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Estimation of Endurance Limits of Welded Joints by the Criterion of Non-propagating Cracks
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ScienceDirect ScienceDirect Procedia Engineering 00 (2017)000–000
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Procedia Engineering 00 (2017)000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Engineering 206 (2017) 479–486
International Conference on Industrial Engineering, ICIE 2017 International Conference on Industrial Engineering, ICIE 2017
Estimation of Endurance Limits of Welded Joints by the Criterion of Estimation of Endurance Limits of Welded Joints by the Criterion of Non-propagating Cracks Non-propagating Cracks K. Molokov, Ya. Domashevskaia* K. Molokov, Ya. Domashevskaia* Far Eastern Federal University, 8, Sukhanova St., Vladivostok 690950, Russia Far Eastern Federal University, 8, Sukhanova St., Vladivostok 690950, Russia
Abstract Abstract The distinctive feature of the weld joint and heat affected zone is a significant discontinuity of mechanical characteristics, the grain diameter, as well of as the the presence various concentrators provokingdiscontinuity the fatigue cracks occurrence. It is known that The distinctive feature weld jointofand heat stress affected zone is a significant of mechanical characteristics, the the main factor influencing the risk of occurrence and speed of microcracks propagating is the stress condition as wellthat as grain diameter, as well as the presence of various stress concentrators provoking the fatigue cracks occurrence. Itlevel is known structural and mechanical discontinuity of metal weld joints. the main factor influencing the risk of occurrence and speed of microcracks propagating is the stress condition level as well as This article suggests a model and an approach thejoints. assessment of a reduced stress concentration factor for welded joints with structural and mechanical discontinuity of metalfor weld defects suchsuggests as undercuts, reeds stress in terms of crackstress propagating under factor the influence of ajoints stationary This article a model andand an other approach forconcentrators the assessment of a reduced concentration for welded with variable load. defects such as undercuts, reeds and other stress concentrators in terms of crack propagating under the influence of a stationary The results of the suggested model have been checked on carbon and ferrite-pearlite steels. The authors have manage to obtain variable load. the fluctuating stresses valuesmodel on the criteria non-propagating cracks depending on the size the defect based the The results of the suggested have been of checked on carbon and ferrite-pearlite steels. The of authors have and manage to on obtain endurance limitstresses criterion.values The sizes of criteria non-propagating cracks appearing the area ofonstress concentration action depending the fluctuating on the of non-propagating cracks independing the size of the defect andand based on the on the diameter of the circular hole of in non-propagating the plate have been calculated. Comparing model mentioned action in G.M.and Charzynski's endurance limit criterion. The sizes cracks appearing in the areatoofthe stress concentration depending σ / 3 macro monograph, the suggested methodology gives an asymptotic approximation endurance limit to the value 1 on the diameter of the circular hole in the plate have been calculated. Comparing to the model mentioned in G.M. Charzynski's concentratorsthe withsuggested increasingmethodology stress as a hole. Theanmodel has beenapproximation built for an infinite plate and in the macro monograph, gives asymptotic endurance limitdoes to not the consider value σchanges 1 / 3 stress field duewith to the limited size effects. concentrators increasing stress as a hole. The model has been built for an infinite plate and does not consider changes in the © 2017 The Authors. Published byeffects. Elsevier B.V. stress field to the Published limited sizeby © 2017 Thedue Authors. Elsevier Ltd. committee of the International Conference on Industrial Engineering. Peer-review under responsibility of the scientific © 2017 The under Authors. Published by B.V.committee of the International Conference on Industrial Engineering Peer-review responsibility of Elsevier the scientific Keywords: welded joint; endurance limit stress; defect; stress concentrator; breakloose macrocrack; fatigue crack; stressEngineering. intensity factor. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Keywords: welded joint; endurance limit stress; defect; stress concentrator; breakloose macrocrack; fatigue crack; stress intensity factor.
* Corresponding author. Tel.: +7-423-265-2429; fax: +7-423-243-2315. E-mail address: [email protected] * Corresponding author. Tel.: +7-423-265-2429; fax: +7-423-243-2315. E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering . 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering .
