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Numerical and experimental investigations of the influence of corner singularities on 3D fatigue crack propagation
Engineering Fracture Mechanics 72 (2005) 2095–2105 www.elsevier.com/locate/engfracmech
Numerical and experimental investigations of the influence of corner singularities on 3D fatigue crack propagation M. Heyder *, K. Kolk, G. Kuhn Institute of Applied Mechanics, University of Erlangen-Nuremberg, Egerlandstr. 5, 91058 Erlangen, Germany Received 12 March 2004; received in revised form 3 November 2004; accepted 10 January 2005 Available online 21 March 2005
Abstract In the case of surface breaking cracks, the typical square-root stress singularity is generally not sustainable and a 3D corner singularity in the vicinity of the intersection of crack front and free surface has to be considered. Only the crack front intersection under a special angle cr ensures a valid square-root stress singularity and the applicability of the classical SIF-concept. In this paper, the theory of the numerical determination of the intersection angle cr is briefly described and the influence of the 3D corner singularity on fatigue crack growth is experimentally investigated. Therefore, 3D fatigue crack propagation experiments under pure mode-I are performed. Transparent specimens of PMMA are used, in order to be able to observe and to document accurate sequences of real 3D crack front evolution profiles via in situ photographic measurement. 2005 Elsevier Ltd. All rights reserved. PACS: 46.30.N; 62.20.M Keywords: 3D corner singularity; Crack front angle; Crack propagation
1. Introduction Stable 3D fatigue crack growth is considered in this paper. This process can be described in terms of linear elastic fracture mechanics as the crack already grows under relatively small external loads compared to static loading conditions. Hence, the crack or the crack near-field, respectively, is characterized by the classical stress intensity factors (SIFs) or the energy release rate, which is directly linked to the SIFs in this context. These SIFs are related to the well-known square-root stress singularity. But this kind of *
Corresponding author. Tel.: +49 9131 8528516; fax: +49 9131 8528503. E-mail address: [email protected] (M. Heyder).
0013-7944/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2005.01.006
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singularity is generally not present at non-smooth parts of the crack front and at the intersection of the crack front and a free surface. At such points one has to deal with a 3D corner singularity [2,1,7]. The arising question is, which influences of this corner singularity have to be regarded [16]. A first answer can be found e.g. for mode-I in [1] and for mixed-mode in [5]. Based on both investigations, one may conclude, that the crack front will be shaped ensuring the classical square-root singularity or at least a bounded energy release rate. If the square-root singularity really occurs at the intersection of the crack front and the free surface, it will be linked to a special crack front intersection angle cr. This question is investigated both numerically and experimentally under mode-I conditions. For a given geometrical situation, the solution of a quadratic eigenvalue problem provides the singular exponents of the 3D singularity, cf. Section 2. The experiments as described in Section 3 have to be performed precisely. Only a careful comparison between experimental observations, e.g. well documented sequences of incremental crack front shapes, and corresponding numerical analyses enable an identification of the influences of the 3D corner singularity. In order to be able to observe and to store the crack front propagation by photographic in situ measurements, specimens of the transparent material PMMA have been used. Experiments using different geometries are analyzed to verify if the characteristic angle cr is unique. This is realized in terms of different cross-sections and varying boundary conditions.
2. Singularities along a crack front It is well-known, that the stress field shows a singular behavior in the neighborhood of a crack front within the framework of linear elasticity. In the context of 2D, assumptions (plane strain or plane stress) are made so that the crack front is reduced to a single crack tip and the classical square-root stress singularity is always present. As no information is available along the crack front, one has to switch from 2D to 3D to involve the crack front into the analysis. As a consequence, two certain kinds of singularities have to be distinguished. Along a smooth part of a crack front a wedge singularity is present. But at non-smooth parts of the crack front and particularly at the intersection of the crack front with the free surface, a 3D corner singularity has to be taken into account. 2.1. Wedge singularity The wedge singularity is equivalent to the classical square-root singularity with a multiplicity of three. For sake of completeness, the stresses of the asymptotic crack near-field are given in Eq. (1) in terms of local crack front coordinates, cf. Fig. 1, rij ðr; U; P Þ ¼
III X pffiffi K M ðP Þ M pffiffiffiffiffiffiffi fij ðUÞ þ T ij þ Oð rÞ 2pr M¼I
ð1Þ
for arbitrary smooth crack fronts [12]. The intensity of this singularity is characterized by the stress intensity factors KM (M = I, II, III). The remaining terms are the angular functions fijM ðUÞ and the elastic T-stresses Tij. Both fracture mechanics parameters (KM(P) and Tij(P)) calculated very accurately are from the numerical stresses in the crack near-field via an optimized extrapolation method based on a regression analysis [8]. The stress field is provided by a special BEM formulation—the 3D Dual Discontinuity Method (DDM)—a powerful tool for linear-elastic stress concentration problems [15]. Alternatively, if a FEM analysis is performed, the SIFs can be determined e.g. by the virtual crack closure method (VCCM) [3] or a hpversion FEA as well as a mathematical splitting scheme [6].
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Fig. 1. Local crack front coordinate system.
