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Crack front curvature: Influence and effects on the crack tip fields in bi-dimensional specimens
International Journal of Fatigue 44 (2012) 41–50
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Crack front curvature: Influence and effects on the crack tip fields in bi-dimensional specimens D. Camas ⇑, J. Garcia-Manrique, A. Gonzalez-Herrera Departamento de Ingeniería Civil, de Materiales y Fabricación, Universidad de Málaga, Escuela de Ingenierías, C/Doctor Ortiz Ramos, s/n., Campus de Teatinos, E-29071, Málaga, Spain
a r t i c l e
i n f o
Article history: Received 24 December 2011 Received in revised form 21 May 2012 Accepted 22 May 2012 Available online 4 June 2012 Keywords: Finite element analysis Crack front curvature Plastic zone size Out-of-plane stress Fracture mechanic
a b s t r a c t Crack front curvature is evidence in most experimental crack advance test. When classical linear elastic fracture mechanic theory deals with bi-dimensional crack configurations, it ignores the three-dimensional effects of crack propagation. Issues as the influence of the specimen thickness and the crack front curvature are not considered. Previous numerical studies have shed light on out-of-plane plastic zone development or stress state. Nevertheless, these numerical studies are based on the assumption that the crack front is ideally straight; despite it is well known that the crack front has some kind of curvature. In the present work, a CT aluminium specimen has been modelled three-dimensionally and several calculations have been made considering a huge combination of different single load levels, specimen thicknesses and crack front curvatures. Due to the abrupt transition from plane strain to plane stress, an ultrafine mesh along the thickness has been applied. The analysis of the evolution of the plastic zone and the stress state along the thickness provides information about the combined influence of these parameters on fracture mechanics. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Engineers and designers need to predict fatigue life in structures and components under cyclic loads. Nonetheless, the complexity of the problem limits the possibility of dealing in a rational predictive way with this phenomenon. The characterisation of fracture and fatigue crack propagation is based on the classical linear elastic fracture mechanics, which only considers two plane states. The main advantage of the plane theories is the simplicity compared to the three-dimensional equations. The yielded zone size close to the crack front influences the stress and strain state in the crack front. It is a factor of fundamental importance in the fracture and fatigue behaviour of metallic materials [1–3]. Several efforts have been made to determine the plastic zone size since 1960. Irwin [4] and Dugdale [5] proposed different models to estimate plastic zone size ahead of the crack. In the middle 1980s, Banks and Garlick [6] made an attempt to quantify the shape and the size of the plastic zone using the von Mises yield criterion. These previous analysis were not strictly correct because they were based on a purely elastic analysis. The redistribution of stresses due to yielding was neglected. Dodds et al. [7] studied the shape of the plastic zone using the stress
⇑ Corresponding author. Tel.: +34 951952440; fax: +34 951952601. E-mail addresses: [email protected] (D. Camas), [email protected] (J. Garcia-Manrique), [email protected] (A. Gonzalez-Herrera). 0142-1123/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijfatigue.2012.05.012
distribution obtained by finite element analysis. These yielded zones are related to two extreme stress states: plane stress or plane strain, but no clear information about out-of-plane yielded areas or stress state was provided. Dealing with plates of finite thickness, the dominant state of stress is selected according to relative thickness. For thin specimens, the stress state is considered to be plane stress. Otherwise, it is considered to be plane strain. The ASTM E399 [8] indicates the specific condition in which a fracture toughness specimen has an acceptable plane strain state. The simplicity of bi-dimensional (2D) theory has become limitation for fracture and fatigue mechanics, so it is necessary to develop three-dimensional (3D) theories. Recently, Bellet [9,10] showed experimental evidence and proved that the commonly 2D or planar methods for fracture and fatigue assessment of notched bodies may lead to inaccurate predictions when applied to certain 3D geometries. There are only few analytical 3D solutions available in the literature due to the complexity of the stress field. Sternberg and Sadowsky [11] obtained an approximate solution for 3D stress distribution near a circular hole in an infinite plate of arbitrary thickness by using series expansion and taking finite terms into account. Based on the assumption of a generalised plane-strain theory, which assumes that the through-the-thickness extensional strain is uniform in the thickness direction, Kotousov and Wang [12] developed analytical solutions for the 3D stress distribution around typical stress concentrators in an isotropic plate of
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Nomenclature a b d d5%
crack length specimen thickness distance to the exterior face of the gravity centre of the plastic size area distance from the surface of the specimen which produces a fall in the value of the stress corresponding to the plane strain state of 5%
arbitrary thickness. However, no theoretical solution near a 3D crack front in an elastic–plastic solid is available. Due to these difficulties, detailed 3D finite element analyses are essential to characterise the stress and strain variations near a crack front in elastic–plastic solids and also ascertain the range of dominance of 2D plane strain and plane stress solutions. To this end, some works performed elastic–plastic 3D analysis of thin plates under mode I [13,14] and mixed mode (mode I + II) [15] loading conditions, obtaining several parameters as plastic zone sizes, stresses, deformations and constraint variations along the thickness. Their results revealed strong thickness variations of stress fields very near the crack front. The main problem in 3D studies is that there is a very abrupt transition in the stress and strain fields in a thin part of the external surface. This transition can be properly captured only if an ultrafine mesh along the thickness is done. Recent studies with fine meshes along the thickness have shown an important influence of the specimen thickness and load applied [16,17]. Besides, unexpected through the thickness plastic zone size evolutions were realised. These results were not observed in previous studies due to the lower number of elements employed through the thickness. Previous numerical studies that analyse the plastic zone size and shape and the stress distribution near the crack front are often based on the assumption that the crack front is straight. It is well known, that the crack front in through cracks presents some kind of curvature when the crack grows under fatigue loading conditions. Some efforts have been made in order to explain and characterise the crack front curvature. One explanation is the crack closure effect introduced by Elber [18] in the first 1970s. Apparently, crack closure is a 3D mechanistic phenomenon. Experimental observations [19,20] and numerical analysis [16] shows that fatigue cracks open first at midthickness and later at the surface, which often leads to curved crack fronts for through cracks [21]. Branco et al. [22] analysed numerically, the influence of the crack closure in the stable crack shape growth and compared the results when the crack closure was not considered. While without crack closure the crack shape remained straight with only a smooth propagation delay near the surface, the consideration of the crack closure was found to produce a significant tunnelling effect. Other one is the stress intensity factor distribution along the thickness and the presence of the corner point singularity or vertex point. Usually, the stress field is described by a unique stress intensity factor value. But as early as 1977, Benthem [23] presented the 3D corner singularity at the intersection of the crack front and the surface. Bazˇant and Estenssoro [24] used a FE analysis to propose that this singularity produce the deviation of the crack front from the orthogonal direction near the surface. However to date, the influence size of the corner point singularity is not known. Pook [25] suggested that this size is a function of load and concluded that crack front intersect a free surface at a critical angle, which is function of the loading type and the Poisson’s ratio. Hutarˇ et al. [26] presented a very interesting methodology based on generalised stress intensity factor to describe the crack behaviour
K rc rpD Ya
stress intensity factor radius of curvature Dugdale’s plastic zone size yielded area
close to the free surface. They studied the evolution of the crack shape due to the effect of the free surface and the influence of the specimen thickness. Hence, fatigue crack propagation rate decreases leading also to a curved crack front. Therefore, the purpose of the present study was to analyse the influence of the crack front curvature in the stress fields along the thickness of through cracks and its relation with the load applied and the specimen thickness. Hence, a comprehensive numerical study by means of 3D FEA of the stress fields along the crack front for different load cases, with different specimen thickness and crack front curvatures is made. This work was performed by means of the simulation and calculation of a huge number of load cases combining different crack curvatures. A single load is applied to the specimen with an ultrafine mesh along the thickness. The analysis was focussed in the transition of the response of different parameters – as stress state, plastic zone, etc. – along the thickness. An especial attention is paid to the comparison with previously obtained result with straight cracks [17].
2. FEA modelling methodology In this study, a CT specimen geometry (W = 50 mm, a = 20 mm) has been modelled three-dimensionally, with elastic–plastic behaviour of the material. A combination of three different thicknesses (b = 3, 6 and 12 mm), four different crack front curvatures and a range of loads corresponding to stress intensity factors from K = 10 to 40 MPa m1/2 have been considered. It is important to note that the stress intensity factor arises from a two dimensional linearly elastic analysis. Therefore, there are some limitations on their validity as the small scale yielding argument and that cracks must stand perpendicular to the surfaces. Nevertheless, experience has shown that these limitations can be managed [27]. In the present study, the remote far-field stress intensity factor K is just used to express the load magnitude, due to the general use of this parameter on practical engineering problems involving cracks. The relation between the load applied and the stress intensity factor K is determined by next expression [28]:
0 a 2 þ Wa P pffiffiffiffiffiffi @ K¼ 0:886 þ 4:64 3 W b W 1 Wa 2 a 2 a 3 a 4 13:32 þ 14:72 5:6 W W W
ð1Þ
where P is the load applied, b is the specimen thickness, W is the distance from the loading line to the back face and a is the crack length. The crack front curvatures are defined in terms of the radius of curvature and calculations range from straight crack to a radius 4/3 times the specimen thickness. The total number of different cases studied is 84. This allows studying a huge range of behaviour including low and high loads. Fig. 1 illustrates a 3D mesh employed for a crack front curvature case. It was simulated with the
D. Camas et al. / International Journal of Fatigue 44 (2012) 41–50
Fig. 1. Finite element model.