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering. 10.1016/j.proeng.2017.10.504
K.Domashevskaia Molokov et al. / /Procedia (2017) 000–000 479–486 K. Molokov, Ya. ProcediaEngineering Engineering206 00 (2017)
2480
1. Goal setting Despite the fact that the level of production of the welding allows obtaining quality welded joints, in practice, we cannot always ensure the absence of technological weld defects, especially in large-sized metal-structures. There is a big possibility to miss forbidden defect on its geometrical characteristics [1-10]. Values of fluctuating stresses of symmetrical cycle external load to the structural element can be represented as endurance limit stress of the element with stress concentration or without it [9]. Limit of endurance of the structural element without stress concentration is equivalent to the endurance limit of the material σ fr . For a structural element with stress concentrator external load value can either lead or not lead to the development of a crack to critical values Consequently, we have a load value, which provokes the crack appearance in a concentrator, but it is not enough to continue the crack movement to critical dimensions [11-15]. Thus, the minimum value of the variable load, which is already not enough for the development of crack to the critical dimensions, can be considered the endurance limit of the structural element with the stress concentrator, or the "threshold" endurance limit, by the criterion of non-propagating cracks. This "threshold" endurance level will depend on the form and size of the concentrator. Having built a model, with which you can calculate "threshold" endurance levels for concentrators with different forms and sizes, and if you know the material endurance level you can calculate the effective stress concentration coefficients. Endurance limit stress of the material of the ferrite-pearlite steel class can be calculated taking into account structural-mechanical characteristics [6]. It is proposed in works [1,6,8] on the basis of semiempirical models and structural-mechanical approach to determine the limit of endurance. Authors [7,8] studied the defect size of stress concentrator, which size didn't influence on endurance limit [16]. In these works, authors determine the dependency, which is underlying of engineering approach of evaluation, construction endurance, according to the tests of smooth and notched specimens in air. It allows calculating the radius of a front of elementary propagating semicircular crack, based only on well-known endurance limit of the symmetric cycle [17,18]. 2. Development mathematical model Development mathematical model for the description of propagated cracks from stress concentrators in circular notches or holes will allow transferring the results of the study (taking into account the correction) to various welded joints. As far as external load decreases, tending to 1 / 3 for concentrator with theoretical stress concentration factor equal to 3, it is logical to assume, that even for physically large stress concentrators the appearance of a crack will not be provoked by this concentrator. Thus, for the considered concentrator with a theoretical coefficient of stress concentration equal т the cracks will not propagate at stresses σ 1 / α т . This is a known limit case for any stress macro concentrator [19]. The results of effective stress concentration coefficients in model [8] are good conform with the experimental data for the zone of geometrically small stress concentrators, here the dependence is presented in the following form [8]:
th 1k 1
4
1 Fd F0
(1)
where, σ th 1k is threshold stress or endurance limit on an external load, in which the crack is propagated beyond the zone of the action of the concentrator, MPa; σ 1 is endurance limit undamaged material in the absence of stress concentration, MPa; σ 1 / σ th 1k is an effective coefficient of stress concentration. Other arguments, which are including in (1), was depend from concentrator form which was closed hole in diameter d from 40 to 500 mkm and depth h taking into account the depth of the drill exit [6]. The cross-sectional area of this defect, taking into account the exit of the drill, was calculated by formula:
et al. / Procedia Engineering 20600(2017) K. Molokov, K. Ya.Molokov Domashevskaia / Procedia Engineering (2017)479–486 000–000
Fd d h
d2 4 3
4813
(2)
and the calculated area of the crack formed in the region of the concentrator, including the cross-sectional area of the artificial defect formed by drilling, is: F0
2
a02 ,
(3)
where, a 0 is radius of the initial semicircular crack initiated by the artificial defect, m:
81 a0 10 1 3
2,3
(4)
Calculation by dependency (1)–(4) gives good correlation with the experimental data only for limited sizes of artificial defects. It is mentioned the degradation of correlation with experiment already in the radius of holes 1.5 mm and more in [8]. The model (1) gives overestimated values of the effective stress concentration coefficient. Let's consider infinite plate with stress concentrator in the form of a through hole. We will take into account only circumferential stresses σθ , which are formed by stress concentrator (fig. 1).
Fig. 1. To the minimum size of a crack from a macro concentrator: a - circumferential stresses with a uniaxial tension of a plate with a circular hole; b - schematization of the damaged area at the edge of the circular hole.
Let us take the first approximation, that radial stresses is not essential in the crack initiation, perpendicular to the external load in the adjacent of plastic area hole. Nevertheless, if we compare plastic zones under plane strain, where all 3 stress components and uniaxial stress state are taken into account, then the dimensions of this region along the X axis significantly differ both at the level of yield strength σ т , and at the level of the endurance limit of the diagrams. We introduce the correction K r to refine the radius vector, because of a significant divergence of diagrams σθ and σi . Now we shall transfer this correction as a correction to the radius hole R a . This transfer is correct because the ratio of any point of the stress diagram under consideration to the radius of this hole is a constant value for a particular external load and the form of the concentrator in question. The diagram of the circumferential stresses in direction of external load σ 0 is expressed through the radius hole R a and radius vector r , which is represented as r R a l thk , then from the Kirsch solution in the polar coordinate system:
