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The Effect of Negative Stress Triaxialities on Ductile Damage and Fracture Behavior in Metal Sheets
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Structural Integrity Procedia 00 (2018) 000–000 Structural IntegrityProcedia Procedia13 (2018)91–96 000–000 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
The The Effect Effect of of Negative Negative Stress Stress Triaxialities Triaxialities on on Ductile Ductile Damage Damage and and Fracture Behavior in Metal Sheets XV Portuguese Conference on Fracture, PCF 2016, February 2016, Paço de Arcos, Portugal Fracture Behavior in 10-12 Metal Sheets a,∗ a a Marco Schmidt u nig a,∗, Steffen Gerkea , Michael Br¨ a Marco Schmidt , Steffen Gerke , Michael Br¨ u nig Thermo-mechanical modeling of a high pressure turbine a Institut a Institut
blade of an
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
airplane gas turbine engine
Abstract P. Brandãoa, V. Infanteb, A.M. Deusc* Abstract The paper deals with an anisotropic continuum damage and fracture model and a series of biaxial experiments with focus on a Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av.biaxial Roviscoexperiments Pais, 1, 1049-001 Lisboa, The paper dealstriaxialities. with an anisotropic continuum model and series of with focusand on negative stress The continuum modeldamage is basedand on fracture the assumption that adifferent damage mechanisms are present Portugal negative stress triaxialities. The continuum model is based on the assumption that different damage mechanisms are present and b have IDMEC, to be taken into account depending on stress triaxiality andTécnico, Lode parameter. Therefore, onset and evaluation Department of Mechanical Engineering, Instituto Superior Universidade de Lisboa,modeling Av. RoviscoofPais, 1, 1049-001 Lisboa, have to be taken into account depending on stressdamage triaxiality and Lode parameter. Therefore, modeling of onset and evaluation Portugaland a stress-state-dependent damage rule. To identify the of damage are based on a stress-state-dependent condition c of damage are based onofabiaxial stress-state-dependent damage condition and Universidade aaluminum stress-state-dependent damage rule.1,To identify the CeFEMA, Department Mechanical Engineering, Instituto Superior Técnico, de Lisboa, Av.been Rovisco Pais, 1049-001 Lisboa, corresponding parameters experiments with specimens taken from sheets have performed and results of corresponding parameters biaxial experiments with specimens taken from aluminum sheets have been performed and results of Portugal corresponding numerical simulations are discussed in detail. The experimental behavior has been analyzed with a digital image corresponding numerical simulations are discussed in detail. The experimental behavior has been analyzed with a digital image correlation system to compare the strain fields with those obtained by numerical simulations. In addition, fracture modes are correlation to electron comparemicroscopy. the strain fields those obtained by In addition, fracturecut-off modes are detected by system scanning Basedwith on the experimental andnumerical numericalsimulations. results a stress-state-dependent value Abstract detected by scanning electron microscopy. Based on the experimental and numerical results a stress-state-dependent cut-off value for negative stress triaxialities is proposed. for negative stress triaxialities is proposed. During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, c especially 2018The TheAuthors. Authors. Published by Elsevier © 2018 Published byturbine Elsevier B.V. B.V. high pressure (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent c 2018 The the Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. Peer-review under responsibility ofcreep. the ECF22 organizers. degradation, one of which is A model using the finite element method (FEM) was developed, in order to be able to predict Peer-review under responsibility of the ECF22 organizers. the creep behaviour of HPT blades. Flight (FDR) for a specific aircraft, provided by a commercial aviation Keywords: Continuum damage model; Ductile metal data sheets;records Stress-state-dependence; Experiments; Numerical simulations company,Continuum were useddamage to obtain thermal data for three different flight Numerical cycles. Insimulations order to create the 3D model model; Ductile and metalmechanical sheets; Stress-state-dependence; Experiments; Keywords: 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.
