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Evaluation of Size Effect in Low Cycle Fatigue for Q&T rotor steel
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3rd 3rd International International Symposium Symposium on on Fatigue Fatigue Design Design and and Material Material Defects, Defects, FDMD FDMD 2017, 2017, 19-22 19-22 September September 2017, 2017, Lecco, Italy Lecco, Italy
Evaluation of Size Effect in Low Cycle Fatigue for Q&T rotor steel
of Size Effect in Low Cycle Fatigue for2016, Q&T rotor steel Portugal XVEvaluation Portuguese Conference on Fracture, PCF 2016, 10-12 February Paço de Arcos, a, b, b b S.P. S.P. Zhu Zhua, b, *, *, S. S. Foletti Folettib,, S. S. Beretta Berettab
Thermo-mechanical modeling of aScience high pressure turbine blade of an School and School of of Mechatronics Mechatronics Engineering, Engineering, University University of of Electronic Electronic Science and Technology Technology of of China, China, Chengdu Chengdu 611731, 611731, China China Department of Mechanical Engineering, Politecnico di Milano, Via La Masa 1, 20156 Milan, Italy Department of Mechanical Engineering, gas Politecnico di Milano, Via La Masa 1, 20156 Milan, Italy airplane turbine engine a a
Abstract Abstract
b b
P. Brandãoa, V. Infanteb, A.M. Deusc*
This modeling of effect of steel cycle fatigue (LCF) Three types of a presents This study study presents aa probabilistic probabilistic of size size effect Superior of 30NiCrMoV12 30NiCrMoV12 steel in in aa low lowde cycle fatigue (LCF) regime. regime. typesLisboa, of fatigue fatigue Department of Mechanicalmodeling Engineering, Instituto Técnico, Universidade Lisboa, Av. Rovisco Pais, 1,Three 1049-001 tests have been conducted on three geometrical specimens. Results indicate that nearly all of the LCF lifetime of 30NiCrMoV12 consists tests have been conducted on three geometrical specimens. Results indicate that nearly all of the LCF lifetime of 30NiCrMoV12 consists of of the the Portugal development multiple short which shown statistical aspects their coalescence. gives rise b development ofDepartment multiple surface surface short cracks, cracks, which has hasInstituto shown Superior statisticalTécnico, aspects of of their mutual mutualdeinteractions interactions and coalescence. This gives Lisboa, rise to to IDMEC,of of Mechanical Engineering, Universidade Lisboa, Av.and Rovisco Pais, 1, This 1049-001 an Portugal weakest an approach approach of of LCF LCF damage damage based based on on statistical statistical physics. physics. A A statistical statistical Weibull’s Weibull’s weakest link link model model has has been been verified verified and and aa procedure procedure for for surface surface c multiple fracture simulation then by the processes initiation, coalescence CeFEMA, Department Engineering, Instituto Superior Técnico,of Universidade de Lisboa, and Av. Rovisco Pais,of 1049-001surface Lisboa, multiple fracture simulationofis isMechanical then established established by including including the random random processes of initiation, propagation propagation and coalescence of1,dispersed dispersed surface cracks. Using this procedure, the life of a component can be predicted from the criterion of critical cracks formation by coalescence Portugal cracks. Using this procedure, the life of a component can be predicted from the criterion of critical cracks formation by coalescence of of dispersed dispersed defects. defects. © 2017 The Elsevier Copyright ©Authors. 2017 ThePublished Authors. by Published Elsevier B.V. Abstract © 2017 The Authors. Published by ElsevierbyB.V. B.V. Peer-review under responsibility of the Scientific Committee of the 3rd International Symposium on Fatigue Design and Material Defects. Peer-review under responsibility of Peer-review under responsibility of the the Scientific Scientific Committee Committee of of the the 3rd 3rd International International Symposium Symposium on on Fatigue Fatigue Design Design and and Material Material Defects. Defects.
During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, parts to undergo different types of time-dependent degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict the creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation 1. Introduction 1. company, Introduction were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were Fatigue and and fracture fracture tests tests are are normally normally conducted conducted on on small small test test specimens specimens of of structural materials materials used used for for aircrafts aircrafts and and highhighFatigue obtained. The data that was gathered was fed into the FEM model and differentstructural simulations were run, first with a simplified 3D speed trains. predicting of scale or as speed train the speed trains. In Inblock predicting the fatigue life of full full scale components components or structures, structures, such as high high speedobtained train railway railway axles, the specimen specimen rectangular shape,the in fatigue order tolife better establish the model, and then withsuch the real 3D mesh fromaxles, the blade scrap. The size effect is when the testing of standard specimens reference other words, size effectexpected is critical criticalbehaviour when utilizing utilizing the laboratory laboratory testingwas of small small standard specimens as the reference basis. Inblade. other Therefore words, fatigue fatigue overall in terms of displacement observed, in particular at as thethe trailing edgebasis. of theIn such a testing on large specimens for those structures is not always possible due to financial considerations. Therefore, characterizing model be specimens useful in the of structures predictingisturbine bladepossible life, given setfinancial of FDR considerations. data. testing oncan large forgoal those not always duea to Therefore, characterizing the the effect effect of of specimen specimen size size on on fatigue fatigue life life is is needed needed and and corresponding corresponding methods methods are are lacking, lacking, especially especially aa robust robust probabilistic probabilistic method method for the of size 2016 The Authors. by Elsevier B.V. life. for©quantifying quantifying the effect effectPublished of specimen specimen size on on fatigue fatigue life. According to the German FKM guideline launched for of Peer-review responsibility of the Scientific Committee of PCFassessment 2016. According tounder the German FKM guideline launched for the the strength strength assessment of structures structures under under different different loading loading conditions, conditions, the the treatment treatment of of size size effect effect is is purely purely empirical, empirical, which which is is determined determined from from empirical empirical design design curves curves or or formulae formulae [1-3]. [1-3]. The The application application of this to cross-sectional shapes and stress is not High Pressure Blade; Creep; Finite Method; 3D Model; Simulation. of Keywords: this size size effect effect to other other Turbine cross-sectional shapes andElement stress distributions distributions is often often not explained explained as as well well as as the the robustness robustness of of the the low low cycle fatigue (LCF) resistance against specimen size [4]. One of the commonly-used way to characterize the size effect by cycle fatigue (LCF) resistance against specimen size [4]. One of the commonly-used way to characterize the size effect by means means of the the weakest-link weakest-link theory theory and and the the statistics statistics of of extremes. extremes. Therefore, Therefore, to to explain explain the the specimen specimen size size effect effect toward toward increasing increasing reliability reliability of of of fatigue fatigue critical critical components, components, metallurgical metallurgical and and mechanical mechanical details details must must be be considered considered in in fatigue fatigue modelling modelling and and lifing lifing procedure. procedure. Note Note from from [5, [5, 6] 6] that that LCF LCF life life of of ductile ductile steels steels is is generally generally dominated dominated by by crack crack propagation propagation life life rather rather than than crack crack initiation initiation life. life. Namely, the effect of specimen size on the fatigue propagation rate, particularly on the scatter in small fatigue crack Namely, the effect of specimen size on the fatigue propagation rate, particularly on the scatter in small fatigue crack growth, growth, has has been been viewed viewed as as the the main main factor factor influencing influencing the the fatigue fatigue life. life. The The challenge challenge will will be be to to use use not not only only the the governing governing factors, factors, such such as as Keywords: size low cycle crack statistics of extremes; crack Keywords: size effect; effect; low pressure cycle fatigue; fatigue; crack growth; growth; extremes; crack coalescence coalescence especially the high turbine (HPT)statistics blades.ofSuch conditions cause these
* Corresponding author. Tel.: +351 218419991.
* Corresponding Corresponding [email protected] E-mail address: address: [email protected] [email protected] E-mail address: * author. E-mail 2452-3216 © © 2017 The The Authors. Published Published by by Elsevier Elsevier B.V. B.V. 2452-3216 2452-32162017 © 2016 Authors. The Authors. Published by Elsevier B.V. Peer-review under responsibility of Scientific Committee of International Symposium on Fatigue Design and Material Defects. Peer-review under responsibility of the the of Scientific Committee of the the 3rd 3rdof International Peer-review under responsibility the Scientific Committee PCF 2016.Symposium on Fatigue Design and Material Defects.
