PANM 17. Karel Mikeš Comparison of crack propagation criteria in linear elastic fracture mechanics

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1 PANM 17 Karel Mikeš Comparison of crack propagation criteria in linear elastic fracture mechanics In: Jan Chleboun and Petr Přikryl and Karel Segeth and Jakub Šístek and Tomáš Vejchodský (eds.): Programs and Algorithms of Numerical Mathematics, Proceedings of Seminar. Dolní Maxov, June 8-13, Institute of Mathematics AS CR, Prague, pp Persistent URL: Terms of use: Institute of Mathematics AS CR, 2015 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 Programs and Algorithms of Numerical Matematics 17 J. Chleboun, P. Přikryl, K. Segeth, J. Šístek, T. Vejchodský (Eds.) Institute of Mathematics AS CR, Prague 2015 COMPARISON OF CRACK PROPAGATION CRITERIA IN LINEAR ELASTIC FRACTURE MECHANICS Karel Mikeš Faculty of Civil Engineering, Czech Technical University in Prague Thákurova 7, Prague 6, Czech Republic Abstract In linear fracture mechanics, it is common to use the local Irwin criterion or the equivalent global Griffith criterion for decision whether the crack is propagating or not. In both cases, a quantity called the stress intensity factor can be used. In this paper, four methods are compared to calculate the stress intensity factor numerically; namely by using the stress values, the shape of a crack, nodal reactions and the global energetic method. The most accurate global energetic method is used to simulate the crack propagation in opening mode. In mixed mode, this method is compared with the frequently used maximum circumferential stress criterion. 1. Introduction The description of crack propagation is one of the most important ingredients of linear elastic fracture mechanics (LEFM). The main questions are: At which loading level will the crack propagation begin and in which direction will the crack propagate? The aim of this paper is to compare numerical implementations of most frequently used crack propagation criteria for opening mode and mixed mode in 2D. 2. Stress intensity factor concept The stress intensity factor is a quantity used in LEFM to describe the asymptotic singular stress field near the crack tip. The stress in the vicinity of the crack tip is unbounded and grows in inverse proportion to the square root of distance from the tip. Under plane stress, the asymptotic stress field is described by σ x (r, θ) = K I 2πr cos θ 2 σ y (r, θ) = K I 2πr cos θ 2 ( 1 sin θ 3θ sin 2 2 ( 1 + sin θ 3θ sin 2 2 τ xy (r, θ) = K I sin θ 2πr 2 cos θ 3θ cos 2 2 K II cos θ 2πr ) K II sin θ 2πr 2 ) + K II sin θ 2πr 2 cos θ 2 ( 2 cos θ 3θ cos 2 2 3θ cos ( 1 sin θ 3θ sin 2 2 ),(1) 2, (2) ), (3)

3 where K I and K II are the stress intensity factors for modes I and II which represent the loading and geometry conditions, r is the distance from the crack tip and θ is the polar angle; see e.g. [7]. 3. Crack propagation in mode I (opening mode) In mode I, the crack is opening without sliding. Therefore, we can assume that the crack will propagate in the original direction and we have to decide when the propagation starts Local Irwin criterion This concept was introduced by Irwin [4] in The stress intensity factor is used to decide about the crack propagation. The propagation will start when the value of the stress intensity factor K I reaches its critical value so-called fracture toughness denoted by K c Global Griffith criterion This criterion was introduced by Griffith [3] in The crack will grow if a sufficient amount of energy is released by its propagation. The criterion is based on the strain energy release rate, defined as G(u, a) = 1 t W e (u, a), (4) a where W e (u, a) is the elastic strain energy considered as a function of the imposed displacement u and the crack length a. The beam thickness is denoted by t; see Figure 1. Under plane stress and in mode I, both criteria are equivalent [2]. We can write G(u, a) = K I 2 E. (5) The rules for crack propagation according to the local Irwin and global Griffith criteria are summarized in Table 1, where K c resp. G f are material properties called fracture toughness [Nm 3/2 ] and fracture energy [Nm 1 ], resp. Local criterion Global criterion Crack behaviour K I < K c G < G f no crack propagation K I = K c G = G f crack propagation K I > K c G > G f inadmissible (in statics) Table 1: Crack propagation rules according to the local and global criteria 143

