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69
УДК 539.421.2
Краткий обзор трехмерных эффектов вблизи фронта трещины
Z. He1, A. Kotousov1, F. Berto23, R. Branco4
1 Университет Аделаиды, Аделаида, SA 5005, Австралия 2 Падуанский университет, Виченца, 36100, Италия 3 Норвежский университет естественных и технических наук, Тронхейм, 7491, Норвегия 4 Политехнический институт Коимбры, Коимбра, 3045-601, Португалия
История и перспективы развития механики разрушения показывают необходимость дальнейшего глубокого изучения трехмерных задач о трещинах. Трехмерные решения проливают свет на явления разрушения и усталости и обосновывают применение классических сингулярных решений плоских теорий упругости, которые, по сути, являются приближенными теориями даже в случае точного решения определяющих уравнений данных теорий. В настоящей работе представлен краткий обзор последних результатов исследований трехмерных эффектов вблизи фронта трещины. В частности, обсуждаются связанные и сингулярные моды, которые не учитываются в большинстве экспериментальных и теоретических исследований. Описан новый экспериментальный метод оценки коэффициента интенсивности напряжений, возникающих в поле поперечных смещений на поверхности вблизи вершины трещины. Предложен новый закон подобия для хрупкого разрушения. Обзор в основном посвящен хрупкому разрушению. Пластичность и другие эффекты не обсуждаются.
Ключевые слова: особенность в угловой точке (вершине), связанные моды, влияние толщины пластины, коэффициент интенсивности напряжений, распространение усталостной трещины, линейная механика упругого разрушения
A brief review of recent three-dimensional studies of brittle fracture
Z. He1, A. Kotousov1, F. Berto2,3, and R. Branco4
1 School of Mechanical Engineering, The University of Adelaide, Adelaide, SA 5005, Australia 2 Department of Management and Engineering, University of Padova, Vicenza, 36100, Italy
3 Department of Engineering Design and Materials, Norwegian University of Science and Technology, Trondheim, 7491, Norway 4 Department of Mechanical Engineering, Polytechnic Institute of Coimbra, Coimbra, 3045-601, Portugal
3D crack problems are area where a further intensive research is required. 3D solutions can shed more light on fracture and fatigue phenomena, provide a more accurate evaluation of strength and fatigue life or justify the application of the classical solutions of plane theories of elasticity. These, in fact, are approximate theories even when the governing equations of these theories are solved exactly. The current paper aims to provide a brief summary of the latest investigations of 3D effects associated with crack geometries and brittle fracture. In particular, we present an overview of the coupled fracture modes and 3D vertex singularities, which are currently largely ignored in experimental and theoretical studies. We also describe a recently developed experimental method for the evaluation of the stress intensity factors. This review is concerned with the situation generally described in the literature as small scale plasticity. Large plastic deformations and other non-linear effects are beyond the scope of this article.
Keywords: 3D corner (vertex) singularity, coupled modes, plate thickness effect, stress intensity factor, fatigue crack growth, linear elastic fracture mechanics
1. Introduction
Classical singular solutions of plane theory of elasticity form a foundation of contemporary fracture mechanics. These solutions are currently adopted in many design procedures, standards and failure assessment codes. Relative simplicity is the main reason behind the popularity of these solutions as the three-dimensional (3D) equations of elasti-
© He Z., Kotousov A., Berto F., Branco R., 2016
city are not very amenable to analytical approaches [1]. Likewise, two-dimensional (2D) computational models are currently dominating the numerical analysis of plate components weakened by cracks and other strong stress concentrators. This is because such models are far more computationally efficient, much easier to develop, implement and verify in comparison with the corresponding 3D counterparts. As
a result, the vast majority of experimental studies and analysis of fracture tests have also relied on the theoretical framework of the plane theory of elasticity [2-4].