Currently big effort is made to use all capacities of materials in order to save resources and costs. This requires more ©Currently 2016 The big Authors. byuse Elsevier B.V. effortPublished is made to all capacities of materials in order to save resources and costs. This requires more in-depth knowledge of the material andScientific more sophisticated models to reflect its behavior sufficiently accurate Peer-review under responsibility of the Committee ofmaterial PCF 2016. in-depth knowledge of the material and more sophisticated material models to reflect its behavior sufficiently accurate and to ensure safety at the same time. In this context focus is given on the deterioration behavior of ductile metals and to ensure safety at the same time. In this context focus is given on the deterioration behavior of ductile metals which are ofHigh interest inTurbine several engineering disciplines. Thus,3D corresponding phenomenological material models have Keywords: Pressure Creep; Finite Element Method; Model; Simulation. which are of interest in severalBlade; engineering disciplines. Thus, corresponding phenomenological material models have to be capable to reflect besides the elastic-plastic behavior also the damage and failure processes. Especially the to be capable to reflect besides the elastic-plastic behavior also the damage and failure processes. Especially the damage behavior strongly depends on the stress state: Void nucleation, growth and coalescence occur at high positive damage behavior strongly depends on the stress state: Void nucleation, growth and coalescence occur at high positive stress triaxialities whereas micro-shear cracks appear at stress triaxialities around zero and below. stress triaxialities whereas micro-shear cracks appear at stress triaxialities around zero and below. The influence of positive stress triaxialities on the damage and fracture behavior could be investigated with uniaxial The influence of positive stress triaxialities on the damage and fracture behavior could be investigated with uniaxial tension tests of unnotched and notched specimens, see for example Bao and Wierzbicki (2004), Br¨unig et al. (2008), tension tests of unnotched and notched specimens, see for example Bao and Wierzbicki (2004), Br¨unig et al. (2008), Gao et al. (2010) and Dunand and Mohr (2011). In addition, further uniaxial test specimens with special geometry Gao et al. (2010) and Dunand and Mohr (2011). In addition, further uniaxial test specimens with special geometry Corresponding author. Tel.: +351 218419991. ∗*Corresponding author. Tel.: +49-89-60043413 ; ∗ Corresponding E-mail address: [email protected] author. Tel.: +49-89-60043413 ;
fax: +49-89-60044549. fax: +49-89-60044549. 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.2018.12.016
Marco Schmidt et al. / Procedia Structural Integrity 13 (2018) 91–96 M. Schmidt et al. / Structural Integrity Procedia 00 (2018) 000–000
92 2
have been proposed to study shear mechanisms in critical regions and thus achieving stress triaxialities around zero, see Bao and Wierzbicki (2004), Br¨unig et al. (2008) and Driemeier et al. (2010). Unfortunately this type of specimens tends to rotations within the shear region leading to rather tension dominated stress states. Consequently, biaxial test specimens were developed, (Makinde et al. (1992), Kuwabara (2007), Gerke et al. (2017)) to cover a wide range of stress triaxialities with one specimen geometry. Due to the geometry of the test specimens shear tension as well as shear compression loads should be possible within a biaxial test setup. The presented paper outlines the basic concept of a phenomenological anisotropic damage and failure model. Biaxial experiments and corresponding numerical simulations reflect the influence of negative stress triaxialities on the damage behavior. The experimental evaluation is carried out by digital image correlation for the analysis of the strain fields and by scanning electron microscope for the investigations of the fracture surface. In conclusion, a cut-off value for negative stress triaxialities below which no damage occurs is discussed, see also Br¨unig et al. (2018). 2. Continuum damage and fracture model The anisotropic damage and failure model for ductile metals is based on the additive decomposition of the strain rate tensor into an elastic, a plastic and a damage part (Br¨unig (2003)). For this purpose, damaged and fictitious undamaged configurations are introduced, which are coupled by damage tensors. The plastic material behavior is characterized by a stress-state-dependent yield condition, which indicates the onset of plastic flow, and by a stress-state-dependent flow rule. A similar approach is used to model the damage behavior. The damage condition (1) f da = αI1 + β J2 − σ = 0
characterizes onset of damage. It is formulated in terms of the first and second deviatoric invariant of the Kirchhoff stress tensor, I1 and J2 , and the damage threshold σ. The parameters α and β in Eq. (1) depend on the stress triaxiality η=
σm I1 = √ σeq 3 3J2
where σm = I1 /3 means the hydrostatic stress and σeq = on the Lode parameter ω=
(2) √ 3J2 represents the von Mises equivalent stress as well as
2T 2 − T 1 − T 3 T1 − T3
(3)
with the principal stress components T 1 ≥ T 2 ≥ T 3 . The parameters α and β are based on numerical calculations of different single-pore models (Br¨unig et al. (2013)) as well as on results of biaxial experiments and associated numerical simulations (Br¨unig et al. (2016)). In particular, the parameter α is taken to be −0.15 for ηcut < η < 0 , (4) α(η) = 0.33 for η>0 where ηcut is the cut-off value for negative stress triaxialities below which no damage occurs. The parameter β(η, ω) = β0 (η, ω = 0) + βω (ω) ≥ 0
(5)
β0 (η) = −1.28η + 0.85
(6)
βω (ω) = −0.017ω3 − 0.065ω2 − 0.078ω
(7)
with
and
is a non-negative function depending on the stress triaxiality (Eq. (2)) and the Lode parameter (Eq. (3)). Besides the damage condition, the damage rule 1 da ˙ ¯ ¯ (8) H = µ˙ α¯ √ 1 + βN + δM 3
Marco Schmidt et al. / Procedia Structural Integrity 13 (2018) 91–96 M. Schmidt et al. / Structural Integrity Procedia 00 (2018) 000–000
93 3
(b) 1 F2 [kN ]
(a) F2
F1
F1
0.5 LF Exp. Sim. -3:1 -7:1
u2 [mm]
0
F2
0
0.4
0.8
1.2
Fig. 1. (a) Z-specimen with force directions; (b) Force-displacement-curves of F1 : F2 = −3 : 1 and F1 : F2 = −7 : 1.
