7; (d) – n = 1.
These loading levels correspond to 10-8 – 10-3 m/cycle crack growth rate.
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A. Vshivkov et al. / Procedia Structural Integrity 13 (2018) 1189–1194 Author name / Structural Integrity Procedia 00 (2018) 000–000
5. Conclusion A series of experiments was carry out. For different biaxial coefficients the strain field at the fatigue crack tip was measured. The results of the measurements have a good agreement with numerical simulation. The theoretical calculation of plastic deformation based on the elastic solution and the secant modulus of elasticity is carried out. According to the results of calculations, the error in determining the plastic deformation through the elastic solution does not exceed 30% for each component of the strain tensor, the maximum shear strain, and the intensity of plastic deformations. This allows us to conclude that it is possible to use equation (1) for the theoretical calculation of the deformation field at the fatigue crack tip and the subsequent calculation of energy dissipation. 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 1648-590148]. References Yates, J.R., Zanganeh, M., Tomlinson, R.A., Brown, M.W., DiazGarrido F.A., 2008. Crack paths under mixed mode loading. Engineering Fracture Mechanics 75, 3-4, 319-330. Mokhtarishirazabad, M., Lopez-Crespo, P., Moreno, B., Lopez-Moreno, A., Zanganeh M., 2017. Optical and analytical investigation of overloads in biaxial fatigue cracks. International Journal of Fatigue 100, 2, 583-590. Izumi, Y., Sakagami, T., Yasumura, K., Shiozawa, D., 2014. A new approach for evaluating stress intensity factor based on thermoelastic stress analysis. APCFS/SIF, 47-51. Short, J. S., Hoeppner, D. W., 1989. A Global/local theory of fatigue crack propagation. Engineering Fracture mechanics, 33, 2, 175-184. 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, 1-14. 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. Ranganathan, N., Chalon, F., Meo, S., 2008. Some aspects of the energy based approach to fatigue crack propagation Original research article. International Journal of Fatigue 30, 10–11, 1921-1929. 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.