K. Molokov et al. / Procedia Engineering 206 (2017) 479–486 K. Molokov, Ya. Domashevskaia / Procedia Engineering 00 (2017) 000–000
482 4
0 Ra / K r 2 2
Ra / K r lthk
2 4 Ra / K r 3 Ra / K r lthk
(5)
where l thk is length of the strained crack formed in the area gaining damage, which is caused by the macro concentrator, m. All cracks, which are bigger l thk will be propagated under all other equal conditions, so we call lthk the threshold size of the propagated crack for the element with the stress concentrator. Let's suppose that a crack exists in the area of damage O p (see fig. 1b), then the criterion for its propagation will be condition:
lthk
fr
(6)
and under the influence of a symmetric external load variable, the stress in the area of the concentrator will vary close to the symmetrical cycle [5], then:
lthk
1
(7)
The border - as a condition for propagating cracks - in this case, can be written by substituting to solution (5) endurance limit σ 1 instead of , and instead of l thk its expression through external stresses σ 0 . Then, from the condition of friction cracks, linear fracture mechanics (LMR) [13,17,21], for an edge crack, we have:
lthk
1 K thr 1,12 0
2
(8)
where K thr is threshold coefficient of stress intensity under with an operating cycle with asymmetry rос . In this concrete case we consider the symmetric cycle, then K thr K th 1 . If we find the change in the correction K r as a function of the radius vector r or R a and solve (5) with respect to σ 0 , the σ 0 is the endurance limit, we will have the endurance limit for the structural element with the concentrator R a . This value of the endurance limit will be the threshold σ th 1k and reflect the criterion of the formed and propagating crack for the region O p . And, depending on (1), we put:
th 1k 0,8 1
(9)
then the radius of the resulting semicircular crack will obey the dependence (4) for all values of the hole radius R a d / 2 for d h in (2). In this case, the correction K r for the hole radius can be calculated as:
Kr
a0 Ra
(10)
where a 0 can be calculated using (4) - this is true for steels with a temporary strength of 350 to 1400 MPa; R a is the hole radius obtained under the same condition (9). For determining R a by the corresponding condition (9), we
r and solve the resulting equation (5) with respect to r. We take K r 1 , σθ 0,8σ 1 , and assume R a / K r l thk get:
et al. / Procedia Engineering 20600 (2017) 479–486 K. Molokov,K. Ya.Molokov Domashevskaia / Procedia Engineering (2017) 000–000
r
Ra
2
Ra 7 2
4835
(11)
Now, taking into account (8), we have: 2
K th 1 1 Ra 1,12 0,8 1
7 1 1
(12)
7 1 1
(13)
or Ra
K th2 1
1,12 0,8 1 2
Substituting (13) and (4) in (10), we find the correction for the radius of the hole:
Kr
103 812,3
7 1 1 1,12 0,8
1
0,3
2
K th2 1
(14)
After substituting (7), (14) and (8) in (5) and solving it numerically with respect to σ 0 , where σ 0 is threshold limit endurance σ th 1k , or the criterion of propagated cracks after beginning in the macro concentrator:
0 thrk
(15)
where σ'0 and σ thrk is the external load with a given cycle asymmetry and the threshold endurance limit of nonpropagating cracks for this asymmetry, respectively. For calculation of endurance limit of undamaged material we use dependency, that takes into account the characteristics of its structure. For example, for the heat-effected zone and the zone of thermal influence (ZTI) is characterized by the structural heterogeneity, the endurance limit of which is σ 1 calculated in accordance with the well-known work [3]: 2 0, 7 T 2 1 0, 7 T l0 1 K th 1
1/ 2
(16)
where l0 is the maximum size of the microcrack, which does not affect the endurance limit; σ t is the yield strength of the material at the level of 0.2% by the conditional stretch diagram; K th 1 is the threshold stress intensity factor for asymmetry rос 1 . 3. Comparison of theoretical results with experimental data
Using the dependence (5), we constructed curves for steel, threshold stresses of the endurance limit σ th 1k of a symmetric cycle, depending on the defect size in the form of a circular hole (fig. 2). These curves limit the area from above when the stresses of the external load σ '0 do not lead to the propagation of a crack, provoked by the macro concentrator. On the same graph, for comparison, the results of experimental data have been taken from [6, 8] are
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K. Molokov et al. / Procedia Engineering 206 (2017) 479–486 K. Molokov, Ya. Domashevskaia / Procedia Engineering 00 (2017) 000–000
presented. Many experimental studies are devoted to the cracks appearance near holes which diameters are insignificant and approach micro defects. But there are not enough the experimental data for large hole diameters: such tests are very laborious. Therefore, the questions of the theoretical estimation of the cracks initiation, as well as their stopping beyond the zone of high stresses action from the concentrator, with known characteristics of external loading remain relevant. In the graphs, the threshold stresses are given in normalized values to the endurance limit of an undamaged material without stress concentrators (fig. 2). Also, the question for the developing crack in the region of high-stress concentration is also of interest, what is the maximum length of the crack that will reach when exiting this area before it stops. We will call this length the threshold length of a crack originating in a known concentrator, or simply the threshold length of the crack of the concentrator. Its calculation will answer whether it is possible to continue the operation of the structure or according to the technical requirements of operation, a crack of such length. In fig. 3, the dependence of the change in the threshold length of the crack concentrator depending on the size of the concentrator is proposed. It is caused by stresses from the concentrator and is non-developing at the threshold endurance limit σ th 1k of an element with stress concentration.
Fig. 2. Curves of the dependence of the effective concentration coefficient (the reciprocal of ( σ th 1k / σ 1 ) as a function of the radius of the circular hole. Here ⋇, ⊡ and - are the experimental data from [6] (analog - steel 50) and [5] (steel 10), respectively; ─ The theory according to (5) for steel 50 (the curve from below) and steel 10 (the curve from above); --- theory (1); ⨀ and � - steel 22K and St3, respectively.
