- Email: [email protected]
Biaxial experiments on the effect of non-proportional loading paths on damage and fracture behavior of ductile metals
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Structural Integrity Procedia 00 (2018) 000–000 Structural IntegrityProcedia Procedia13 (2018)57–62 000–000 Procedia Structural Integrity Structural Integrity 0000(2018) (2016) 000–000
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ECF22 - Loading and Environmental effects on Structural Integrity ECF22 - Loading and Environmental effects on Structural Integrity
Biaxial Biaxial experiments experiments on on the the effect effect of of non-proportional non-proportional loading loading paths paths on fracture of XV Portuguese Conference and on Fracture, PCF behavior 2016, 10-12 February 2016,metals Paço de Arcos, Portugal on damage damage and fracture behavior of ductile ductile metals a Moritz Zistla,∗ unigaa a,∗, Steffen Gerkea , Michael Br¨ Moritz Zistl , Steffen Gerke , Michael Br¨ unig Thermo-mechanical modeling of a high pressure turbine a
blade of an
Institut f¨ur Mechanik und Statik, Universit¨at der Bundeswehr M¨unchen, Werner-Heisenberg-Weg 39, 85579 Neubiberg, Germany f¨ur Mechanik und Statik, Universit¨at der Bundeswehr M¨unchen, Werner-Heisenberg-Weg 39, 85579 Neubiberg, Germany
a Institut
airplane gas turbine engine
Abstract P. Brandãoa, V. Infanteb, A.M. Deusc* Abstract The paper deals with a series of new experiments to study the effect of non-proportional loading paths on damage and fracture a Department of aMechanical Instituto Técnico, Universidade de Lisboa,loading Av. Rovisco Pais, 1,damage 1049-001 Lisboa, The paper with series of Engineering, new experiments toSuperior study the effect of non-proportional paths onmodel and fracture behavior ofdeals ductile metals. In this context, a thermodynamically consistent anisotropic continuum damage is presented. It Portugal behavior of ductile metals. In this context, a thermodynamically consistent anisotropic continuum damage model is presented. It b takesIDMEC, into account the effect of stress Engineering, state on damage conditions well asUniversidade on the evolution of damage strains. branches Department of Mechanical Instituto Superioras Técnico, de Lisboa, Av. Rovisco Pais,Different 1, 1049-001 Lisboa, takes into account the effect of stress state on damage conditions as well as on the evolution of damage strains. Different branches of the damage criteria corresponding to various ductile damage Portugal and fracture mechanisms depending on stress state are considered. c of damageDepartment criteria corresponding toEngineering, various ductile andTécnico, fracture mechanisms on stressPais, state are considered. CeFEMA, of Mechanical Instituto Superior Universidade dedepending Lisboa, Av. parameters Rovisco 1049-001 Lisboa, Thethe two-dimensionally loaded X0-specimen covering adamage wide range of stress triaxialities and Lode in1,the tension and The two-dimensionally loaded X0-specimen covering a wide range of stress triaxialities and Lode parameters in the tension and Portugal shear stress domains is being used. These tests are driven under different non-proportional loading paths. The formation of strain shear stress domains is being used. These tests are driven under different non-proportional loading paths. The formation of strain fields of the specimens is recorded by digital image correlation technique. Furthermore, scanning electron microscope analysis of fields of the surfaces specimens is recorded by digital image correlation technique. Furthermore, scanning electron microscope analysis of the fracture clearly shows various failure modes corresponding to these loading conditions. theAbstract fracture surfaces clearly shows various failure modes corresponding to these loading conditions. 2018The TheAuthors. Authors. Published by Elsevier ©c During 2018 Published by Elsevier B.V. B.V. their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, c 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. Peer-review under responsibility of the ECF22 organizers. especially under the high pressure turbine Such conditions cause these parts to undergo different types of time-dependent Peer-review responsibility of the(HPT) ECF22blades. organizers. degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict Keywords: Keywords: the creep behaviour ofexperiments; HPT blades. Flight data loading recordspaths; (FDR) forimage a specific aircraft, provided a commercial aviation Damage and fracture; biaxial non-proportional digital correlation; scanning electronby microscopy Damage and fracture; biaxial non-proportional loadingdata paths; correlation; company, were used to experiments; obtain thermal and mechanical fordigital threeimage different flightscanning cycles. electron In ordermicroscopy to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a 1. overall Introduction 1. model Introduction can be useful in the goal of predicting turbine blade life, given a set of FDR data.