2.2. 3D corner singularity Compared to the wedge singularity, the 3D corner singularity is in general not a-priori known for every singular point mentioned above. Hence, the value of the singularity has to be determined by a singularity analysis, that traces back to [1,2] and efficient numerical procedures have been published recently [4,10]. The outline of such a scheme is as follows. Around the singular point O with its coordinates xO 2 IR3 an e-neighborhood XeO 2 IR3 is considered. XeO is defined by the cut-set of an e-ball Xe := {x 2 IR3:jx xOj < e} and the domain X 2 IR3 as shown in Fig. 2. It is assumed, that the 3D body X consists of homogeneous, isotropic and linear-elastic material. To extract the 3D corner singularities the displacement field is utilized. Therefore, the corresponding elastic solution in the vicinity of the singular point O related to a spherical coordinate system, cf. Fig. 2, is asymptotically expanded in the form 1 X ui ðq; h; u; OÞ ¼ K L ðOÞqaL giL ðh; u; OÞ ð2Þ L¼1
Fig. 2. Vicinity at the singular point O.
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under the assumption that no logarithmic terms occur, which is true for most practical applications. aL denotes the asymptotic exponents satisfying aL > 0.5 to keep the elastic energy bounded, cf. [4], and giL ðh; u; OÞ are the corresponding angular functions. The asymptotic exponents are now linked to generalized intensity factors K L . The variables u and h belong to the bounded 2D domain Xuh oXeO with Xuh {u · h:u 2 [0, 2p] ^ h 2 [0, p]}. Among other techniques, the asymptotic exponents aL are obtained by the solution of a quadratic eigenvalue problem. After a linearization and spectral transformation it is solved with the help of the mathematical package ARPACK [13]. Finally, the eigenvalues aL and eigenvectors giL are known simultaneously. Because the asymptotic behavior is focused, the interval 0.5 < aL < 1 is considered excluding the rigid body motion modes (a = 0 and 1) as they are known. aL depends on the geometrical situation around the singular point described by XeO and on material parameters. For single, homogeneous and isotropic materials it only depends on the Poissons ratio m. 2.2.1. General remarks For aL = 0.5 the intensity factor K L may correspond with one of the classical stress intensity factors KM. In general, one cannot distinguish between mode-I, II, III any more but only between symmetric and antisymmetric modes. The symmetric mode corresponds to mode-I and the antisymmetric modes can be either pure mode-II or mode-III or a combination of both. The classical SIF-concept only fails at some special points with aL 5 0.5. But to be still able to apply this concept for the description of the behavior in the crack near-field the SIFs are numerically defined at these particular points [14]. Following (2) leads to u OðqaL Þ and therefore r OðqaL 1 Þ. This means, if aL is greater than 0.5 and less than 1.0 the stresses are still singular but weaker compared to the square-root singularity. Consequently, KM(P) tends to zero as P tends to the singular point O, see also [6]. If aL is less than 0.5 the opposite situation is present and KM(P) tends to infinity. Therefore, the exponents aL have to be known to determine the classical SIFs asymptotically. It is advantageous for the simulation of crack propagation that, if the singular point O is located at the ends of a crack front, the crack front angle c can be determined by the assumption of aL = 0.5, [9]. As only mode-I is considered in this paper, the particular exponent aL is related to the symmetrical crack opening. The corresponding crack front angle is denoted by cr and only KI is existing. 2.2.2. Consequence for real crack propagation In case of surface breaking cracks it has been experimentally observed, cf. Section 3.2, that the crack front always intersects the free surface having a special angle—the crack front angle cr. This angle depends only on the Poissons ratio m and on the geometrical situation around the singular point. The intersecting angle c is shown in Fig. 3 and defined between the inward directed normal vector n and the tangent t related to the crack front at point O. As only mode-I is considered in this paper, the crack surface coincides with the plane spanned by the vectors n and t, if the inclination angle d is equal to 90. Due to the violation of the wedge singularity only at some special points, it is assumed, that the crack front will change its shape to ensure a valid square-root singularity along the whole crack front. To verify this assumption, the present singularities have to be known. The first interesting question concerning real crack propagation is, if a manufactured initial crack front (e.g. a machined notch) intersecting the free surface under an angle cm 5 cr is going to change its crack front angle to approach cr. Moreover, does this process of adjusting the angle c happen abruptly or continuously? The second question deals with the problem, if the angle cr can be influenced by changing other parameters, e.g. the orientation of the free surface relative to the crack front/surface. To answer these questions, experimental investigations on specimens with different cross-sections and suddenly changing boundary conditions (creation of new free surfaces intersecting the crack front) are performed. By conducting these experiments, the behavior of the crack front angle c has to be observed. If the
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Fig. 3. Definition of the crack front angle c.
resultant angle c is always practically equal to cr, the crack front will be shaped ensuring the classical square-root singularity (aL = 0.5) even in the vicinity of the intersection of the crack front and the free surface.