Fig. 3. Curves of change in the permissible effective coefficient of stress concentration from the maximum threshold crack formed in the concentrator for various steels: ⊡ - steel 50; ⨀-steel 10; � - steel St3; ⋇ - steel 22K.
K. Molokov et al. / Procedia Engineering 206 (2017) 479–486 K. Molokov, Ya. Domashevskaia / Procedia Engineering 00 (2017) 000–000
485 7
A comparison of the theoretical curves in fig. 2 with experimental data shows adjustment of the results of our model with macro-concentrators of small size. For large radiuses Ra , circular holes of the experimental values on the graphs are not given, but the nature of the curves according to the proposed method is adequate in comparison with the model (1). The model (1) shows that for some steels already at Ra 10 mm the cracks will develop with an external load of 1 / 3 . This circumstance contradicts existing and known approved ideas: the crack does not develop at stresses below the endurance limit of the undamaged material. While (5) shows that as the radius of the circular hole increases, th 1k approaches asymptotically to 1 / 3 , and this is an adequate behavior and does not contradict known facts in LMR. 4. The discussion of the results
In order to use the diagrams of the limiting propagation of cracks (see fig. 2 and fig. 3), it is necessary to know the external load σ '0 , the limit of endurance of the undamaged material σ fr and the radius of the macro concentrator R a . From the known value of R a , we draw a vertical line up to the cross with the curve of a certain material (see fig. 2) and find the value of the ratio σ th 1k / σ 1 along the vertical axis. Therefore, intersection K σ 1 , where K is the value found in the vertical axis, and it will be the ultimate external load at which the crack is not yet propagated. If the external load is higher than σ th 1k , then for a given concentrator the crack generated in it will be propagated and reach a critical value. The next stage is the determination of the maximum length of the crack, which is non-propagable at a given external load. For this you need to use the diagram of the maximum threshold crack (see fig. 3). We postpone the value of the ratio σ th 1k / σ 1 . Further, before crossing with the curve of the corresponding material, draw a horizontal line and note the threshold crack length L thk . 5. Conclusions
It should be noted that for small values of L thk, , under the visual test the crack can not be seen when L L thk and th 1k / 1 0,8 , and it should be considered that cracks of this size do not affect the endurance limit. This case refers to physically small stress concentrators. Taking into account the obtained results, it can be concluded that for some materials, for example, brittle materials and high-strength steels with low ductility, the threshold crack size may be larger than the critical size of the macrocrack, i.e. Lthk Lc . In this case, it is necessary to rely on the value of Lc , and the situation of brittle failure from the macro concentrator becomes paramount. For high-strength steels, Lc is usually small, and for
relatively large concentrators, such as Ra / Lc 1 3 , the cracks can reach their critical size even under external
loads which are a little above 1 / 3 - in the case of a concentrator in the form of a circular hole. Thus, relatively large concentrators represent a great danger for high-strength steels or steels with low ductility, if there is no wellverified periodic diagnostics of the structure. References [1] A.V. Gridasov, Influence of the type and geometric parameters of the welded joint to endurance of welded structures of thermal power facilities, Research on improving the efficiency of shipbuilding and ship repair. 45 (2005) 254–257. [2] T.L. Anderson, Fracture mechanics: fundamentals and applications, 2-nd ed, London, 1995, 682 p. [3] S.S. Manson, G.R. Halford, Fatigue and durability of structural materials, ASM International, 2006, 456 p. [4] G.M. Charzyński, Deformation. Destruction. Reliability: Problems of deformation and fracture of steel, Methods for assessing the strength of power equipment and pipelines, Lenand, Moscow, 2014, 544 p. [5] B.P. Kishkin, Structural strength of materials, State University, Moscow, 1976, 184 p. [6] G.V. Matokhin, K.P. Gorbachev, A.Y. Vorobyov, A framework for assessing the strength and durability of welded structures, FESTU, Vladivostok, 2008, 270 p.