commercial FEA code ANSYS. Due to symmetry, only one quarter of the specimen needs to be modelled in 3D. The most critical area corresponds to the crack front. In this area there is a great gradient in the stress and strain field. Therefore, in order to capture properly this effect, the size of the elements around the crack tip must be small enough. In order not to penalise the computational cost, it is necessary to make a huge transition from this zone to the most remote ones. For this reason, the specimen has been divided in two different volumes. In the first one, around the crack tip, a homogeneous and uniform undistorted mesh with hexahedral elements is made. And the second one is meshed with tetrahedral elements which allow the transition. In the present study, a unique eight-node element with linear base function has been used. At each node, there are three degrees of freedom corresponding to the translations in the x, y and z directions. The tetrahedral elements are obtained degenerating the element, condensing faces to lines. At first sight, it may seem that employing quadratic elements could allow using less number of elements decreasing the computational cost. But since this is a nonlinear analysis, a better accuracy at less expense is obtained if a fine mesh of linear elements is used rather than a comparable coarse mesh of quadratic elements [29]. Besides, this kind of element has been employed due to the huge experience obtained by the present authors in previous works in which the accuracy of the model has been analysed [17,30–32]. The plastic zone size is the scale reference to relate to the minimum element size. In this work Dugdale’s plastic zone size has been used as a reference, expressed as:
r pD ¼
p KI 8a ry
and the length of the crack front, 80 and 120 divisions have been considered. In the vicinity of the surface of the specimen, there is a transition area in the plastic zone. Close to the mid plane, where plane strain conditions are dominant, the plastic zone is homogeneous. In order to visualise with a higher resolution the transition area of the plastic zone, the length along the thickness of the element next to the surface of the specimen is the half of the length of the element next to the mid plane of the specimen. Outside this region, the number of through-thickness divisions was reduced to speed up the simulations. Accordingly, a transition in the through-thickness mesh has been done (see detail in Fig. 1). A good compromise between accuracy and computational cost was achieved by employing hexahedral prism shaped elements within the plastic zone and tetrahedral elements outside. In addition, the use of 8-node tetrahedral elements facilitated the transition between both zones. In straight-crack-front three-dimensional models, all the crack front nodes are located at a distance equal to the crack front length. Nevertheless, when the crack front has any sort of curvature it is necessary to set a point along the thickness to determine the crack front length. This crack front length defines the value of the stress intensity factor (K). In the present study, the gravity centre of the line which constitutes the crack front has been considered to characterise the crack length (see Fig. 2). The material modelled has been an Al-2024-T351 aluminium alloy that shows weak hardening (E = 73.5 GPa, m = 0.33, r yield = 425 MPa, K0 = 685 MPa, n0 = 0.073, being K0 and n0 parameters in the Ramberg–Osgood yielding model). In previous works, this material has been numerically modelled in different ways. Theses previous 2D studies were made with both isotropic and kinematic hardening models. Nevertheless, no differences were found in the results when there is a weak work hardening rule, as in this case [31]. So, the results presented in this paper have been calculated with a model that employs an isotropic hardening rule with H/E = 0.003 that fits well the actual behaviour of this aluminium alloy and have a smaller computational cost. The values H and E represent the slopes of the plastic and the elastic line, respectively. It has been modelled by a three-linear representation.
2 ð2Þ
where a is a constraint factor equal to 1 for plane stress and 3 for plane strain. As in this work a 3D model is considered, a has been taken equal to 1 in order to be higher than any yielded zone size along the thickness. In all the different cases studied, the volume meshed with hexahedral elements is divided in the same number of elements. This volume is proportional to the Dugdale’s plastic zone, so the size of the elements of this area is proportional to the applied load. Previous studies developed by the present authors [30] concluded that the FEA results are strongly dependent on the minimum element size around the crack tip. The minimum element size has been established following a recommendation proposed previously, adapted to reduce the computational cost. The minimum element size is 184 times smaller than Dugdale’s plastic zone size. As the crack opening load is not required and the plastic wake does not need to be simulated, a lesser requirement of a number of divisions of the plastic zone equal to 76 is adopted. This value is the consequence of a previous mesh refinement study [17]. Once the minimum element size is established, and in order to keep an acceptable element shape ratio, in this work the number of thickness divisions adopted has been 40. Only where the element shape ratio was not acceptable because of the small load value
43
Fig. 2. CT specimen dimensions and crack length definition.
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3. Results The influence of the crack front curvature can be observed analysing different parameters. There are a huge number of results to be reported, nevertheless only the most representatives are going to be shown. In this way, the plastic zone and the stress field at the crack tip have been considered as the most significants and will be focused on. Furthermore, qualitative observations will be quantified in the subsequent sections.
3.1. Plastic zone size As mentioned before, the fracture and fatigue behaviour of a metallic material is strongly influenced by a small yielded area close to the crack front. Therefore, it is necessary to know properly the shape of the yielded area. There are previous studies by means of FEA that analyses the shape of this yielded area. Most of them are in 2D, but also there are some 3D studies which consider the crack ideally straight [13,33]. However, it is well known that the crack front shape when the crack growths in fatigue is not straight, but presents any curvature. An accurate vision of the plastic zone in the area close to the crack front can be obtained analysing the plastic strain suffered by the elements. In Fig. 3, a comparison of the yielded area around the crack front for two different crack shapes geometries for a 3 mm specimen thickness is shown. Both have the same applied load corresponding to a stress intensity factor K = 20 MPa m1/2, but different crack front curvature: straight and 4 mm. The whole yielded region is represented. Firstly, as it has been reported previously [16,17] with straight cracks, it can be appreciated that the plastic zone size of both cases shows essential differences with the theoretical model. This one predicts that the plastic zone size at the surface of the specimen, under plane stress conditions, is three times greater than the plastic strain in the mid plane, under plane strain conditions. That is the classical ‘‘bone shape’’ plastic zone. Nevertheless, results obtained are different. Comparisons between the theoretical plastic zone in plane stress and plane strain states, and the numerical results are shown in Fig. 4. The numerical results presented in this figure have been obtained considering a straight crack front. The results shown in Fig. 4a and b correspond to 3 mm and 12 mm specimen thicknesses, respectively. It can be observed that comparing numerical and theoretical results, sizes and shapes are different. Analysing Fig. 4a can be deduced that the reason why neither plane stress or plane strain plastic zone shape and size has been achieved is the small thickness of the specimen with respect the load applied, according to the ASTM E399 [8]. However, in Fig. 4b can be observed the same behaviour when, in this case,
the relation between the specimen thickness and the load applied is suitable for achieving both states. It is noteworthy that for both cases, shapes and sizes numerically obtained are really similar in both states, at the mid-plane and at the surface. In order to study the influence of the crack front curvature in the plastic zone size, it is of interest to analyse the yielded areas. In Fig. 5, the evolution of the yielded areas at the surface (Fig. 5a) and in the mid-plane (Fig. 5b) with the crack front curvature is represented. In these figures, the specimen thickness is 3 mm and K = 25 MPa m1/2. As it can be observed, the yielded area in the mid-plane is not influenced by the crack front curvature, while the yielded area at the surface of the specimen gets larger as the radius of the crack front curvature is reduced. Furthermore, the yielded area depends of the load applied. In Fig. 6 the influence of both parameters, the crack front curvature and the load applied, in the yielded area is analysed. Data are normalised by the corresponding area to the Dugdale’s plastic zone in plane stress and plane strain respectively. In Fig. 6a, the evolution of the yielded area on the surface of the specimen is illustrated. It should be noted that while the straight crack front case presents a linear behaviour with the load applied, a strong influence of the crack front curvature in the lowest load range can be observed. When the radius of curvature decreases, the yielded area increases. By increasing the value of the applied load, the results converge to the straight crack front ones, minimising the effect of the curvature. The behaviour of these parameters in the midplane can be observed in Fig. 6b. In this case, the yielded area is only affected by the applied load. Therefore, there is a huge combined influence of both parameters: the load applied and the crack front curvature. In Fig. 7, the ratio between the yielded area at the surface and at the mid plane is represented versus the stress intensity factor for the 3 mm and 12 mm specimen thickness. The radius of curvature for both cases is represented as the relation between the radius of curvature and the specimen thickness. The first issue to note is that the ratio is almost equal to unity in the applied load range for the straight crack front case. Hence, the yielded area at the surface is equal to the yielded area in the mid-plane in all the applied load range for this case. This behaviour may be explained by the corner point singularity and the decrease of the stress intensity factor as a corner point is approached [32]. Secondly, there is a strong dependence with the crack front curvature in the lowest loads range. The curvature causes the yielded area in the surface to be much greater than in the mid plane, but for the greater load range, the values converge. As it can be observed, for the 3 mm specimen thickness the yielded areas at the surface and at the mid plane are equals for stress intensity factor values greater than 30 MPa m1/2, while for smaller ones there is a
Fig. 3. Plastic zone close to the crack front.