3. Experimental investigations The influence of the additional 3D corner singularity on the shape of the crack front, especially in the vicinity of the crack front intersection with the free surface, is going to be examined. For this purpose, a test series with four-point-bending specimens of PMMA has been carried out. The advantage of PMMA is its (a) brittle behavior, so that linear fracture mechanics can be used for description and its (b) transparency, which enables a permanent photographic in situ documentation of the crack front shape, as explained in Section 3.1. The corresponding elastic material parameters are the Poissons ratio m 0.365 and the Youngs modulus E 3.6 GPa. The size effect of the specimens is studied in Section 3.2 based on a rectangular cross-section. With specimens having a trapezoidal cross-section, the position of the free surface relative to the crack front is changed. If the crack front angle is influenced by this modification will be answered in Section 3.3. After that the creation of new free surfaces intersecting the crack front is discussed in Section 3.4 to study the related new crack front angles. 3.1. Experimental setup The experimental setup is visualized in Fig. 4. To document the shape of the crack front, a camera (VOSSCCD-1300, 1280 · 1024 pixels) and a lightning are placed ahead of the polished front of the specimen. The backside of the specimen is blackened to enhance the contrast of the taken picture. All experimental investigations are performed on a servo-hydraulic machine (Schenk PC 400-N, PL 100, PM 100-RN, S56) with a frequency of f ¼ 6:9 4 Hz (25 000 load cycles per hour). This frequency is low enough to avoid heating of the specimen due to friction at the crack front faces. The measuring of the actual crack front shape is carried out by an automatic crack front detection algorithm. Thereby, the contrast of the bright crack front versus the dark background is used. Starting from the darkest area (5 · 5 pixel) of the taken picture (this is at any rate part of the background), this area is enlarged until the contrast of the adjoined pixel is smaller then a user defined value. The lower boundary of this located area is the crack front shape. In combination with a measuring unit, the coordinates of the
Ku ¨ HLER
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Fig. 4. Experimental setup of a four-point-bending specimen.
crack front are calculated and interpolation points (the distance between two successive points is dÆÆ = 20 pixel) are printed into an input-file, e.g. for usage in FEM/BEM-preprocessors [11]. 3.2. Rectangular cross-section The influence of the width of the specimens on the shape of the crack front is checked by specimens with a rectangular cross-section. In this test series different dimensions (five specimens per size) are investigated. Starting from a standard model with the dimensions length = 220 mm · width = 20 mm · height = 60 mm, this size of the specimens is firstly scaled only in the width (Fig. 5a) and secondly in all directions (Fig. 5b), each with the factor 32 and 2. Starting from an initial straight notch in the middle of the specimens, the propagating crack profile is monitored as described in the last section. A typical picture of these crack fronts is shown in Fig. 5c. As a result, all documented crack fronts have the same body structure and after scaling with respect to the width the same shape, too. Independently of the size of the specimens, the value of the crack front angle c was found to be c 14 ± 3 which is shown in Fig. 6. Obviously, there is no size effect concerning the crack front angle.
Fig. 5. Specimens with rectangular cross-sections and angle c 14.
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Fig. 6. Crack front shapes of the specimens with rectangular cross-section.
3.3. Trapezoidal cross-section A variable orientation of the free surface with respect to the crack surface while ensuring the same inclination angle (d = 90) is realized by a trapezoidal cross-section. But it should be distinguished, if the crack grows into the narrower or wider part of the specimen as indicated in Fig. 7a. Five specimens per trapezoidal angle (2b = ±30/60/90) are investigated. Fig. 7b shows one half of a specimen with 2b = 90 and Fig. 7d a specimen with 2b = 60. As an interesting fact of this test, the trapezoidal angle b does not have any dependence on the angle c which is always practically 14. The only difference between the positive and negative b-specimens is a different bending behavior of the crack front. In the case of b > 0 the crack front shows a complete convex shape. But for negative values of b there is a transition between convex and concave zones. The crack front is also convex in the middle, whereas it is concave in the vicinity of the free surface to approach the unique crack front angle. Therefore, the area where the tangential angle c appears, is much smaller compared to b > 0-specimens. A magnified cutout of the 2b = 90-specimen is shown in Fig. 7c, where the characteristic angle cr 14 has been measured, too. Based on these investigations of the trapezoidal specimens it has been shown, that the crack front will always find a shape that ensures the square-root stress singularity at the vicinity of a free surface. 3.4. Crack propagation rate According to the last two sections, it seems that the crack front angle is unique, independently of the global geometry. But does the process of adjusting this special crack front angle cr happen continuously
Fig. 7. Specimens with different trapezoidal cross-sections and angles b, but with the same crack front angle c 14.
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(a)
(b)
Fig. 8. Specimens with trapezoidal cross-sections and new free intersecting surfaces.
or abruptly? This question will be answered by disturbing an existing crack surface by a new free surface. It will be exemplary shown by the help of the trapezoidal specimen. There are three alternatives for this new free surface. The newly created intersecting angle cn is (a) greater, (b) equal or (c) smaller than the characteristic angle cr. In the first case (cn > cr), the classical SIF tends to infinity (KI ! 1) at the new end of the crack front. Therefore, an abrupt crack propagation at the new free surface could be expected as demonstrated in Fig. 8a and studied in Section 3.4.1. In the second case, the SIF is well defined and no significant influence concerning the crack propagation rate is observed. The third case leads to a SIF tending to zero (KI ! 0), so that no crack propagation will occur at the new free surfaces. The crack should only propagate in the middle of both parts of the specimen as shown in Fig. 8b to approach the characteristic angle cr, c.f. Section 3.4.2. 3.4.1. Crack propagation rate for cn > cr In this case, the inclined left side of the trapezoidal cross-section is cropped, so that the new free surface intersects the initial crack front under cn > cr. The consequence of this is KI ! 1, that could lead to an abrupt crack propagation—even at very low loading conditions. Therefore, the force was reduced to 75% of the force to create the initial crack, so that the stress intensity factor at the non-influenced crack front (middle and right part of the specimen) falls below the lower threshold Kth for fatigue crack propagation (KI < Kth). In Fig. 9, the fatigue crack propagation can be observed only at the influenced left side of the specimen. Against ones expectations, there is no abrupt crack
Fig. 9. Crack propagation after cropping the left side of the specimen.