describes the formation of macroscopic strains caused by growth of micro-defects on the micro-scale, where N=
1 √
2 J2
˜ devT
1 devS˜ M = devS˜
and
with
˜ ˜ − 2 J2 1 devS˜ = devTdev T 3
(9)
are normalized deviatoric stress-based tensors. In addition, the equivalent damage strain rate µ˙ characterizes the amount of irreversible damage strain rates, and the stress-state-dependent parameters a¯ , β¯ and δ¯ are kinematic variables characterizing the ratio between isotropic and anisotropic damage deformations. With 0 for ηcut < η ≤ 0 0.5714η for 0 < η ≤ 1.75 α(η) ¯ = (10) 1 for η > 1.75 the volumetric portion corresponding to evaluation of micro-voids is weighted, and with ¯ ω) = β¯ 0 (η) + fβ (η)β¯ ω (ω), β(η,
(11)
containing 0.87 0.97875 − 0.32625η β¯ 0 (η) = 0
for for for
ηcut < η ≤ 1/3 1/3 < η ≤ 3 , η> 3
fβ = −0.0252 + 0.0378η
and
1 − ω2 β¯ ω (ω) = 0
for for
ηcut < η ≤ 2/3 , η > 2/3
(12) (13)
(14)
here taken to be the isochoric portion corresponding to development of micro-shear-cracks is balanced. In this context the parameter δ¯ is neglected based on results of numerical simulations of biaxial experiments with different loading conditions (Br¨unig et al. (2016)). 3. Biaxial experiments and numerical simulations To validate the continuum damage model and to investigate the influence of negative stress triaxialities on ductile damage behavior, an experimental series with various biaxial load ratios is performed. The experiments are carried out with a biaxial testing machine that uses four independent electro-mechanically driven cylinders with a maximum load of ±20kN. The test specimen (Fig. 1(a)) is characterized by a tension-compression-axis (F1 ) and a shear-axis (F2 ). In addition, the central region has notches in two directions to localize the strains and damage. The used material is an aluminum alloy of the 2017 series and was selected for its pronounced ductile behavior.
Marco Schmidt et al. / Procedia Structural Integrity 13 (2018) 91–96 M. Schmidt et al. / Structural Integrity Procedia 00 (2018) 000–000
94 4 (a1)
(b1)
0.00
-0.45
0.28
(a2)
0.00
(b2)
0.00
0.26
-0.49
0.00
Fig. 2. (a) 1. Principal strain; (b) 2. Principal strain; (1) F1 : F2 = −3 : 1; (2) F1 : F2 = −7 : 1.
The influence of negative stress triaxialities on the damage behavior has been investigated experimentally by different, throughout the experiment constant loading ratios F1 : F2 . Shear loading caused by F2 is superimposed by compression loading caused by F1 i.e. with more elevated negative load factor F1 (from 0 to −8) more compression is superimposed which causes more negative stress triaxialities. The experimental evaluation was made with a digital image correlation system. The corresponding displacement measure u2 of the force-displacement-curves (see Fig. 1(b)) was extracted by two points on the surface of the specimen with a distance of 6mm whereas one evaluation point was on the right and one on the left side of the notch. The experimental results are compared with the corresponding numerical simulations performed with Ansys Classic. In this present paper, a detailed evaluation of the load cases F1 : F2 = −3 : 1 and F1 : F2 = −7 : 1 is considered. Fig. 1(b) shows the load-displacement curves of both load cases in shear direction. Both load cases show a very good agreement between experiment and numerical simulation. With the load combination F1 : F2 = −3 : 1 a maximum force of 910N and a maximum displacement of 1.18mm is achieved. With increasing compression, load case F1 : F2 = −7 : 1 shows that the maximum force drops to 510N and a displacement of 1.05mm is achieved. The experimental curves of both load cases indicate similar behavior at the end of the test. The high compression makes it difficult to perform the experiments due to shear and friction mechanisms. Furthermore, there are very good agreements in the comparison of the principal strains between the experiments and numerical simulations, as one can see in Fig. 2. The left column shows the first principal strain and the right column shows the second one. For each comparison, the DIC evaluation of the experiment is on the left and the evaluation of the numerical simulation on the right. The strains are evaluated at 23 of the maximum displacement u2 and it can be seen in both load cases that a strain band develops for the first and second principal strain from bottom left to top right. Besides the good agreement between experiment and numerical simulation, the localization of the strains can be easily recognized. This localization gives an idea of the orientation of the crack leading to failure of the specimen. In the load case F1 : F2 = −3 : 1 the first principal strain reaches a maximum of 0.28 and the second one a minimum of −0.45. For the load case F1 : F2 = −7 : 1 the maximum of the first principal strain is 0.26 whereas the minimum of the second one is −0.49. In order to assess the influence of the stress state on damage behavior, the stress triaxiality and the Lode parameter of the numerical calculations were evaluated in a cut plane passing though the center of the notch at 32 of the final displacement u2 , see Fig. 3. At load case F1 : F2 = −3 : 1 the distribution of stress triaxiality and the Lode parameter in the middle of the fracture surface is relatively homogeneous. It becomes more inhomogeneous towards the outside. The minimum stress triaxiality of η = −0.39 is not in the center of the cut plane and has the associated Lode parameter
Marco Schmidt et al. / Procedia Structural Integrity 13 (2018) 91–96 M. Schmidt et al. / Structural Integrity Procedia 00 (2018) 000–000 (a1)
(b1)
95 5
(c1)
10 m
-0.66 (a2)
0.50
-1.0 (b2)
1.0 (c2)
10 m
Fig. 3. (a) Stress triaxiality; (b) Lode parameter; (1) F1 : F2 = −3 : 1; (2) F1 : F2 = −7 : 1.
