Application of the equivalent material concept to fracture of U-notched solids under small scale yielding

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ECF22 - Loading and Environmental effects on Structural Integrity ECF22 - Loading and Environmental effects on Structural Integrity

Application of the equivalent material concept to fracture of UApplication of the equivalent material concept to fracture of Unotchedonsolids small scale yielding XV Portuguese Conference Fracture,under PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal notched solids under small scale yielding a*, A.R. Torabib F.J. Gómezof Thermo-mechanical modeling high pressure turbine blade of an b F.J. Gómeza*, a A.R. Torabi Advanced Material Simulation, S.L., c/Asturias 3. Bilbao E-48015. Spain airplane gas turbine engine Simulation, S.L., c/Asturias 3. BilbaoofE-48015. Fracture Research Laboratory,Advanced Faculty ofMaterial New Sciences and Technologies, University Tehran, Spain P.O. Box 14395-1561, Tehran, Iran a

b b

a

Fracture Research Laboratory, Faculty of New Sciences and Technologies, University of Tehran, P.O. Box 14395-1561, Tehran, Iran

P. Brandãoa, V. Infanteb, A.M. Deusc*

Abstract AbstractaDepartment of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Thisb paper studies the applicability of the equivalent materialPortugal concept developed by the author to the fracture of elastoplastic IDMEC, Department of Mechanical of Engineering, Instituto Superiorconcept Técnico,developed Universidade Lisboa, Av.to Roviscofracture Pais, 1, of 1049-001 Lisboa, This paper applicability the equivalent material bydethe author elastoplastic material duestudies to the the presence of U-notches. The approach of equivalent material concept consists of the simplifying the study of an Portugal material due to the presence ofitU-notches. The approach of equivalent material consists of simplifying the study Lisboa, of an c elastoplastic reducing to theEngineering, linear elastic case Superior with a maximum stress concept suchde asLisboa, in a tensile test the deformation CeFEMA,material Department of Mechanical Instituto Técnico, Universidade Av. Rovisco Pais, 1, 1049-001energy elastoplastic material reducingThis it toidea the linear elastic casethe with a maximum stress such as to in aestablish tensile test the deformation energy is equal to the real material. combined with cohesive zone model allows a procedure to predict the Portugal is equaloftoU-notched the real material. combinedhas with thesuccessfully cohesive zone model to establish a procedure to all predict the failure elements.This The idea methodology been applied to allows five elastoplastic materials, and in of them, failure ofofU-notched methodology been has successfully applied to fiveanalysis elastoplastic materials, and inmethodology all of them, the level plasticity elements. regarding The the load of plastic has collapse been determined. This verifies the proposed theAbstract level of plasticity regardinglimits the load of plastic collapse haswithin been small determined. This analysis verifies the proposed methodology and establish some application when the failure occurs scale yielding and establish some application limits when the failure occurs within small scale yielding their operation, modern aircraft B.V. engine components are subjected to increasingly demanding operating conditions, © During 2018 The Authors. Published by Elsevier © especially 2018 The Authors. Published by Elsevier(HPT) B.V. blades. Such conditions cause these parts to undergo different types of time-dependent the high pressure © 2018 The Authors. Publishedturbine by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. Peer-review under responsibility the ECF22 organizers. degradation, one of which isofcreep. model using the finite element method (FEM) was developed, in order to be able to predict Peer-review under responsibility of theAECF22 organizers. the creep behaviour of HPT blades. Flight data Equivalent records (FDR) forConcept. a specific aircraft, provided by a commercial aviation Keywords: U-notches; failure criteria; cohesive zone model; Material company, were used to criteria; obtain cohesive thermal zone and model; mechanical data Material for three different flight cycles. In order to create the 3D model Keywords: U-notches; failure Equivalent Concept. 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 1. rectangular Introduction block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The 1. overall Introduction expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a Stresscan concentrators ingoal structural elements such as life, U-shaped notches weak points with high risk of brittle model be useful in the of predicting turbine blade given a set of FDRare data.