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Fig. 10. Crack propagation after notching in the left part of the specimen.
propagation. This can be explained with the fact, that there is not enough energy in one load cycle to create the new whole crack surface. 3.4.2. Crack propagation rate for cn < cr To examine the case cn < cr, the specimens with the trapezoidal cross-section are notched lengthwise, to create a new free surface in the middle of the already existing crack front (Fig. 8b). The new angle is cn 0, so that crack propagation should only occur in the middle of both parts of the specimen. The following fatigue crack propagation shows a continuous crack propagation in the middle of each side of the specimen to approach the characteristic angle c 14 at the new surface (Fig. 10).
4. Verification of the measured crack front angles All investigated specimens, cf. Sections 3.2, 3.3 and 3.4, have a different cross-section and therefore a different global geometry. Moreover, in each case the crack front is shaped uniquely. But the crack front angle c has practically always the same value cr. The reason for that behavior is, that the geometrical situation in the vicinity of the intersection of the crack front and the free surface is the same. Namely, the crack surface is perpendicular to a smooth free surface and only the symmetrical crack opening mode is activated. A singularity analysis as mentioned in Section 2.2 will be applied to verify, if c = cr corresponds to the assumed square-root stress singularity aL = 0.5. The surface Xuh is spanned by (u, h) 2 [0; 180] · [0; 180] and the crack intersects this plane at u = d = 90 and h 2 [h*; 180]. For h* = 90 and m = 0 the singular exponent aL is analytically known and equal to 0.5. The relative error of the numerically determined value anum is approximately 106. But for m 5 0, the inverse problem of seekL ing the angle h* that ensures aL = 0.5 has to be solved. The crack front angle cr is related to h* by cr = 90 h*. The corresponding distribution in dependence of the Poissons ratio m is shown in Fig. 11.
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Fig. 11. Crack front angle cr ensuring a valid square-root singularity under mode-I.
The solid curve represents the plot of 2m cr ¼ 90 arctan m
ð3Þ
which was proposed by [16]. This equation is only valid for the considered geometrical situation at the singular point. The numerical results obtained from the inverse problem of the 3D singularity analysis are marked by the dotted line. Both results are in a very good agreement for m 2 [0; 0.15]. In the remaining range of m the numerical solution leads to slightly larger crack front angles. Based on our results, Eq. (3) is a first rough estimate for the crack front angle in this special case—symmetrical crack opening of a crack intersecting the free surface perpendicularly. Additionally, the numerical crack front angle for m = 0.3 is in good agreement to a reference angle of cref r 11 which can be found in [1]. In case of the investigated material PMMA with a measured m = 0.365, the angle cr has a value of 14.049. This result matches with the experimentally determined crack front angles. Hence, at least for mode-I the assumption is confirmed, that the crack front is shaped ensuring a valid square-root stress singularity in the vicinity of the intersection of the crack front and the free surface. As this type of singularity only holds at smooth parts of a crack front at first glance, now it can be stated that the square-root singularity holds along the whole crack front.
5. Conclusion The influence of the 3D corner singularity in case of surface breaking cracks has been investigated both numerically and experimentally. Therefore, different specimens with a single edge crack loaded under four point bending are considered. The vicinity of the intersection of the crack front and the free surface is analyzed, because of its three-dimensional singular behavior. In this area, a practically constant crack front angle could be experimentally observed. The process of adjusting this angle occurs continuously. The value of the crack front angle cr(m) has been verified and confirmed by a 3D singularity analysis. This agreement indirectly confirms the assumptions of linear elasticity. In addition, the observed crack front angle corresponds to the classical square-root stress singularity. Hence, at least for mode-I, the crack front is shaped,
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that this type of singularity holds along the whole crack front. The analogous effect of the 3D singularity in case of mixed-mode is addressed to a forthcoming paper.