2452-3216 Copyright 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of the 3rd International Symposium on Fatigue Design and Material Defects. 10.1016/j.prostr.2017.11.101
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local microstructure, local stress/strain, and damage physics, but also statistical approaches to understand the size effect in a LCF regime. According to this, this paper attempts to quantify the specimen size effect on fatigue lives of 30NiCrMoV12 steel from the viewpoint of mechanical-probabilistic modelling and numerical simulation of fatigue crack growth. In this paper, an alternative procedure for practical fatigue design by considering size effect will be critically investigated. 2. Experimental 2.1. Material Low cycle fatigue tests have been conducted on specimens with three different geometries of similar shape, which are designed according to ASTM standard E606 [7] and prepared by electro-chemical polishing [8]. Figure 1 shows one of the specimen geometries used for the LCF tests. Three different specimen geometries have been manufactured to study statistical size effects. The diameter (𝐷𝐷𝐷𝐷0 ) and the gauge length (𝐿𝐿𝐿𝐿0 ) of the specimen are listed in Table 1. The specimens were made of 30NiCrMoV12 steel, which is commonly used for railway axles due to high mechanical properties and high performance. Its heat treatment includes normalization at 900℃ for uniforming and grain size refinement, then a quench treatment at 850℃ and cooling in oil bath, finally a tempering process taking at 625℃ for 12 hours in order to increase toughness. Monotonic mechanical properties of 30NiCrMoV12 are reported in Table 2. Table 2. Mechanical properties of 30NiCrMoV12
Table 1. Three specimen geometries Geometry
𝑫𝑫𝟎𝟎
G1: Standard
8
𝐿𝐿𝐿𝐿0 /𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
3
8
Yield stress 𝜎𝜎𝜎𝜎𝑦𝑦𝑦𝑦
36
Elongation at fracture
G2: Small 𝑳𝑳𝟎𝟎
Elastic modulus 𝐸𝐸𝐸𝐸
𝐷𝐷𝐷𝐷0 /𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
G3: Large
20
14
Ultimate tensile strength 𝜎𝜎𝜎𝜎𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈
Figure 1. Standard specimen geometry for LCF tests
197𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
878𝑀𝑀𝑀𝑀𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
1045𝑀𝑀𝑀𝑀𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 21.6%
2.2. Testing For the investigation of LCF lifetime behaviour, MTS fatigue testing machines with maximum loads of 100 kN and 250 kN have been used for tension-compression testing. The strain range was controlled by three extensometers. All LCF tests were carried out at strain amplitude levels between 0.35% and 0.8% at load ratio 𝑅𝑅𝑅𝑅 = −1. Fatigue lifetime of the specimen is defined according to the load drop criterion. In this analysis, three types of LCF tests were performed, including typical LCF tests and replica tests on smooth/notched specimens. The latter were conducted to observe the crack length for crack growth/evolution modelling on the specimen surface, particularly, some tension-compression tests were interrupted at fixed cycles and plastic replicas of the specimen surfaces were made. The quantitative crack length and numbers were determined by optical microscopy. All replica tests were performed on smooth and notched standard specimens at strain amplitude 𝜀𝜀𝜀𝜀𝑎𝑎𝑎𝑎 = 0.5%. For the notched standard specimen, two micro holes of 100𝜇𝜇𝜇𝜇𝑚𝑚𝑚𝑚 diameter were drilled with an angular distance of 120° and 1.5 millimeter far from the centre line of the specimen. Replicas were taken during three tests on the standard specimens, three tests on the big specimen and six interrupted tests for the small specimens. In this study, the surface cracks with length longer than 20𝜇𝜇𝜇𝜇𝑚𝑚𝑚𝑚 were measured for statistical analysis. 2.3. Results
In LCF analysis, a so-called strain-life approach, like the Coffin-Manson (CM) equation, is often used to describe the relationship between strain and the number of cycles to failure under uniaxial loadings, which relates the local elastic-plastic behaviour by ∆𝜀𝜀𝜀𝜀𝑡𝑡𝑡𝑡 2
=
∆𝜀𝜀𝜀𝜀𝑒𝑒𝑒𝑒 2
+
∆𝜀𝜀𝜀𝜀𝑝𝑝𝑝𝑝 2
=
𝜎𝜎𝜎𝜎𝑓𝑓𝑓𝑓′ 𝐸𝐸𝐸𝐸
𝑏𝑏𝑏𝑏
�2𝑁𝑁𝑁𝑁𝑓𝑓𝑓𝑓 � + 𝜀𝜀𝜀𝜀𝑓𝑓𝑓𝑓′ �2𝑁𝑁𝑁𝑁𝑓𝑓𝑓𝑓 �
𝑐𝑐𝑐𝑐
(1)
where ∆𝜀𝜀𝜀𝜀𝑡𝑡𝑡𝑡 , ∆𝜀𝜀𝜀𝜀𝑒𝑒𝑒𝑒 and ∆𝜀𝜀𝜀𝜀𝑝𝑝𝑝𝑝 are the total strain range, elastic strain range and plastic strain range, respectively; 𝜎𝜎𝜎𝜎𝑓𝑓𝑓𝑓′ is the fatigue strength coefficient; 𝐸𝐸𝐸𝐸 is the Young’s modulus; 𝑁𝑁𝑁𝑁𝑓𝑓𝑓𝑓 is the number of cycles to failure; 𝑏𝑏𝑏𝑏 is the fatigue strength exponent; 𝜀𝜀𝜀𝜀𝑓𝑓𝑓𝑓′ is the fatigue ductility coefficient; 𝑐𝑐𝑐𝑐 is the fatigue ductility exponent. When using Eq. (1) for LCF analysis, the Ramberg-Osgood (RO) equation described the stress-strain behaviour of the material ∆𝜀𝜀𝜀𝜀𝑡𝑡𝑡𝑡 2
=
∆𝜀𝜀𝜀𝜀𝑒𝑒𝑒𝑒 2
+
∆𝜀𝜀𝜀𝜀𝑝𝑝𝑝𝑝 2
=
∆𝜎𝜎𝜎𝜎 2𝐸𝐸𝐸𝐸
+�
1
∆𝜎𝜎𝜎𝜎 𝑛𝑛𝑛𝑛′ � 2𝐾𝐾𝐾𝐾′
(2)
where ∆𝜎𝜎𝜎𝜎 is the tress range; 𝐾𝐾𝐾𝐾 ′ and 𝑛𝑛𝑛𝑛′ are the cyclic strength coefficient and cyclic strain hardening exponent, respectively. Using Eq. (1) and Eq. (2), cyclic response and fatigue behaviour of three specimen sizes can be plotted as shown in Figure 2.
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S.P. Zu et al. / Structural Integrity Procedia 00 (2017) 000–000 (a)
(b)
Coffin-Manson curve (standard) Coffin-Manson curve (small)
700
600
Standard spec (Ø8mm) Small spec (Ø3mm)
-2
Big spec (Ø14mm)
500
Total strain [mm/mm]
10
s [Mpa]
400
300
200
Ramberg-Osgood curve (standard) Ramberg-Osgood curve (small)
100
Standard spec (Ø8mm) Small spec (Ø3mm)
10
-3
10
3
10
2N
f
4
10
0
5
Big spec (Ø14mm)
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Total strain [mm/mm]
[cycle]
Figure 2. LCF tests of three different specimens of 30NiCrMoV12
3. Size effect in LCF based on weakest-link modeling 3.1. Weakest-link theory The weakest link theory was originally developed by Weibull to describe the tensile fracture of brittle materials, which is based on the Weibull probability distribution of failure. Specifically, due to the randomly distributed material defects (non-homogeneities, inclusions, precipitates) in a material per volume unit, the theory states that fatigue crack initiates where the most dangerous defect or the weakest link exists [4, 9, 10]. Thus, a stochastic distribution of defects within specimens/components leads to scatter in the fatigue behaviour of the material. Through relating the effect of load and cross-sectional area with the fatigue life, a classic form of the Weibull distribution for the failure probability can be expressed as 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓 (𝜎𝜎𝜎𝜎, 𝛺𝛺𝛺𝛺) = 1 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 �−
1 ∫ 𝛺𝛺𝛺𝛺0 𝛺𝛺𝛺𝛺
𝑓𝑓𝑓𝑓(𝜎𝜎𝜎𝜎)𝑑𝑑𝑑𝑑𝛺𝛺𝛺𝛺�
where 𝛺𝛺𝛺𝛺0 is the reference volume or surface; 𝑓𝑓𝑓𝑓(𝜎𝜎𝜎𝜎) is a function of the risk of rupture with three parameters 𝑓𝑓𝑓𝑓(𝜎𝜎𝜎𝜎) = �
𝜎𝜎𝜎𝜎−𝜎𝜎𝜎𝜎𝑢𝑢𝑢𝑢 𝑚𝑚𝑚𝑚 𝜎𝜎𝜎𝜎0
�
(3)
(4)
where 𝑚𝑚𝑚𝑚 and 𝜎𝜎𝜎𝜎0 are the shape parameter and scale parameter of the Weibull distribution, respectively; 𝜎𝜎𝜎𝜎𝑢𝑢𝑢𝑢 is the threshold stress. By using Eq. (3) for describing the life (as done by Wormsen et al. [10]), the distribution of experimental results can be derived for specimens with different geometries. By increasing the component volume or surface, the probability of failure increases due to the higher probability of finding a critical defect. Thus, a relation can be obtained for two different specimen sizes under a similar failure probability 𝑁𝑁𝑁𝑁2 𝑁𝑁𝑁𝑁1
𝐴𝐴𝐴𝐴
= � 1� 𝐴𝐴𝐴𝐴2
1 𝑚𝑚𝑚𝑚
(5)
where 𝑁𝑁𝑁𝑁1 is the fatigue life for a specimen with the known surface or cross-sectional area 𝐴𝐴𝐴𝐴1 , 𝑁𝑁𝑁𝑁2 is the estimated fatigue life for the specimen with determined surface or cross-sectional area 𝐴𝐴𝐴𝐴2 . In the current research, 𝐴𝐴𝐴𝐴1 and 𝐴𝐴𝐴𝐴2 can be calculated over the gage section of different specimens. Adopting Eq. (5) and taking the distribution of fatigue life of small specimens as a reference, the distribution of the life for large and standard specimens can be expressed by a function of the probability of failure of small specimens 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓,𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 as 𝜂𝜂𝜂𝜂𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑛𝑛𝑛𝑛𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑑𝑑𝑑𝑑
𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓,𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 1 − �1 − 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓,𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �
𝜂𝜂𝜂𝜂𝑙𝑙𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒
𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓,𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒 = 1 − �1 − 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓,𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �
(6)
(7)
where 𝜂𝜂𝜂𝜂𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 and 𝜂𝜂𝜂𝜂𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒 are the ratio between gauge surfaces of the standard/large specimens and the small ones, respectively. Under a log-normal life distribution of different sizes of specimens, Figure 3 plots life distributions of the standard and large specimens according to Eq. (6) and Eq. (7). Comparing with experimental mean and scatter of the life cycles at 𝜖𝜖𝜖𝜖𝑎𝑎𝑎𝑎 = 0.5%, a consequence of the defects distribution based on statistic of extremes cannot explain well the tested life distributions in Figure 3.