Accurate and realistic modeling of the deformation and fracture behavior of ductile sheet metals is of interest ©Accurate 2016 The and Authors. Published by Elsevier realistic modeling of theB.V. deformation and fracture behavior of ductile sheet metals is of interest inPeer-review several engineering disciplines. With increasing inelastic deformations damage and fracture mechanisms occur in under responsibility of the Scientific Committee of PCF 2016. in several engineering disciplines. With increasing inelastic deformations damage and fracture mechanisms occur in the material. It can be noted that they depend on the stress state (Bao and Wierzbicki (2004); Gao et al. (2010)). the material. It can be noted that they depend on the stress state (Bao and Wierzbicki (2004); Gao et al. (2010)). Under tension stress conditions damage ductile metalsSimulation. is mainly caused by nucleation, growth and Keywords: High dominated Pressure Turbine Blade; Creep; Finite Element in Method; 3D Model; Under tension dominated stress conditions damage in ductile metals is mainly caused by nucleation, growth and coalescence of voids whereas the formation of micro-shear-cracks is the predominant damage mechanism under shear coalescence of voids whereas the formation of micro-shear-cracks is the predominant damage mechanism under shear and compression dominated stress states. In addition, the damage behavior depends as well strongly on the loading and compression dominated stress states. In addition, the damage behavior depends as well strongly on the loading path of the material sample. Therefore, this path dependency has to be studied experimentally. In the present paper, a path of the material sample. Therefore, this path dependency has to be studied experimentally. In the present paper, a continuum damage model using functions for different damage modes will be presented and detailed results of biaxial continuum damage model using functions for different damage modes will be presented and detailed results of biaxial experiments presented and discussed. Biaxial experiments with the recently developed cruciform X0-specimen will experiments presented and discussed. Biaxial experiments with the recently developed cruciform X0-specimen will be shown with focus on the effect of non-proportional loading paths. Digital image correlation technique is used be shown with focus on the effect of non-proportional loading paths. Digital image correlation technique is used Corresponding author. Tel.: +351 218419991. ∗*Corresponding author. Tel.: +49-89-6004-3413 ∗ Corresponding E-mail address: [email protected] author. Tel.: +49-89-6004-3413
; fax: +49-89-6004-4549. ; fax: +49-89-6004-4549. E-mail address: [email protected] E-mail address: [email protected] 2452-3216 2016 The Authors. Published Elsevier B.V. c© 2210-7843 2018 The Authors. Published byby Elsevier B.V. cunder 2210-7843 2018 The responsibility Authors.of Published by organizers. Elsevier B.V. Peer-review under of the Scientific Committee of PCF 2016. Peer-review responsibility the ECF22 Peer-review under responsibility of the ECF22 organizers. 2452-3216 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. 10.1016/j.prostr.2018.12.010
58 2
Moritz Zistl et al. / Procedia Structural Integrity 13 (2018) 57–62 M. Zistl et al. / Structural Integrity Procedia 00 (2018) 000–000
to analyze the formation of strain fields in critical notched regions of the specimen where damage and fracture are expected to localize. Fracture mechanisms on the micro-scale are visualized by scanning electron microscope analysis of fracture surfaces. 2. Continuum damage model Large inelastic deformations and the evolution of damage in ductile metals can be modeled by the continuum framework presented by Br¨unig (2003). It introduces damaged and fictitious undamaged configurations. The thermodynamically consistent framework is based on kinematic description of damage leading to the definition of damage strain tensors. The effective undamaged configurations are considered to formulate equations modeling elastic-plastic behavior of the undamaged matrix material. The onset of damage is described by the damage criterion f da = αI1 + β J2 − σ = 0.
(1)
It is formulated in terms of the stress invariants I1 = trT and J2 = 1/2 devT·devT of the Kirchhoff stress tensor T as well as in the damage threshold σ. The damage mode parameters α and β correspond to different stress-statedependent damage mechanisms on the micro-level and depend on the stress triaxiality and the Lode parameter (Br¨unig et al. (2013)). In addition, the evolution of macroscopic deformations of the material caused by stress-state-dependent damage processes acting on the micro-scale are described by the damage strain rate tensor 1 da ˙ ¯ H = µ˙ α¯ √ 1 + β N . 3
(2)
√ ˜ is the normalized deviatoric stress tensor. The stress-state-dependent parameters α¯ and β¯ Here N = 1/(2 J2 ) devT are also chosen to be functions of to the stress triaxiality and the Lode parameter, see Br¨unig et al. (2013) for further ¯ are kinematic variables describing the portion of volumetric details. These stress-state-dependent parameters (α¯ and β) and isochoric damage-based deformations and µ˙ represents the equivalent damage strain rate measure characterizing √ the amount of increase in damage. The damage rule (2) takes into account a volumetric part (α¯ 1/ 3 1) corresponding to isotropic growth of voids as well as a deviatoric part (β¯ N) corresponding to anisotropic evolution of micro-shearcracks. Therefore, both basic damage mechanisms acting on the micro-level can be modeled by the proposed damage rule (2).