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[7] K.A. Molokov, Estimating fatigue endurance limit in the zone of stress concentrators, Youth and scientific and technical progress: Materials of regional scientific and engineering, Conf. Part 2, FESTU, Vladivostok, 2006, pp. 295–300. [8] G.M. Charzyński, The calculation basis on fatigue and endurance, LENAND, Moscow, 2016, 168 p. [9] Concentration of stresses, Digest of articles. Issue 1, Naukova Dumka, Kiev, 1965, 347 p. [10] N.A. Makhutov, Deformation criteria for destruction and calculation of structural elements for strength, Mechanical Engineering, Moscow, 1981, 282 p. [11] H. Nisitani, M. Endo, Unified treatment of deep and shallow swimmers, Basic Questions in Fatigue, ASTM, Philadelphia, 1988, pp. 136– 153. [12] N.E. Frost, D.S. Dugdale, Fatigue tests of notched mild steel plates with measurements of fatigue cracks, Journ. Mech. Ph. Of Solids. 5 (1957) 182–192. [13] P. Lukas, L. Kunz, B. Weiss, R. Sticker, Notch size effect in fatigue, Fatigue Fract. Engng. Struct. 12(3) (1989) 175–186. [14] K. Hatanaka, S. Shimizu, A. Nagae, Size effect on rotating bending fatigue, Bulletin JSME. 26(218) (1983) 1–8. [15] R.P. Skeleton, C.J. Rees, G.A. Webster, Loading in high-temperature low-cyclic fatigue, Fatigue Fract. Eng. Mater. Struct. 19(2) (1996) 287–297. [16] R.M. Schneiderovich, Stability under static and re-static loads, Mechanical engineering, Moscow, 1968, 343 p. [17] D. Broek, The Practical Use of Fracture Mechanics, Kluwer Academic Publishers, 1988. 532 p. [18] K. Tanaka, Y. Nakai, Prediction of fatigue threshold of notched components, Trans. ASME: J.Eng. Mater. And Technol. 106(2) (1984) 192– 199. [19] R. Peterson, Coefficients of stress concentration, The World, Moscow, 1977, 302 p. [20] Y. Murakami, Theory of Elasticity and Stress Concentration, John Wiley & Sons, Ltd, 2017, 446 p. [21] L.M. Kachanov, Fundamentals of fracture mechanics, Science Moscow, 1974, 312 p.
ScienceDirect ScienceDirect Procedia Engineering 00 (2017)000–000
Available online at www.sciencedirect.com
ScienceDirect
Procedia Engineering 00 (2017)000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Engineering 206 (2017) 479–486
International Conference on Industrial Engineering, ICIE 2017 International Conference on Industrial Engineering, ICIE 2017
Estimation of Endurance Limits of Welded Joints by the Criterion of Estimation of Endurance Limits of Welded Joints by the Criterion of Non-propagating Cracks Non-propagating Cracks K. Molokov, Ya. Domashevskaia* K. Molokov, Ya. Domashevskaia* Far Eastern Federal University, 8, Sukhanova St., Vladivostok 690950, Russia Far Eastern Federal University, 8, Sukhanova St., Vladivostok 690950, Russia
Abstract Abstract The distinctive feature of the weld joint and heat affected zone is a significant discontinuity of mechanical characteristics, the grain diameter, as well of as the the presence various concentrators provokingdiscontinuity the fatigue cracks occurrence. It is known that The distinctive feature weld jointofand heat stress affected zone is a significant of mechanical characteristics, the the main factor influencing the risk of occurrence and speed of microcracks propagating is the stress condition as wellthat as grain diameter, as well as the presence of various stress concentrators provoking the fatigue cracks occurrence. Itlevel is known structural and mechanical discontinuity of metal weld joints. the main factor influencing the risk of occurrence and speed of microcracks propagating is the stress condition level as well as This article suggests a model and an approach thejoints. assessment of a reduced stress concentration factor for welded joints with structural and mechanical discontinuity of metalfor weld defects suchsuggests as undercuts, reeds stress in terms of crackstress propagating under factor the influence of ajoints stationary This article a model andand an other approach forconcentrators the assessment of a reduced concentration for welded with variable load. defects such as undercuts, reeds and other stress concentrators in terms of crack propagating under the influence of a stationary The results of the suggested model have been checked on carbon and ferrite-pearlite steels. The authors have manage to obtain variable load. the fluctuating stresses valuesmodel on the criteria non-propagating cracks depending on the size the defect based the The results of the suggested have been of checked on carbon and ferrite-pearlite steels. The of authors have and manage to on obtain endurance limitstresses criterion.values The sizes of criteria non-propagating cracks appearing the area ofonstress concentration action depending the fluctuating on the of non-propagating cracks independing the size of the defect andand based on the on the diameter of the circular hole of in non-propagating the plate have been calculated. Comparing model mentioned action in G.M.and Charzynski's endurance limit criterion. The sizes cracks appearing in the areatoofthe stress concentration depending σ / 3 macro monograph, the suggested methodology gives an asymptotic approximation endurance limit to the value 1 on the diameter of the circular hole in the plate have been calculated. Comparing to the model mentioned in G.M. Charzynski's concentratorsthe withsuggested increasingmethodology stress as a hole. Theanmodel has beenapproximation built for an infinite plate and in the macro monograph, gives asymptotic endurance limitdoes to not the consider value σchanges 1 / 3 stress field duewith to the limited size effects. concentrators increasing stress as a hole. The model has been built for an infinite plate and does not consider changes in the © 2017 The Authors. Published byeffects. Elsevier B.V. stress field to the Published limited sizeby © 2017 Thedue Authors. Elsevier Ltd. committee of the International Conference on Industrial Engineering. Peer-review under responsibility of the scientific © 2017 The under Authors. Published by B.V.committee of the International Conference on Industrial Engineering Peer-review responsibility of Elsevier the scientific Keywords: welded joint; endurance limit stress; defect; stress concentrator; breakloose macrocrack; fatigue crack; stressEngineering. intensity factor. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Keywords: welded joint; endurance limit stress; defect; stress concentrator; breakloose macrocrack; fatigue crack; stress intensity factor.