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4.5
Small specimen by testing 4
Standard specimen by testing Large specimen by testing
3.5
Standard specimen by Eq. (6) Large specimen by Eq. (7)
3
PDF
2.5 2 1.5 1 0.5 0 2.5
3
3.5
4
Log
10
4.5
(2N)
Figure 3. Expected life based only on size effect due to statistic of extremes
3.2. Fatigue crack growth life Note from Figure 4 that the inspection of fracture surfaces with SEM show that most cracks initiated at the surface of the specimen. For 30NiCrMoV12 steel, its LCF life is obviously dominated by crack propagation life rather than crack initiation life. Thus, the effect of specimen geometrical size on the fatigue crack propagation rate is critical essential for a theoretical understanding of fatigue lives. (a)
(b)
(c)
Figure 4. Multiple surface cracks of (a) small, (b) standard and (c) large specimen
Fatigue crack growth in the presence of a marked plastic strain range under LCF conditions can be well described by using Tomkins model [11]. Through assuming the amount of crack growth per cycle under tension-compression loadings to be the amount of irreversible shear decohesion which occurs at the crack tip, Tomkins [11] correlates crack growth to the plastic strain amplitude in an exponential law 𝐺𝐺𝐺𝐺 = 𝐺𝐺𝐺𝐺𝑖𝑖𝑖𝑖 𝑒𝑒𝑒𝑒 𝑘𝑘𝑘𝑘𝑙𝑙𝑙𝑙𝑁𝑁𝑁𝑁 � 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘𝑙𝑙𝑙𝑙 = 𝑘𝑘𝑘𝑘𝑙𝑙𝑙𝑙0 𝜀𝜀𝜀𝜀𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎
(8)
where 𝐺𝐺𝐺𝐺 and 𝐺𝐺𝐺𝐺𝑖𝑖𝑖𝑖 are the crack length and initial crack length, respectively; 𝜀𝜀𝜀𝜀𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 is the plastic strain amplitude, 𝑘𝑘𝑘𝑘𝑙𝑙𝑙𝑙0 and 𝑑𝑑𝑑𝑑 are material parameters that establish the short cracks growth in a material. Accordingly, a simplified form can be expressed as 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
𝑠𝑠𝑠𝑠𝑁𝑁𝑁𝑁
(9)
= 𝑘𝑘𝑘𝑘𝑙𝑙𝑙𝑙 𝐺𝐺𝐺𝐺
The number of cycles is obtained by integrating Eq. (9) from an initial crack length 𝐺𝐺𝐺𝐺𝑖𝑖𝑖𝑖 to a final one 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓 . Parameter of Eq. (9) was calibrated from experiments, as shown in Figure 5. Two scale factors can be obtained to shift the distribution found with the statics of extremes for the standard/large specimen from the reference small one 𝛾𝛾𝛾𝛾𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝛾𝛾𝛾𝛾𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒 =
∆𝑁𝑁𝑁𝑁𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ∆𝑁𝑁𝑁𝑁𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙
∆𝑁𝑁𝑁𝑁𝑙𝑙𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒
∆𝑁𝑁𝑁𝑁𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙
=
=
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓 ,𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎�
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖 �
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓 ,𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙�
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖 �
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓 ,𝑙𝑙𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒�
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖 �
�
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓 ,𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙�
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖 �
�
�
�
(10)
(11)
Combining Eq. (6) to Eq. (11), the life scale factors at 𝜖𝜖𝜖𝜖𝑎𝑎𝑎𝑎 = 0.5% due to the propagation size effect can be calculated from an 𝜋𝜋𝜋𝜋𝐷𝐷𝐷𝐷 initial crack length 𝐺𝐺𝐺𝐺𝑖𝑖𝑖𝑖 = 100𝜇𝜇𝜇𝜇𝑚𝑚𝑚𝑚 to a final crack length 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓 = 0, namely, 𝛾𝛾𝛾𝛾𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 1.18048 and 𝛾𝛾𝛾𝛾𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒 = 1.27523. 2 As pointed out by Koyama et al. [5], the specimen size effect can be quantified by the sum of the effect of statistical distribution
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of the defects and the effect of crack propagation. From a crack growth point of view, a larger specimen would lead to a longer life, which shift the dotted/dash lines in Figure 3 to the right-hand of the x-axis. According to this, a comparison between the curves of experimental distribution of the life and the ones due to the effects of statistical defects and crack propagation can be made by using Eq. (10) and Eq. (11), as shown in Figure 6 and results in Table 3. 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄
𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄
Figure 5 Crack growth measurement near the micro hole 4.5
Comparison between experimental data end the sum of the size effects Small specimen experimental
4
Table 3 Mean and scatter of the life cycles at 𝜖𝜖𝜖𝜖𝑎𝑎𝑎𝑎 = 0.5% due to the combination of propagation and statistical distribution of defects
Standard specimen experimental Large specimen experimental
3.5
Standard specimen superposition Large specimen superposition
3
PDF
Standard spec.
2 1.5
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 (𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)
Mean
3.7021
3.7441
Std.
0.0779
1
0 2.5
0.1620
Mean Std.
Large spec.
0.5
3
3.5
4
Log
10
Extracted /
Experimental / 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 (𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)
2.5
3.6807 0.1557
3.6382 0.1759
𝜇𝜇𝜇𝜇% error 1.15% 1.13%
4.5
(2N)
Figure 6. Comparison between extracted distribution and experimental ones
4. Numerical modeling of multiple fracture As aforementioned, the weakest-link theory assumed that the critical defects do not interact which are sparsely distributed. This assumption only works under limited number of potentially critical defect sites. As noticed from Figure 4, the presence of multiple surface cracks scatted over the gauge section is the main manifestation of damages in 30NiCrMoV12 steel under cyclic loadings. In particular, its fracture was induced by the processes of initiation, growth and coalescence of cracks. Combining with the replica tests as mentioned in Section 2, a numerical Monte Carlo simulation is performed for multiple fracture evaluation and to compare experimental data on crack length distribution evolution, as well as their experimental lives, with the simulated ones. 4.1. Initial prerequisites of the model The simulation conducted in this analysis includes the process of random crack initiation, propagation and coalescence of surface microcracks occurring simultaneously. According to available experimental data on replica tests under cyclic loadings, the initial prerequisites and assumptions are taken as follows. (1) Uniform material properties are assumed over the entire damaged surface, namely, the simulation surface area; (2) Uniaxial stress state is loaded and time-independent during the crack evolution; (3) All cracks on the surface are oriented normally to the stress action direction, like the x-axis in this analysis; (4) A constant increment of the load cycles is assigned as the count number of iterations; (5) The positions of all cracks on the surface are randomly distributed according to Poisson’s law. Particularly, the probability that n cracks are located on the area A with the crack density 𝜆𝜆𝜆𝜆 𝐺𝐺𝐺𝐺(𝑛𝑛𝑛𝑛) =
(−𝜆𝜆𝜆𝜆𝐴𝐴𝐴𝐴)𝑛𝑛𝑛𝑛 𝑠𝑠𝑠𝑠!