Stress in structural elements such as U-shaped notches are weak load points high risk of brittle failure andconcentrators an integrity assessment methodology is needed to evaluate the maximum thatwith resist. failure and anAuthors. integrityPublished assessment methodology © 2016 The by Elsevier B.V. is needed to evaluate the maximum load that resist. Peer-review under responsibility of the Scientific Committee of PCF 2016.

Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation.

* Corresponding author. Tel.: +34 629269931. * Corresponding Tel.: +34 629269931. E-mail address:author. [email protected] E-mail address: [email protected] 2452-3216 © 2018 The Authors. Published by Elsevier B.V. 2452-3216 © 2018 Authors. Published Elsevier B.V. Peer-review underThe responsibility of theby ECF22 organizers. Peer-review underauthor. responsibility the ECF22 organizers. * Corresponding Tel.: +351of218419991. E-mail address: [email protected]

2452-3216 © 2016 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the Scientific Committee of PCF 2016. 2452-3216  2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. 10.1016/j.prostr.2018.12.045

F.J. Gómez et al. / Procedia (2018) 267–272 Author name / Structural Integrity Structural Procedia Integrity 00 (2018)13000–000

268 2

Nomenclature E ft KI KIC KICR KICR* KIR lch Lr R SED SEDnecking

Elastic modulus Fracture strength Stress intensity factor Fracture toughness Critical notch stress intensity factor Non-dimensional critical notch stress intensity factor Notch stress intensity factor Characteristic length Ratio between the maximum load and plastic collapse load Notch radius Strain energy density Strain energy density under necking

 f u f  f max u y

Strain Failure strain Strain under necking Fictitious failure strain Stress Fictitious fracture strength Elastic stress at the tip of the notch Ultimate tensile strength Elastic limit

In a cracked structural component and linear elastic material, Fracture Mechanics states that the maximum load is reached when the stress intensity factor is equal to the fracture toughness of the material. This criterion is still valid in elastoplastic materials when the plastic zone is limited to a region close to the crack tip (Irwin 1957). In Unotched solids, in linear elasticity, there is no tensional singularity, however, the approximate expression of the stress field at the tip of the notch given by Creager and Paris (1967) permits to stablish similar assessment based on the notch stress intensity factor (Glinka 1985). In elastoplastic materials, Creager and Paris formulation is no longer valid. One possibility to overcome this limitation is to apply tensional corrections as suggested by Neuber (1958) or Glinka (1987), valid under small scale yielding. A.R. Torabi, one of the authors of the present communication, has proposed the Equivalent Material Concept (EMC), based on the strain energy density, to replace the real material with an elastoplastic behavior by a fictitious equivalent linear elastic material (Torabi 2012 and 2013). The EMC can be combined with various failure criteria such as the cohesive zone model or a phenomenological formulation to develop a procedure for predicting the maximum load of components with U-notches. 2. Failure criteria in linear elastic materials In a linear elastic material, the maximum load that supports a cracked solid in mode I is obtained when the stress intensity factor, KI, which depends on the geometry and the stress, reaches the value of the fracture toughness, KIC: 𝐾𝐾𝐼𝐼 = 𝐾𝐾𝐼𝐼𝐼𝐼 (𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑙𝑙)

(1)

Dealing with U-shaped notches, the tensional field can be approximated by Creager and Paris formulation (Creager and Paris 1967) depending on a single factor, KIR, and again a similar criterion can be established: the maximum load is obtained when the notch stress intensity factor reaches a critical value, KICR, depending on the material but also the notch radius.