* Corresponding author. Tel.: +7-423-265-2429; fax: +7-423-243-2315. E-mail address: [email protected] * Corresponding author. Tel.: +7-423-265-2429; fax: +7-423-243-2315. E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering . 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering .
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering. 10.1016/j.proeng.2017.10.504
K.Domashevskaia Molokov et al. / /Procedia (2017) 000–000 479–486 K. Molokov, Ya. ProcediaEngineering Engineering206 00 (2017)
2480
1. Goal setting Despite the fact that the level of production of the welding allows obtaining quality welded joints, in practice, we cannot always ensure the absence of technological weld defects, especially in large-sized metal-structures. There is a big possibility to miss forbidden defect on its geometrical characteristics [1-10]. Values of fluctuating stresses of symmetrical cycle external load to the structural element can be represented as endurance limit stress of the element with stress concentration or without it [9]. Limit of endurance of the structural element without stress concentration is equivalent to the endurance limit of the material σ fr . For a structural element with stress concentrator external load value can either lead or not lead to the development of a crack to critical values Consequently, we have a load value, which provokes the crack appearance in a concentrator, but it is not enough to continue the crack movement to critical dimensions [11-15]. Thus, the minimum value of the variable load, which is already not enough for the development of crack to the critical dimensions, can be considered the endurance limit of the structural element with the stress concentrator, or the "threshold" endurance limit, by the criterion of non-propagating cracks. This "threshold" endurance level will depend on the form and size of the concentrator. Having built a model, with which you can calculate "threshold" endurance levels for concentrators with different forms and sizes, and if you know the material endurance level you can calculate the effective stress concentration coefficients. Endurance limit stress of the material of the ferrite-pearlite steel class can be calculated taking into account structural-mechanical characteristics [6]. It is proposed in works [1,6,8] on the basis of semiempirical models and structural-mechanical approach to determine the limit of endurance. Authors [7,8] studied the defect size of stress concentrator, which size didn't influence on endurance limit [16]. In these works, authors determine the dependency, which is underlying of engineering approach of evaluation, construction endurance, according to the tests of smooth and notched specimens in air. It allows calculating the radius of a front of elementary propagating semicircular crack, based only on well-known endurance limit of the symmetric cycle [17,18]. 2. Development mathematical model Development mathematical model for the description of propagated cracks from stress concentrators in circular notches or holes will allow transferring the results of the study (taking into account the correction) to various welded joints. As far as external load decreases, tending to 1 / 3 for concentrator with theoretical stress concentration factor equal to 3, it is logical to assume, that even for physically large stress concentrators the appearance of a crack will not be provoked by this concentrator. Thus, for the considered concentrator with a theoretical coefficient of stress concentration equal т the cracks will not propagate at stresses σ 1 / α т . This is a known limit case for any stress macro concentrator [19]. The results of effective stress concentration coefficients in model [8] are good conform with the experimental data for the zone of geometrically small stress concentrators, here the dependence is presented in the following form [8]:
th 1k 1
4
1 Fd F0
(1)
where, σ th 1k is threshold stress or endurance limit on an external load, in which the crack is propagated beyond the zone of the action of the concentrator, MPa; σ 1 is endurance limit undamaged material in the absence of stress concentration, MPa; σ 1 / σ th 1k is an effective coefficient of stress concentration. Other arguments, which are including in (1), was depend from concentrator form which was closed hole in diameter d from 40 to 500 mkm and depth h taking into account the depth of the drill exit [6]. The cross-sectional area of this defect, taking into account the exit of the drill, was calculated by formula:
et al. / Procedia Engineering 20600(2017) K. Molokov, K. Ya.Molokov Domashevskaia / Procedia Engineering (2017)479–486 000–000
Fd d h
d2 4 3
4813
(2)
and the calculated area of the crack formed in the region of the concentrator, including the cross-sectional area of the artificial defect formed by drilling, is: F0
2
a02 ,
(3)
where, a 0 is radius of the initial semicircular crack initiated by the artificial defect, m:
81 a0 10 1 3
2,3
(4)
Calculation by dependency (1)–(4) gives good correlation with the experimental data only for limited sizes of artificial defects. It is mentioned the degradation of correlation with experiment already in the radius of holes 1.5 mm and more in [8]. The model (1) gives overestimated values of the effective stress concentration coefficient. Let's consider infinite plate with stress concentrator in the form of a through hole. We will take into account only circumferential stresses σθ , which are formed by stress concentrator (fig. 1).
Fig. 1. To the minimum size of a crack from a macro concentrator: a - circumferential stresses with a uniaxial tension of a plate with a circular hole; b - schematization of the damaged area at the edge of the circular hole.