(12)
𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒(−𝜆𝜆𝜆𝜆𝐴𝐴𝐴𝐴)
(6) The initial length 𝐺𝐺𝐺𝐺 of the crack is assigned by a two-parameter Weibull distribution; 𝐺𝐺𝐺𝐺(𝐺𝐺𝐺𝐺) = 1 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 �− �
𝑎𝑎𝑎𝑎−𝑎𝑎𝑎𝑎0 𝛽𝛽𝛽𝛽 𝛼𝛼𝛼𝛼
� �
(13)
where 𝛽𝛽𝛽𝛽 and 𝛼𝛼𝛼𝛼 are the Weibull shape parameter and scale parameter, which were obtained from measurements on the plastic replicas, and 𝐺𝐺𝐺𝐺0 is the resolution of crack identification. (7) In crack propagation, the crack length increment for any crack within a given number of iterations is prescribed; (8) In crack coalescence, quasi-collinear cracks in close proximity interact through the influence zones at both tips, which shows the changes in their propagation paths when their tips approach one another [12]. The influence zone is defined by a circle of diameter 𝑟𝑟𝑟𝑟𝑝𝑝𝑝𝑝 according to the size of local plastic deformation zone at the crack tip [13]
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𝑟𝑟𝑟𝑟𝑝𝑝𝑝𝑝 =
𝛥𝛥𝛥𝛥𝐽𝐽𝐽𝐽𝑒𝑒𝑒𝑒𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ∙𝐸𝐸𝐸𝐸 ′
373
(14)
2𝜋𝜋𝜋𝜋𝜎𝜎𝜎𝜎02
where ∆𝐽𝐽𝐽𝐽𝑒𝑒𝑒𝑒𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 is the effective cyclic J-Integral, 𝜎𝜎𝜎𝜎0 is the flow stress and 𝐸𝐸𝐸𝐸 ′ is the cyclic Young’s Modulus under plane strain conditions, 𝐸𝐸𝐸𝐸 ′ = 𝐸𝐸𝐸𝐸 ⁄(1 − 𝑣𝑣𝑣𝑣 2 ) and 𝑣𝑣𝑣𝑣 is the Poisson’s ratio. The criterion of Murakami [14] on judging defects interaction/coalescence is adopted, by the distance between the tips 𝑑𝑑𝑑𝑑 is less than the minimum of the two crack length (15)
𝑑𝑑𝑑𝑑 < 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑛𝑛𝑛𝑛(𝐺𝐺𝐺𝐺𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐𝑘𝑘𝑘𝑘1 , 𝐺𝐺𝐺𝐺𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐𝑘𝑘𝑘𝑘2 )
(9) An unloaded material zone during crack propagation and coalescence is formed around each crack, of which the shape of the zone is defined as a circle of diameter equal to the crack length. In these zones, the crack growth is arrested if its tip comes into the unloading zone from the nearest crack. Based on abovementioned starting prerequisites, a simulation flow chart is given in Figure 7, the basic inputs obtained from experimental testing for multiple fracture simulation of different geometrical specimens include the following parameters: (1) Crack density on the surface; (2) Damage surface size; (3) Distribution of crack length and crack growth rate (without regard for the crack coalescence) on the surface; (4) Fracture and failure criterion. Start Input source data 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝐺𝐺𝐺𝐺𝐺𝐺 2𝐸𝐸𝐸𝐸𝐸 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝜆𝜆𝜆𝜆, 𝛼𝛼𝛼𝛼, 𝛽𝛽𝛽𝛽, 𝐴𝐴𝐴𝐴, 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓 , 𝑘𝑘𝑘𝑘𝑙𝑙𝑙𝑙 , 𝜀𝜀𝜀𝜀𝑎𝑎𝑎𝑎 , 𝜎𝜎𝜎𝜎0 , 𝐸𝐸𝐸𝐸 For 𝑚𝑚𝑚𝑚 = 1 to total simulation number (102 ) Monte Carlo sampling
Random crack initiation Crack position: Poisson’s law Crack length: Weibull distribution For 𝑗𝑗 = 1 iteration Δ𝑁𝑁𝑁𝑁/𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
Crack coalescence analysis Unloaded and influence zone identification Crack propagation type Calculation of crack tip distance Merge cracks or change crack tip angles 𝑗𝑗 = 𝑗𝑗 + 1
Crack propagation Increment of crack length 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 Crack growth rate: random variable vs. random process Track the largest critical crack 𝐺𝐺𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚,𝑗𝑗 If maximum crack length 𝐺𝐺𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚,𝑗𝑗 > 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓
No
Yes
Calculation of life 𝑁𝑁𝑁𝑁𝑓𝑓𝑓𝑓,𝑖𝑖𝑖𝑖 = 2𝐸𝐸𝐸𝐸𝐸 + 𝑗𝑗∆𝑁𝑁𝑁𝑁 Continue?
Life distribution 𝑵𝑵𝒇𝒇 End
Figure 7. Main flow chart of multiple fracture simulation
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4.2. Simulation results and discussions During simulation, the material damaging under cyclic loading can be characterized by crack initiation, propagation and coalescence, which are dispersed over the surface with different lengths as shown in Figure 8. Two mechanisms of crack propagation, namely, crack length increment due to its own growth for those cracks without coalescence and sudden crack length increment due to the neighbouring cracks coalescence through interacting by their tips, are involved in this simulation. Accordingly, the largest crack exhibits jump-like growth due to merging with other cracks in the plastic radius. Through following the procedure in Section 4.1, an example probability plot of experimental and simulated lives on small specimens is presented in Figure 9. Note that the simulated life distribution covers well with the lives of small specimens except that of G2.13 and G2.14, in which their fracture surfaces in Figure 10 have shown the worst condition on multiple surface crack interaction and propagation. (a)
(b)
Figure 8. Scheme of crack distribution and unloaded zones (a) Surface crack simulation at 𝜀𝜀𝜀𝜀𝑎𝑎𝑎𝑎 = 0.5% (circles mark the unloaded zones, square marks the crack coalescence) and (b) Replica on specimen G 2.12 at 3E3 cycles (a)
3
(b)
Log-Normal probability plot Small spec. life comparison between simulated and experimental life
Log-normal distributions of life comparison between small specimens
8
Simulated point
Experimental point Fitted curve
2
Experimental point
7
95% confidence band 5% confidence band
Experimental distribution Simulated distribution
6
Simulated point
1
𝑮𝑮𝑮𝑮. 𝟏𝟏𝟏𝟏
𝑮𝑮𝑮𝑮. 𝟏𝟏𝟏𝟏
-1
PDF
z=
1
(q)
5
0
4 3 2
-2
1 -3
3
3.1
3.2
3.3
3.4
3.5
Log
10
3.6
3.7
3.8
3.9
4
0
3
3.1
3.2
3.3
Cycle
3.4
3.5
Log
10
3.6
3.7
3.8
3.9
4
Cycle
Figure 9. Comparison between experimental and simulated lives of small specimen: (a) probability plot and (b) life distribution
(a) G2.13 (b) G2.14 Figure 10. Fracture surface of small specimens
5. Conclusions In this study, we proposed a probabilistic modelling and numerical simulation procedure for evaluating the effects of specimen size on fatigue life with three different geometrical specimens. Specifically, the specimen size effect is quantified from two categorizes, namely, size effect due to the effects of statistical defects and crack propagation, which provides a reference for fatigue
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life estimation under different specimen sizes. Moreover, a simulation procedure for surface multiple fracture is established by including the random processes of initiation, propagation and coalescence of dispersed surface cracks, which accounts for the observed experimental scatter in fatigue lifetime of small specimen of 30NiCrMoV12 steel. Using this procedure, the life of a component can be predicted from the criterion of critical cracks formation by coalescence of dispersed defects. Acknowledgments Dr. S.P. Zhu acknowledges support for his period of study at Politecnico di Milano by the Polimi International Fellowship Grant scheme. The authors acknowledged support by Lucchini RS (Italy) for supplying the heat treated steel. References [1] O. Hertel, M. Vormwald, “Statistical and geometrical size effects in notched members based on weakest-link and short-crack modelling,” Eng. Fract. Mech., vol. 95, pp. 72–83, 2012. [2] M. Makkonen, “Statistical size effect in the fatigue limit of steel,” Int. J. Fatigue, vol. 23, no. 5, pp. 395–402, 2001. [3] S.P. Zhu, S. Foletti, S. Beretta, “Probabilistic framework for multiaxial LCF assessment under material variability,” Int. J. Fatigue, in press, https://doi.org/10.1016/j.ijfatigue.2017.06.019, 2017. [4] T. Tomaszewski, J. Sempruch, T. Piatkowski, “Verification of selected models of the size effect based on high-cycle fatigue testing on mini specimens made of EN AW-6063 Aluminum alloy,” J. Theor. Appl. Mech., vol. 52, no. 4, pp. 883–894, 2014. [5] M. Koyama, H. Li, Y. Hamano, and T. Sawaguchi, “Mechanical-probabilistic evaluation of size effect of fatigue life using data obtained from single smooth specimen : An example using Fe-30Mn-4Si-2Al seismic damper alloy,” Eng. Fail. Anal., vol. 72, pp. 34–47, 2017. [6] Y. Murakami, K. J. Miller, “What is fatigue damage ? A view point from the observation of low cycle fatigue process,” Int. J. Fatigue, vol. 27, pp. 991–1005, 2005. [7] “ASTM Standard E606. Standard Practice for Strain-Controlled Fatigue Testing,” Annu. B. ASTM Stand. 3, ASTM Int., 2004. [8] T. S. Hahn, A. R. Marder, “Effect of electropolishing variables on the current density- voltage relationship,” Metallography, vol. 21, pp. 365–375, 1988. [9] F. W. Zok, “On weakest link theory and Weibull statistics,” J. Am. Ceram. Soc., vol. 100, pp. 1265–1268, 2017. [10] A. Wormsen, B. Sjödin, G. Härkegård, and A. Fjeldstad, “Non-local stress approach for fatigue assessment based on weakest-link theory and statistics of extremes,” Fatigue Fract. Eng. Mater. Struct., vol. 30, no. 12, pp. 1214–1227, 2007. [11] Tomkins B, “Fatigue crack propagation - an analysis,” Philos. Mag., vol. 18, no. 155, pp. 1041–1066, 1968. [12] H.Y. Yoon, S.C. Lee, “Probabilistic distribution of fatigue crack growth life considering effect of crack coalescence,” JSME Int. J. Ser. A Solid Mech. Mater. Eng., vol. 46, no. 4, pp. 607–612, 2003. [13] S. Rabbolini, S. Beretta, S. Foletti, M. E. Cristea, “Crack closure effects during low cycle fatigue propagation in line pipe steel: An analysis with digital image correlation,” Eng. Fract. Mech., vol. 148, pp. 441–456, 2015. [14] Y. Murakami, Metal fatigue: effects of small defects and nonmetallic inclusions. Elsevier, 2002.