F.J. Gómez et al. / Procedia Structural Integrity 13 (2018) 267–272 Author name / Structural Integrity Procedia 00 (2018) 000–000 𝑅𝑅 ( 𝐾𝐾𝐼𝐼𝑅𝑅 = 𝐾𝐾𝐼𝐼𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚, 𝑅𝑅)

269 3

(2)

Gómez and Elices (2006) checked the validity of the expression (2) in linear elastic materials and showed that introducing a non-dimensional formulation based on the fracture toughness and the characteristic length of the cohesive zone model, leads a function with a weak material dependency. 𝐾𝐾𝐼𝐼𝑅𝑅

𝐾𝐾𝐼𝐼𝐼𝐼

𝑅𝑅∗ ≈ 𝐾𝐾𝐼𝐼𝐼𝐼 ( 𝐾𝐾

𝑙𝑙𝑐𝑐ℎ = ( 𝑓𝑓𝐼𝐼𝐼𝐼 ) 𝑡𝑡

𝑅𝑅

𝑙𝑙𝑐𝑐ℎ

2

(3)

)

(4)

where ft is the fracture strength. Gómez and Elices (2006) collected failure data of alumina, silicon nitride, monocrystalline and polycrystalline silica, zirconia partially stabilized with magnesia, zirconia partially stabilized with yttria, tetragonal zirconia fully stabilized with yttria, and PMMA at -60°C and demonstrated how the expression (3) reduces the dependency of the material. The experimental results were fitted to the following expression which constitutes itself a phenomenological failure criterion (Gomez et al 2005): 𝐾𝐾𝐼𝐼𝑅𝑅

𝐾𝐾𝐼𝐼𝐼𝐼

=√

1+0.47392(𝑅𝑅⁄𝑙𝑙𝑐𝑐ℎ )+2.1382(𝑅𝑅⁄𝑙𝑙𝑐𝑐ℎ )2 +𝜋𝜋/4(𝑅𝑅⁄𝑙𝑙𝑐𝑐ℎ )3 1+(𝑅𝑅⁄𝑙𝑙𝑐𝑐ℎ )2

(5)

Failure due to U-notches can be explained in a similar way by applying criteria such as the critical strain energy density (Lazzarin and Berto 2005), maximum stress, mean stress (Seweryn and Lukaszewick 2002, Susmel and Taylor 2008) or the theory of cohesive zone model (Gomez et al 2005). 3. Equivalent material concept All materials considered in Gómez and Elices (2006) had linear elastic behavior until failure. In elastoplastic materials, U-notches can be analyzed in a similar way combining the failure criteria with simplifications or rules that substitute the real elastoplastic behavior by another linear elastic fictitious one. Within this type of approach is the Equivalent Material Concept (EMC) (Torabi 2012 and 2013), which establishes that the modulus of elasticity E and the fracture toughness of the fictitious material as the same than the real material and the fracture stress can be obtained assuming that the energy density developed in a tensile test at maximum load is the same in the real case and the fictitious one. (𝑆𝑆𝑆𝑆𝑆𝑆)𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 =

𝜎𝜎𝑓𝑓2

2𝐸𝐸

(6)

where SEDnecking is the strain energy density developed in a tensile test under maximum load (Figure 1) and f the fictitious fracture strength of the equivalent material. 4. Failure criteria in elastoplastic materials To verify the validity of the proposed methodology, failure data of five different materials have been compiled and revised: PMMA, polycarbonate, aluminum alloy, vessel steels and structural steel. Gómez and Elices (2000) studied the failure of PMMA due to notches at room temperature, carrying out an extensive experimental program of fracture tests with different radii, notch depths, sizes and types of solicitation. The fictitious fracture strength of PMMA has been calculated by applying expression (6) to the stress-strain curve measured at room temperature. The non-dimensional critical notch stress intensity factors calculated with the properties of Table 1 are shown in Figure 2. The ratio between the maximum load and the plastic collapse load, Lr, (FITNET 2007) has been calculated for the largest radius and the value obtained appears in Table 1.

F.J.name Gómez et al. / Procedia (2018) 267–272 Author / Structural IntegrityStructural Procedia Integrity 00 (2018)13000–000

270 4

Fig. 1. Stress-strain curve in real and fictitious materials.