References [1] Bazant ZP, Estenssoro LF. Surface singularity and crack propagation. Int J Solids Struct 1979;15:405–26. [2] Benthem JP. State of stress at the vertex of a quarter-infinite crack in half-space. Int J Solids Struct 1977;13:479–92. [3] Buchholz FG. Finite element analysis of 3D mixed-mode fracture problem by virtual crack closure integral methods. In: Murthy K, Buchholz, editors. Fracture mechanics. Bangalore: Interline Publishing; 1994. p. 7–12. [4] Dimitrov A, Andra¨ A, Schnack E. Efficient computation of order an mode of corner singularities in 3D-elasticity. Int J Num Meth Engng 2001;52:805–27. [5] Dimitrov A, Buchholz FG, Schnack E. Effects in crack propagation. Proceedings of the fifth world congress on computational mechanics. Vienna: Austria; 2002. [6] Fawaz SA, Andersson B. Accurate stress intensity factor solutions for corner cracks at a hole. Engng Fract Mech 2004;71(9–10): 1235–54. [7] Ghahremani F. Numerical variation method for extracting 3D singularities. Int J Solids Struct 1977;27:1371–86. [8] Kolk K, Kuhn G. 3D crack growth simulation. Proceedings of the second European conference on computational mechanics. Cracow: Poland; 2001. [9] Kolk K, Kuhn G. A predictor–corrector procedure for the determination of the shape of 3D crack fronts. Proceedings of the fifth world congress on computational mechanics. Vienna: Austria; 2002. [10] Kolk K, Mishuris G, Kuhn G. The determination of 3D singularities with known 2D singularities––a p-like method. in preparation. [11] Kroiss T, Heyder M, Kuhn G. A software for 2D/3D crack detection, student research project. University of Erlangen– Nuremberg. 2004. [12] Leblond J, Torlai O. The stress field near the front of an arbitrarily shaped crack in a three-dimensional elastic body. Int J Elasticity 1992;29:97–131. [13] Lehoucq RB, Sorenson DC, Yang C. ARPACK users guide solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods. Available from: ftp://ftp.caam.rice.edu/pub/software/ARPACK. 1998. [14] Nakamura T, Parks DM. Antisymmetrical 3-D stress field near the crack front of a thin elastic plate. Int J Solids Struct 1989;25:1411–20. [15] Partheymu¨ller P, Haas M, Kuhn G. Comparison of the basic and the discontinuity formulation of the 3D-dual boundary element method. Engng Anal Bound Elements 2000;24:777–88. [16] Pook LP. Some implications of corner point singularities. Engng Fract Mech 1994;48:367–78.
Numerical and experimental investigations of the influence of corner singularities on 3D fatigue crack propagation M. Heyder *, K. Kolk, G. Kuhn Institute of Applied Mechanics, University of Erlangen-Nuremberg, Egerlandstr. 5, 91058 Erlangen, Germany Received 12 March 2004; received in revised form 3 November 2004; accepted 10 January 2005 Available online 21 March 2005
Abstract In the case of surface breaking cracks, the typical square-root stress singularity is generally not sustainable and a 3D corner singularity in the vicinity of the intersection of crack front and free surface has to be considered. Only the crack front intersection under a special angle cr ensures a valid square-root stress singularity and the applicability of the classical SIF-concept. In this paper, the theory of the numerical determination of the intersection angle cr is briefly described and the influence of the 3D corner singularity on fatigue crack growth is experimentally investigated. Therefore, 3D fatigue crack propagation experiments under pure mode-I are performed. Transparent specimens of PMMA are used, in order to be able to observe and to document accurate sequences of real 3D crack front evolution profiles via in situ photographic measurement. 2005 Elsevier Ltd. All rights reserved. PACS: 46.30.N; 62.20.M Keywords: 3D corner singularity; Crack front angle; Crack propagation
1. Introduction Stable 3D fatigue crack growth is considered in this paper. This process can be described in terms of linear elastic fracture mechanics as the crack already grows under relatively small external loads compared to static loading conditions. Hence, the crack or the crack near-field, respectively, is characterized by the classical stress intensity factors (SIFs) or the energy release rate, which is directly linked to the SIFs in this context. These SIFs are related to the well-known square-root stress singularity. But this kind of *
Corresponding author. Tel.: +49 9131 8528516; fax: +49 9131 8528503. E-mail address: [email protected] (M. Heyder).
0013-7944/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2005.01.006
2096
M. Heyder et al. / Engineering Fracture Mechanics 72 (2005) 2095–2105
singularity is generally not present at non-smooth parts of the crack front and at the intersection of the crack front and a free surface. At such points one has to deal with a 3D corner singularity [2,1,7]. The arising question is, which influences of this corner singularity have to be regarded [16]. A first answer can be found e.g. for mode-I in [1] and for mixed-mode in [5]. Based on both investigations, one may conclude, that the crack front will be shaped ensuring the classical square-root singularity or at least a bounded energy release rate. If the square-root singularity really occurs at the intersection of the crack front and the free surface, it will be linked to a special crack front intersection angle cr. This question is investigated both numerically and experimentally under mode-I conditions. For a given geometrical situation, the solution of a quadratic eigenvalue problem provides the singular exponents of the 3D singularity, cf. Section 2. The experiments as described in Section 3 have to be performed precisely. Only a careful comparison between experimental observations, e.g. well documented sequences of incremental crack front shapes, and corresponding numerical analyses enable an identification of the influences of the 3D corner singularity. In order to be able to observe and to store the crack front propagation by photographic in situ measurements, specimens of the transparent material PMMA have been used. Experiments using different geometries are analyzed to verify if the characteristic angle cr is unique. This is realized in terms of different cross-sections and varying boundary conditions.