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3rd 3rd International International Symposium Symposium on on Fatigue Fatigue Design Design and and Material Material Defects, Defects, FDMD FDMD 2017, 2017, 19-22 19-22 September September 2017, 2017, Lecco, Italy Lecco, Italy
Evaluation of Size Effect in Low Cycle Fatigue for Q&T rotor steel
of Size Effect in Low Cycle Fatigue for2016, Q&T rotor steel Portugal XVEvaluation Portuguese Conference on Fracture, PCF 2016, 10-12 February Paço de Arcos, a, b, b b S.P. S.P. Zhu Zhua, b, *, *, S. S. Foletti Folettib,, S. S. Beretta Berettab
Thermo-mechanical modeling of aScience high pressure turbine blade of an School and School of of Mechatronics Mechatronics Engineering, Engineering, University University of of Electronic Electronic Science and Technology Technology of of China, China, Chengdu Chengdu 611731, 611731, China China Department of Mechanical Engineering, Politecnico di Milano, Via La Masa 1, 20156 Milan, Italy Department of Mechanical Engineering, gas Politecnico di Milano, Via La Masa 1, 20156 Milan, Italy airplane turbine engine a a
Abstract Abstract
b b
P. Brandãoa, V. Infanteb, A.M. Deusc*
This modeling of effect of steel cycle fatigue (LCF) Three types of a presents This study study presents aa probabilistic probabilistic of size size effect Superior of 30NiCrMoV12 30NiCrMoV12 steel in in aa low lowde cycle fatigue (LCF) regime. regime. typesLisboa, of fatigue fatigue Department of Mechanicalmodeling Engineering, Instituto Técnico, Universidade Lisboa, Av. Rovisco Pais, 1,Three 1049-001 tests have been conducted on three geometrical specimens. Results indicate that nearly all of the LCF lifetime of 30NiCrMoV12 consists tests have been conducted on three geometrical specimens. Results indicate that nearly all of the LCF lifetime of 30NiCrMoV12 consists of of the the Portugal development multiple short which shown statistical aspects their coalescence. gives rise b development ofDepartment multiple surface surface short cracks, cracks, which has hasInstituto shown Superior statisticalTécnico, aspects of of their mutual mutualdeinteractions interactions and coalescence. This gives Lisboa, rise to to IDMEC,of of Mechanical Engineering, Universidade Lisboa, Av.and Rovisco Pais, 1, This 1049-001 an Portugal weakest an approach approach of of LCF LCF damage damage based based on on statistical statistical physics. physics. A A statistical statistical Weibull’s Weibull’s weakest link link model model has has been been verified verified and and aa procedure procedure for for surface surface c multiple fracture simulation then by the processes initiation, coalescence CeFEMA, Department Engineering, Instituto Superior Técnico,of Universidade de Lisboa, and Av. Rovisco Pais,of 1049-001surface Lisboa, multiple fracture simulationofis isMechanical then established established by including including the random random processes of initiation, propagation propagation and coalescence of1,dispersed dispersed surface cracks. Using this procedure, the life of a component can be predicted from the criterion of critical cracks formation by coalescence Portugal cracks. Using this procedure, the life of a component can be predicted from the criterion of critical cracks formation by coalescence of of dispersed dispersed defects. defects. © 2017 The Elsevier Copyright ©Authors. 2017 ThePublished Authors. by Published Elsevier B.V. Abstract © 2017 The Authors. Published by ElsevierbyB.V. B.V. Peer-review under responsibility of the Scientific Committee of the 3rd International Symposium on Fatigue Design and Material Defects. Peer-review under responsibility of Peer-review under responsibility of the the Scientific Scientific Committee Committee of of the the 3rd 3rd International International Symposium Symposium on on Fatigue Fatigue Design Design and and Material Material Defects. Defects.
During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, parts to undergo different types of time-dependent degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict the creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation 1. Introduction 1. company, Introduction were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were Fatigue and and fracture fracture tests tests are are normally normally conducted conducted on on small small test test specimens specimens of of structural materials materials used used for for aircrafts aircrafts and and highhighFatigue obtained. The data that was gathered was fed into the FEM model and differentstructural simulations were run, first with a simplified 3D speed trains. predicting of scale or as speed train the speed trains. In Inblock predicting the fatigue life of full full scale components components or structures, structures, such as high high speedobtained train railway railway axles, the specimen specimen rectangular shape,the in fatigue order tolife better establish the model, and then withsuch the real 3D mesh fromaxles, the blade scrap. The size effect is when the testing of standard specimens reference other words, size effectexpected is critical criticalbehaviour when utilizing utilizing the laboratory laboratory testingwas of small small standard specimens as the reference basis. Inblade. other Therefore words, fatigue fatigue overall in terms of displacement observed, in particular at as thethe trailing edgebasis. of theIn such a testing on large specimens for those structures is not always possible due to financial considerations. Therefore, characterizing model be specimens useful in the of structures predictingisturbine bladepossible life, given setfinancial of FDR considerations. data. testing oncan large forgoal those not always duea to Therefore, characterizing the the effect effect of of specimen specimen size size on on fatigue fatigue life life is is needed needed and and corresponding corresponding methods methods are are lacking, lacking, especially especially aa robust robust probabilistic probabilistic method method for the of size 2016 The Authors. by Elsevier B.V. life. for©quantifying quantifying the effect effectPublished of specimen specimen size on on fatigue fatigue life. According to the German FKM guideline launched for of Peer-review responsibility of the Scientific Committee of PCFassessment 2016. According tounder the German FKM guideline launched for the the strength strength assessment of structures structures under under different different loading loading conditions, conditions, the the treatment treatment of of size size effect effect is is purely purely empirical, empirical, which which is is determined determined from from empirical empirical design design curves curves or or formulae formulae [1-3]. [1-3]. The The application application of this to cross-sectional shapes and stress is not High Pressure Blade; Creep; Finite Method; 3D Model; Simulation. of Keywords: this size size effect effect to other other Turbine cross-sectional shapes andElement stress distributions distributions is often often not explained explained as as well well as as the the robustness robustness of of the the low low cycle fatigue (LCF) resistance against specimen size [4]. One of the commonly-used way to characterize the size effect by cycle fatigue (LCF) resistance against specimen size [4]. One of the commonly-used way to characterize the size effect by means means of the the weakest-link weakest-link theory theory and and the the statistics statistics of of extremes. extremes. Therefore, Therefore, to to explain explain the the specimen specimen size size effect effect toward toward increasing increasing reliability reliability of of of fatigue fatigue critical critical components, components, metallurgical metallurgical and and mechanical mechanical details details must must be be considered considered in in fatigue fatigue modelling modelling and and lifing lifing procedure. procedure. Note Note from from [5, [5, 6] 6] that that LCF LCF life life of of ductile ductile steels steels is is generally generally dominated dominated by by crack crack propagation propagation life life rather rather than than crack crack initiation initiation life. life. Namely, the effect of specimen size on the fatigue propagation rate, particularly on the scatter in small fatigue crack Namely, the effect of specimen size on the fatigue propagation rate, particularly on the scatter in small fatigue crack growth, growth, has has been been viewed viewed as as the the main main factor factor influencing influencing the the fatigue fatigue life. life. The The challenge challenge will will be be to to use use not not only only the the governing governing factors, factors, such such as as Keywords: size low cycle crack statistics of extremes; crack Keywords: size effect; effect; low pressure cycle fatigue; fatigue; crack growth; growth; extremes; crack coalescence coalescence especially the high turbine (HPT)statistics blades.ofSuch conditions cause these
* Corresponding author. Tel.: +351 218419991.
* Corresponding Corresponding [email protected] E-mail address: address: [email protected] [email protected] E-mail address: * author. E-mail 2452-3216 © © 2017 The The Authors. Published Published by by Elsevier Elsevier B.V. B.V. 2452-3216 2452-32162017 © 2016 Authors. The Authors. Published by Elsevier B.V. Peer-review under responsibility of Scientific Committee of International Symposium on Fatigue Design and Material Defects. Peer-review under responsibility of the the of Scientific Committee of the the 3rd 3rdof International Peer-review under responsibility the Scientific Committee PCF 2016.Symposium on Fatigue Design and Material Defects.