Nisitani and Hyakutake (1985) examined the fracture of symmetrical polycarbonate notched geometries with several radii, notch depths and thickness. The fictitious strength has been determined using expression (6). Fracture toughness does not appear in the previous article and has been taken from Kinloch and Young (1983). The nondimensional notch stress intensity factors, shown in Figure 2, have been calculated with expression (7), where max is the stress at the root of the notch. The maximum value of Lr is collected in Table 1. 𝐾𝐾𝐼𝐼𝑅𝑅 =

𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚 2

√𝜋𝜋𝜋𝜋

(7)

Lee at al, (2002) analysed the behaviour of notched geometries of A508 steel at -196ºC. Prismatic specimens of 10x10x55 mm were tested with a radius varying between 0.06 and 0.28 mm. The plastic stress-strain curve of the material does not appear in the original work and has been estimated from the elastic limit, y, and the maximum stress, u, following the work of Kamaya (2016). The corresponding notch stress intensity factors have been obtained from expression (7). The mechanical properties of the steel and the maximum values of L r are shown in Table 1. Fuentes et al (2018) studied the influence of the notch radius on the fracture toughness of an Al7075-T651 aluminum alloy at directions TL and LT. The experimental program consisted of compact specimens of W = 40 mm, with notch radii of 0.15, 0.2, 0.5, 1.0, 2.0 mm. The stress-strain curve of the material and the values of the plastic collapse loads can be found in the original work and the fictitious fracture strength has been calculated using expression (6). The notch stress intensity factors of the geometries have been calculated using the expression (7) and appear in Figure 2. The same authors, Madrazo et al (2014), have analyzed the failure of notched geometries of S355 steel at -196 ° C. As in the previous case, compact specimens have been tested with W = 50 mm and notch radii: 0, 0.15, 0.25, 0.5, 1.0, 2.0 mm. The plastic stress-strain curve did not appear in the original work and it has been estimated from the approximation suggested by Kamaya (2016) in steels. The generalized stress intensity factors have been calculated with the expression (6). The fracture toughness, the fictitious fracture strength and the maximum value of Lr appear in Table 1. Figure 2 shows the non-dimensional notch stress intensity factor for the five materials analyzed. In all cases, the new values coincide with the non-dimensional curve obtained in linear elastic materials, with the fitting proposed in the expression (5) and with the predictions of the cohesive crack model. Table 1 shows the fracture and mechanical properties used and the maximum values of Lr. This physical magnitude establishes the degree of plasticity achieved at failure; in all cases it is less than 0.97.

Author name / Structural Integrity Procedia 00 (2018) 000–000

5

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271

Table 1. Materials properties and maximum Lr values. Material PMMA (Gómez) PC (Nisitani) Acero -196°C (Lee) Aluminio Al7075-T651-TL (Madrazo) Aluminio Al7075-T651-LT (Madrazo) Acero S355 -196°C (Madrazo)

KIC (MPa m0.5)

y (MPa) 43.9 58 918

1.04 2.2 46.7

539

27

554

27

853.5

31.3

f (MPa) 142 94.5 3291 2709 2727 3865

Lr <0.87 <0.97 <0.66 <0.84 <0.66 <0.87

10

Cohesive model Expression (5)

K

R

6

IC

/K

IC

8

4

2

0 0.01

0.1

R/l

1

10

100

ch

Alumina

SiC

Mg-PSZ

2.5Y-TZP

Alumina + 7% Zr

Si policrystalline

Y-PSZ

3Y-TZP

Si N

Si monocrystalline

2Y-TZP

PMMA -60ºC

3

4

PMMA RT

Al7075-T651

PC

Steel S355 -196C

Steel A508 -196C Fig. 2. Non-dimensional generalized stress intensity factor in elastoplastic materials (red) and linear elastic materials (black)