3. Biaxial experiments with the X0-specimen The experimental program has been developed by Gerke et al. (2017) to propose two-dimensional tests revealing the effect of stress state on inelastic behavior in ductile metals. The experiments are performed using the biaxial test machine type LFM-BIAX 20 kN shown in Fig. 1. The specimens are loaded by four individually driven cylinders with loading up to ± 20 kN. The cruciform X0specimen (Fig. 2) is characterized by crosswise arranged notches with a central opening. The investigated material is the aluminum alloy AlSiMgMn EN AW-6082. Specimens are taken from sheets with 4 mm thickness. In the center of this specimen four notches in thickness direction have been milled (b) which will lead to high stresses and the localization of irreversible deformations in these regions. The notched parts have the length of 6 mm and their reduced thickness is 2 mm whereas the notch radii are 3 mm in plane (c) and 2 mm in thickness (d) directions, respectively. The loading conditions are shown in Fig. 3. The specimen will be separately loaded in two directions with F1 and F2 . First results with this specimen have been presented by Gerke et al. (2018). There for different loading ratios damage and fracture mechanisms caused by proportional and one non-proportional loading path are considered and the results are compared and discussed.
Moritz Zistl et al. / Procedia Structural Integrity 13 (2018) 57–62 M. Zistl et al. / Structural Integrity Procedia 00 (2018) 000–000
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Figure 1. Biaxial testing machine without DIC system
(b)
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In the present paper, an additional path is presented with special focus on non-proportional loading scenarios and the experimental results will be compared with those based on previously presented loading histories. During the proportional experiments, the load ratio is kept constant during the entire tests. The non-proportional ones start with respective successive loadings and end after an axis-switch with the final proportional path up to final fracture guaranteeing the same final load ratios F1/F2 to obtain comparable results. The non-proportional scenario (NP 1/0s+1) investigated by Gerke et al. (2018) is first loading by only F1 (black arrows, loading ratio 1/0) followed by F2 (green arrow, 1/+1) whereas in the new, second scenario (NP 1/−1s+1) in the first step both axes are simultaneous loaded by F1 and F2 = −F1 (black arrows, 1/-1) followed by a switch in axis 2 with only loading by F2 up to F2 = F1 (green arrows, 1/+1) whereas F is kept constant, followed by the final proportional loading path up to final fracture, see Fig. 4. 1 During the biaxial experiments, the displacement of the specimens’ surface are analyzed by digital image correlation (DIC) technique. The displacements in 1-direction u1.1 and u1.2 are shown in Fig. 3. The relative movement of these points is given by ∆uref.1 = u1.1 − u1.2 and for the 2-direction accordingly. After the experiments scanning electron microscopy (SEM) of fracture surfaces is conducted to reveal the stress-state-dependent ductile damage and fracture processes on the micro-level.
Moritz Zistl et al. / Procedia Structural Integrity 13 (2018) 57–62 M. Zistl et al. / Structural Integrity Procedia 00 (2018) 000–000
60 4
NP 1/0s+1 2 F2.1
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1 F1.1
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F1.2
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Figure 3. Definition of displacements and loading conditions of the X0-specimen
8.00
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Figure 4. Force F1 vs. force F2 ; solid lines proportional and dashed lines Figure 5. Load-displacement curves for the final load ratio F1 = F2 = 1/+1 non-proportional experiments
4. Experimental results In the present paper, experimental results of different loading paths for the representative final load ratio F1/F2 = 1/+1 (tension loading) have been selected and will be discussed in detail. With this load ratio experimental data for different stress histories will be obtained leading to different damage and fracture mechanisms on the micro-level. Focus will be on the effect of the loading path on deformation, damage and fracture behavior. Therefore the test results for the biaxially loaded X0-specimen obtained from proportional loading paths (P) will be compared with those based on non-proportional ones (NP) and among themselves. Load-displacement curves for the proportional and two different non-proportional loading paths are shown in Fig. 5. In particular, in the case of proportional loading of the X0-specimen the maximum load is F1 = 7.3 kN and the displacements at final fracture are ∆uref.1 = 0.27 mm in axis 1. For the non-proportional loading path 1/0s+1 the maximum load is 3% higher and the displacement at fracture in axis 1 is 44% larger compared to the proportional loading path whereas a decrease in final displacement in axis 2 of 63% has been observed in the biaxial tests. For the case of nonproportional loading 1/−1s+1 a slightly bigger increase in maximum load (5%) has been measured and the displacement at fracture in axis 1 is 22% larger compared to the proportional loading path whereas the displacement at fracture in axis 2 is 75% smaller. Furthermore, Fig. 6 shows the principal strains in the top notched part of the X0-specimen shortly before final fracture occurs. In particular, the first principal strain based on the proportional loading path develops in broad strain bands with elliptic shaped strain maxima of 15%. For the non-proportional path (1/0s+1) in similar broad strain bands
Moritz Zistl et al. / Procedia Structural Integrity 13 (2018) 57–62 M. Zistl et al. / Structural Integrity Procedia 00 (2018) 000–000