2452-3216 Copyright 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of the 3rd International Symposium on Fatigue Design and Material Defects. 10.1016/j.prostr.2017.11.101
S.P. Zhu et al. / Procedia Structural Integrity 7 (2017) 368–375
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local microstructure, local stress/strain, and damage physics, but also statistical approaches to understand the size effect in a LCF regime. According to this, this paper attempts to quantify the specimen size effect on fatigue lives of 30NiCrMoV12 steel from the viewpoint of mechanical-probabilistic modelling and numerical simulation of fatigue crack growth. In this paper, an alternative procedure for practical fatigue design by considering size effect will be critically investigated. 2. Experimental 2.1. Material Low cycle fatigue tests have been conducted on specimens with three different geometries of similar shape, which are designed according to ASTM standard E606 [7] and prepared by electro-chemical polishing [8]. Figure 1 shows one of the specimen geometries used for the LCF tests. Three different specimen geometries have been manufactured to study statistical size effects. The diameter (𝐷𝐷𝐷𝐷0 ) and the gauge length (𝐿𝐿𝐿𝐿0 ) of the specimen are listed in Table 1. The specimens were made of 30NiCrMoV12 steel, which is commonly used for railway axles due to high mechanical properties and high performance. Its heat treatment includes normalization at 900℃ for uniforming and grain size refinement, then a quench treatment at 850℃ and cooling in oil bath, finally a tempering process taking at 625℃ for 12 hours in order to increase toughness. Monotonic mechanical properties of 30NiCrMoV12 are reported in Table 2. Table 2. Mechanical properties of 30NiCrMoV12
Table 1. Three specimen geometries Geometry
𝑫𝑫𝟎𝟎
G1: Standard
8
𝐿𝐿𝐿𝐿0 /𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
3
8
Yield stress 𝜎𝜎𝜎𝜎𝑦𝑦𝑦𝑦
36
Elongation at fracture
G2: Small 𝑳𝑳𝟎𝟎
Elastic modulus 𝐸𝐸𝐸𝐸
𝐷𝐷𝐷𝐷0 /𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
G3: Large
20
14
Ultimate tensile strength 𝜎𝜎𝜎𝜎𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈
Figure 1. Standard specimen geometry for LCF tests
197𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
878𝑀𝑀𝑀𝑀𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
1045𝑀𝑀𝑀𝑀𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 21.6%
2.2. Testing For the investigation of LCF lifetime behaviour, MTS fatigue testing machines with maximum loads of 100 kN and 250 kN have been used for tension-compression testing. The strain range was controlled by three extensometers. All LCF tests were carried out at strain amplitude levels between 0.35% and 0.8% at load ratio 𝑅𝑅𝑅𝑅 = −1. Fatigue lifetime of the specimen is defined according to the load drop criterion. In this analysis, three types of LCF tests were performed, including typical LCF tests and replica tests on smooth/notched specimens. The latter were conducted to observe the crack length for crack growth/evolution modelling on the specimen surface, particularly, some tension-compression tests were interrupted at fixed cycles and plastic replicas of the specimen surfaces were made. The quantitative crack length and numbers were determined by optical microscopy. All replica tests were performed on smooth and notched standard specimens at strain amplitude 𝜀𝜀𝜀𝜀𝑎𝑎𝑎𝑎 = 0.5%. For the notched standard specimen, two micro holes of 100𝜇𝜇𝜇𝜇𝑚𝑚𝑚𝑚 diameter were drilled with an angular distance of 120° and 1.5 millimeter far from the centre line of the specimen. Replicas were taken during three tests on the standard specimens, three tests on the big specimen and six interrupted tests for the small specimens. In this study, the surface cracks with length longer than 20𝜇𝜇𝜇𝜇𝑚𝑚𝑚𝑚 were measured for statistical analysis. 2.3. Results
In LCF analysis, a so-called strain-life approach, like the Coffin-Manson (CM) equation, is often used to describe the relationship between strain and the number of cycles to failure under uniaxial loadings, which relates the local elastic-plastic behaviour by ∆𝜀𝜀𝜀𝜀𝑡𝑡𝑡𝑡 2
=
∆𝜀𝜀𝜀𝜀𝑒𝑒𝑒𝑒 2
+
∆𝜀𝜀𝜀𝜀𝑝𝑝𝑝𝑝 2
=
𝜎𝜎𝜎𝜎𝑓𝑓𝑓𝑓′ 𝐸𝐸𝐸𝐸
𝑏𝑏𝑏𝑏
�2𝑁𝑁𝑁𝑁𝑓𝑓𝑓𝑓 � + 𝜀𝜀𝜀𝜀𝑓𝑓𝑓𝑓′ �2𝑁𝑁𝑁𝑁𝑓𝑓𝑓𝑓 �
𝑐𝑐𝑐𝑐
(1)
where ∆𝜀𝜀𝜀𝜀𝑡𝑡𝑡𝑡 , ∆𝜀𝜀𝜀𝜀𝑒𝑒𝑒𝑒 and ∆𝜀𝜀𝜀𝜀𝑝𝑝𝑝𝑝 are the total strain range, elastic strain range and plastic strain range, respectively; 𝜎𝜎𝜎𝜎𝑓𝑓𝑓𝑓′ is the fatigue strength coefficient; 𝐸𝐸𝐸𝐸 is the Young’s modulus; 𝑁𝑁𝑁𝑁𝑓𝑓𝑓𝑓 is the number of cycles to failure; 𝑏𝑏𝑏𝑏 is the fatigue strength exponent; 𝜀𝜀𝜀𝜀𝑓𝑓𝑓𝑓′ is the fatigue ductility coefficient; 𝑐𝑐𝑐𝑐 is the fatigue ductility exponent. When using Eq. (1) for LCF analysis, the Ramberg-Osgood (RO) equation described the stress-strain behaviour of the material ∆𝜀𝜀𝜀𝜀𝑡𝑡𝑡𝑡 2
=
∆𝜀𝜀𝜀𝜀𝑒𝑒𝑒𝑒 2
+
∆𝜀𝜀𝜀𝜀𝑝𝑝𝑝𝑝 2
=
∆𝜎𝜎𝜎𝜎 2𝐸𝐸𝐸𝐸
+�
1
∆𝜎𝜎𝜎𝜎 𝑛𝑛𝑛𝑛′ � 2𝐾𝐾𝐾𝐾′
(2)
where ∆𝜎𝜎𝜎𝜎 is the tress range; 𝐾𝐾𝐾𝐾 ′ and 𝑛𝑛𝑛𝑛′ are the cyclic strength coefficient and cyclic strain hardening exponent, respectively. Using Eq. (1) and Eq. (2), cyclic response and fatigue behaviour of three specimen sizes can be plotted as shown in Figure 2.
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S.P. Zu et al. / Structural Integrity Procedia 00 (2017) 000–000 (a)
(b)
Coffin-Manson curve (standard) Coffin-Manson curve (small)
700
600
Standard spec (Ø8mm) Small spec (Ø3mm)
-2
Big spec (Ø14mm)
500
Total strain [mm/mm]
10
s [Mpa]
400
300
200
Ramberg-Osgood curve (standard) Ramberg-Osgood curve (small)
100
Standard spec (Ø8mm) Small spec (Ø3mm)
10
-3
10
3
10
2N
f
4
10
0
5
Big spec (Ø14mm)
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Total strain [mm/mm]
[cycle]
Figure 2. LCF tests of three different specimens of 30NiCrMoV12
3. Size effect in LCF based on weakest-link modeling 3.1. Weakest-link theory The weakest link theory was originally developed by Weibull to describe the tensile fracture of brittle materials, which is based on the Weibull probability distribution of failure. Specifically, due to the randomly distributed material defects (non-homogeneities, inclusions, precipitates) in a material per volume unit, the theory states that fatigue crack initiates where the most dangerous defect or the weakest link exists [4, 9, 10]. Thus, a stochastic distribution of defects within specimens/components leads to scatter in the fatigue behaviour of the material. Through relating the effect of load and cross-sectional area with the fatigue life, a classic form of the Weibull distribution for the failure probability can be expressed as 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓 (𝜎𝜎𝜎𝜎, 𝛺𝛺𝛺𝛺) = 1 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 �−
1 ∫ 𝛺𝛺𝛺𝛺0 𝛺𝛺𝛺𝛺
𝑓𝑓𝑓𝑓(𝜎𝜎𝜎𝜎)𝑑𝑑𝑑𝑑𝛺𝛺𝛺𝛺�
where 𝛺𝛺𝛺𝛺0 is the reference volume or surface; 𝑓𝑓𝑓𝑓(𝜎𝜎𝜎𝜎) is a function of the risk of rupture with three parameters 𝑓𝑓𝑓𝑓(𝜎𝜎𝜎𝜎) = �
𝜎𝜎𝜎𝜎−𝜎𝜎𝜎𝜎𝑢𝑢𝑢𝑢 𝑚𝑚𝑚𝑚 𝜎𝜎𝜎𝜎0
�
(3)
(4)
where 𝑚𝑚𝑚𝑚 and 𝜎𝜎𝜎𝜎0 are the shape parameter and scale parameter of the Weibull distribution, respectively; 𝜎𝜎𝜎𝜎𝑢𝑢𝑢𝑢 is the threshold stress. By using Eq. (3) for describing the life (as done by Wormsen et al. [10]), the distribution of experimental results can be derived for specimens with different geometries. By increasing the component volume or surface, the probability of failure increases due to the higher probability of finding a critical defect. Thus, a relation can be obtained for two different specimen sizes under a similar failure probability 𝑁𝑁𝑁𝑁2 𝑁𝑁𝑁𝑁1
𝐴𝐴𝐴𝐴
= � 1� 𝐴𝐴𝐴𝐴2
1 𝑚𝑚𝑚𝑚
(5)
where 𝑁𝑁𝑁𝑁1 is the fatigue life for a specimen with the known surface or cross-sectional area 𝐴𝐴𝐴𝐴1 , 𝑁𝑁𝑁𝑁2 is the estimated fatigue life for the specimen with determined surface or cross-sectional area 𝐴𝐴𝐴𝐴2 . In the current research, 𝐴𝐴𝐴𝐴1 and 𝐴𝐴𝐴𝐴2 can be calculated over the gage section of different specimens. Adopting Eq. (5) and taking the distribution of fatigue life of small specimens as a reference, the distribution of the life for large and standard specimens can be expressed by a function of the probability of failure of small specimens 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓,𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 as 𝜂𝜂𝜂𝜂𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑛𝑛𝑛𝑛𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑑𝑑𝑑𝑑
𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓,𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 1 − �1 − 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓,𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �
𝜂𝜂𝜂𝜂𝑙𝑙𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒
𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓,𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒 = 1 − �1 − 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓,𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 �
(6)
(7)
where 𝜂𝜂𝜂𝜂𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 and 𝜂𝜂𝜂𝜂𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒 are the ratio between gauge surfaces of the standard/large specimens and the small ones, respectively. Under a log-normal life distribution of different sizes of specimens, Figure 3 plots life distributions of the standard and large specimens according to Eq. (6) and Eq. (7). Comparing with experimental mean and scatter of the life cycles at 𝜖𝜖𝜖𝜖𝑎𝑎𝑎𝑎 = 0.5%, a consequence of the defects distribution based on statistic of extremes cannot explain well the tested life distributions in Figure 3.