of ω = 0.23. The stress triaxiality of η = −0.29 and the Lode parameter of ω = 0 are achieved in the center of the cut plane. With increasing compression, it can be seen with load case F1 : F2 = −7 : 1 that the distribution of the stress triaxiality and the Lode parameter becomes more inhomogeneous. The minimum value of the stress triaxiality for this load case is η = −0.55 and the corresponding Lode parameter of ω = 0.26. Scanning electron microscope images of the fracture surfaces, see Fig. 3(c) were taken to visualize the relationship between stress triaxiality and damage mechanisms. In the load case F1 : F2 = −3 : 1, in addition to significant shear mechanisms, some small voids which have already been compressed can also be detected in Fig.3(c1). With increasing compression, voids are hardly visible, as the load case with the load factor F1 : F2 = −7 : 1 shows in Fig. 3(c2). These voids are more compressed due to the remarkable superimposed compression and only shear mechanisms can be seen. It should therefore be noted that damage mechanisms still occur even for these high negative stress triaxialities. In summary, a connection between the stress triaxialities of the numerical simulations and the damage mechanisms from the experiments can be seen, thereby confirming that a good approximation of the damage material behavior is achieved with the presented anisotropic continuum damage model.
4. Cut-off value for negative stress triaxialities The minima of the stress triaxialities in both load cases discussed above reached a high negative value. Therefore, the cut-off value below which no damage should occur has to be discussed. In Br¨unig et al. (2018) the function for the cut-off value ηcut = −0.6 + 0.27ω for
0≤ω≤1
(15)
was proposed and the stress state in the center of the fracture surface was evaluated. With this function the cut-off value of ηcut = −0.54 is obtained for the present load case F1 : F2 = −3 : 1 with the Lode parameter ω = 0.23. The minimum stress triaxiality of this load case was η = −0.39 and therefore does not fall below this limit value. For the second load case, the cut-off value is ηcut = −0.53. The minimum stress triaxiality reached η = −0.55, which is slightly below the cut-off value. As a consequence of the results, the stress-state-dependent function for the cut-off value remains a good approach and should be further investigated for other specimens and load cases.
96 6
Marco Schmidt et al. / Procedia Structural Integrity 13 (2018) 91–96 M. Schmidt et al. / Structural Integrity Procedia 00 (2018) 000–000
5. Conclusion The main objective of this paper was to evaluate the influence of negative stress triaxialities on the damage and fracture behavior of ductile metals. For this purpose, a biaxial test specimen made of an aluminum alloy was tested experimentally and analyzed numerically under various shear-compression loading conditions. The numerical simulations are based on the presented anisotropic damage model with stress-state-dependent criteria. In addition to the evaluation of the force-displacement curves, the strain fields were visualized with the help of a digital image correlation system and compared with results of corresponding numerical simulations. To analyze the relationship between stress state and damage mechanisms, stress triaxialities and Lode parameters were evaluated numerically and compared to the fracture surfaces of the failed specimens, which were recorded with a scanning electron microscope. Furthermore, a stress-state-dependent function of the cut-off value for negative stress triaxialities, below which no damage occurs, is discussed on the basis of the experimental and numerical results. 6. Acknowledgments Financial support from the Deutsche Forschungsgemeinschaft DFG (German Research Foundation - project number 281419279) is gratefully acknowledged. The SEM pictures have been performed at the Institut f¨ur Werkstoffe des Bauwesens at the Universit¨at der Bundeswehr M¨unchen and the special support of Wolfgang Saur is gratefully acknowledged. References Bao, Y., Wierzbicki, T., 2004. On the fracture locus in the equivalent strain and stress triaxiality space. International Journal of Mechanical Sciences 46, 81–98. Br¨unig, M., 2003. An anisotropic ductile damage model based on irreversible thermodynamics. International Journal of Plasticity 19, 1679–1713. Br¨unig, M., Chyra, O., Albrecht, D., Driemeier, L., Alves, M., 2008. A ductile damage criterion at various stress triaxialities. International Journal of Plasticity 24, 1731–1755. Br¨unig, M., Gerke, S., Hagenbrock, V., 2013. Micro-mechanical studies on the effect of stress triaxiality and Lode parameter on ductile damage. International Journal of Plasticity 50, 49–65. Br¨unig, M., Gerke, S., Schmidt, M., 2016. Biaxial experiments and phenomenological modeling of stress-state-dependent ductile damage and fracture. International Journal of Fracture 200, 63–76. Br¨unig, M., Gerke, S., Schmidt, M., 2018. Damage and failure at negative stress triaxialities: Experiments, modeling and numerical simulations. International Journal of Plasticity 102, 70–82. Driemeier, L., Br¨unig, M., Micheli, G., Alves, M., 2010. Experiments on stress-triaxiality dependence of material behavior of aluminum alloys. Mechanics of Materials 42, 207–217. Dunand, M., Mohr, D., 2011. On the predictive capabilities of the shear modified Gurson and the modified Mohr-Coulomb fracture models over a wide range of stress triaxialities and Lode angles. Journal of the Mechanics and Physics of Solids 59, 1374–1394. Gao, X., Zhang, G., Roe, C., 2010. A study on the effect of stress state on ductile fracture. International Journal of Damage Mechanics 19, 75–94. Gerke, S., Preawpan, A., Br¨unig, M., 2017. New biaxially loaded specimens for the analysis of damage and fracture in sheet metals. International Journal of Solids and Structures 110-111, 209–218. Kuwabara, T., 2007. Advances in experiments on metal sheet and tubes in support of constitutive modeling and forming simulations. International Journal of Plasticity 23, 385–419. Makinde, A., Thibodeau, L., Neale, K., 1992. Development of an apparatus for biaxial testing using cruciform specimens. Experimental Mechanics 32, 138–144.