D. Camas et al. / International Journal of Fatigue 44 (2012) 41–50
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Fig. 4. Comparison of the theoretical plastic zone in plane stress and plane strain states, with the numerical plastic zone obtained in the surface and in the mid-plane for a 3 mm (a) and 12 mm (b) specimen thickness and K = 20 MPa m1/2.
huge influence of the crack front curvature. For the 12 mm specimen thickness, the convergence does not occur in the applied load range and is expected for higher loads [17]. Finally, the evolution along the thickness of the yielded area is analysed. Hence, two different parameters are studied: the yielded area and the plastic zone along the thickness, where the plastic zone is the yielded area in the crack plane. In this way, the evolution of the yielded area in perpendicular planes to the crack plane is analysed (defined this yielded area, Ya, as drawn in Figs. 4 and 5). As it has been reported in a previous work [17], the plastic zone evolution along the thickness depends on the specimen thickness and the load applied. Three different behaviours according to the load level were found. The first one (type I) refers to cases where the transition zone occupies part of the thickness. The transition zone is the area of the plastic zone size that extends from the point in which the typical values in the interior of the specimen begins to increase, to the surface. The second type (type II) refers to cases where the transition zone occupies all the specimen thickness, increasing gradually the value of the yielded area at the mid-plane. The last one (type III) refers to cases where the maximum plastic zone size is at the mid-plane. Furthermore, higher load levels do not increase the plastic size in the interior zone, but it does in the exterior. This is resumed in Fig. 8a. As well, in Fig. 8 the typical evolutions of the plastic zone size along the thickness with the crack front curvature for the three different types of behaviour described before are represented. The results shown in this figure correspond to a 3 mm specimen thickness. It can be observed that for the first two behaviours, decreasing the radius of curvature implies an increase of the plastic zone size in the transition zone. In the third one, the plastic zone sizes are almost the same for the considered radii. In order to analyse the dependence of the different parameters, it is necessary to quantify the results obtained. The position of the gravity centre of the plastic zone size area has been considered for this purpose. The parameter d is defined as the distance from the surface of the specimen in orthogonal direction to the gravity centre (see Fig. 9). It is a way to characterise numerically the evolution of the plastic zone size along the thickness and the influence of the specimen thickness, the load applied and the crack front curvature. The parameter d tries to represent the dominance of the plane strain state over the whole crack behaviour. The results obtained
for the 3 mm and 12 mm specimen thickness are shown in Fig. 10. For the 3 mm specimen thickness and the load range applied, the three different behaviours can be observed. In the first one, increasing the load applied and decreasing the radius of the crack front curvature involves a displacement of the gravity centre toward the surface of the specimen due to the extension of the plastic zone in this area. In the second type, increasing the load applied involves a displacement of the gravity centre toward the interior of the specimen due to the increase of the value of the plastic size at the mid-plane. In the last one, higher load levels do not increase the plastic size in the interior zone, but it does in the exterior. Traditionally, the ASTM E399 equation [8] discretises the plane state behaviour of the specimen in two states, but reality is a continuum. Analysing the evolution of the plastic zone size along the thickness by means the d parameter, was found that the behaviour is more complex as it can be seen in Fig. 10 [17]. Besides, it seems that the requirement of the standard is conservative and should be considerate at least all the type I. The type III corresponds to load levels in which the state of the specimen can be considered as plane stress due to the distribution of the yielded area along the thickness is almost constant. Finally, the type II corresponds to a transition type in which neither of the plane states can be considered. Analysing the influence of the crack front curvature, in the last two types, it can be observed that there is no influence of the crack front curvature, while it is important in the first type. This can be better appreciated in Fig. 10b, where the 12 mm specimen thickness results are represented. For this specimen thickness, only the first type is present for the load range considered. The influence of the crack front curvature is clear even at the highest applied load. Finally, the evolution of the yielded area (Ya) along the thickness is analysed. According to classical linear elastic fracture mechanic, the transition between the yielded area related to plane stress state and the yielded area related to plane strain conditions is linear and from a certain point, constant. But no clear information about this has been provided. The evolution of the yielded areas along the thickness for two different cases is represented in Fig. 11. Fig. 11a corresponds to a type I case, in particular for a 3 mm specimen thickness and K = 15 MPa m1/2. Fig. 11b
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Ya surface / Ya rpD
(a) 1,4 1,2 1,0 0,8 0,6 0,4 0,2
5
10
15
20
25
30
35
40
45
35
40
45
K (MPa·m 1/2 )
Ya mid-plane / Ya rpD
(b) 1,0 0,8 0,6 0,4 0,2 0,0
5
10
15
20
25
30
K (MPa·m 1/2 ) Straight crack front
rc=10mm
rc=6mm
rc=4mm
Fig. 6. Evolution of the yielded area with the load applied and the crack front curvature (a) in the surface of the specimen and (b) in the mid-plane.
Fig. 5. Evolution of the yielded areas in the surface (a) and in the mid-plane (b) with the crack front curvature for a 3 mm specimen thickness and K = 25 MPa m1/2.
corresponds to a type III case, in particular for a 3 mm specimen thickness and K = 35 MPa m1/2. It should be noted that the evolution of the yielded areas and the plastic zone size along the thickness depends on three different parameters as the specimen thickness, the load applied and the crack front curvature as it has been shown in the present section. The qualitative analysis of Fig. 11 in terms of the typology previously defined gives us the chance to study the influence of the curvature on the plastic zone along the thickness in the two extreme cases. Starting with type III (Fig. 11b), corresponding to a high load for a particular thickness, it can be observed how the yielded area is quite constant along the thickness, with a value approximate to the 60% of the theoretical plastic zone size in plane stress (area corresponding to rpD). A slight influence of the curvature (lesser than 15%) is present at the exterior portion of the thickness (at one third from the exterior). Nevertheless, when we observe type I behaviour, we can see how the yielded area is greater than the yielded area in the interior at a 10% of the exterior surface. With the straight crack front, it is about a 20% only, however when the curvature increases the difference is higher. The yielded area at the interior drops barely while at
the exterior it increases remarkably, reaching values similar to the theoretical value with plane stress condition. This behaviour represents an opposite mechanism to reduce the curvature when it is too high. With a straight crack front, it has been argued [32] that, as a consequence of the elastic–plastic strain field resulting from the strong different constraint conditions at the interior and at the exterior of the crack tip, a smaller K value is present near the surface. It is supported by the evidence that a smaller plastic zone than the expected value with plane stress condition is obtained at the surface (as also shown in Figs. 3, 4, 8 and 11 in the present paper). According to these results if the curvature is excessive, the exterior of the specimen support an additional load which would lead to reduce the curvature. A balance between both mechanisms combined with the effect of crack closure in fatigue would configure the shape of the real crack. It must be remarked that this typology (type I) can be considered the more general behaviour. Excepting thin specimen at high load rates, most of the specimens respond to this pattern in a broad range of load.
3.2. Stress state along the thickness In order to evaluate the transition of the plane strain state to plane stress along the crack front, the values of the stress rz have been obtained, where rz is defined as the stress in the orthogonal direction to the surface of the specimen. In Fig. 12, the characteristic evolution of the stress rz along the crack front is shown. The position of the crack front normalised by the specimen thickness is represented in the x-axis. In the y-axis the stress rz expressed in N/m2 is represented. This value is not
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Ya surface / Ya mid-plane
(a)
6,0
Gravity centre
5,0 4,0 3,0
mid plane
surface
2,0 1,0 0,0
Fig. 9. Position of the gravity centre of the plastic size along the crack front, parameter d.
5
10
15
20
25
30
35
40
45
K (MPa·m 1/2 )
(b) 7,0 Ya surface / Ya mid-plane
d
6,0 5,0 4,0 3,0 2,0 1,0 0,0
5
10
15
20
25
30
35
40
45
K (MPa·m 1/2 ) Straight crack front
rc/b=10/3
rc/b=2
rc/b=4/3
Fig. 7. Evolution of the yielded area at the surface and at the mid-plane ratio with the applied load and the crack front curvature: (a) 3 mm and (b) 12 mm specimen thickness.
(b) 0,7
0,6
Plastic zone size / rpD
Plastic zone size / rpD
(a) 0,7
normalised because it is governed by the von Mises yield criterion and is independent of the load level. The value in the surface of the specimen is not null as it was expected. That is because the values of the stresses are computed in the integration points of the element and then, by interpolation, the value in the nodes are calculated. The high value of the stress gradient in this area is the cause of this numerical artefact. Nevertheless, as the trend is clear, this error is not relevant for the purpose of this analysis which seeks to determine the dominance of the plane strain state along the specimen thickness. Strictly, the only position where plane strain state is present is at the mid plane of the specimen. It corresponds to the maximum value of rz. Nevertheless, it can be seen that this value decreases very slowly and remains very close to the maximum for a broad band of the specimen thickness. Very close to the external surface the value of rz start to drop to zero. Therefore, in order to evaluate and quantify this effect, a parameter d5% has been defined as the distance from the surface of the specimen which produces a fall in the value of the stress corresponding to the plane strain state of 5%. This parameter is just employed to analyse the influence of
III
0,5 0,4
II
0,3 0,2 0,1
I 0,2
0,4
0,6
0,8
0,5 0,4 0,3 0,2 0,1 0,0 0
0 0
0,6
1
0,2
position along the crack / b
0,6
0,8
1
(d) 0,7 Plastic zone size / rpD
Plastic zone size / rpD
(c) 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0,0
0,4
position in thickness / b
0
0,2
0,4
0,6
0,8
1
0,6 0,5 0,4 0,3 0,2 0,1 0,0
position in thickness / b Straight crack front
0
0,2
0,4
0,6
0,8
1
position in thickness / b rc=10mm
rc=6mm
rc=4mm
Fig. 8. Evolution along the thickness of the plastic zone size with the crack front curvature in a 3 mm specimen thickness: (a) schematic typology, (b) type I, (c) type II, and (d) type III behaviour.