2. Singularities along a crack front It is well-known, that the stress field shows a singular behavior in the neighborhood of a crack front within the framework of linear elasticity. In the context of 2D, assumptions (plane strain or plane stress) are made so that the crack front is reduced to a single crack tip and the classical square-root stress singularity is always present. As no information is available along the crack front, one has to switch from 2D to 3D to involve the crack front into the analysis. As a consequence, two certain kinds of singularities have to be distinguished. Along a smooth part of a crack front a wedge singularity is present. But at non-smooth parts of the crack front and particularly at the intersection of the crack front with the free surface, a 3D corner singularity has to be taken into account. 2.1. Wedge singularity The wedge singularity is equivalent to the classical square-root singularity with a multiplicity of three. For sake of completeness, the stresses of the asymptotic crack near-field are given in Eq. (1) in terms of local crack front coordinates, cf. Fig. 1, rij ðr; U; P Þ ¼
III X pffiffi K M ðP Þ M pffiffiffiffiffiffiffi fij ðUÞ þ T ij þ Oð rÞ 2pr M¼I
ð1Þ
for arbitrary smooth crack fronts [12]. The intensity of this singularity is characterized by the stress intensity factors KM (M = I, II, III). The remaining terms are the angular functions fijM ðUÞ and the elastic T-stresses Tij. Both fracture mechanics parameters (KM(P) and Tij(P)) calculated very accurately are from the numerical stresses in the crack near-field via an optimized extrapolation method based on a regression analysis [8]. The stress field is provided by a special BEM formulation—the 3D Dual Discontinuity Method (DDM)—a powerful tool for linear-elastic stress concentration problems [15]. Alternatively, if a FEM analysis is performed, the SIFs can be determined e.g. by the virtual crack closure method (VCCM) [3] or a hpversion FEA as well as a mathematical splitting scheme [6].
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Fig. 1. Local crack front coordinate system.
2.2. 3D corner singularity Compared to the wedge singularity, the 3D corner singularity is in general not a-priori known for every singular point mentioned above. Hence, the value of the singularity has to be determined by a singularity analysis, that traces back to [1,2] and efficient numerical procedures have been published recently [4,10]. The outline of such a scheme is as follows. Around the singular point O with its coordinates xO 2 IR3 an e-neighborhood XeO 2 IR3 is considered. XeO is defined by the cut-set of an e-ball Xe := {x 2 IR3:jx xOj < e} and the domain X 2 IR3 as shown in Fig. 2. It is assumed, that the 3D body X consists of homogeneous, isotropic and linear-elastic material. To extract the 3D corner singularities the displacement field is utilized. Therefore, the corresponding elastic solution in the vicinity of the singular point O related to a spherical coordinate system, cf. Fig. 2, is asymptotically expanded in the form 1 X ui ðq; h; u; OÞ ¼ K L ðOÞqaL giL ðh; u; OÞ ð2Þ L¼1
Fig. 2. Vicinity at the singular point O.
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under the assumption that no logarithmic terms occur, which is true for most practical applications. aL denotes the asymptotic exponents satisfying aL > 0.5 to keep the elastic energy bounded, cf. [4], and giL ðh; u; OÞ are the corresponding angular functions. The asymptotic exponents are now linked to generalized intensity factors K L . The variables u and h belong to the bounded 2D domain Xuh oXeO with Xuh {u · h:u 2 [0, 2p] ^ h 2 [0, p]}. Among other techniques, the asymptotic exponents aL are obtained by the solution of a quadratic eigenvalue problem. After a linearization and spectral transformation it is solved with the help of the mathematical package ARPACK [13]. Finally, the eigenvalues aL and eigenvectors giL are known simultaneously. Because the asymptotic behavior is focused, the interval 0.5 < aL < 1 is considered excluding the rigid body motion modes (a = 0 and 1) as they are known. aL depends on the geometrical situation around the singular point described by XeO and on material parameters. For single, homogeneous and isotropic materials it only depends on the Poissons ratio m. 2.2.1. General remarks For aL = 0.5 the intensity factor K L may correspond with one of the classical stress intensity factors KM. In general, one cannot distinguish between mode-I, II, III any more but only between symmetric and antisymmetric modes. The symmetric mode corresponds to mode-I and the antisymmetric modes can be either pure mode-II or mode-III or a combination of both. The classical SIF-concept only fails at some special points with aL 5 0.5. But to be still able to apply this concept for the description of the behavior in the crack near-field the SIFs are numerically defined at these particular points [14]. Following (2) leads to u OðqaL Þ and therefore r OðqaL 1 Þ. This means, if aL is greater than 0.5 and less than 1.0 the stresses are still singular but weaker compared to the square-root singularity. Consequently, KM(P) tends to zero as P tends to the singular point O, see also [6]. If aL is less than 0.5 the opposite situation is present and KM(P) tends to infinity. Therefore, the exponents aL have to be known to determine the classical SIFs asymptotically. It is advantageous for the simulation of crack propagation that, if the singular point O is located at the ends of a crack front, the crack front angle c can be determined by the assumption of aL = 0.5, [9]. As only mode-I is considered in this paper, the particular exponent aL is related to the symmetrical crack opening. The corresponding crack front angle is denoted by cr and only KI is existing. 2.2.2. Consequence for real crack propagation In case of surface breaking cracks it has been experimentally observed, cf. Section 3.2, that the crack front always intersects the free surface having a special angle—the crack front angle cr. This angle depends only on the Poissons ratio m and on the geometrical situation around the singular point. The intersecting angle c is shown in Fig. 3 and defined between the inward directed normal vector n and the tangent t related to the crack front at point O. As only mode-I is considered in this paper, the crack surface coincides with the plane spanned by the vectors n and t, if the inclination angle d is equal to 90. Due to the violation of the wedge singularity only at some special points, it is assumed, that the crack front will change its shape to ensure a valid square-root singularity along the whole crack front. To verify this assumption, the present singularities have to be known. The first interesting question concerning real crack propagation is, if a manufactured initial crack front (e.g. a machined notch) intersecting the free surface under an angle cm 5 cr is going to change its crack front angle to approach cr. Moreover, does this process of adjusting the angle c happen abruptly or continuously? The second question deals with the problem, if the angle cr can be influenced by changing other parameters, e.g. the orientation of the free surface relative to the crack front/surface. To answer these questions, experimental investigations on specimens with different cross-sections and suddenly changing boundary conditions (creation of new free surfaces intersecting the crack front) are performed. By conducting these experiments, the behavior of the crack front angle c has to be observed. If the
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Fig. 3. Definition of the crack front angle c.
resultant angle c is always practically equal to cr, the crack front will be shaped ensuring the classical square-root singularity (aL = 0.5) even in the vicinity of the intersection of the crack front and the free surface.