Let us take the first approximation, that radial stresses is not essential in the crack initiation, perpendicular to the external load in the adjacent of plastic area hole. Nevertheless, if we compare plastic zones under plane strain, where all 3 stress components and uniaxial stress state are taken into account, then the dimensions of this region along the X axis significantly differ both at the level of yield strength σ т , and at the level of the endurance limit of the diagrams. We introduce the correction K r to refine the radius vector, because of a significant divergence of diagrams σθ and σi . Now we shall transfer this correction as a correction to the radius hole R a . This transfer is correct because the ratio of any point of the stress diagram under consideration to the radius of this hole is a constant value for a particular external load and the form of the concentrator in question. The diagram of the circumferential stresses in direction of external load σ 0 is expressed through the radius hole R a and radius vector r , which is represented as r R a l thk , then from the Kirsch solution in the polar coordinate system:
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0 Ra / K r 2 2
Ra / K r lthk
2 4 Ra / K r 3 Ra / K r lthk
(5)
where l thk is length of the strained crack formed in the area gaining damage, which is caused by the macro concentrator, m. All cracks, which are bigger l thk will be propagated under all other equal conditions, so we call lthk the threshold size of the propagated crack for the element with the stress concentrator. Let's suppose that a crack exists in the area of damage O p (see fig. 1b), then the criterion for its propagation will be condition:
lthk
fr
(6)
and under the influence of a symmetric external load variable, the stress in the area of the concentrator will vary close to the symmetrical cycle [5], then:
lthk
1
(7)
The border - as a condition for propagating cracks - in this case, can be written by substituting to solution (5) endurance limit σ 1 instead of , and instead of l thk its expression through external stresses σ 0 . Then, from the condition of friction cracks, linear fracture mechanics (LMR) [13,17,21], for an edge crack, we have:
lthk
1 K thr 1,12 0
2
(8)
where K thr is threshold coefficient of stress intensity under with an operating cycle with asymmetry rос . In this concrete case we consider the symmetric cycle, then K thr K th 1 . If we find the change in the correction K r as a function of the radius vector r or R a and solve (5) with respect to σ 0 , the σ 0 is the endurance limit, we will have the endurance limit for the structural element with the concentrator R a . This value of the endurance limit will be the threshold σ th 1k and reflect the criterion of the formed and propagating crack for the region O p . And, depending on (1), we put:
th 1k 0,8 1
(9)
then the radius of the resulting semicircular crack will obey the dependence (4) for all values of the hole radius R a d / 2 for d h in (2). In this case, the correction K r for the hole radius can be calculated as:
Kr
a0 Ra
(10)
where a 0 can be calculated using (4) - this is true for steels with a temporary strength of 350 to 1400 MPa; R a is the hole radius obtained under the same condition (9). For determining R a by the corresponding condition (9), we
r and solve the resulting equation (5) with respect to r. We take K r 1 , σθ 0,8σ 1 , and assume R a / K r l thk get:
et al. / Procedia Engineering 20600 (2017) 479–486 K. Molokov,K. Ya.Molokov Domashevskaia / Procedia Engineering (2017) 000–000
r
Ra
2
Ra 7 2
4835
(11)
Now, taking into account (8), we have: 2
K th 1 1 Ra 1,12 0,8 1
7 1 1
(12)
7 1 1
(13)
or Ra
K th2 1
1,12 0,8 1 2
Substituting (13) and (4) in (10), we find the correction for the radius of the hole:
Kr
103 812,3
7 1 1 1,12 0,8
1
0,3
2
K th2 1
(14)
After substituting (7), (14) and (8) in (5) and solving it numerically with respect to σ 0 , where σ 0 is threshold limit endurance σ th 1k , or the criterion of propagated cracks after beginning in the macro concentrator:
0 thrk
(15)
where σ'0 and σ thrk is the external load with a given cycle asymmetry and the threshold endurance limit of nonpropagating cracks for this asymmetry, respectively. For calculation of endurance limit of undamaged material we use dependency, that takes into account the characteristics of its structure. For example, for the heat-effected zone and the zone of thermal influence (ZTI) is characterized by the structural heterogeneity, the endurance limit of which is σ 1 calculated in accordance with the well-known work [3]: 2 0, 7 T 2 1 0, 7 T l0 1 K th 1
1/ 2
(16)
where l0 is the maximum size of the microcrack, which does not affect the endurance limit; σ t is the yield strength of the material at the level of 0.2% by the conditional stretch diagram; K th 1 is the threshold stress intensity factor for asymmetry rос 1 . 3. Comparison of theoretical results with experimental data
Using the dependence (5), we constructed curves for steel, threshold stresses of the endurance limit σ th 1k of a symmetric cycle, depending on the defect size in the form of a circular hole (fig. 2). These curves limit the area from above when the stresses of the external load σ '0 do not lead to the propagation of a crack, provoked by the macro concentrator. On the same graph, for comparison, the results of experimental data have been taken from [6, 8] are
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presented. Many experimental studies are devoted to the cracks appearance near holes which diameters are insignificant and approach micro defects. But there are not enough the experimental data for large hole diameters: such tests are very laborious. Therefore, the questions of the theoretical estimation of the cracks initiation, as well as their stopping beyond the zone of high stresses action from the concentrator, with known characteristics of external loading remain relevant. In the graphs, the threshold stresses are given in normalized values to the endurance limit of an undamaged material without stress concentrators (fig. 2). Also, the question for the developing crack in the region of high-stress concentration is also of interest, what is the maximum length of the crack that will reach when exiting this area before it stops. We will call this length the threshold length of a crack originating in a known concentrator, or simply the threshold length of the crack of the concentrator. Its calculation will answer whether it is possible to continue the operation of the structure or according to the technical requirements of operation, a crack of such length. In fig. 3, the dependence of the change in the threshold length of the crack concentrator depending on the size of the concentrator is proposed. It is caused by stresses from the concentrator and is non-developing at the threshold endurance limit σ th 1k of an element with stress concentration.