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(a) 0,60
Yielded area/Area rpD
(a) 1,2 I
0,55
d/b
0,50 0,45 0,40
II
0,35
III 3mm
10
15
20
25
K (MPa·m
30 1/2
35
40
45
0,6 0,4 0,2
0,2
0,4
0,6
0,8
1,0
0,8
1,0
position in crack/b
)
(b) 1,2 12mm
I
0,55
Yielded area/Area rpD
(b) 0,60 0,50
d/b
0,8
0,0 0,0
0,30 5
1,0
0,45 0,40
1,0 0,8 0,6 0,4 0,2
0,35 0,0 0,0
0,30 5
10
15
20
25
30
35
40
0,2
0,4
K (MPa·m 1/2 )
straight
Straight crack front
rc/b=10/3
rc/b=2
rc/b=4/3
0,6
position in crack/b
45
rc=10mm
rc=6mm
rc=4mm
Fig. 11. Evolution of the yielded area along the thickness with the crack front curvature for (a) K = 15 MPa m1/2 and (b) K = 35 MPa m1/2 in a 3 mm specimen thickness.
Fig. 10. Evolution of the gravity centre position of the plastic zone size along the crack with the load applied and the crack front curvature: (a) 3 mm and (b) 12 mm specimen thickness.
1,2E+09 1,0E+09
σ z (N/m2)
the load applied, the specimen thickness and the crack front curvature on the evolution of the dominance of the plane strain state along the thickness (see Fig. 12). This criterion of 5% has been chosen in order to see clearly this evolution and as a compromise between accuracy and numerical stability. Lower values should be better in order to determine with more accuracy the zone with similar conditions to plane strain state (mid-side) but implies more numerical errors and uncertainty, affecting the observed phenomenon. On the other hand, higher values imply that the zone identified is in a transitional situation and it would be meaningless for the purpose of our study. This value d5% has been proved to be useful to capture this transition [17]. In Fig. 13, the values obtained for a 3 mm specimen thickness are shown. For each crack front curvature case, there is a linear relationship between the distances from the surface at which starts to dominate the plane strain state and the load level applied. Furthermore, decreasing the radius of the crack front curvature implies a decrease of the dominance of the plane strain state. To analyse the influence of the specimen thickness and crack front curvature, the values obtained for the different specimen thicknesses and two radii of curvature are shown in Fig. 14. In this figure is represented the absolute distance where starts the dominance of plane strain state. Only two different radii of curvature are represented, the straight crack front and a radius of curvature normalised by the specimen thickness equal to 2. For the straight crack front, the graphs are overlapped. Only a small deviation occurs for values above K = 30 MPa m1/2. So the transition occurs at a distance of the specimen surface virtually independent of the specimen thickness, only related to the load level. Nevertheless, for the crack front with curvature, the deviation is significant in
Plane strain stress
8,0E+08
stress 5% lesser
6,0E+08 4,0E+08 2,0E+08 0,0E+00 0,0
Mid plane
d5%
0,2
0,4
0,6
0,8
position along the crack/b
1,0
Surface
Fig. 12. Stress in z direction evolution along the normalised crack front and d5% parameter.
all the load range applied. So the statement that the transition occurs at a distance of the specimen surface independent of the specimen thickness and only related to the load level is only true when the crack front considered is straight. In other case, it will depend on the load level, the specimen thickness and the crack front curvature. 4. Conclusions In the present work, the effect of the crack front curvature has been study by means of FE calculations. The influence of the three-dimensional effect of the specimen thickness on the behaviour of the crack front has been shown. It has been evaluated and
D. Camas et al. / International Journal of Fatigue 44 (2012) 41–50
0,8
d5% / b
0,6 0,4 0,2 0,0 5
10
15
20
25
30
35
40
45
K (MPa·m1/2) rc=10mm rc=4mm
Straight crack front rc=6mm
d5%
Fig. 13. Normalised distance to the specimen surface of the plane strain state.
1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0,0 5
10
15
20
25
30
35
40
45
K (MPa·m1/2 ) 3mm staight
6mm straight
12mm straight
3mm rc/b=2
6mm rc/b=2
12mm rc/b=2
Fig. 14. Evolution of the distance to the specimen surface of the plane strain state with the load applied for a straight crack front and a crack front with curvature.
quantified in terms of the yielded area, the plastic zone size along the thickness and in terms of the stress state transition from interior to the exterior of the specimen. As a general conclusion, these results evidence that different K values combined with different specimen thickness and crack front curvatures lead to different behaviours of the crack front. The previously proposed typology definition [17] has been useful in the present study to characterise qualitatively these behaviours. Regarding the analysis of the results obtained in this work, the following conclusions can be remarked: – The plastic zone depends on the load applied, the specimen thickness and the crack front curvature. The yielded area along the thickness and the plastic zone size at the crack plane depend strongly on these three parameters. – The yielded area at the surface increases when the radius of curvature decreases, while the yielded area in the mid plane is only influenced by the load applied and the specimen thickness. The crack front curvature has a strong influence in the ratio between the yielded area at the surface with the yielded area at the mid plane in the lowest loads range. Then, the ratio trends to converge when the applied load increases independently of the crack front curvature. The transition zone in the plastic zone size extends when the radius of curvature decreases. – Regarding the position of the gravity centre of the plastic zone size, considered as a representative parameter of the evolution of the plastic zone shape with the load, three different behaviours have been found. The load ranges under these behaviours
49
are dependent on the specimen thickness. The crack front curvature has a strong influence in the first type of behaviour (type I), but not in the other two. – Finally, in terms of the transition from plane strain to plane stress, it should be noted that for each crack front curvature, there is a linear behaviour of the distances from the surface at which starts to dominate the plane strain state with load applied. There is a strong influence of the crack front curvature, decreasing the dominance of plane strain state when the radius of the crack front decreases. Finally, when the crack front is straight the transition occurs at a distance of the specimen surface independent of the specimen thickness and only related to the load level. Nevertheless, when the crack front presents some kind of curvature this is no longer true. Turning our sight to fatigue problem, it is well known the effect of the crack closure phenomenon in the crack growth rate. This phenomenon is clearly influenced by the effect of the plastic wake developed during the crack growth. In a previous study [16] was shown that the effect of the crack closure is more prominent close to surface than in the mid-plane and there is a ‘‘transition zone’’ in between. The influence of the load applied, the specimen thickness and the crack front curvature on the plastic zone near the crack front has been shown in this study, so it is expected that these parameters will have influence on the crack closure values achieved at the surface and in the size of the ‘‘transition zone’’. In particular, it has been shown the huge influence of the crack front curvature on the yielded area close to the surface, so some influence is expected. This is an interesting cross-related effect to be studied in future. Fatigue crack closure influences the crack growth shape at the same time that this shape changes crack closure behaviour along the thickness.
References [1] Brown MV, de los Rios ER, Miller KJ. A critical comparison of proposed parameters for high-strain fatigue crack growth. ASTM STP 1988;924:233–59. [2] Wang GS, Blom AF. A strip model for fatigue crack growth predictions under general load conditions. Eng Fract Mech 1991;40:507–33. [3] Wang GS. The plasticity aspect of fatigue crack growth. Eng Fract Mech 1993;46:909–30. [4] Irwin GR. Plastic zone near a crack and fracture toughness. In: Proceedings of the 7th sagamore research conference on mechanics & metals behaviour of sheet material, vol. 4. New York; 1960. p. 463–78. [5] Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Solids 1960;8:100–4. [6] Banks TM, Garlick A. The form of crack tip plastic zones. Eng Fract Mech 1984;19:571–81. [7] Dodds Jr RH, Anderson TL, Kirk MT. A framework to correlate a/W effects on elastic–plastic fracture toughness (Jc). Int J Fract 1991;48:1–22. [8] E399-09e1. Standard test method for linear-elastic plane-strain fracture toughness KIc of metallic materials. In: American society for testing and materials, Book of Standards, vol. 03.01. Philadelphia; 2010. [9] Bellet D, Taylor D, Marco S, Mazzeo E, Guillois J, Pircher T. The fatigue behaviour of three-dimensional stress concentrations. Int J Fatigue 2005;27:207–21. [10] Bellet D, Taylor D. The effect of crack shape on the fatigue limit of threedimensional stress concentrations. Int J Fatigue 2006;28:114–23. [11] Sternberg E, Sadowsky MA. Three-dimensional solution for the stress concentration around a circular hole in a plate of arbitrary thickness. J Appl Mech 1949;16:27–36. [12] Kotousov A, Wang CH. Three-dimensional stress constraint in an elastic plate with a notch. Int J Solids Struct 2002;39:4311–26. [13] Nakamura T, Parks DM. Three-dimensional crack front fields in a thin ductile plate. J Mech Phys Solids 1990;38:787–812. [14] Newman Jr JC, Bigelow CA, Shivakumar KN. Three-dimensional elastic–plastic finite-element analyses of constraint variations in cracked bodies. Eng Fract Mech 1993;46:1–13. [15] Subramanya HY, Viswanath S, Narasimhan R. A three-dimensional numerical study of mixed mode (I and II) crack tip fields in elastic–plastic solids. Int J Fract 2005;136:167–85. [16] Gonzalez-Herrera A, Zapatero J. Tri-dimensional numerical modeling of plasticity induced crack closure. Eng Fract Mech 2008;75:4513–28.