3. Experimental investigations The influence of the additional 3D corner singularity on the shape of the crack front, especially in the vicinity of the crack front intersection with the free surface, is going to be examined. For this purpose, a test series with four-point-bending specimens of PMMA has been carried out. The advantage of PMMA is its (a) brittle behavior, so that linear fracture mechanics can be used for description and its (b) transparency, which enables a permanent photographic in situ documentation of the crack front shape, as explained in Section 3.1. The corresponding elastic material parameters are the Poissons ratio m 0.365 and the Youngs modulus E 3.6 GPa. The size effect of the specimens is studied in Section 3.2 based on a rectangular cross-section. With specimens having a trapezoidal cross-section, the position of the free surface relative to the crack front is changed. If the crack front angle is influenced by this modification will be answered in Section 3.3. After that the creation of new free surfaces intersecting the crack front is discussed in Section 3.4 to study the related new crack front angles. 3.1. Experimental setup The experimental setup is visualized in Fig. 4. To document the shape of the crack front, a camera (VOSSCCD-1300, 1280 · 1024 pixels) and a lightning are placed ahead of the polished front of the specimen. The backside of the specimen is blackened to enhance the contrast of the taken picture. All experimental investigations are performed on a servo-hydraulic machine (Schenk PC 400-N, PL 100, PM 100-RN, S56) with a frequency of f ¼ 6:9 4 Hz (25 000 load cycles per hour). This frequency is low enough to avoid heating of the specimen due to friction at the crack front faces. The measuring of the actual crack front shape is carried out by an automatic crack front detection algorithm. Thereby, the contrast of the bright crack front versus the dark background is used. Starting from the darkest area (5 · 5 pixel) of the taken picture (this is at any rate part of the background), this area is enlarged until the contrast of the adjoined pixel is smaller then a user defined value. The lower boundary of this located area is the crack front shape. In combination with a measuring unit, the coordinates of the
Ku ¨ HLER
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Fig. 4. Experimental setup of a four-point-bending specimen.
crack front are calculated and interpolation points (the distance between two successive points is dÆÆ = 20 pixel) are printed into an input-file, e.g. for usage in FEM/BEM-preprocessors [11]. 3.2. Rectangular cross-section The influence of the width of the specimens on the shape of the crack front is checked by specimens with a rectangular cross-section. In this test series different dimensions (five specimens per size) are investigated. Starting from a standard model with the dimensions length = 220 mm · width = 20 mm · height = 60 mm, this size of the specimens is firstly scaled only in the width (Fig. 5a) and secondly in all directions (Fig. 5b), each with the factor 32 and 2. Starting from an initial straight notch in the middle of the specimens, the propagating crack profile is monitored as described in the last section. A typical picture of these crack fronts is shown in Fig. 5c. As a result, all documented crack fronts have the same body structure and after scaling with respect to the width the same shape, too. Independently of the size of the specimens, the value of the crack front angle c was found to be c 14 ± 3 which is shown in Fig. 6. Obviously, there is no size effect concerning the crack front angle.
Fig. 5. Specimens with rectangular cross-sections and angle c 14.
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Fig. 6. Crack front shapes of the specimens with rectangular cross-section.
3.3. Trapezoidal cross-section A variable orientation of the free surface with respect to the crack surface while ensuring the same inclination angle (d = 90) is realized by a trapezoidal cross-section. But it should be distinguished, if the crack grows into the narrower or wider part of the specimen as indicated in Fig. 7a. Five specimens per trapezoidal angle (2b = ±30/60/90) are investigated. Fig. 7b shows one half of a specimen with 2b = 90 and Fig. 7d a specimen with 2b = 60. As an interesting fact of this test, the trapezoidal angle b does not have any dependence on the angle c which is always practically 14. The only difference between the positive and negative b-specimens is a different bending behavior of the crack front. In the case of b > 0 the crack front shows a complete convex shape. But for negative values of b there is a transition between convex and concave zones. The crack front is also convex in the middle, whereas it is concave in the vicinity of the free surface to approach the unique crack front angle. Therefore, the area where the tangential angle c appears, is much smaller compared to b > 0-specimens. A magnified cutout of the 2b = 90-specimen is shown in Fig. 7c, where the characteristic angle cr 14 has been measured, too. Based on these investigations of the trapezoidal specimens it has been shown, that the crack front will always find a shape that ensures the square-root stress singularity at the vicinity of a free surface. 3.4. Crack propagation rate According to the last two sections, it seems that the crack front angle is unique, independently of the global geometry. But does the process of adjusting this special crack front angle cr happen continuously
Fig. 7. Specimens with different trapezoidal cross-sections and angles b, but with the same crack front angle c 14.