NP 1/0s+1
P
0.15
61 5
NP 1/-1s+1
0.00 -0.08
0.00 Figure 6. Principal strains in the upper notched part of the X0-specimen for the final load ratio F1 = F2 = 1/+1
NP 1/-1s+1
NP 1/0s+1
P
1
1
1
1
1
1
1
1
1
1 1
1
1
1
Figure 7. Fracture modes for the final load ratio F1 = F2 = 1/+1
with maxima now in a tilted band of strain with values about 15%. In the case of the alternative non-proportional loading (1/−1s+1) the principal strains are more localized (the band is smaller) and only maximum values of about 10% have been reached in the experiments. For the second principal strains the localized bands are again widely spread with about -5% for proportional path. For the non-proportional (1/0s+1) paths greater absolute values can be seen, again superimposed with a tilted band and maximum values of around -8%. For the non-proportional case (1/−1s+1) only smaller and more localized strains developed with minima of about -5%. The localized bands of principal strains illustrated in Fig. 6 differ slightly from the fracture lines shown in Fig. 7. In all cases nearly orthogonal orientated fracture lines can be observed forming a relief-like plane and therefore no remarkable influence of the loading paths on these results can be noted. Ultimately, detailed analysis of the fracture surfaces by scanning electron microscopy (SEM) was carried out, see Fig. 8. In the case of the proportional loading path tension loading occurred in the critical notched parts of the X0-specimen causing near hydrostatic tension stress states with the corresponding high positive triaxiality. On the microscale this leads to the growth of voids with very big voids. During the non-proportional loading path (1/0s+1) the critical notched parts of the X0-specimen are firstly loaded by tension-shear and after the change in the loading path (see Fig. 4) by superimposed tension. This leads in the first loading stage to simultaneous growth of voids and formation of micro-shear-cracks. During the final proportional part of the loading history (see Fig. 4) the growth of voids continues. The fracture mode for the non-proportional loading path (1/0s+1) is characterized by voids and slightly smeared-out dimples. Compared to the proportional loading history, slightly smaller and more voids are visualized by the SEM analysis. Furthermore, during the non-proportional loading path (1/−1s+1) the notched region of the X0-
Moritz Zistl et al. / Procedia Structural Integrity 13 (2018) 57–62 M. Zistl et al. / Structural Integrity Procedia 00 (2018) 000–000
62 6 P
NP 1/-1s+1
NP 1/0s+1
20 µm
20 µm
20 µm
Figure 8. Fracture surfaces by scanning electron microscopy for the final load ratio F1 = F2 = 1/+1
specimen is firstly deformed by the shear which is then continuously replaced by tension loading. During shear loading formation of micro-shear cracks occurs which is then, after the change in the loading path (see Fig. 4), superimposed by tension leading to additional growth of voids. Compared to the proportional loading path, the voids in the fracture surface are remarkably smaller in size and number. Therefore, the SEM analysis clearly demonstrates that the loading path remarkably affects the damage and fracture processes on the micro-scale and the mechanisms occurring firstly are the predominate ones in the final fracture process. 5. Conclusions In this paper a series of experiments with the biaxially loaded X0-specimen has been presented. The experiments have shown the effect of the loading path on damage and fracture mechanisms of ductile metals. These mechanisms are considered in the presented phenomenological continuum model, which is based on different branches of ductile damage and fracture criteria corresponding to various stress-state-dependent mechanisms. In the notched parts of the specimens shear and tension behavior occurs during loading leading to different localized deformations. The SEM analyses revealed various stress-state-dependent damage and fracture mechanisms in the specimen during the deformation path. In addition, proportional and different non-proportional loading paths lead to different failure processes on the micro- and the macro-scale demonstrating the remarkable effect of loading histories on evolution of damage leading to final fracture. The results will be used to validate the proposed continuum damage model. Acknowledgements The project has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project number 322157331, this financial support is greatfully acknowledged. The SEM images of the fracture surfaces presented in this paper were performed at the Institut f¨ur Werkstoffe im Bauwesen, Bundeswehr University Munich and the support of Wolfgang Saur greatfully acknowledged. References Y. Bao and T. Wierzbicki, 2004. On the fracture locus in the equivalent strain and stress triaxiality space. International Journal of Mechanical Sciences, Vol. 46, 81–98. M. Br¨unig, 2003. An anisotropic ductile damage model based on irreversible thermodynamics. International Journal of Plasticity, Vol. 19, 1679– 1713, 2003. M. Br¨unig and S. Gerke and V. Hagenbrock, 2013. Micro-mechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage. International Journal of Plasticity, Vol. 50, 49–65, 2013. X. Gao, G. Zhang, C. Roe, 2010. A study on the effect of the stress state on ductile fracture. International Journal of Damage Mechanics 19, 75–94, 2010. S. Gerke, P. Adulyasak and M. Br¨unig, 2017. New biaxially loaded specimens for the analysis of damage and fracture in sheet metals. International Journal of Solids and Structures, Vol. 110, 209–218. S. Gerke, M. Zistl, A. Bhardwaj, M. Br¨unig, 2018. Experiments with the X0-specimen on the effect of non-proportional loading paths on damage and fracture mechanisms in aluminum alloys. International Journal of Solids and Structures, (submitted).