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[17] Camas D, Garcia-Manrique J, Gonzalez-Herrera A. Numerical study of the thickness transition in bi-dimensional specimen cracks. Int J Fatigue 2011;33:921–8. [18] Elber W. The significance of fatigue crack closure, damage tolerance in aircraft structures. ASTM STP 1971;486:230–47. [19] Ewalds HL, Furnee RT. Crack closure measurements along the crack front in center cracked specimens. Int J Fracture 1978;14:R53–5. [20] Sunder R, Dash PH. Measurement of fatigue crack closure through electron microscopy. Int J Fatigue 1982;4:97–105. [21] Schijve J. Fatigue of structures and materials. second ed. The Netherlands: Springer; 2009. [22] Branco R, Rodrigues DM, Antunes FV. Influence of through-thickness crack shape on plasticity induced crack closure. Fatigue Fract Eng Mater Struct 2008;31:209–20. [23] Benthem JP. State of stress at the vertex of a quarter-infinite crack in a halfspace. Int J Solids Struct 1977;13:479–92. [24] Bazˇant ZP, Estenssoro LF. Surface singularity and crack propagation. Int J Solids Struct 1979;15:405–26. [25] Pook LP. Some implications of corner point singularities. Eng Fract Mech 1994;48:367–78.
[26] Hutarˇ P, Náhlík L, Knésl Z. The effect of a free surface on fatigue crack behaviour. Int J Fatigue 2010;32:1265–9. [27] Pook L. Metal fatigue. What it is, why it matters. The Netherlands: Springer; 2009. [28] Murakami Y. Stress intensity factors handbook. United States: Pergamon Books; 1987. [29] ANSYS. User’s manual, v. 13;2009. [30] Gonzalez-Herrera A, Zapatero J. Influence of minimum element size to determine crack closure stress by the finite element method. Eng Fract Mech 2005;72:337–55. [31] Zapatero J, Gonzalez-Herrera A. Advances in the numerical modelling of fatigue crack closure using finite elements. In: Lignelli AF, editor. Fatigue Crack Growth: Mech Behav Pred. New York: Nova Science Publishers; 2009. p. 83–124. [32] Garcia-Manrique J, Camas D, Lopez-Crespo P, Gonzalez-Herrera A. Stress intensity factor analysis of through thickness effects. Int J Fatigue 2012. http:// dx.doi.org/10.1016/j.ijfatigue.2011.12.012. [33] Roychowdhury S, Dodds RH. A numerical investigation of 3-D small-scale yielding fatigue crack growth. Eng Fract Mech 2003;70:2363–83.
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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Crack front curvature: Influence and effects on the crack tip fields in bi-dimensional specimens D. Camas ⇑, J. Garcia-Manrique, A. Gonzalez-Herrera Departamento de Ingeniería Civil, de Materiales y Fabricación, Universidad de Málaga, Escuela de Ingenierías, C/Doctor Ortiz Ramos, s/n., Campus de Teatinos, E-29071, Málaga, Spain
a r t i c l e
i n f o
Article history: Received 24 December 2011 Received in revised form 21 May 2012 Accepted 22 May 2012 Available online 4 June 2012 Keywords: Finite element analysis Crack front curvature Plastic zone size Out-of-plane stress Fracture mechanic
a b s t r a c t Crack front curvature is evidence in most experimental crack advance test. When classical linear elastic fracture mechanic theory deals with bi-dimensional crack configurations, it ignores the three-dimensional effects of crack propagation. Issues as the influence of the specimen thickness and the crack front curvature are not considered. Previous numerical studies have shed light on out-of-plane plastic zone development or stress state. Nevertheless, these numerical studies are based on the assumption that the crack front is ideally straight; despite it is well known that the crack front has some kind of curvature. In the present work, a CT aluminium specimen has been modelled three-dimensionally and several calculations have been made considering a huge combination of different single load levels, specimen thicknesses and crack front curvatures. Due to the abrupt transition from plane strain to plane stress, an ultrafine mesh along the thickness has been applied. The analysis of the evolution of the plastic zone and the stress state along the thickness provides information about the combined influence of these parameters on fracture mechanics. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Engineers and designers need to predict fatigue life in structures and components under cyclic loads. Nonetheless, the complexity of the problem limits the possibility of dealing in a rational predictive way with this phenomenon. The characterisation of fracture and fatigue crack propagation is based on the classical linear elastic fracture mechanics, which only considers two plane states. The main advantage of the plane theories is the simplicity compared to the three-dimensional equations. The yielded zone size close to the crack front influences the stress and strain state in the crack front. It is a factor of fundamental importance in the fracture and fatigue behaviour of metallic materials [1–3]. Several efforts have been made to determine the plastic zone size since 1960. Irwin [4] and Dugdale [5] proposed different models to estimate plastic zone size ahead of the crack. In the middle 1980s, Banks and Garlick [6] made an attempt to quantify the shape and the size of the plastic zone using the von Mises yield criterion. These previous analysis were not strictly correct because they were based on a purely elastic analysis. The redistribution of stresses due to yielding was neglected. Dodds et al. [7] studied the shape of the plastic zone using the stress
⇑ Corresponding author. Tel.: +34 951952440; fax: +34 951952601. E-mail addresses: [email protected] (D. Camas), [email protected] (J. Garcia-Manrique), [email protected] (A. Gonzalez-Herrera). 0142-1123/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijfatigue.2012.05.012
distribution obtained by finite element analysis. These yielded zones are related to two extreme stress states: plane stress or plane strain, but no clear information about out-of-plane yielded areas or stress state was provided. Dealing with plates of finite thickness, the dominant state of stress is selected according to relative thickness. For thin specimens, the stress state is considered to be plane stress. Otherwise, it is considered to be plane strain. The ASTM E399 [8] indicates the specific condition in which a fracture toughness specimen has an acceptable plane strain state. The simplicity of bi-dimensional (2D) theory has become limitation for fracture and fatigue mechanics, so it is necessary to develop three-dimensional (3D) theories. Recently, Bellet [9,10] showed experimental evidence and proved that the commonly 2D or planar methods for fracture and fatigue assessment of notched bodies may lead to inaccurate predictions when applied to certain 3D geometries. There are only few analytical 3D solutions available in the literature due to the complexity of the stress field. Sternberg and Sadowsky [11] obtained an approximate solution for 3D stress distribution near a circular hole in an infinite plate of arbitrary thickness by using series expansion and taking finite terms into account. Based on the assumption of a generalised plane-strain theory, which assumes that the through-the-thickness extensional strain is uniform in the thickness direction, Kotousov and Wang [12] developed analytical solutions for the 3D stress distribution around typical stress concentrators in an isotropic plate of
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Nomenclature a b d d5%
crack length specimen thickness distance to the exterior face of the gravity centre of the plastic size area distance from the surface of the specimen which produces a fall in the value of the stress corresponding to the plane strain state of 5%
arbitrary thickness. However, no theoretical solution near a 3D crack front in an elastic–plastic solid is available. Due to these difficulties, detailed 3D finite element analyses are essential to characterise the stress and strain variations near a crack front in elastic–plastic solids and also ascertain the range of dominance of 2D plane strain and plane stress solutions. To this end, some works performed elastic–plastic 3D analysis of thin plates under mode I [13,14] and mixed mode (mode I + II) [15] loading conditions, obtaining several parameters as plastic zone sizes, stresses, deformations and constraint variations along the thickness. Their results revealed strong thickness variations of stress fields very near the crack front. The main problem in 3D studies is that there is a very abrupt transition in the stress and strain fields in a thin part of the external surface. This transition can be properly captured only if an ultrafine mesh along the thickness is done. Recent studies with fine meshes along the thickness have shown an important influence of the specimen thickness and load applied [16,17]. Besides, unexpected through the thickness plastic zone size evolutions were realised. These results were not observed in previous studies due to the lower number of elements employed through the thickness. Previous numerical studies that analyse the plastic zone size and shape and the stress distribution near the crack front are often based on the assumption that the crack front is straight. It is well known, that the crack front in through cracks presents some kind of curvature when the crack grows under fatigue loading conditions. Some efforts have been made in order to explain and characterise the crack front curvature. One explanation is the crack closure effect introduced by Elber [18] in the first 1970s. Apparently, crack closure is a 3D mechanistic phenomenon. Experimental observations [19,20] and numerical analysis [16] shows that fatigue cracks open first at midthickness and later at the surface, which often leads to curved crack fronts for through cracks [21]. Branco et al. [22] analysed numerically, the influence of the crack closure in the stable crack shape growth and compared the results when the crack closure was not considered. While without crack closure the crack shape remained straight with only a smooth propagation delay near the surface, the consideration of the crack closure was found to produce a significant tunnelling effect. Other one is the stress intensity factor distribution along the thickness and the presence of the corner point singularity or vertex point. Usually, the stress field is described by a unique stress intensity factor value. But as early as 1977, Benthem [23] presented the 3D corner singularity at the intersection of the crack front and the surface. Bazˇant and Estenssoro [24] used a FE analysis to propose that this singularity produce the deviation of the crack front from the orthogonal direction near the surface. However to date, the influence size of the corner point singularity is not known. Pook [25] suggested that this size is a function of load and concluded that crack front intersect a free surface at a critical angle, which is function of the loading type and the Poisson’s ratio. Hutarˇ et al. [26] presented a very interesting methodology based on generalised stress intensity factor to describe the crack behaviour
K rc rpD Ya
stress intensity factor radius of curvature Dugdale’s plastic zone size yielded area
close to the free surface. They studied the evolution of the crack shape due to the effect of the free surface and the influence of the specimen thickness. Hence, fatigue crack propagation rate decreases leading also to a curved crack front. Therefore, the purpose of the present study was to analyse the influence of the crack front curvature in the stress fields along the thickness of through cracks and its relation with the load applied and the specimen thickness. Hence, a comprehensive numerical study by means of 3D FEA of the stress fields along the crack front for different load cases, with different specimen thickness and crack front curvatures is made. This work was performed by means of the simulation and calculation of a huge number of load cases combining different crack curvatures. A single load is applied to the specimen with an ultrafine mesh along the thickness. The analysis was focussed in the transition of the response of different parameters – as stress state, plastic zone, etc. – along the thickness. An especial attention is paid to the comparison with previously obtained result with straight cracks [17].