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ECF22 - Loading and Environmental effects on Structural Integrity ECF22 - Loading and Environmental effects on Structural Integrity
The The Effect Effect of of Negative Negative Stress Stress Triaxialities Triaxialities on on Ductile Ductile Damage Damage and and Fracture Behavior in Metal Sheets XV Portuguese Conference on Fracture, PCF 2016, February 2016, Paço de Arcos, Portugal Fracture Behavior in 10-12 Metal Sheets a,∗ a a Marco Schmidt u nig a,∗, Steffen Gerkea , Michael Br¨ a Marco Schmidt , Steffen Gerke , Michael Br¨ u nig Thermo-mechanical modeling of a high pressure turbine a Institut a Institut
blade of an
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
airplane gas turbine engine
Abstract P. Brandãoa, V. Infanteb, A.M. Deusc* Abstract The paper deals with an anisotropic continuum damage and fracture model and a series of biaxial experiments with focus on a Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av.biaxial Roviscoexperiments Pais, 1, 1049-001 Lisboa, The paper dealstriaxialities. with an anisotropic continuum model and series of with focusand on negative stress The continuum modeldamage is basedand on fracture the assumption that adifferent damage mechanisms are present Portugal negative stress triaxialities. The continuum model is based on the assumption that different damage mechanisms are present and b have IDMEC, to be taken into account depending on stress triaxiality andTécnico, Lode parameter. Therefore, onset and evaluation Department of Mechanical Engineering, Instituto Superior Universidade de Lisboa,modeling Av. RoviscoofPais, 1, 1049-001 Lisboa, have to be taken into account depending on stressdamage triaxiality and Lode parameter. Therefore, modeling of onset and evaluation Portugaland a stress-state-dependent damage rule. To identify the of damage are based on a stress-state-dependent condition c of damage are based onofabiaxial stress-state-dependent damage condition and Universidade aaluminum stress-state-dependent damage rule.1,To identify the CeFEMA, Department Mechanical Engineering, Instituto Superior Técnico, de Lisboa, Av.been Rovisco Pais, 1049-001 Lisboa, corresponding parameters experiments with specimens taken from sheets have performed and results of corresponding parameters biaxial experiments with specimens taken from aluminum sheets have been performed and results of Portugal corresponding numerical simulations are discussed in detail. The experimental behavior has been analyzed with a digital image corresponding numerical simulations are discussed in detail. The experimental behavior has been analyzed with a digital image correlation system to compare the strain fields with those obtained by numerical simulations. In addition, fracture modes are correlation to electron comparemicroscopy. the strain fields those obtained by In addition, fracturecut-off modes are detected by system scanning Basedwith on the experimental andnumerical numericalsimulations. results a stress-state-dependent value Abstract detected by scanning electron microscopy. Based on the experimental and numerical results a stress-state-dependent cut-off value for negative stress triaxialities is proposed. for negative stress triaxialities is proposed. During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, c especially 2018The TheAuthors. Authors. Published by Elsevier © 2018 Published byturbine Elsevier B.V. B.V. high pressure (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent c 2018 The the Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. Peer-review under responsibility ofcreep. the ECF22 organizers. degradation, one of which is A model using the finite element method (FEM) was developed, in order to be able to predict Peer-review under responsibility of the ECF22 organizers. the creep behaviour of HPT blades. Flight (FDR) for a specific aircraft, provided by a commercial aviation Keywords: Continuum damage model; Ductile metal data sheets;records Stress-state-dependence; Experiments; Numerical simulations company,Continuum were useddamage to obtain thermal data for three different flight Numerical cycles. Insimulations order to create the 3D model model; Ductile and metalmechanical sheets; Stress-state-dependence; Experiments; Keywords: 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.