Application of plane theory of elasticity to planar crack or angular corner geometries leads to the concept of stress singularity and stress intensity factor, which are the cornerstone of linear elastic fracture mechanics. In 1952, Williams [5] was the first who demonstrated that the in-plane elastic stress components at the tip of a sharp corner with opening angle y (Fig. 1, a) can be singular. In a special case of a crack geometry (the opening angle equals to 2n, Fig. 1, b), the in-plane stress components near the crack tip can be written in the power series as [5, 6]:
to
°aß = X [cy/2-1/aß (n, 9)], (1)
n=1
where r and 9 are polar coordinates as shown in Fig. 1, b, the origin of the coordinate system is located at the crack tip, and the crack edges coincide with the lines 9 = ±n, /aß are dimensionless functions and (a, ß) = (r, 9). With r approaching zero, the first term (n = 1) in Eq. (1) tends to infinity, while the higher order terms (n > 2) remain finite. Thus, the stress state in the close vicinity to the crack tip is dominated by the first term of the asymptotic expansion (1), which is proportional to Irwin [7, 8] rewrote
this singular term in the following form:
K
°aß=-[^gaß (9\ (2)
-\l2nr
where gaß are dimensionless functions of the angular position 9 only and K is the stress intensity factor, which depends on the geometry of the problem as well as loading conditions. Subscripts I, II and III are often used to denote the type of loading or fracture modes. A number of very powerful analytical and numerical methods for the calculation of the stress intensity factor have been developed in the past. Many texts and handbooks [9-11 ] provide explicit relationships of K for a wide range of geometries and loading conditions.
The fracture criterion in linear elastic fracture mechanics is formulated in terms of the stress intensity factor. It states that the crack initiation occurs when the stress intensity factor reaches a critical value Kc, which is called as fracture toughness [8]. The value of Kc has to be determin-
ed experimentally. Linear elastic fracture mechanics is one of the most successful developments of continuum mechanics and, nowadays, it is widely utilized for analysis of problems involving cracks, such as "brittle fracture, fatigue crack growth, stress-corrosion cracking, dynamic fracture and, creep- and visco-elastic fracture" [12]. The application of the classical linear elastic fracture mechanics to the failure assessment of cracked components is based on the following loose argument [13].
Consider a cracked plate made of reasonably brittle material. Three characteristic zones can be identified (Fig. 2): the process zone, where the material behavior is highly nonlinear; K-dominance region, where the linear elastic asymptotic stress field of the form Kris expected to be accurate; and the area of general stress state. Fracture normally starts in the process zone. When this process zone is fully encapsulated by the K-dominance region, then all nonlinear deformations and microphenomena in the process zone are controlled by the asymptotic field (2), where K is the only parameter affecting the stress state in the K-dominance region as well as in the process zone. Subsequently, the conditions for the failure initiation will be a function of the magnitude of the stress intensity factor only and nothing else [14, 15].
A direct consequence of the selection of K as a single fracture controlling parameter in linear elastic fracture mechanics is the scaling law of brittle fracture. In general, the question of scaling occupies a central position in physics and engineering. Without understanding how strength changes with the size of a specimen or a structure, it is virtually impossible to apply any test results obtained at the laboratory specimen scale to predict and avoid fractures at smaller or larger scales: the scales of actual engineering applications. Linear elastic fracture mechanics predicts that the strength increases/reduces as the inverse square-root of the scale factor. We illustrate this by considering a classical example: the scaled pair of single edge cracked specimens [16]. As shown in Fig. 3, the first specimen is a long strip of finite width w, weakened by an edge crack of length a, and subjected to a remote uniform stress aCr, which causes fracture of the first specimen; while the second specimen has width Aw, crack of length Xa, and subjected to applied stress
Fig. 1. Plate corner geometry characterized by a notch opening angle y (a); special case of the geometry at y = 2n (crack) (b)
Fig. 3. Scaled pair of single edge cracked specimens
a2r, which is critical for the second specimen. Thus, for both specimens at fracture, linear elastic fracture mechanics predicts:
Kl = aCr yfnaY (a/w) = Kc, (3a)
Kl = acr Y (Xa/ (Xw)) = Kc, (3b)
where Y is the dimensionless stress intensity magnification factor. Equating both sides of (3 a) and (3b) yields:
acr
(4)
where X is the scale factor (the ratio of in-plane sizes of the specimens) and a is the ratio of the critical stresses which cause fracture for the first and second specimens.