2. FEA modelling methodology In this study, a CT specimen geometry (W = 50 mm, a = 20 mm) has been modelled three-dimensionally, with elastic–plastic behaviour of the material. A combination of three different thicknesses (b = 3, 6 and 12 mm), four different crack front curvatures and a range of loads corresponding to stress intensity factors from K = 10 to 40 MPa m1/2 have been considered. It is important to note that the stress intensity factor arises from a two dimensional linearly elastic analysis. Therefore, there are some limitations on their validity as the small scale yielding argument and that cracks must stand perpendicular to the surfaces. Nevertheless, experience has shown that these limitations can be managed [27]. In the present study, the remote far-field stress intensity factor K is just used to express the load magnitude, due to the general use of this parameter on practical engineering problems involving cracks. The relation between the load applied and the stress intensity factor K is determined by next expression [28]:
0 a 2 þ Wa P pffiffiffiffiffiffi @ K¼ 0:886 þ 4:64 3 W b W 1 Wa 2 a 2 a 3 a 4 13:32 þ 14:72 5:6 W W W
ð1Þ
where P is the load applied, b is the specimen thickness, W is the distance from the loading line to the back face and a is the crack length. The crack front curvatures are defined in terms of the radius of curvature and calculations range from straight crack to a radius 4/3 times the specimen thickness. The total number of different cases studied is 84. This allows studying a huge range of behaviour including low and high loads. Fig. 1 illustrates a 3D mesh employed for a crack front curvature case. It was simulated with the
D. Camas et al. / International Journal of Fatigue 44 (2012) 41–50
Fig. 1. Finite element model.
commercial FEA code ANSYS. Due to symmetry, only one quarter of the specimen needs to be modelled in 3D. The most critical area corresponds to the crack front. In this area there is a great gradient in the stress and strain field. Therefore, in order to capture properly this effect, the size of the elements around the crack tip must be small enough. In order not to penalise the computational cost, it is necessary to make a huge transition from this zone to the most remote ones. For this reason, the specimen has been divided in two different volumes. In the first one, around the crack tip, a homogeneous and uniform undistorted mesh with hexahedral elements is made. And the second one is meshed with tetrahedral elements which allow the transition. In the present study, a unique eight-node element with linear base function has been used. At each node, there are three degrees of freedom corresponding to the translations in the x, y and z directions. The tetrahedral elements are obtained degenerating the element, condensing faces to lines. At first sight, it may seem that employing quadratic elements could allow using less number of elements decreasing the computational cost. But since this is a nonlinear analysis, a better accuracy at less expense is obtained if a fine mesh of linear elements is used rather than a comparable coarse mesh of quadratic elements [29]. Besides, this kind of element has been employed due to the huge experience obtained by the present authors in previous works in which the accuracy of the model has been analysed [17,30–32]. The plastic zone size is the scale reference to relate to the minimum element size. In this work Dugdale’s plastic zone size has been used as a reference, expressed as:
r pD ¼
p KI 8a ry
and the length of the crack front, 80 and 120 divisions have been considered. In the vicinity of the surface of the specimen, there is a transition area in the plastic zone. Close to the mid plane, where plane strain conditions are dominant, the plastic zone is homogeneous. In order to visualise with a higher resolution the transition area of the plastic zone, the length along the thickness of the element next to the surface of the specimen is the half of the length of the element next to the mid plane of the specimen. Outside this region, the number of through-thickness divisions was reduced to speed up the simulations. Accordingly, a transition in the through-thickness mesh has been done (see detail in Fig. 1). A good compromise between accuracy and computational cost was achieved by employing hexahedral prism shaped elements within the plastic zone and tetrahedral elements outside. In addition, the use of 8-node tetrahedral elements facilitated the transition between both zones. In straight-crack-front three-dimensional models, all the crack front nodes are located at a distance equal to the crack front length. Nevertheless, when the crack front has any sort of curvature it is necessary to set a point along the thickness to determine the crack front length. This crack front length defines the value of the stress intensity factor (K). In the present study, the gravity centre of the line which constitutes the crack front has been considered to characterise the crack length (see Fig. 2). The material modelled has been an Al-2024-T351 aluminium alloy that shows weak hardening (E = 73.5 GPa, m = 0.33, r yield = 425 MPa, K0 = 685 MPa, n0 = 0.073, being K0 and n0 parameters in the Ramberg–Osgood yielding model). In previous works, this material has been numerically modelled in different ways. Theses previous 2D studies were made with both isotropic and kinematic hardening models. Nevertheless, no differences were found in the results when there is a weak work hardening rule, as in this case [31]. So, the results presented in this paper have been calculated with a model that employs an isotropic hardening rule with H/E = 0.003 that fits well the actual behaviour of this aluminium alloy and have a smaller computational cost. The values H and E represent the slopes of the plastic and the elastic line, respectively. It has been modelled by a three-linear representation.
2 ð2Þ
where a is a constraint factor equal to 1 for plane stress and 3 for plane strain. As in this work a 3D model is considered, a has been taken equal to 1 in order to be higher than any yielded zone size along the thickness. In all the different cases studied, the volume meshed with hexahedral elements is divided in the same number of elements. This volume is proportional to the Dugdale’s plastic zone, so the size of the elements of this area is proportional to the applied load. Previous studies developed by the present authors [30] concluded that the FEA results are strongly dependent on the minimum element size around the crack tip. The minimum element size has been established following a recommendation proposed previously, adapted to reduce the computational cost. The minimum element size is 184 times smaller than Dugdale’s plastic zone size. As the crack opening load is not required and the plastic wake does not need to be simulated, a lesser requirement of a number of divisions of the plastic zone equal to 76 is adopted. This value is the consequence of a previous mesh refinement study [17]. Once the minimum element size is established, and in order to keep an acceptable element shape ratio, in this work the number of thickness divisions adopted has been 40. Only where the element shape ratio was not acceptable because of the small load value
43
Fig. 2. CT specimen dimensions and crack length definition.
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3. Results The influence of the crack front curvature can be observed analysing different parameters. There are a huge number of results to be reported, nevertheless only the most representatives are going to be shown. In this way, the plastic zone and the stress field at the crack tip have been considered as the most significants and will be focused on. Furthermore, qualitative observations will be quantified in the subsequent sections.
3.1. Plastic zone size As mentioned before, the fracture and fatigue behaviour of a metallic material is strongly influenced by a small yielded area close to the crack front. Therefore, it is necessary to know properly the shape of the yielded area. There are previous studies by means of FEA that analyses the shape of this yielded area. Most of them are in 2D, but also there are some 3D studies which consider the crack ideally straight [13,33]. However, it is well known that the crack front shape when the crack growths in fatigue is not straight, but presents any curvature. An accurate vision of the plastic zone in the area close to the crack front can be obtained analysing the plastic strain suffered by the elements. In Fig. 3, a comparison of the yielded area around the crack front for two different crack shapes geometries for a 3 mm specimen thickness is shown. Both have the same applied load corresponding to a stress intensity factor K = 20 MPa m1/2, but different crack front curvature: straight and 4 mm. The whole yielded region is represented. Firstly, as it has been reported previously [16,17] with straight cracks, it can be appreciated that the plastic zone size of both cases shows essential differences with the theoretical model. This one predicts that the plastic zone size at the surface of the specimen, under plane stress conditions, is three times greater than the plastic strain in the mid plane, under plane strain conditions. That is the classical ‘‘bone shape’’ plastic zone. Nevertheless, results obtained are different. Comparisons between the theoretical plastic zone in plane stress and plane strain states, and the numerical results are shown in Fig. 4. The numerical results presented in this figure have been obtained considering a straight crack front. The results shown in Fig. 4a and b correspond to 3 mm and 12 mm specimen thicknesses, respectively. It can be observed that comparing numerical and theoretical results, sizes and shapes are different. Analysing Fig. 4a can be deduced that the reason why neither plane stress or plane strain plastic zone shape and size has been achieved is the small thickness of the specimen with respect the load applied, according to the ASTM E399 [8]. However, in Fig. 4b can be observed the same behaviour when, in this case,
the relation between the specimen thickness and the load applied is suitable for achieving both states. It is noteworthy that for both cases, shapes and sizes numerically obtained are really similar in both states, at the mid-plane and at the surface. In order to study the influence of the crack front curvature in the plastic zone size, it is of interest to analyse the yielded areas. In Fig. 5, the evolution of the yielded areas at the surface (Fig. 5a) and in the mid-plane (Fig. 5b) with the crack front curvature is represented. In these figures, the specimen thickness is 3 mm and K = 25 MPa m1/2. As it can be observed, the yielded area in the mid-plane is not influenced by the crack front curvature, while the yielded area at the surface of the specimen gets larger as the radius of the crack front curvature is reduced. Furthermore, the yielded area depends of the load applied. In Fig. 6 the influence of both parameters, the crack front curvature and the load applied, in the yielded area is analysed. Data are normalised by the corresponding area to the Dugdale’s plastic zone in plane stress and plane strain respectively. In Fig. 6a, the evolution of the yielded area on the surface of the specimen is illustrated. It should be noted that while the straight crack front case presents a linear behaviour with the load applied, a strong influence of the crack front curvature in the lowest load range can be observed. When the radius of curvature decreases, the yielded area increases. By increasing the value of the applied load, the results converge to the straight crack front ones, minimising the effect of the curvature. The behaviour of these parameters in the midplane can be observed in Fig. 6b. In this case, the yielded area is only affected by the applied load. Therefore, there is a huge combined influence of both parameters: the load applied and the crack front curvature. In Fig. 7, the ratio between the yielded area at the surface and at the mid plane is represented versus the stress intensity factor for the 3 mm and 12 mm specimen thickness. The radius of curvature for both cases is represented as the relation between the radius of curvature and the specimen thickness. The first issue to note is that the ratio is almost equal to unity in the applied load range for the straight crack front case. Hence, the yielded area at the surface is equal to the yielded area in the mid-plane in all the applied load range for this case. This behaviour may be explained by the corner point singularity and the decrease of the stress intensity factor as a corner point is approached [32]. Secondly, there is a strong dependence with the crack front curvature in the lowest loads range. The curvature causes the yielded area in the surface to be much greater than in the mid plane, but for the greater load range, the values converge. As it can be observed, for the 3 mm specimen thickness the yielded areas at the surface and at the mid plane are equals for stress intensity factor values greater than 30 MPa m1/2, while for smaller ones there is a
Fig. 3. Plastic zone close to the crack front.