Currently big effort is made to use all capacities of materials in order to save resources and costs. This requires more ©Currently 2016 The big Authors. byuse Elsevier B.V. effortPublished is made to all capacities of materials in order to save resources and costs. This requires more in-depth knowledge of the material andScientific more sophisticated models to reflect its behavior sufficiently accurate Peer-review under responsibility of the Committee ofmaterial PCF 2016. in-depth knowledge of the material and more sophisticated material models to reflect its behavior sufficiently accurate and to ensure safety at the same time. In this context focus is given on the deterioration behavior of ductile metals and to ensure safety at the same time. In this context focus is given on the deterioration behavior of ductile metals which are ofHigh interest inTurbine several engineering disciplines. Thus,3D corresponding phenomenological material models have Keywords: Pressure Creep; Finite Element Method; Model; Simulation. which are of interest in severalBlade; engineering disciplines. Thus, corresponding phenomenological material models have to be capable to reflect besides the elastic-plastic behavior also the damage and failure processes. Especially the to be capable to reflect besides the elastic-plastic behavior also the damage and failure processes. Especially the damage behavior strongly depends on the stress state: Void nucleation, growth and coalescence occur at high positive damage behavior strongly depends on the stress state: Void nucleation, growth and coalescence occur at high positive stress triaxialities whereas micro-shear cracks appear at stress triaxialities around zero and below. stress triaxialities whereas micro-shear cracks appear at stress triaxialities around zero and below. The influence of positive stress triaxialities on the damage and fracture behavior could be investigated with uniaxial The influence of positive stress triaxialities on the damage and fracture behavior could be investigated with uniaxial tension tests of unnotched and notched specimens, see for example Bao and Wierzbicki (2004), Br¨unig et al. (2008), tension tests of unnotched and notched specimens, see for example Bao and Wierzbicki (2004), Br¨unig et al. (2008), Gao et al. (2010) and Dunand and Mohr (2011). In addition, further uniaxial test specimens with special geometry Gao et al. (2010) and Dunand and Mohr (2011). In addition, further uniaxial test specimens with special geometry Corresponding author. Tel.: +351 218419991. ∗*Corresponding author. Tel.: +49-89-60043413 ; ∗ Corresponding E-mail address: [email protected] author. Tel.: +49-89-60043413 ;
fax: +49-89-60044549. fax: +49-89-60044549. 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.2018.12.016
Marco Schmidt et al. / Procedia Structural Integrity 13 (2018) 91–96 M. Schmidt et al. / Structural Integrity Procedia 00 (2018) 000–000
92 2
have been proposed to study shear mechanisms in critical regions and thus achieving stress triaxialities around zero, see Bao and Wierzbicki (2004), Br¨unig et al. (2008) and Driemeier et al. (2010). Unfortunately this type of specimens tends to rotations within the shear region leading to rather tension dominated stress states. Consequently, biaxial test specimens were developed, (Makinde et al. (1992), Kuwabara (2007), Gerke et al. (2017)) to cover a wide range of stress triaxialities with one specimen geometry. Due to the geometry of the test specimens shear tension as well as shear compression loads should be possible within a biaxial test setup. The presented paper outlines the basic concept of a phenomenological anisotropic damage and failure model. Biaxial experiments and corresponding numerical simulations reflect the influence of negative stress triaxialities on the damage behavior. The experimental evaluation is carried out by digital image correlation for the analysis of the strain fields and by scanning electron microscope for the investigations of the fracture surface. In conclusion, a cut-off value for negative stress triaxialities below which no damage occurs is discussed, see also Br¨unig et al. (2018). 2. Continuum damage and fracture model The anisotropic damage and failure model for ductile metals is based on the additive decomposition of the strain rate tensor into an elastic, a plastic and a damage part (Br¨unig (2003)). For this purpose, damaged and fictitious undamaged configurations are introduced, which are coupled by damage tensors. The plastic material behavior is characterized by a stress-state-dependent yield condition, which indicates the onset of plastic flow, and by a stress-state-dependent flow rule. A similar approach is used to model the damage behavior. The damage condition (1) f da = αI1 + β J2 − σ = 0
characterizes onset of damage. It is formulated in terms of the first and second deviatoric invariant of the Kirchhoff stress tensor, I1 and J2 , and the damage threshold σ. The parameters α and β in Eq. (1) depend on the stress triaxiality η=
σm I1 = √ σeq 3 3J2
where σm = I1 /3 means the hydrostatic stress and σeq = on the Lode parameter ω=
(2) √ 3J2 represents the von Mises equivalent stress as well as
2T 2 − T 1 − T 3 T1 − T3
(3)
with the principal stress components T 1 ≥ T 2 ≥ T 3 . The parameters α and β are based on numerical calculations of different single-pore models (Br¨unig et al. (2013)) as well as on results of biaxial experiments and associated numerical simulations (Br¨unig et al. (2016)). In particular, the parameter α is taken to be −0.15 for ηcut < η < 0 , (4) α(η) = 0.33 for η>0 where ηcut is the cut-off value for negative stress triaxialities below which no damage occurs. The parameter β(η, ω) = β0 (η, ω = 0) + βω (ω) ≥ 0
(5)
β0 (η) = −1.28η + 0.85
(6)
βω (ω) = −0.017ω3 − 0.065ω2 − 0.078ω
(7)
with
and
is a non-negative function depending on the stress triaxiality (Eq. (2)) and the Lode parameter (Eq. (3)). Besides the damage condition, the damage rule 1 da ˙ ¯ ¯ (8) H = µ˙ α¯ √ 1 + βN + δM 3
Marco Schmidt et al. / Procedia Structural Integrity 13 (2018) 91–96 M. Schmidt et al. / Structural Integrity Procedia 00 (2018) 000–000
93 3
(b) 1 F2 [kN ]
(a) F2
F1
F1
0.5 LF Exp. Sim. -3:1 -7:1
u2 [mm]
0
F2
0
0.4
0.8
1.2
Fig. 1. (a) Z-specimen with force directions; (b) Force-displacement-curves of F1 : F2 = −3 : 1 and F1 : F2 = −7 : 1.