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4.5
Small specimen by testing 4
Standard specimen by testing Large specimen by testing
3.5
Standard specimen by Eq. (6) Large specimen by Eq. (7)
3
2.5 2 1.5 1 0.5 0 2.5
3
3.5
4
Log
10
4.5
(2N)
Figure 3. Expected life based only on size effect due to statistic of extremes
3.2. Fatigue crack growth life Note from Figure 4 that the inspection of fracture surfaces with SEM show that most cracks initiated at the surface of the specimen. For 30NiCrMoV12 steel, its LCF life is obviously dominated by crack propagation life rather than crack initiation life. Thus, the effect of specimen geometrical size on the fatigue crack propagation rate is critical essential for a theoretical understanding of fatigue lives. (a)
(b)
(c)
Figure 4. Multiple surface cracks of (a) small, (b) standard and (c) large specimen
Fatigue crack growth in the presence of a marked plastic strain range under LCF conditions can be well described by using Tomkins model [11]. Through assuming the amount of crack growth per cycle under tension-compression loadings to be the amount of irreversible shear decohesion which occurs at the crack tip, Tomkins [11] correlates crack growth to the plastic strain amplitude in an exponential law 𝐺𝐺𝐺𝐺 = 𝐺𝐺𝐺𝐺𝑖𝑖𝑖𝑖 𝑒𝑒𝑒𝑒 𝑘𝑘𝑘𝑘𝑙𝑙𝑙𝑙𝑁𝑁𝑁𝑁 � 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘𝑙𝑙𝑙𝑙 = 𝑘𝑘𝑘𝑘𝑙𝑙𝑙𝑙0 𝜀𝜀𝜀𝜀𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎
(8)
where 𝐺𝐺𝐺𝐺 and 𝐺𝐺𝐺𝐺𝑖𝑖𝑖𝑖 are the crack length and initial crack length, respectively; 𝜀𝜀𝜀𝜀𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 is the plastic strain amplitude, 𝑘𝑘𝑘𝑘𝑙𝑙𝑙𝑙0 and 𝑑𝑑𝑑𝑑 are material parameters that establish the short cracks growth in a material. Accordingly, a simplified form can be expressed as 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
𝑠𝑠𝑠𝑠𝑁𝑁𝑁𝑁
(9)
= 𝑘𝑘𝑘𝑘𝑙𝑙𝑙𝑙 𝐺𝐺𝐺𝐺
The number of cycles is obtained by integrating Eq. (9) from an initial crack length 𝐺𝐺𝐺𝐺𝑖𝑖𝑖𝑖 to a final one 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓 . Parameter of Eq. (9) was calibrated from experiments, as shown in Figure 5. Two scale factors can be obtained to shift the distribution found with the statics of extremes for the standard/large specimen from the reference small one 𝛾𝛾𝛾𝛾𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝛾𝛾𝛾𝛾𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒 =
∆𝑁𝑁𝑁𝑁𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ∆𝑁𝑁𝑁𝑁𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙
∆𝑁𝑁𝑁𝑁𝑙𝑙𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒
∆𝑁𝑁𝑁𝑁𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙
=
=
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓 ,𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎�
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖 �
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓 ,𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙�
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖 �
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓 ,𝑙𝑙𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒�
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖 �
�
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓 ,𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙�
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 �𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖 �
�
�
�
(10)
(11)
Combining Eq. (6) to Eq. (11), the life scale factors at 𝜖𝜖𝜖𝜖𝑎𝑎𝑎𝑎 = 0.5% due to the propagation size effect can be calculated from an 𝜋𝜋𝜋𝜋𝐷𝐷𝐷𝐷 initial crack length 𝐺𝐺𝐺𝐺𝑖𝑖𝑖𝑖 = 100𝜇𝜇𝜇𝜇𝑚𝑚𝑚𝑚 to a final crack length 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓 = 0, namely, 𝛾𝛾𝛾𝛾𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 1.18048 and 𝛾𝛾𝛾𝛾𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙𝑙𝑙𝑒𝑒𝑒𝑒 = 1.27523. 2 As pointed out by Koyama et al. [5], the specimen size effect can be quantified by the sum of the effect of statistical distribution
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of the defects and the effect of crack propagation. From a crack growth point of view, a larger specimen would lead to a longer life, which shift the dotted/dash lines in Figure 3 to the right-hand of the x-axis. According to this, a comparison between the curves of experimental distribution of the life and the ones due to the effects of statistical defects and crack propagation can be made by using Eq. (10) and Eq. (11), as shown in Figure 6 and results in Table 3. 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄
𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄
Figure 5 Crack growth measurement near the micro hole 4.5
Comparison between experimental data end the sum of the size effects Small specimen experimental
4
Table 3 Mean and scatter of the life cycles at 𝜖𝜖𝜖𝜖𝑎𝑎𝑎𝑎 = 0.5% due to the combination of propagation and statistical distribution of defects
Standard specimen experimental Large specimen experimental
3.5
Standard specimen superposition Large specimen superposition
3
Standard spec.
2 1.5
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 (𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)
Mean
3.7021
3.7441
Std.
0.0779
1
0 2.5
0.1620
Mean Std.
Large spec.
0.5
3
3.5
4
Log
10
Extracted /
Experimental / 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙10 (𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)
2.5
3.6807 0.1557
3.6382 0.1759
𝜇𝜇𝜇𝜇% error 1.15% 1.13%
4.5
(2N)
Figure 6. Comparison between extracted distribution and experimental ones
4. Numerical modeling of multiple fracture As aforementioned, the weakest-link theory assumed that the critical defects do not interact which are sparsely distributed. This assumption only works under limited number of potentially critical defect sites. As noticed from Figure 4, the presence of multiple surface cracks scatted over the gauge section is the main manifestation of damages in 30NiCrMoV12 steel under cyclic loadings. In particular, its fracture was induced by the processes of initiation, growth and coalescence of cracks. Combining with the replica tests as mentioned in Section 2, a numerical Monte Carlo simulation is performed for multiple fracture evaluation and to compare experimental data on crack length distribution evolution, as well as their experimental lives, with the simulated ones. 4.1. Initial prerequisites of the model The simulation conducted in this analysis includes the process of random crack initiation, propagation and coalescence of surface microcracks occurring simultaneously. According to available experimental data on replica tests under cyclic loadings, the initial prerequisites and assumptions are taken as follows. (1) Uniform material properties are assumed over the entire damaged surface, namely, the simulation surface area; (2) Uniaxial stress state is loaded and time-independent during the crack evolution; (3) All cracks on the surface are oriented normally to the stress action direction, like the x-axis in this analysis; (4) A constant increment of the load cycles is assigned as the count number of iterations; (5) The positions of all cracks on the surface are randomly distributed according to Poisson’s law. Particularly, the probability that n cracks are located on the area A with the crack density 𝜆𝜆𝜆𝜆 𝐺𝐺𝐺𝐺(𝑛𝑛𝑛𝑛) =
(−𝜆𝜆𝜆𝜆𝐴𝐴𝐴𝐴)𝑛𝑛𝑛𝑛 𝑠𝑠𝑠𝑠!