D. Camas et al. / International Journal of Fatigue 44 (2012) 41–50
45
Fig. 4. Comparison of the theoretical plastic zone in plane stress and plane strain states, with the numerical plastic zone obtained in the surface and in the mid-plane for a 3 mm (a) and 12 mm (b) specimen thickness and K = 20 MPa m1/2.
huge influence of the crack front curvature. For the 12 mm specimen thickness, the convergence does not occur in the applied load range and is expected for higher loads [17]. Finally, the evolution along the thickness of the yielded area is analysed. Hence, two different parameters are studied: the yielded area and the plastic zone along the thickness, where the plastic zone is the yielded area in the crack plane. In this way, the evolution of the yielded area in perpendicular planes to the crack plane is analysed (defined this yielded area, Ya, as drawn in Figs. 4 and 5). As it has been reported in a previous work [17], the plastic zone evolution along the thickness depends on the specimen thickness and the load applied. Three different behaviours according to the load level were found. The first one (type I) refers to cases where the transition zone occupies part of the thickness. The transition zone is the area of the plastic zone size that extends from the point in which the typical values in the interior of the specimen begins to increase, to the surface. The second type (type II) refers to cases where the transition zone occupies all the specimen thickness, increasing gradually the value of the yielded area at the mid-plane. The last one (type III) refers to cases where the maximum plastic zone size is at the mid-plane. Furthermore, higher load levels do not increase the plastic size in the interior zone, but it does in the exterior. This is resumed in Fig. 8a. As well, in Fig. 8 the typical evolutions of the plastic zone size along the thickness with the crack front curvature for the three different types of behaviour described before are represented. The results shown in this figure correspond to a 3 mm specimen thickness. It can be observed that for the first two behaviours, decreasing the radius of curvature implies an increase of the plastic zone size in the transition zone. In the third one, the plastic zone sizes are almost the same for the considered radii. In order to analyse the dependence of the different parameters, it is necessary to quantify the results obtained. The position of the gravity centre of the plastic zone size area has been considered for this purpose. The parameter d is defined as the distance from the surface of the specimen in orthogonal direction to the gravity centre (see Fig. 9). It is a way to characterise numerically the evolution of the plastic zone size along the thickness and the influence of the specimen thickness, the load applied and the crack front curvature. The parameter d tries to represent the dominance of the plane strain state over the whole crack behaviour. The results obtained
for the 3 mm and 12 mm specimen thickness are shown in Fig. 10. For the 3 mm specimen thickness and the load range applied, the three different behaviours can be observed. In the first one, increasing the load applied and decreasing the radius of the crack front curvature involves a displacement of the gravity centre toward the surface of the specimen due to the extension of the plastic zone in this area. In the second type, increasing the load applied involves a displacement of the gravity centre toward the interior of the specimen due to the increase of the value of the plastic size at the mid-plane. In the last one, higher load levels do not increase the plastic size in the interior zone, but it does in the exterior. Traditionally, the ASTM E399 equation [8] discretises the plane state behaviour of the specimen in two states, but reality is a continuum. Analysing the evolution of the plastic zone size along the thickness by means the d parameter, was found that the behaviour is more complex as it can be seen in Fig. 10 [17]. Besides, it seems that the requirement of the standard is conservative and should be considerate at least all the type I. The type III corresponds to load levels in which the state of the specimen can be considered as plane stress due to the distribution of the yielded area along the thickness is almost constant. Finally, the type II corresponds to a transition type in which neither of the plane states can be considered. Analysing the influence of the crack front curvature, in the last two types, it can be observed that there is no influence of the crack front curvature, while it is important in the first type. This can be better appreciated in Fig. 10b, where the 12 mm specimen thickness results are represented. For this specimen thickness, only the first type is present for the load range considered. The influence of the crack front curvature is clear even at the highest applied load. Finally, the evolution of the yielded area (Ya) along the thickness is analysed. According to classical linear elastic fracture mechanic, the transition between the yielded area related to plane stress state and the yielded area related to plane strain conditions is linear and from a certain point, constant. But no clear information about this has been provided. The evolution of the yielded areas along the thickness for two different cases is represented in Fig. 11. Fig. 11a corresponds to a type I case, in particular for a 3 mm specimen thickness and K = 15 MPa m1/2. Fig. 11b
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Ya surface / Ya rpD
(a) 1,4 1,2 1,0 0,8 0,6 0,4 0,2
5
10
15
20
25
30
35
40
45
35
40
45
K (MPa·m 1/2 )
Ya mid-plane / Ya rpD
(b) 1,0 0,8 0,6 0,4 0,2 0,0
5
10
15
20
25
30
K (MPa·m 1/2 ) Straight crack front
rc=10mm
rc=6mm
rc=4mm
Fig. 6. Evolution of the yielded area with the load applied and the crack front curvature (a) in the surface of the specimen and (b) in the mid-plane.
Fig. 5. Evolution of the yielded areas in the surface (a) and in the mid-plane (b) with the crack front curvature for a 3 mm specimen thickness and K = 25 MPa m1/2.
corresponds to a type III case, in particular for a 3 mm specimen thickness and K = 35 MPa m1/2. It should be noted that the evolution of the yielded areas and the plastic zone size along the thickness depends on three different parameters as the specimen thickness, the load applied and the crack front curvature as it has been shown in the present section. The qualitative analysis of Fig. 11 in terms of the typology previously defined gives us the chance to study the influence of the curvature on the plastic zone along the thickness in the two extreme cases. Starting with type III (Fig. 11b), corresponding to a high load for a particular thickness, it can be observed how the yielded area is quite constant along the thickness, with a value approximate to the 60% of the theoretical plastic zone size in plane stress (area corresponding to rpD). A slight influence of the curvature (lesser than 15%) is present at the exterior portion of the thickness (at one third from the exterior). Nevertheless, when we observe type I behaviour, we can see how the yielded area is greater than the yielded area in the interior at a 10% of the exterior surface. With the straight crack front, it is about a 20% only, however when the curvature increases the difference is higher. The yielded area at the interior drops barely while at
the exterior it increases remarkably, reaching values similar to the theoretical value with plane stress condition. This behaviour represents an opposite mechanism to reduce the curvature when it is too high. With a straight crack front, it has been argued [32] that, as a consequence of the elastic–plastic strain field resulting from the strong different constraint conditions at the interior and at the exterior of the crack tip, a smaller K value is present near the surface. It is supported by the evidence that a smaller plastic zone than the expected value with plane stress condition is obtained at the surface (as also shown in Figs. 3, 4, 8 and 11 in the present paper). According to these results if the curvature is excessive, the exterior of the specimen support an additional load which would lead to reduce the curvature. A balance between both mechanisms combined with the effect of crack closure in fatigue would configure the shape of the real crack. It must be remarked that this typology (type I) can be considered the more general behaviour. Excepting thin specimen at high load rates, most of the specimens respond to this pattern in a broad range of load.
3.2. Stress state along the thickness In order to evaluate the transition of the plane strain state to plane stress along the crack front, the values of the stress rz have been obtained, where rz is defined as the stress in the orthogonal direction to the surface of the specimen. In Fig. 12, the characteristic evolution of the stress rz along the crack front is shown. The position of the crack front normalised by the specimen thickness is represented in the x-axis. In the y-axis the stress rz expressed in N/m2 is represented. This value is not
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D. Camas et al. / International Journal of Fatigue 44 (2012) 41–50
Ya surface / Ya mid-plane
(a)
6,0
Gravity centre
5,0 4,0 3,0
mid plane
surface
2,0 1,0 0,0
Fig. 9. Position of the gravity centre of the plastic size along the crack front, parameter d.