4 3 Výpočetní modely Výpočty pro jednotlivé modely byly prováděny pomocí metody konečných prvků fyzikální za rozměr předpokladu jednotek, rovinné proto napjatosti. není nutné K uvádět, výpočtům zda se byl jedná použit o milimetry program OOFEM. nebo 3.3. Simulation centimetry, Jedná in ale se o opening veličiny objektově mode můžeme orientovaný, považovat volně za šiřitelný bezrozměrné. software určený pro řešení multifyzikálních problémů metodou konečných prvků, který je Four methods have been used to calculate the stress intensity factor or the strain vyvíjen na katedře mechaniky Stavební fakulty ČVUT [4]. Pomocí tohoto energy release rate in opening mode; F/2 namely by using (i) the stress values, (ii) the programu crack opening, byly (iii) pro nodal řešený forces model and získány (iv) the release hodnoty of strain posunů energy. a uzlových The first reakcí three pro methods jednotlivé have auzly locala character pro jednotlivé and deal prvky withbyly the values vypočteny near the hodnoty crack tip deformací to calculate napětí. the stress Pro intensity následné factor. zpracování The fourth těchto method hodnot evaluates a výpočty the change faktoru ofintenzity the energy a napětí of the whole byly vytvořeny beam whenvlastní the crack programy is extended. v jazyce Three Delphi different Object types Pascal. of triangular h finite elements have been used for each method; namely (i) three-node element with 3.1 linearnosník a approximation s vrubem of displacement, (ii) six-node element with quadratic approximation and (iii) six-node quadratic element with singular shape functions on the Prvním edges starting zkoumaným from the modelem L/2 crack tip; byl see prostý Figure nosník 2. Numerical se svislou simulations trhlinou (vrubem) have been na performed spodním using okraji theuprostřed open-source rozpětí, finite element namáhaný codetříbodovým OOFEM [6]. ohybem, The relative viz Obrázek of all methods 3.1. Poměr in a three-point byl error (a) zvolen bending 4, protože test pro withtento geometry poměr (b) according jsou v to literatuře Figure 1 have Obrázek been evaluated 3.2: (a) geometrie by comparing modelované the computed poloviny value nosníku, of the stress (b) intensity použitá síť factor dostupné přibližné vzorce, které bude možné využít pro ověření správnosti with the,,exact values obtained by using approximate analytic formulas available numericky Pro in [2], výpočet [5] and získaných metodou [7]. The výsledků. konečných relative errors Dále prvků ofzavedeme allbyla methods použita veličinu for all relativně types of, dosti která elements jemná představuje trojúhelníková shown inpoměr Table délky 2. síť, která trhliny byla ku v výšce okolí nosníku. kořene trhliny ještě výrazně zjemněna, are viz Obrázek 3.2 (b). Celkem byly vytvořeny tři sítě lišící se typem použitých prvků. Nejprve síť obsahující nejjednodušší trojúhelníkové tříuzlové prvky, které aproximují pole posunů pomocí lineárních funkcí. Poté lepší šestiuzlové trojúhelníkové prvky, které k aproximaci pole posunů využívají kvadratické funkce. A na závěr upravené kvadratické prvky, u kterých byl na úsečkách obsahující kořen trhliny posunut mezilehlý uzel z jedné poloviny do jedné čtvrtiny blíže ke kořeni trhliny, viz Obrázek 3.3. Tento posun vynutí u bázových funkcí nenulových v kořeni trhliny singularitu typu r 1/2. Tedy stejnou, jaká se vyskytuje v asymptotickém poli napětí v okolí kořene trh- liny. Předpokládáme, že tato úprava umožní ještě lépe vystihnout hodnoty napětí v Obrázek okolí Figure kořene 3.1: Geometrie 1: trhliny. Geometry Tento a zatížení of thezpůsob three-point modelu umožnil nosníku bending pro s test každou vrubem použitou metodu porovnat vliv typu prvků sítě. Ke generování sítí byl použit generátor T3D [5]. Protože nosník je osově symetrický, bylo pro zjednodušení výpočtu možné modelovat pouze jednu symetrickou polovinu nosníku, viz Obrázek 3.2 (a). Tím se snížil počet neznámých na polovinu a výpočet probíhal výrazně rychleji. Dalším zjednodušením bylo, že nosník nebyl zatěžován silou, ale byl mu v místě působení síly předepsán svislý konstantní posun. Velikost zatěžující síly, která by odpovídala předepsanému posunu, byla získána jako svislá reakce zatěžovaného uzlu. Geometrie nosníku byla zvolena následovně: výška Figure Obrázek 2:, Used délka 3.3: finite Lineární, elements:, tloušťka kvadratický linear, (left), délka a quadratic upravený trhliny (middle), kvadratický, velikost modified prvek základního quadratic trojúhelníkového (right) prvku,, velikost trojúhelníkového prvku v okolí singular kořene Přibližné trhliny vzorce. Při pro výpočtu výpočet metodou faktoru intenzity konečných napětí prvků není podstatný Pro nosník s vrubem s poměrem délky ku výšce L = 4 je v díle [3] k dispozici h následující vzorec:

5 Linear Quadratic Singular quadratic Values of stress > 30 % % % Shape of a crack < 10 % 2 % 1 % Node reactions 5-10 % < 5 % < 5 % Energetic method 5 % 2 % 0.5 % Nano and Macro Mechanics 2013 Faculty of Civil Engineering, CTU in Prague, 2013, 19 th September Table E = 202: GPa, The Poisson relative ratio errors ν = 0,2 ofand thefracture computed toughness stress K c intensity = 4 MNm 3/2 factor, we Kobtain I for the different Forcedisplacement methods and diagram, different see in finite Fig. 2. elements F [kn] a 0 = 50 mm a = 100 mm a = 350 mm a = 150 mm a = 200 mm a = 250 mm a = 300 mm a = 400 mm a = 450 mm a = 480 mm u [mm] Fig. 2 Force-displacement diagram of the three-point bending test Figure 3: Force-displacement diagram of the three-point bending test MCSC F [kn] MSERR The global energy method using the modified quadratic elements is the most accurate but also the most time-consuming one. Figure 3 shows the load-displacement diagram obtained with this method for a beam of the following geometry: height h = 0.5 m, length L = 2 m, thickness t = 0.2 m, initial crack length a 0 = 0.05 m, elastic modulus E = 20 GPa, Poisson ratio ν = 0.2 and fracture toughness K c = 4 MNm 3/2. 4. Crack propagation in mixed mode The mixed mode represents a combination of tensile opening and in-plane shear. The direction of crack propagation is not known in advance and has to be determined by a suitable criterion CRACK PROPAGATION IN MIXED MODE In opening mode in a plane there is a combination of an opening and a shear. The direction of the crack propagation has to be determined by another criterion. u [m]

6 4.1. Maximum circumferential stress criterion (MCSC) This criterion determines the crack propagation direction based on the maximal circumferential stress σ θ, which is defined as Substituting from (1) (3), we obtain σ θ (r, θ) = σ y cos 2 θ + σ x sin 2 θ 2τ xy sin θ cos θ. (6) σ θ (r, θ) = K I 2πr cos 3 θ 2 3 K II 2πr cos 2 θ 2 sin θ 2. (7) The angle θ with maximal circumferential stress (MCSC1) is obtained by solving the equation with the conditions sin θ + K II K I (3 cos θ 1) = 0 (8) θ ( π, π); K I > 0; K II sin θ 2 < 0, (9) where the ratio K II /K I is obtained by fitting (1) (3) to the values of the stress field in a number of Gauss points near the crack tip. Another approach (referred to as MCSC2) is based on substituting the values of the stress field at Gauss points into the original definition of circumferential stress (5). After smooth of these data by a polynomial function, the angle θ that maximizes this function can be found. Both approaches give almost the same results; see Figure 4. The first method (MCSC1) turned out to be numerically preferable and therefore is used in the following examples Maximum strain energy release rate criterion (MSERRC) This criterion determines the crack propagation in the direction that leads to the maximum strain energy release rate defined in (4). Numerical realization consists in simulation of a number of sufficiently small crack extensions in several different directions. For each direction, the strain energy release rate is evaluated by subtracting the final strain energy from the original one and dividing by the increment of the crack area. The obtained values are smoothed using a polynomial function, for which the maximum is then found. This criterion predicts, in most cases, similar crack trajectories to the MCSC. However, application to the three-point bending test with an eccentric initial crack leads to a different crack path; see Figure