272 6

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5. Conclusions This work has proposed a procedure to estimate the maximum load of U-notched components in elastoplastic materials where Linear Elastic Fracture Mechanics is not directly applicable. The Equivalent Material Concept and a failure theory, as the cohesive crack model or phenomenological failure criteria has been combined. The application of the proposed methodology requires the material stress-strain curve and the fracture toughness. The approach used has been validated successfully in five materials: PMMA, polycarbonate, A504 steel, Al7075T651 aluminum and S355 steel. The results obtained are valid under mode I loading and small scale yielding conditions. To quantify this limitation, it has been suggested to use the ratio between the maximum load and the plastic collapse one. In the materials and geometries studied, this value was below 0.97. References Creager, M., Paris, P.C., 1967. Elastic field equations for blunt cracks with reference to stress corrosion cracking. International Journal of Fracture Mechanics 3, 247–252. FITNET, 2007. European Fitness-for-Service Network, EU´s Framework 5, Proposal No. GTC1-2001-43049, Contract No. G1RT-CT-200105071. Fuentes, D., Cicero, s., Berto, F., Torabi, A.R., Madrazo, V., Azizi, P., 2018. Estimation of Fracture Loads in AL7075-T651 Notched Specimens Using the Equivalent Material Concept Combined with the Strain Energy Density Criterion and with the Theory of Critical Distances. Metals 8, 87; doi:10.3390/met8020087. Glinka, G., 1985. Energy density approach to calculation of inelastic strain-stress near notches and cracks. Engineering Fracture Mechanics 22, 485–508. Glinka, G., Newport, A., 1987. Universal features of elastic notch-tip stress fields. International Journal of Fatigue 9, 143–150. Gómez, F.J., Elices, M., 2006. Fracture loads for ceramic samples with rounded notches. Engineering Fracture Mechanics 73, 880–894. Gómez, F.J., Elices, M., Planas, J., 2005. The cohesive crack concept: application to PMMA at –60˚C. Engineering Fracture Mechanics 72, 1268–1285. Gómez, F.J., Elices, M., Valiente, A., 2000. Cracking in PMMA containing U-shaped notches, Fatigue Fracture Engineering Material and Structures 23, 795–803. Irwin, G.R., 1957. Analysis of Stresses and Strain Near the End of a Crack Traversing Plate. Journal of Applied Mechanics 24, 361–364. Kamaya, M., 2016. Ramberg-Osgood type stress-strain curve estimation using yield and ultimate strengths for failure assessments. International Journal of Pressure Vessels and Piping 137, 1–12. Kinloch, A.J., Young, R.J., 1983. Fracture Behaviour of Polymers, Elsevier Applied Science Publishers. Lazzarin, P., Berto, F., 2005. Some expressions for the strain energy in a finite volume surrounding the root of blunt V-notches. International Journal of Fracture 135, 161–185. Lee, B.W., Jang, J., Kwon, D., 2002. Evaluation of fracture toughness using small notched specimens. Materials Science and Engineering A334, 207–214. Madrazo, V., Cicero, S., García, T., 2014. Assessment of notched structural steel components using failure assessment diagrams and the theory of critical distances. Engineering Failure Analysis 36, 104–120. Neuber, H., 1958. Theory of Notch Stresses: Principles for Exact Calculation of Strength with Reference to Structural form and Material, second ed., Springer Verlag, Berlin. Nisitani, H., Hyakutake, H., 1985. Condition for determining the static yield and fracture of a polycarbonate plate specimen with notches. Engineering Fracture Mechanics 22, 359–368. Seweryn, A., Lukaszewicz, A., 2002. Verification of brittle fracture criteria for elements with V-shaped notches. Engineering Fracture Mechanics 69, 1487–1510. Susmel, L., Taylor, D., 2008. The theory of critical distances to predict static strength of notched brittle components subjected to mixed-mode loading. Engineering Fracture Mechanics 75, 534–550. Torabi, A.R., 2012. Estimation of tensile load-bearing capacity of ductile metallic materials weakened by a V-notch: the equivalent material concept. Material Science and Engineering A 536, 249–255. Torabi, A.R., 2013. Ultimate bending strength evaluation of U-notched ductile steel samples under large-scale yielding conditions. International Journal of Fracture 180, 261–268.