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ECF22 - Loading and Environmental effects on Structural Integrity ECF22 - Loading and Environmental effects on Structural Integrity
Biaxial Biaxial experiments experiments on on the the effect effect of of non-proportional non-proportional loading loading paths paths on fracture of XV Portuguese Conference and on Fracture, PCF behavior 2016, 10-12 February 2016,metals Paço de Arcos, Portugal on damage damage and fracture behavior of ductile ductile metals a Moritz Zistla,∗ unigaa a,∗, Steffen Gerkea , Michael Br¨ Moritz Zistl , Steffen Gerke , Michael Br¨ unig Thermo-mechanical modeling of a high pressure turbine a
blade of an
Institut f¨ur Mechanik und Statik, Universit¨at der Bundeswehr M¨unchen, Werner-Heisenberg-Weg 39, 85579 Neubiberg, Germany f¨ur Mechanik und Statik, Universit¨at der Bundeswehr M¨unchen, Werner-Heisenberg-Weg 39, 85579 Neubiberg, Germany
a Institut
airplane gas turbine engine
Abstract P. Brandãoa, V. Infanteb, A.M. Deusc* Abstract The paper deals with a series of new experiments to study the effect of non-proportional loading paths on damage and fracture a Department of aMechanical Instituto Técnico, Universidade de Lisboa,loading Av. Rovisco Pais, 1,damage 1049-001 Lisboa, The paper with series of Engineering, new experiments toSuperior study the effect of non-proportional paths onmodel and fracture behavior ofdeals ductile metals. In this context, a thermodynamically consistent anisotropic continuum damage is presented. It Portugal behavior of ductile metals. In this context, a thermodynamically consistent anisotropic continuum damage model is presented. It b takesIDMEC, into account the effect of stress Engineering, state on damage conditions well asUniversidade on the evolution of damage strains. branches Department of Mechanical Instituto Superioras Técnico, de Lisboa, Av. Rovisco Pais,Different 1, 1049-001 Lisboa, takes into account the effect of stress state on damage conditions as well as on the evolution of damage strains. Different branches of the damage criteria corresponding to various ductile damage Portugal and fracture mechanisms depending on stress state are considered. c of damageDepartment criteria corresponding toEngineering, various ductile andTécnico, fracture mechanisms on stressPais, state are considered. CeFEMA, of Mechanical Instituto Superior Universidade dedepending Lisboa, Av. parameters Rovisco 1049-001 Lisboa, Thethe two-dimensionally loaded X0-specimen covering adamage wide range of stress triaxialities and Lode in1,the tension and The two-dimensionally loaded X0-specimen covering a wide range of stress triaxialities and Lode parameters in the tension and Portugal shear stress domains is being used. These tests are driven under different non-proportional loading paths. The formation of strain shear stress domains is being used. These tests are driven under different non-proportional loading paths. The formation of strain fields of the specimens is recorded by digital image correlation technique. Furthermore, scanning electron microscope analysis of fields of the surfaces specimens is recorded by digital image correlation technique. Furthermore, scanning electron microscope analysis of the fracture clearly shows various failure modes corresponding to these loading conditions. theAbstract fracture surfaces clearly shows various failure modes corresponding to these loading conditions. 2018The TheAuthors. Authors. Published by Elsevier ©c During 2018 Published by Elsevier B.V. B.V. their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, c 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. Peer-review under responsibility of the ECF22 organizers. especially under the high pressure turbine Such conditions cause these parts to undergo different types of time-dependent Peer-review responsibility of the(HPT) ECF22blades. organizers. degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict Keywords: Keywords: the creep behaviour ofexperiments; HPT blades. Flight data loading recordspaths; (FDR) forimage a specific aircraft, provided a commercial aviation Damage and fracture; biaxial non-proportional digital correlation; scanning electronby microscopy Damage and fracture; biaxial non-proportional loadingdata paths; correlation; company, were used to experiments; obtain thermal and mechanical fordigital threeimage different flightscanning cycles. electron In ordermicroscopy to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a 1. overall Introduction 1. model Introduction can be useful in the goal of predicting turbine blade life, given a set of FDR data.