7 Nano and Macro Mechanics 2013 Faculty of Civil Engineering, CTU in Prague, 2013, 19 th September MCSC1 MCSC2 MSERRC Nano and Macro Mechanics 2013 Faculty of Civil Engineering, CTU in Prague, 2013, 19 th September Fig. 3 Comparing of the different criterion in the three-point bending test Figure 4: Comparison of crack paths according to different criteria for the three-point bending test with an eccentric initial crack 4.3. Comparative example Another example is taken MCSC1 over from [4]. It is a rectangular panel with two holes and two initial 4.3. cracks Comparative submitted to a vertical example MCSC2 tensile. In this example both criterions lead to almost the same crack paths The and last results example are similar ismserrc to taken the results from from [1]. [4], Itsee is in a Fig. rectangular 4 and 5. panel with two holes and two initial cracks subjected to tension in the vertical direction. In this example, both criteria lead to almost the same crack paths and the results are similar to those from [1]; see Figure 5. The load-displacement diagrams are depicted in Figure 6. Both criteria predict the same behaviour but the curve obtained with MCSC is not smooth. This means that MCSC is less accurate when used to decide whether the crack is propagating Fig. 3 Comparing or not. of the different criterion in the three-point bending test MSERRC Conclusion Comparative MCSC example Another In opening example is mode, taken over the best from results [4]. It is are a rectangular obtainedpanel by the with global two holes energy and two method. initial In cracks the mixed submitted mode to Fig. a that 4 vertical Comparing arises tensile. of in the In different this three-point example criterion both bending criterions a vertical test, tensile lead the to test MSERRC almost the same criterion crack paths and results are similar to the results from [4], see in Fig. 4 and 5. Fig. 5 Comparing of the different criterion in a vertical tensile test from [4] 5. CONCLUSION With using the global energetic method, we obtained very good results in mode I. In the three-point bending test in mixed mode, criterion based on this method leads to the different crack path then the criterions using MSERRC the circumferential stress. But both criterions give the similar results in other examples in mixed mode. MCSC REFERENCES Fig. 4 Comparing of the different criterion in a vertical tensile test [1] JIRÁSEK, M. and ZEMAN, J. Přetváření a porušování materiálů. Praha: ČVUT, ISBN Figure 5: Comparison of crack paths according to different criteria in a vertical tensile test. Simulation in OOFEM (left) against results taken from [1] (right) Fig. 5 Comparing of the different criterion in a vertical tensile test from [4] 5. CONCLUSION 147 With using the global energetic method, we obtained very good results in mode I. In the three-point bending test in mixed mode, criterion based on this method leads to the different crack path then the criterions using the circumferential stress. But both criterions give the similar results in other examples in mixed mode.

8 a 0 = 300 mm a 0 = 350 mm a 0 = 400 mm a 0 = 450 mm a 0 = 480 mm u [mm] Fig. 2 Force-displacement diagram of the three-point bending test F [kn] MCSC MSERR u [m] 4. CRACK Figure 6: PROPAGATION Comparison of IN load-displacement MIXED MODE diagrams for different criteria In opening mode in a plane there is a combination of an opening and a shear. The direction of the crack propagation has to be determined by another criterion. based on this method leads to a different crack path than the MCSC criterion using the circumferential stress. But both MSERRC and MCSC give similar results in other examples in mixed mode. Both criteria seem to be accurate in prediction of the propagation angle, but to determine whether the crack propagates it is more appropriate to use MSERRC. Acknowledgements This work was supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS14/029/OHK1/1T/11. References [1] Bouchard, P. O., Bay, F., and Chastel. Y.: Numerical modelling of crack propagation: automatic remeshing and comparison of different criteria. Comput. Methods Appl. Mech. Engrg. 192 (2003), [2] Broek, D.: Elementary engineering fracture mechanics. Martinus Nijhoff Publishers, Hague, [3] Griffith, A. A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. 221 (1921),

9 [4] Irwin, G. R.: Analysis of stresses and strains near the end of a crack traversing a plate. J. Appl. Mech. 24 (1957), [5] Jirásek, M. and Zeman, J.: Přetváření a porušování materiálů ČVUT, Praha, [6] Patzák, B.: OOFEM an object-oriented simulation tool for advanced modeling of materials and structures. Acta Polytechnica 52 (2012), [7] Wang, C. H.: Introduction to fracture mechanics. DSTO Aeronautical and Maritime Research Laboratory, Melbourne,