describes the formation of macroscopic strains caused by growth of micro-defects on the micro-scale, where N=
1 √
2 J2
˜ devT
1 devS˜ M = devS˜
and
with
˜ ˜ − 2 J2 1 devS˜ = devTdev T 3
(9)
are normalized deviatoric stress-based tensors. In addition, the equivalent damage strain rate µ˙ characterizes the amount of irreversible damage strain rates, and the stress-state-dependent parameters a¯ , β¯ and δ¯ are kinematic variables characterizing the ratio between isotropic and anisotropic damage deformations. With 0 for ηcut < η ≤ 0 0.5714η for 0 < η ≤ 1.75 α(η) ¯ = (10) 1 for η > 1.75 the volumetric portion corresponding to evaluation of micro-voids is weighted, and with ¯ ω) = β¯ 0 (η) + fβ (η)β¯ ω (ω), β(η,
(11)
containing 0.87 0.97875 − 0.32625η β¯ 0 (η) = 0
for for for
ηcut < η ≤ 1/3 1/3 < η ≤ 3 , η> 3
fβ = −0.0252 + 0.0378η
and
1 − ω2 β¯ ω (ω) = 0
for for
ηcut < η ≤ 2/3 , η > 2/3
(12) (13)
(14)
here taken to be the isochoric portion corresponding to development of micro-shear-cracks is balanced. In this context the parameter δ¯ is neglected based on results of numerical simulations of biaxial experiments with different loading conditions (Br¨unig et al. (2016)). 3. Biaxial experiments and numerical simulations To validate the continuum damage model and to investigate the influence of negative stress triaxialities on ductile damage behavior, an experimental series with various biaxial load ratios is performed. The experiments are carried out with a biaxial testing machine that uses four independent electro-mechanically driven cylinders with a maximum load of ±20kN. The test specimen (Fig. 1(a)) is characterized by a tension-compression-axis (F1 ) and a shear-axis (F2 ). In addition, the central region has notches in two directions to localize the strains and damage. The used material is an aluminum alloy of the 2017 series and was selected for its pronounced ductile behavior.
Marco Schmidt et al. / Procedia Structural Integrity 13 (2018) 91–96 M. Schmidt et al. / Structural Integrity Procedia 00 (2018) 000–000
94 4 (a1)
(b1)
0.00
-0.45
0.28
(a2)
0.00
(b2)
0.00
0.26
-0.49
0.00
Fig. 2. (a) 1. Principal strain; (b) 2. Principal strain; (1) F1 : F2 = −3 : 1; (2) F1 : F2 = −7 : 1.
The influence of negative stress triaxialities on the damage behavior has been investigated experimentally by different, throughout the experiment constant loading ratios F1 : F2 . Shear loading caused by F2 is superimposed by compression loading caused by F1 i.e. with more elevated negative load factor F1 (from 0 to −8) more compression is superimposed which causes more negative stress triaxialities. The experimental evaluation was made with a digital image correlation system. The corresponding displacement measure u2 of the force-displacement-curves (see Fig. 1(b)) was extracted by two points on the surface of the specimen with a distance of 6mm whereas one evaluation point was on the right and one on the left side of the notch. The experimental results are compared with the corresponding numerical simulations performed with Ansys Classic. In this present paper, a detailed evaluation of the load cases F1 : F2 = −3 : 1 and F1 : F2 = −7 : 1 is considered. Fig. 1(b) shows the load-displacement curves of both load cases in shear direction. Both load cases show a very good agreement between experiment and numerical simulation. With the load combination F1 : F2 = −3 : 1 a maximum force of 910N and a maximum displacement of 1.18mm is achieved. With increasing compression, load case F1 : F2 = −7 : 1 shows that the maximum force drops to 510N and a displacement of 1.05mm is achieved. The experimental curves of both load cases indicate similar behavior at the end of the test. The high compression makes it difficult to perform the experiments due to shear and friction mechanisms. Furthermore, there are very good agreements in the comparison of the principal strains between the experiments and numerical simulations, as one can see in Fig. 2. The left column shows the first principal strain and the right column shows the second one. For each comparison, the DIC evaluation of the experiment is on the left and the evaluation of the numerical simulation on the right. The strains are evaluated at 23 of the maximum displacement u2 and it can be seen in both load cases that a strain band develops for the first and second principal strain from bottom left to top right. Besides the good agreement between experiment and numerical simulation, the localization of the strains can be easily recognized. This localization gives an idea of the orientation of the crack leading to failure of the specimen. In the load case F1 : F2 = −3 : 1 the first principal strain reaches a maximum of 0.28 and the second one a minimum of −0.45. For the load case F1 : F2 = −7 : 1 the maximum of the first principal strain is 0.26 whereas the minimum of the second one is −0.49. In order to assess the influence of the stress state on damage behavior, the stress triaxiality and the Lode parameter of the numerical calculations were evaluated in a cut plane passing though the center of the notch at 32 of the final displacement u2 , see Fig. 3. At load case F1 : F2 = −3 : 1 the distribution of stress triaxiality and the Lode parameter in the middle of the fracture surface is relatively homogeneous. It becomes more inhomogeneous towards the outside. The minimum stress triaxiality of η = −0.39 is not in the center of the cut plane and has the associated Lode parameter
Marco Schmidt et al. / Procedia Structural Integrity 13 (2018) 91–96 M. Schmidt et al. / Structural Integrity Procedia 00 (2018) 000–000 (a1)
(b1)
95 5
(c1)
10 m
-0.66 (a2)
0.50
-1.0 (b2)
1.0 (c2)
10 m
Fig. 3. (a) Stress triaxiality; (b) Lode parameter; (1) F1 : F2 = −3 : 1; (2) F1 : F2 = −7 : 1.