Accurate and realistic modeling of the deformation and fracture behavior of ductile sheet metals is of interest ©Accurate 2016 The and Authors. Published by Elsevier realistic modeling of theB.V. deformation and fracture behavior of ductile sheet metals is of interest inPeer-review several engineering disciplines. With increasing inelastic deformations damage and fracture mechanisms occur in under responsibility of the Scientific Committee of PCF 2016. in several engineering disciplines. With increasing inelastic deformations damage and fracture mechanisms occur in the material. It can be noted that they depend on the stress state (Bao and Wierzbicki (2004); Gao et al. (2010)). the material. It can be noted that they depend on the stress state (Bao and Wierzbicki (2004); Gao et al. (2010)). Under tension stress conditions damage ductile metalsSimulation. is mainly caused by nucleation, growth and Keywords: High dominated Pressure Turbine Blade; Creep; Finite Element in Method; 3D Model; Under tension dominated stress conditions damage in ductile metals is mainly caused by nucleation, growth and coalescence of voids whereas the formation of micro-shear-cracks is the predominant damage mechanism under shear coalescence of voids whereas the formation of micro-shear-cracks is the predominant damage mechanism under shear and compression dominated stress states. In addition, the damage behavior depends as well strongly on the loading and compression dominated stress states. In addition, the damage behavior depends as well strongly on the loading path of the material sample. Therefore, this path dependency has to be studied experimentally. In the present paper, a path of the material sample. Therefore, this path dependency has to be studied experimentally. In the present paper, a continuum damage model using functions for different damage modes will be presented and detailed results of biaxial continuum damage model using functions for different damage modes will be presented and detailed results of biaxial experiments presented and discussed. Biaxial experiments with the recently developed cruciform X0-specimen will experiments presented and discussed. Biaxial experiments with the recently developed cruciform X0-specimen will be shown with focus on the effect of non-proportional loading paths. Digital image correlation technique is used be shown with focus on the effect of non-proportional loading paths. Digital image correlation technique is used Corresponding author. Tel.: +351 218419991. ∗*Corresponding author. Tel.: +49-89-6004-3413 ∗ Corresponding E-mail address: [email protected] author. Tel.: +49-89-6004-3413
; fax: +49-89-6004-4549. ; fax: +49-89-6004-4549. E-mail address: [email protected] E-mail address: [email protected] 2452-3216 2016 The Authors. Published Elsevier B.V. c© 2210-7843 2018 The Authors. Published byby Elsevier B.V. cunder 2210-7843 2018 The responsibility Authors.of Published by organizers. Elsevier B.V. Peer-review under of the Scientific Committee of PCF 2016. Peer-review responsibility the ECF22 Peer-review under responsibility of the ECF22 organizers. 2452-3216 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. 10.1016/j.prostr.2018.12.010
58 2
Moritz Zistl et al. / Procedia Structural Integrity 13 (2018) 57–62 M. Zistl et al. / Structural Integrity Procedia 00 (2018) 000–000
to analyze the formation of strain fields in critical notched regions of the specimen where damage and fracture are expected to localize. Fracture mechanisms on the micro-scale are visualized by scanning electron microscope analysis of fracture surfaces. 2. Continuum damage model Large inelastic deformations and the evolution of damage in ductile metals can be modeled by the continuum framework presented by Br¨unig (2003). It introduces damaged and fictitious undamaged configurations. The thermodynamically consistent framework is based on kinematic description of damage leading to the definition of damage strain tensors. The effective undamaged configurations are considered to formulate equations modeling elastic-plastic behavior of the undamaged matrix material. The onset of damage is described by the damage criterion f da = αI1 + β J2 − σ = 0.
(1)
It is formulated in terms of the stress invariants I1 = trT and J2 = 1/2 devT·devT of the Kirchhoff stress tensor T as well as in the damage threshold σ. The damage mode parameters α and β correspond to different stress-statedependent damage mechanisms on the micro-level and depend on the stress triaxiality and the Lode parameter (Br¨unig et al. (2013)). In addition, the evolution of macroscopic deformations of the material caused by stress-state-dependent damage processes acting on the micro-scale are described by the damage strain rate tensor 1 da ˙ ¯ H = µ˙ α¯ √ 1 + β N . 3
(2)
√ ˜ is the normalized deviatoric stress tensor. The stress-state-dependent parameters α¯ and β¯ Here N = 1/(2 J2 ) devT are also chosen to be functions of to the stress triaxiality and the Lode parameter, see Br¨unig et al. (2013) for further ¯ are kinematic variables describing the portion of volumetric details. These stress-state-dependent parameters (α¯ and β) and isochoric damage-based deformations and µ˙ represents the equivalent damage strain rate measure characterizing √ the amount of increase in damage. The damage rule (2) takes into account a volumetric part (α¯ 1/ 3 1) corresponding to isotropic growth of voids as well as a deviatoric part (β¯ N) corresponding to anisotropic evolution of micro-shearcracks. Therefore, both basic damage mechanisms acting on the micro-level can be modeled by the proposed damage rule (2).