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(a)
(b)
Fig. 8. Specimens with trapezoidal cross-sections and new free intersecting surfaces.
or abruptly? This question will be answered by disturbing an existing crack surface by a new free surface. It will be exemplary shown by the help of the trapezoidal specimen. There are three alternatives for this new free surface. The newly created intersecting angle cn is (a) greater, (b) equal or (c) smaller than the characteristic angle cr. In the first case (cn > cr), the classical SIF tends to infinity (KI ! 1) at the new end of the crack front. Therefore, an abrupt crack propagation at the new free surface could be expected as demonstrated in Fig. 8a and studied in Section 3.4.1. In the second case, the SIF is well defined and no significant influence concerning the crack propagation rate is observed. The third case leads to a SIF tending to zero (KI ! 0), so that no crack propagation will occur at the new free surfaces. The crack should only propagate in the middle of both parts of the specimen as shown in Fig. 8b to approach the characteristic angle cr, c.f. Section 3.4.2. 3.4.1. Crack propagation rate for cn > cr In this case, the inclined left side of the trapezoidal cross-section is cropped, so that the new free surface intersects the initial crack front under cn > cr. The consequence of this is KI ! 1, that could lead to an abrupt crack propagation—even at very low loading conditions. Therefore, the force was reduced to 75% of the force to create the initial crack, so that the stress intensity factor at the non-influenced crack front (middle and right part of the specimen) falls below the lower threshold Kth for fatigue crack propagation (KI < Kth). In Fig. 9, the fatigue crack propagation can be observed only at the influenced left side of the specimen. Against ones expectations, there is no abrupt crack
Fig. 9. Crack propagation after cropping the left side of the specimen.
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Fig. 10. Crack propagation after notching in the left part of the specimen.
propagation. This can be explained with the fact, that there is not enough energy in one load cycle to create the new whole crack surface. 3.4.2. Crack propagation rate for cn < cr To examine the case cn < cr, the specimens with the trapezoidal cross-section are notched lengthwise, to create a new free surface in the middle of the already existing crack front (Fig. 8b). The new angle is cn 0, so that crack propagation should only occur in the middle of both parts of the specimen. The following fatigue crack propagation shows a continuous crack propagation in the middle of each side of the specimen to approach the characteristic angle c 14 at the new surface (Fig. 10).
4. Verification of the measured crack front angles All investigated specimens, cf. Sections 3.2, 3.3 and 3.4, have a different cross-section and therefore a different global geometry. Moreover, in each case the crack front is shaped uniquely. But the crack front angle c has practically always the same value cr. The reason for that behavior is, that the geometrical situation in the vicinity of the intersection of the crack front and the free surface is the same. Namely, the crack surface is perpendicular to a smooth free surface and only the symmetrical crack opening mode is activated. A singularity analysis as mentioned in Section 2.2 will be applied to verify, if c = cr corresponds to the assumed square-root stress singularity aL = 0.5. The surface Xuh is spanned by (u, h) 2 [0; 180] · [0; 180] and the crack intersects this plane at u = d = 90 and h 2 [h*; 180]. For h* = 90 and m = 0 the singular exponent aL is analytically known and equal to 0.5. The relative error of the numerically determined value anum is approximately 106. But for m 5 0, the inverse problem of seekL ing the angle h* that ensures aL = 0.5 has to be solved. The crack front angle cr is related to h* by cr = 90 h*. The corresponding distribution in dependence of the Poissons ratio m is shown in Fig. 11.
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Fig. 11. Crack front angle cr ensuring a valid square-root singularity under mode-I.
The solid curve represents the plot of 2m cr ¼ 90 arctan m
ð3Þ
which was proposed by [16]. This equation is only valid for the considered geometrical situation at the singular point. The numerical results obtained from the inverse problem of the 3D singularity analysis are marked by the dotted line. Both results are in a very good agreement for m 2 [0; 0.15]. In the remaining range of m the numerical solution leads to slightly larger crack front angles. Based on our results, Eq. (3) is a first rough estimate for the crack front angle in this special case—symmetrical crack opening of a crack intersecting the free surface perpendicularly. Additionally, the numerical crack front angle for m = 0.3 is in good agreement to a reference angle of cref r 11 which can be found in [1]. In case of the investigated material PMMA with a measured m = 0.365, the angle cr has a value of 14.049. This result matches with the experimentally determined crack front angles. Hence, at least for mode-I the assumption is confirmed, that the crack front is shaped ensuring a valid square-root stress singularity in the vicinity of the intersection of the crack front and the free surface. As this type of singularity only holds at smooth parts of a crack front at first glance, now it can be stated that the square-root singularity holds along the whole crack front.
5. Conclusion The influence of the 3D corner singularity in case of surface breaking cracks has been investigated both numerically and experimentally. Therefore, different specimens with a single edge crack loaded under four point bending are considered. The vicinity of the intersection of the crack front and the free surface is analyzed, because of its three-dimensional singular behavior. In this area, a practically constant crack front angle could be experimentally observed. The process of adjusting this angle occurs continuously. The value of the crack front angle cr(m) has been verified and confirmed by a 3D singularity analysis. This agreement indirectly confirms the assumptions of linear elasticity. In addition, the observed crack front angle corresponds to the classical square-root stress singularity. Hence, at least for mode-I, the crack front is shaped,
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that this type of singularity holds along the whole crack front. The analogous effect of the 3D singularity in case of mixed-mode is addressed to a forthcoming paper.
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