of ω = 0.23. The stress triaxiality of η = −0.29 and the Lode parameter of ω = 0 are achieved in the center of the cut plane. With increasing compression, it can be seen with load case F1 : F2 = −7 : 1 that the distribution of the stress triaxiality and the Lode parameter becomes more inhomogeneous. The minimum value of the stress triaxiality for this load case is η = −0.55 and the corresponding Lode parameter of ω = 0.26. Scanning electron microscope images of the fracture surfaces, see Fig. 3(c) were taken to visualize the relationship between stress triaxiality and damage mechanisms. In the load case F1 : F2 = −3 : 1, in addition to significant shear mechanisms, some small voids which have already been compressed can also be detected in Fig.3(c1). With increasing compression, voids are hardly visible, as the load case with the load factor F1 : F2 = −7 : 1 shows in Fig. 3(c2). These voids are more compressed due to the remarkable superimposed compression and only shear mechanisms can be seen. It should therefore be noted that damage mechanisms still occur even for these high negative stress triaxialities. In summary, a connection between the stress triaxialities of the numerical simulations and the damage mechanisms from the experiments can be seen, thereby confirming that a good approximation of the damage material behavior is achieved with the presented anisotropic continuum damage model.
4. Cut-off value for negative stress triaxialities The minima of the stress triaxialities in both load cases discussed above reached a high negative value. Therefore, the cut-off value below which no damage should occur has to be discussed. In Br¨unig et al. (2018) the function for the cut-off value ηcut = −0.6 + 0.27ω for
0≤ω≤1
(15)
was proposed and the stress state in the center of the fracture surface was evaluated. With this function the cut-off value of ηcut = −0.54 is obtained for the present load case F1 : F2 = −3 : 1 with the Lode parameter ω = 0.23. The minimum stress triaxiality of this load case was η = −0.39 and therefore does not fall below this limit value. For the second load case, the cut-off value is ηcut = −0.53. The minimum stress triaxiality reached η = −0.55, which is slightly below the cut-off value. As a consequence of the results, the stress-state-dependent function for the cut-off value remains a good approach and should be further investigated for other specimens and load cases.
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5. Conclusion The main objective of this paper was to evaluate the influence of negative stress triaxialities on the damage and fracture behavior of ductile metals. For this purpose, a biaxial test specimen made of an aluminum alloy was tested experimentally and analyzed numerically under various shear-compression loading conditions. The numerical simulations are based on the presented anisotropic damage model with stress-state-dependent criteria. In addition to the evaluation of the force-displacement curves, the strain fields were visualized with the help of a digital image correlation system and compared with results of corresponding numerical simulations. To analyze the relationship between stress state and damage mechanisms, stress triaxialities and Lode parameters were evaluated numerically and compared to the fracture surfaces of the failed specimens, which were recorded with a scanning electron microscope. Furthermore, a stress-state-dependent function of the cut-off value for negative stress triaxialities, below which no damage occurs, is discussed on the basis of the experimental and numerical results. 6. Acknowledgments Financial support from the Deutsche Forschungsgemeinschaft DFG (German Research Foundation - project number 281419279) is gratefully acknowledged. The SEM pictures have been performed at the Institut f¨ur Werkstoffe des Bauwesens at the Universit¨at der Bundeswehr M¨unchen and the special support of Wolfgang Saur is gratefully acknowledged. References Bao, Y., Wierzbicki, T., 2004. On the fracture locus in the equivalent strain and stress triaxiality space. International Journal of Mechanical Sciences 46, 81–98. Br¨unig, M., 2003. An anisotropic ductile damage model based on irreversible thermodynamics. International Journal of Plasticity 19, 1679–1713. Br¨unig, M., Chyra, O., Albrecht, D., Driemeier, L., Alves, M., 2008. A ductile damage criterion at various stress triaxialities. International Journal of Plasticity 24, 1731–1755. Br¨unig, M., Gerke, S., Hagenbrock, V., 2013. Micro-mechanical studies on the effect of stress triaxiality and Lode parameter on ductile damage. International Journal of Plasticity 50, 49–65. Br¨unig, M., Gerke, S., Schmidt, M., 2016. Biaxial experiments and phenomenological modeling of stress-state-dependent ductile damage and fracture. International Journal of Fracture 200, 63–76. Br¨unig, M., Gerke, S., Schmidt, M., 2018. Damage and failure at negative stress triaxialities: Experiments, modeling and numerical simulations. International Journal of Plasticity 102, 70–82. Driemeier, L., Br¨unig, M., Micheli, G., Alves, M., 2010. Experiments on stress-triaxiality dependence of material behavior of aluminum alloys. Mechanics of Materials 42, 207–217. Dunand, M., Mohr, D., 2011. On the predictive capabilities of the shear modified Gurson and the modified Mohr-Coulomb fracture models over a wide range of stress triaxialities and Lode angles. Journal of the Mechanics and Physics of Solids 59, 1374–1394. Gao, X., Zhang, G., Roe, C., 2010. A study on the effect of stress state on ductile fracture. International Journal of Damage Mechanics 19, 75–94. Gerke, S., Preawpan, A., Br¨unig, M., 2017. New biaxially loaded specimens for the analysis of damage and fracture in sheet metals. International Journal of Solids and Structures 110-111, 209–218. Kuwabara, T., 2007. Advances in experiments on metal sheet and tubes in support of constitutive modeling and forming simulations. International Journal of Plasticity 23, 385–419. Makinde, A., Thibodeau, L., Neale, K., 1992. Development of an apparatus for biaxial testing using cruciform specimens. Experimental Mechanics 32, 138–144.