3. Biaxial experiments with the X0-specimen The experimental program has been developed by Gerke et al. (2017) to propose two-dimensional tests revealing the effect of stress state on inelastic behavior in ductile metals. The experiments are performed using the biaxial test machine type LFM-BIAX 20 kN shown in Fig. 1. The specimens are loaded by four individually driven cylinders with loading up to ± 20 kN. The cruciform X0specimen (Fig. 2) is characterized by crosswise arranged notches with a central opening. The investigated material is the aluminum alloy AlSiMgMn EN AW-6082. Specimens are taken from sheets with 4 mm thickness. In the center of this specimen four notches in thickness direction have been milled (b) which will lead to high stresses and the localization of irreversible deformations in these regions. The notched parts have the length of 6 mm and their reduced thickness is 2 mm whereas the notch radii are 3 mm in plane (c) and 2 mm in thickness (d) directions, respectively. The loading conditions are shown in Fig. 3. The specimen will be separately loaded in two directions with F1 and F2 . First results with this specimen have been presented by Gerke et al. (2018). There for different loading ratios damage and fracture mechanisms caused by proportional and one non-proportional loading path are considered and the results are compared and discussed.
Moritz Zistl et al. / Procedia Structural Integrity 13 (2018) 57–62 M. Zistl et al. / Structural Integrity Procedia 00 (2018) 000–000
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Figure 1. Biaxial testing machine without DIC system
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In the present paper, an additional path is presented with special focus on non-proportional loading scenarios and the experimental results will be compared with those based on previously presented loading histories. During the proportional experiments, the load ratio is kept constant during the entire tests. The non-proportional ones start with respective successive loadings and end after an axis-switch with the final proportional path up to final fracture guaranteeing the same final load ratios F1/F2 to obtain comparable results. The non-proportional scenario (NP 1/0s+1) investigated by Gerke et al. (2018) is first loading by only F1 (black arrows, loading ratio 1/0) followed by F2 (green arrow, 1/+1) whereas in the new, second scenario (NP 1/−1s+1) in the first step both axes are simultaneous loaded by F1 and F2 = −F1 (black arrows, 1/-1) followed by a switch in axis 2 with only loading by F2 up to F2 = F1 (green arrows, 1/+1) whereas F is kept constant, followed by the final proportional loading path up to final fracture, see Fig. 4. 1 During the biaxial experiments, the displacement of the specimens’ surface are analyzed by digital image correlation (DIC) technique. The displacements in 1-direction u1.1 and u1.2 are shown in Fig. 3. The relative movement of these points is given by ∆uref.1 = u1.1 − u1.2 and for the 2-direction accordingly. After the experiments scanning electron microscopy (SEM) of fracture surfaces is conducted to reveal the stress-state-dependent ductile damage and fracture processes on the micro-level.
Moritz Zistl et al. / Procedia Structural Integrity 13 (2018) 57–62 M. Zistl et al. / Structural Integrity Procedia 00 (2018) 000–000
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Figure 4. Force F1 vs. force F2 ; solid lines proportional and dashed lines Figure 5. Load-displacement curves for the final load ratio F1 = F2 = 1/+1 non-proportional experiments
4. Experimental results In the present paper, experimental results of different loading paths for the representative final load ratio F1/F2 = 1/+1 (tension loading) have been selected and will be discussed in detail. With this load ratio experimental data for different stress histories will be obtained leading to different damage and fracture mechanisms on the micro-level. Focus will be on the effect of the loading path on deformation, damage and fracture behavior. Therefore the test results for the biaxially loaded X0-specimen obtained from proportional loading paths (P) will be compared with those based on non-proportional ones (NP) and among themselves. Load-displacement curves for the proportional and two different non-proportional loading paths are shown in Fig. 5. In particular, in the case of proportional loading of the X0-specimen the maximum load is F1 = 7.3 kN and the displacements at final fracture are ∆uref.1 = 0.27 mm in axis 1. For the non-proportional loading path 1/0s+1 the maximum load is 3% higher and the displacement at fracture in axis 1 is 44% larger compared to the proportional loading path whereas a decrease in final displacement in axis 2 of 63% has been observed in the biaxial tests. For the case of nonproportional loading 1/−1s+1 a slightly bigger increase in maximum load (5%) has been measured and the displacement at fracture in axis 1 is 22% larger compared to the proportional loading path whereas the displacement at fracture in axis 2 is 75% smaller. Furthermore, Fig. 6 shows the principal strains in the top notched part of the X0-specimen shortly before final fracture occurs. In particular, the first principal strain based on the proportional loading path develops in broad strain bands with elliptic shaped strain maxima of 15%. For the non-proportional path (1/0s+1) in similar broad strain bands
Moritz Zistl et al. / Procedia Structural Integrity 13 (2018) 57–62 M. Zistl et al. / Structural Integrity Procedia 00 (2018) 000–000