5
10
15
20
25
30
35
40
45
K (MPa·m 1/2 )
(b) 7,0 Ya surface / Ya mid-plane
d
6,0 5,0 4,0 3,0 2,0 1,0 0,0
5
10
15
20
25
30
35
40
45
K (MPa·m 1/2 ) Straight crack front
rc/b=10/3
rc/b=2
rc/b=4/3
Fig. 7. Evolution of the yielded area at the surface and at the mid-plane ratio with the applied load and the crack front curvature: (a) 3 mm and (b) 12 mm specimen thickness.
(b) 0,7
0,6
Plastic zone size / rpD
Plastic zone size / rpD
(a) 0,7
normalised because it is governed by the von Mises yield criterion and is independent of the load level. The value in the surface of the specimen is not null as it was expected. That is because the values of the stresses are computed in the integration points of the element and then, by interpolation, the value in the nodes are calculated. The high value of the stress gradient in this area is the cause of this numerical artefact. Nevertheless, as the trend is clear, this error is not relevant for the purpose of this analysis which seeks to determine the dominance of the plane strain state along the specimen thickness. Strictly, the only position where plane strain state is present is at the mid plane of the specimen. It corresponds to the maximum value of rz. Nevertheless, it can be seen that this value decreases very slowly and remains very close to the maximum for a broad band of the specimen thickness. Very close to the external surface the value of rz start to drop to zero. Therefore, in order to evaluate and quantify this effect, a parameter d5% has been defined as the distance from the surface of the specimen which produces a fall in the value of the stress corresponding to the plane strain state of 5%. This parameter is just employed to analyse the influence of
III
0,5 0,4
II
0,3 0,2 0,1
I 0,2
0,4
0,6
0,8
0,5 0,4 0,3 0,2 0,1 0,0 0
0 0
0,6
1
0,2
position along the crack / b
0,6
0,8
1
(d) 0,7 Plastic zone size / rpD
Plastic zone size / rpD
(c) 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0,0
0,4
position in thickness / b
0
0,2
0,4
0,6
0,8
1
0,6 0,5 0,4 0,3 0,2 0,1 0,0
position in thickness / b Straight crack front
0
0,2
0,4
0,6
0,8
1
position in thickness / b rc=10mm
rc=6mm
rc=4mm
Fig. 8. Evolution along the thickness of the plastic zone size with the crack front curvature in a 3 mm specimen thickness: (a) schematic typology, (b) type I, (c) type II, and (d) type III behaviour.
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(a) 0,60
Yielded area/Area rpD
(a) 1,2 I
0,55
d/b
0,50 0,45 0,40
II
0,35
III 3mm
10
15
20
25
K (MPa·m
30 1/2
35
40
45
0,6 0,4 0,2
0,2
0,4
0,6
0,8
1,0
0,8
1,0
position in crack/b
)
(b) 1,2 12mm
I
0,55
Yielded area/Area rpD
(b) 0,60 0,50
d/b
0,8
0,0 0,0
0,30 5
1,0
0,45 0,40
1,0 0,8 0,6 0,4 0,2
0,35 0,0 0,0
0,30 5
10
15
20
25
30
35
40
0,2
0,4
K (MPa·m 1/2 )
straight
Straight crack front
rc/b=10/3
rc/b=2
rc/b=4/3
0,6
position in crack/b
45
rc=10mm
rc=6mm
rc=4mm
Fig. 11. Evolution of the yielded area along the thickness with the crack front curvature for (a) K = 15 MPa m1/2 and (b) K = 35 MPa m1/2 in a 3 mm specimen thickness.
Fig. 10. Evolution of the gravity centre position of the plastic zone size along the crack with the load applied and the crack front curvature: (a) 3 mm and (b) 12 mm specimen thickness.
1,2E+09 1,0E+09
σ z (N/m2)
the load applied, the specimen thickness and the crack front curvature on the evolution of the dominance of the plane strain state along the thickness (see Fig. 12). This criterion of 5% has been chosen in order to see clearly this evolution and as a compromise between accuracy and numerical stability. Lower values should be better in order to determine with more accuracy the zone with similar conditions to plane strain state (mid-side) but implies more numerical errors and uncertainty, affecting the observed phenomenon. On the other hand, higher values imply that the zone identified is in a transitional situation and it would be meaningless for the purpose of our study. This value d5% has been proved to be useful to capture this transition [17]. In Fig. 13, the values obtained for a 3 mm specimen thickness are shown. For each crack front curvature case, there is a linear relationship between the distances from the surface at which starts to dominate the plane strain state and the load level applied. Furthermore, decreasing the radius of the crack front curvature implies a decrease of the dominance of the plane strain state. To analyse the influence of the specimen thickness and crack front curvature, the values obtained for the different specimen thicknesses and two radii of curvature are shown in Fig. 14. In this figure is represented the absolute distance where starts the dominance of plane strain state. Only two different radii of curvature are represented, the straight crack front and a radius of curvature normalised by the specimen thickness equal to 2. For the straight crack front, the graphs are overlapped. Only a small deviation occurs for values above K = 30 MPa m1/2. So the transition occurs at a distance of the specimen surface virtually independent of the specimen thickness, only related to the load level. Nevertheless, for the crack front with curvature, the deviation is significant in
Plane strain stress
8,0E+08
stress 5% lesser
6,0E+08 4,0E+08 2,0E+08 0,0E+00 0,0
Mid plane
d5%
0,2
0,4
0,6
0,8
position along the crack/b
1,0
Surface
Fig. 12. Stress in z direction evolution along the normalised crack front and d5% parameter.
all the load range applied. So the statement that the transition occurs at a distance of the specimen surface independent of the specimen thickness and only related to the load level is only true when the crack front considered is straight. In other case, it will depend on the load level, the specimen thickness and the crack front curvature. 4. Conclusions In the present work, the effect of the crack front curvature has been study by means of FE calculations. The influence of the three-dimensional effect of the specimen thickness on the behaviour of the crack front has been shown. It has been evaluated and
D. Camas et al. / International Journal of Fatigue 44 (2012) 41–50
0,8
d5% / b
0,6 0,4 0,2 0,0 5
10
15
20
25
30
35
40
45
K (MPa·m1/2) rc=10mm rc=4mm
Straight crack front rc=6mm
d5%
Fig. 13. Normalised distance to the specimen surface of the plane strain state.
1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0,0 5
10
15
20
25
30
35
40
45
K (MPa·m1/2 ) 3mm staight
6mm straight
12mm straight
3mm rc/b=2
6mm rc/b=2
12mm rc/b=2
Fig. 14. Evolution of the distance to the specimen surface of the plane strain state with the load applied for a straight crack front and a crack front with curvature.
quantified in terms of the yielded area, the plastic zone size along the thickness and in terms of the stress state transition from interior to the exterior of the specimen. As a general conclusion, these results evidence that different K values combined with different specimen thickness and crack front curvatures lead to different behaviours of the crack front. The previously proposed typology definition [17] has been useful in the present study to characterise qualitatively these behaviours. Regarding the analysis of the results obtained in this work, the following conclusions can be remarked: – The plastic zone depends on the load applied, the specimen thickness and the crack front curvature. The yielded area along the thickness and the plastic zone size at the crack plane depend strongly on these three parameters. – The yielded area at the surface increases when the radius of curvature decreases, while the yielded area in the mid plane is only influenced by the load applied and the specimen thickness. The crack front curvature has a strong influence in the ratio between the yielded area at the surface with the yielded area at the mid plane in the lowest loads range. Then, the ratio trends to converge when the applied load increases independently of the crack front curvature. The transition zone in the plastic zone size extends when the radius of curvature decreases. – Regarding the position of the gravity centre of the plastic zone size, considered as a representative parameter of the evolution of the plastic zone shape with the load, three different behaviours have been found. The load ranges under these behaviours
49
are dependent on the specimen thickness. The crack front curvature has a strong influence in the first type of behaviour (type I), but not in the other two. – Finally, in terms of the transition from plane strain to plane stress, it should be noted that for each crack front curvature, there is a linear behaviour of the distances from the surface at which starts to dominate the plane strain state with load applied. There is a strong influence of the crack front curvature, decreasing the dominance of plane strain state when the radius of the crack front decreases. Finally, when the crack front is straight the transition occurs at a distance of the specimen surface independent of the specimen thickness and only related to the load level. Nevertheless, when the crack front presents some kind of curvature this is no longer true. Turning our sight to fatigue problem, it is well known the effect of the crack closure phenomenon in the crack growth rate. This phenomenon is clearly influenced by the effect of the plastic wake developed during the crack growth. In a previous study [16] was shown that the effect of the crack closure is more prominent close to surface than in the mid-plane and there is a ‘‘transition zone’’ in between. The influence of the load applied, the specimen thickness and the crack front curvature on the plastic zone near the crack front has been shown in this study, so it is expected that these parameters will have influence on the crack closure values achieved at the surface and in the size of the ‘‘transition zone’’. In particular, it has been shown the huge influence of the crack front curvature on the yielded area close to the surface, so some influence is expected. This is an interesting cross-related effect to be studied in future. Fatigue crack closure influences the crack growth shape at the same time that this shape changes crack closure behaviour along the thickness.
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