Fig. 2. Curves of the dependence of the effective concentration coefficient (the reciprocal of ( σ th 1k / σ 1 ) as a function of the radius of the circular hole. Here ⋇, ⊡ and - are the experimental data from [6] (analog - steel 50) and [5] (steel 10), respectively; ─ The theory according to (5) for steel 50 (the curve from below) and steel 10 (the curve from above); --- theory (1); ⨀ and � - steel 22K and St3, respectively.
Fig. 3. Curves of change in the permissible effective coefficient of stress concentration from the maximum threshold crack formed in the concentrator for various steels: ⊡ - steel 50; ⨀-steel 10; � - steel St3; ⋇ - steel 22K.
K. Molokov et al. / Procedia Engineering 206 (2017) 479–486 K. Molokov, Ya. Domashevskaia / Procedia Engineering 00 (2017) 000–000
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A comparison of the theoretical curves in fig. 2 with experimental data shows adjustment of the results of our model with macro-concentrators of small size. For large radiuses Ra , circular holes of the experimental values on the graphs are not given, but the nature of the curves according to the proposed method is adequate in comparison with the model (1). The model (1) shows that for some steels already at Ra 10 mm the cracks will develop with an external load of 1 / 3 . This circumstance contradicts existing and known approved ideas: the crack does not develop at stresses below the endurance limit of the undamaged material. While (5) shows that as the radius of the circular hole increases, th 1k approaches asymptotically to 1 / 3 , and this is an adequate behavior and does not contradict known facts in LMR. 4. The discussion of the results
In order to use the diagrams of the limiting propagation of cracks (see fig. 2 and fig. 3), it is necessary to know the external load σ '0 , the limit of endurance of the undamaged material σ fr and the radius of the macro concentrator R a . From the known value of R a , we draw a vertical line up to the cross with the curve of a certain material (see fig. 2) and find the value of the ratio σ th 1k / σ 1 along the vertical axis. Therefore, intersection K σ 1 , where K is the value found in the vertical axis, and it will be the ultimate external load at which the crack is not yet propagated. If the external load is higher than σ th 1k , then for a given concentrator the crack generated in it will be propagated and reach a critical value. The next stage is the determination of the maximum length of the crack, which is non-propagable at a given external load. For this you need to use the diagram of the maximum threshold crack (see fig. 3). We postpone the value of the ratio σ th 1k / σ 1 . Further, before crossing with the curve of the corresponding material, draw a horizontal line and note the threshold crack length L thk . 5. Conclusions
It should be noted that for small values of L thk, , under the visual test the crack can not be seen when L L thk and th 1k / 1 0,8 , and it should be considered that cracks of this size do not affect the endurance limit. This case refers to physically small stress concentrators. Taking into account the obtained results, it can be concluded that for some materials, for example, brittle materials and high-strength steels with low ductility, the threshold crack size may be larger than the critical size of the macrocrack, i.e. Lthk Lc . In this case, it is necessary to rely on the value of Lc , and the situation of brittle failure from the macro concentrator becomes paramount. For high-strength steels, Lc is usually small, and for
relatively large concentrators, such as Ra / Lc 1 3 , the cracks can reach their critical size even under external
loads which are a little above 1 / 3 - in the case of a concentrator in the form of a circular hole. Thus, relatively large concentrators represent a great danger for high-strength steels or steels with low ductility, if there is no wellverified periodic diagnostics of the structure. References [1] A.V. Gridasov, Influence of the type and geometric parameters of the welded joint to endurance of welded structures of thermal power facilities, Research on improving the efficiency of shipbuilding and ship repair. 45 (2005) 254–257. [2] T.L. Anderson, Fracture mechanics: fundamentals and applications, 2-nd ed, London, 1995, 682 p. [3] S.S. Manson, G.R. Halford, Fatigue and durability of structural materials, ASM International, 2006, 456 p. [4] G.M. Charzyński, Deformation. Destruction. Reliability: Problems of deformation and fracture of steel, Methods for assessing the strength of power equipment and pipelines, Lenand, Moscow, 2014, 544 p. [5] B.P. Kishkin, Structural strength of materials, State University, Moscow, 1976, 184 p. [6] G.V. Matokhin, K.P. Gorbachev, A.Y. Vorobyov, A framework for assessing the strength and durability of welded structures, FESTU, Vladivostok, 2008, 270 p.
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