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with maxima now in a tilted band of strain with values about 15%. In the case of the alternative non-proportional loading (1/−1s+1) the principal strains are more localized (the band is smaller) and only maximum values of about 10% have been reached in the experiments. For the second principal strains the localized bands are again widely spread with about -5% for proportional path. For the non-proportional (1/0s+1) paths greater absolute values can be seen, again superimposed with a tilted band and maximum values of around -8%. For the non-proportional case (1/−1s+1) only smaller and more localized strains developed with minima of about -5%. The localized bands of principal strains illustrated in Fig. 6 differ slightly from the fracture lines shown in Fig. 7. In all cases nearly orthogonal orientated fracture lines can be observed forming a relief-like plane and therefore no remarkable influence of the loading paths on these results can be noted. Ultimately, detailed analysis of the fracture surfaces by scanning electron microscopy (SEM) was carried out, see Fig. 8. In the case of the proportional loading path tension loading occurred in the critical notched parts of the X0-specimen causing near hydrostatic tension stress states with the corresponding high positive triaxiality. On the microscale this leads to the growth of voids with very big voids. During the non-proportional loading path (1/0s+1) the critical notched parts of the X0-specimen are firstly loaded by tension-shear and after the change in the loading path (see Fig. 4) by superimposed tension. This leads in the first loading stage to simultaneous growth of voids and formation of micro-shear-cracks. During the final proportional part of the loading history (see Fig. 4) the growth of voids continues. The fracture mode for the non-proportional loading path (1/0s+1) is characterized by voids and slightly smeared-out dimples. Compared to the proportional loading history, slightly smaller and more voids are visualized by the SEM analysis. Furthermore, during the non-proportional loading path (1/−1s+1) the notched region of the X0-
Moritz Zistl et al. / Procedia Structural Integrity 13 (2018) 57–62 M. Zistl et al. / Structural Integrity Procedia 00 (2018) 000–000
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Figure 8. Fracture surfaces by scanning electron microscopy for the final load ratio F1 = F2 = 1/+1
specimen is firstly deformed by the shear which is then continuously replaced by tension loading. During shear loading formation of micro-shear cracks occurs which is then, after the change in the loading path (see Fig. 4), superimposed by tension leading to additional growth of voids. Compared to the proportional loading path, the voids in the fracture surface are remarkably smaller in size and number. Therefore, the SEM analysis clearly demonstrates that the loading path remarkably affects the damage and fracture processes on the micro-scale and the mechanisms occurring firstly are the predominate ones in the final fracture process. 5. Conclusions In this paper a series of experiments with the biaxially loaded X0-specimen has been presented. The experiments have shown the effect of the loading path on damage and fracture mechanisms of ductile metals. These mechanisms are considered in the presented phenomenological continuum model, which is based on different branches of ductile damage and fracture criteria corresponding to various stress-state-dependent mechanisms. In the notched parts of the specimens shear and tension behavior occurs during loading leading to different localized deformations. The SEM analyses revealed various stress-state-dependent damage and fracture mechanisms in the specimen during the deformation path. In addition, proportional and different non-proportional loading paths lead to different failure processes on the micro- and the macro-scale demonstrating the remarkable effect of loading histories on evolution of damage leading to final fracture. The results will be used to validate the proposed continuum damage model. Acknowledgements The project has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project number 322157331, this financial support is greatfully acknowledged. The SEM images of the fracture surfaces presented in this paper were performed at the Institut f¨ur Werkstoffe im Bauwesen, Bundeswehr University Munich and the support of Wolfgang Saur greatfully acknowledged. References Y. Bao and T. Wierzbicki, 2004. On the fracture locus in the equivalent strain and stress triaxiality space. International Journal of Mechanical Sciences, Vol. 46, 81–98. M. Br¨unig, 2003. An anisotropic ductile damage model based on irreversible thermodynamics. International Journal of Plasticity, Vol. 19, 1679– 1713, 2003. M. Br¨unig and S. Gerke and V. Hagenbrock, 2013. Micro-mechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage. International Journal of Plasticity, Vol. 50, 49–65, 2013. X. Gao, G. Zhang, C. Roe, 2010. A study on the effect of the stress state on ductile fracture. International Journal of Damage Mechanics 19, 75–94, 2010. S. Gerke, P. Adulyasak and M. Br¨unig, 2017. New biaxially loaded specimens for the analysis of damage and fracture in sheet metals. International Journal of Solids and Structures, Vol. 110, 209–218. S. Gerke, M. Zistl, A. Bhardwaj, M. Br¨unig, 2018. Experiments with the X0-specimen on the effect of non-proportional loading paths on damage and fracture mechanisms in aluminum alloys. International Journal of Solids and Structures, (submitted).