(12)
𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒(−𝜆𝜆𝜆𝜆𝐴𝐴𝐴𝐴)
(6) The initial length 𝐺𝐺𝐺𝐺 of the crack is assigned by a two-parameter Weibull distribution; 𝐺𝐺𝐺𝐺(𝐺𝐺𝐺𝐺) = 1 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 �− �
𝑎𝑎𝑎𝑎−𝑎𝑎𝑎𝑎0 𝛽𝛽𝛽𝛽 𝛼𝛼𝛼𝛼
� �
(13)
where 𝛽𝛽𝛽𝛽 and 𝛼𝛼𝛼𝛼 are the Weibull shape parameter and scale parameter, which were obtained from measurements on the plastic replicas, and 𝐺𝐺𝐺𝐺0 is the resolution of crack identification. (7) In crack propagation, the crack length increment for any crack within a given number of iterations is prescribed; (8) In crack coalescence, quasi-collinear cracks in close proximity interact through the influence zones at both tips, which shows the changes in their propagation paths when their tips approach one another [12]. The influence zone is defined by a circle of diameter 𝑟𝑟𝑟𝑟𝑝𝑝𝑝𝑝 according to the size of local plastic deformation zone at the crack tip [13]
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𝑟𝑟𝑟𝑟𝑝𝑝𝑝𝑝 =
𝛥𝛥𝛥𝛥𝐽𝐽𝐽𝐽𝑒𝑒𝑒𝑒𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ∙𝐸𝐸𝐸𝐸 ′
373
(14)
2𝜋𝜋𝜋𝜋𝜎𝜎𝜎𝜎02
where ∆𝐽𝐽𝐽𝐽𝑒𝑒𝑒𝑒𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 is the effective cyclic J-Integral, 𝜎𝜎𝜎𝜎0 is the flow stress and 𝐸𝐸𝐸𝐸 ′ is the cyclic Young’s Modulus under plane strain conditions, 𝐸𝐸𝐸𝐸 ′ = 𝐸𝐸𝐸𝐸 ⁄(1 − 𝑣𝑣𝑣𝑣 2 ) and 𝑣𝑣𝑣𝑣 is the Poisson’s ratio. The criterion of Murakami [14] on judging defects interaction/coalescence is adopted, by the distance between the tips 𝑑𝑑𝑑𝑑 is less than the minimum of the two crack length (15)
𝑑𝑑𝑑𝑑 < 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑛𝑛𝑛𝑛(𝐺𝐺𝐺𝐺𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐𝑘𝑘𝑘𝑘1 , 𝐺𝐺𝐺𝐺𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐𝑘𝑘𝑘𝑘2 )
(9) An unloaded material zone during crack propagation and coalescence is formed around each crack, of which the shape of the zone is defined as a circle of diameter equal to the crack length. In these zones, the crack growth is arrested if its tip comes into the unloading zone from the nearest crack. Based on abovementioned starting prerequisites, a simulation flow chart is given in Figure 7, the basic inputs obtained from experimental testing for multiple fracture simulation of different geometrical specimens include the following parameters: (1) Crack density on the surface; (2) Damage surface size; (3) Distribution of crack length and crack growth rate (without regard for the crack coalescence) on the surface; (4) Fracture and failure criterion. Start Input source data 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝐺𝐺𝐺𝐺𝐺𝐺 2𝐸𝐸𝐸𝐸𝐸 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝜆𝜆𝜆𝜆, 𝛼𝛼𝛼𝛼, 𝛽𝛽𝛽𝛽, 𝐴𝐴𝐴𝐴, 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓 , 𝑘𝑘𝑘𝑘𝑙𝑙𝑙𝑙 , 𝜀𝜀𝜀𝜀𝑎𝑎𝑎𝑎 , 𝜎𝜎𝜎𝜎0 , 𝐸𝐸𝐸𝐸 For 𝑚𝑚𝑚𝑚 = 1 to total simulation number (102 ) Monte Carlo sampling
Random crack initiation Crack position: Poisson’s law Crack length: Weibull distribution For 𝑗𝑗 = 1 iteration Δ𝑁𝑁𝑁𝑁/𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
Crack coalescence analysis Unloaded and influence zone identification Crack propagation type Calculation of crack tip distance Merge cracks or change crack tip angles 𝑗𝑗 = 𝑗𝑗 + 1
Crack propagation Increment of crack length 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 Crack growth rate: random variable vs. random process Track the largest critical crack 𝐺𝐺𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚,𝑗𝑗 If maximum crack length 𝐺𝐺𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚,𝑗𝑗 > 𝐺𝐺𝐺𝐺𝑓𝑓𝑓𝑓
No
Yes
Calculation of life 𝑁𝑁𝑁𝑁𝑓𝑓𝑓𝑓,𝑖𝑖𝑖𝑖 = 2𝐸𝐸𝐸𝐸𝐸 + 𝑗𝑗∆𝑁𝑁𝑁𝑁 Continue?
Life distribution 𝑵𝑵𝒇𝒇 End
Figure 7. Main flow chart of multiple fracture simulation
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4.2. Simulation results and discussions During simulation, the material damaging under cyclic loading can be characterized by crack initiation, propagation and coalescence, which are dispersed over the surface with different lengths as shown in Figure 8. Two mechanisms of crack propagation, namely, crack length increment due to its own growth for those cracks without coalescence and sudden crack length increment due to the neighbouring cracks coalescence through interacting by their tips, are involved in this simulation. Accordingly, the largest crack exhibits jump-like growth due to merging with other cracks in the plastic radius. Through following the procedure in Section 4.1, an example probability plot of experimental and simulated lives on small specimens is presented in Figure 9. Note that the simulated life distribution covers well with the lives of small specimens except that of G2.13 and G2.14, in which their fracture surfaces in Figure 10 have shown the worst condition on multiple surface crack interaction and propagation. (a)
(b)
Figure 8. Scheme of crack distribution and unloaded zones (a) Surface crack simulation at 𝜀𝜀𝜀𝜀𝑎𝑎𝑎𝑎 = 0.5% (circles mark the unloaded zones, square marks the crack coalescence) and (b) Replica on specimen G 2.12 at 3E3 cycles (a)
3
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Log-Normal probability plot Small spec. life comparison between simulated and experimental life
Log-normal distributions of life comparison between small specimens
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Experimental point Fitted curve
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95% confidence band 5% confidence band
Experimental distribution Simulated distribution
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𝑮𝑮𝑮𝑮. 𝟏𝟏𝟏𝟏
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Figure 9. Comparison between experimental and simulated lives of small specimen: (a) probability plot and (b) life distribution
(a) G2.13 (b) G2.14 Figure 10. Fracture surface of small specimens
5. Conclusions In this study, we proposed a probabilistic modelling and numerical simulation procedure for evaluating the effects of specimen size on fatigue life with three different geometrical specimens. Specifically, the specimen size effect is quantified from two categorizes, namely, size effect due to the effects of statistical defects and crack propagation, which provides a reference for fatigue
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life estimation under different specimen sizes. Moreover, a simulation procedure for surface multiple fracture is established by including the random processes of initiation, propagation and coalescence of dispersed surface cracks, which accounts for the observed experimental scatter in fatigue lifetime of small specimen of 30NiCrMoV12 steel. Using this procedure, the life of a component can be predicted from the criterion of critical cracks formation by coalescence of dispersed defects. Acknowledgments Dr. S.P. Zhu acknowledges support for his period of study at Politecnico di Milano by the Polimi International Fellowship Grant scheme. The authors acknowledged support by Lucchini RS (Italy) for supplying the heat treated steel. References [1] O. Hertel, M. Vormwald, “Statistical and geometrical size effects in notched members based on weakest-link and short-crack modelling,” Eng. Fract. Mech., vol. 95, pp. 72–83, 2012. [2] M. Makkonen, “Statistical size effect in the fatigue limit of steel,” Int. J. Fatigue, vol. 23, no. 5, pp. 395–402, 2001. [3] S.P. Zhu, S. Foletti, S. Beretta, “Probabilistic framework for multiaxial LCF assessment under material variability,” Int. J. Fatigue, in press, https://doi.org/10.1016/j.ijfatigue.2017.06.019, 2017. [4] T. Tomaszewski, J. Sempruch, T. Piatkowski, “Verification of selected models of the size effect based on high-cycle fatigue testing on mini specimens made of EN AW-6063 Aluminum alloy,” J. Theor. Appl. Mech., vol. 52, no. 4, pp. 883–894, 2014. [5] M. Koyama, H. Li, Y. Hamano, and T. Sawaguchi, “Mechanical-probabilistic evaluation of size effect of fatigue life using data obtained from single smooth specimen : An example using Fe-30Mn-4Si-2Al seismic damper alloy,” Eng. Fail. Anal., vol. 72, pp. 34–47, 2017. [6] Y. Murakami, K. J. Miller, “What is fatigue damage ? A view point from the observation of low cycle fatigue process,” Int. J. Fatigue, vol. 27, pp. 991–1005, 2005. [7] “ASTM Standard E606. Standard Practice for Strain-Controlled Fatigue Testing,” Annu. B. ASTM Stand. 3, ASTM Int., 2004. [8] T. S. Hahn, A. R. Marder, “Effect of electropolishing variables on the current density- voltage relationship,” Metallography, vol. 21, pp. 365–375, 1988. [9] F. W. Zok, “On weakest link theory and Weibull statistics,” J. Am. Ceram. Soc., vol. 100, pp. 1265–1268, 2017. [10] A. Wormsen, B. Sjödin, G. Härkegård, and A. Fjeldstad, “Non-local stress approach for fatigue assessment based on weakest-link theory and statistics of extremes,” Fatigue Fract. Eng. Mater. Struct., vol. 30, no. 12, pp. 1214–1227, 2007. [11] Tomkins B, “Fatigue crack propagation - an analysis,” Philos. Mag., vol. 18, no. 155, pp. 1041–1066, 1968. [12] H.Y. Yoon, S.C. Lee, “Probabilistic distribution of fatigue crack growth life considering effect of crack coalescence,” JSME Int. J. Ser. A Solid Mech. Mater. Eng., vol. 46, no. 4, pp. 607–612, 2003. [13] S. Rabbolini, S. Beretta, S. Foletti, M. E. Cristea, “Crack closure effects during low cycle fatigue propagation in line pipe steel: An analysis with digital image correlation,” Eng. Fract. Mech., vol. 148, pp. 441–456, 2015. [14] Y. Murakami, Metal fatigue: effects of small defects and nonmetallic inclusions. Elsevier, 2002.