One would expect the scale law as given by Eq. (4) to agree very well with experimental studies as linear elastic fracture mechanics is one of the most developed engineering disciplines with well justified limitations. However, the linear elastic fracture mechanics predictions were found unsatisfactory through several hundreds of test results for appropriately brittle and quasi-brittle materials reported over the past fifty years [16]. In particular, significant variations in the apparent fracture toughness for the same material are often observed in experimental studies conducted with different specimen configurations, crack sizes, and loading conditions [17-21].
Recognizing the large discrepancies between the experimental results and theoretical predictions utilizing the 2D theories of elasticity, it is necessary to keep in mind the fact that these are approximate theories even when the governing equations of these theories are solved exactly [22]. The stress state near an actual crack tip is always three-dimensional, and an account for the 3D effects can shed more light on fracture phenomena including the scaling law, such as given by Eq. (4).
The 3D crack problems have been studied for a long time [23-31]. The first systematic study was carried out by Hartranft and Sih [32] who investigated the 3D stress states near the crack tip and highlighted the variation of the stress intensity factor across the plate thickness. Kong et al. [33]
showed that the tri-axial stress has a significant influence on mixed-mode fracture behavior of materials. Yuan and Brocks [34] demonstrated that the crack front in a plate specimen can be significantly affected by the out-of-plane stress components. A number of authors came up to a conclusion that the 3D stress state region is located within one-half of the plate thickness from the crack tip [35-38]. A retrospective review of theoretical, numerical and experimental investigations of 3D effects at cracks has been provided by Pook [39, 40] and Kotousov et al. [29].
This paper will focus on the recent studies of 3D crack problems. We start this review with a brief description of a numerical approach, which is normally utilized to conduct 3D analysis of crack problems. Further, we will highlight important features of the coupled modes and corner singularity, which cannot be recovered within the plane stress or plane strain theoretical framework of the theory of elasticity. Further, we provide an overview of some interesting results related to the displacement field near the front of a through-the-thickness crack in a linear elastic isotropic plate. These results were utilized to develop a new method for the evaluation of the stress intensity factors from the transverse displacements near a crack tip controlled by 3D stress states. Finally we present a new scaling law. It arises due to a mismatch in singular behavior between the primary (or global) fracture modes and local singular stress states near the crack front, in particular, the stress states associated with the 3D corner singularity.
2. Finite element method for 3D analysis of crack problems and fatigue crack growth modeling
3D analysis of crack problems usually utilizes the finite element method. One of the first comprehensive numerical studies was conducted by Nakamura and Parks [24, 25]. The boundary layer concept, see Fig. 4, was applied to develop a finite element model. It represented a circular cylinder encapsulating a straight crack front with a variable mesh density. The radius of the cylinder was five times the plate thickness, which is large enough to guarantee that (i) the boundary conditions applied at the outer surface of the cylinder correspond to those of a far-field 2D plane stress solution (e.g. Williams' solution), and (ii) the in-plane geometry of the finite element model does not affect the 3D stress state near the crack front. The straight crack front was located at the centre of the cylinder, as shown in Fig. 4. This modeling approach was verified and extended in many other articles devoted to the 3D aspects of fracture mechanics [41-47]. In particular, all these studies agree that the region of 3D effects is confined to approximately one-half of the plate thickness around the crack front, before converging into a 2D plane stress field at a radial distance equal roughly to the plate thickness.
Numerical techniques were also successfully employed to predict the crack front shape evolution and fatigue life
0
Fig. 4. Geometry and mesh division of the 3D finite element model (a); coordinate system with the origin at the centre of the disk (b)
[48-51]. For example, Branco and Antunes [52] developed a versatile automatic crack growth procedure to study the shape evolution of fatigue crack fronts in plates based on the 3D finite element adaptive remeshing technique developed by Smith and Cooper [53]. This procedure was improved and widely adopted in a number of subsequent papers [54-57]. The crack front shape evolution in this procedure is governed by the effective stress intensity factor, which takes into account the plasticity induced crack closure effect. The procedure comprises four main steps: (i) development of representative 3D finite element model; (ii) calculation of the effective stress intensity factors along the crack front; (iii) calculation of the crack front advance, normally by using Paris law; and (iv) determination of a new crack front. The process is repeated until a specified crack front shape or final fracture is achieved. The evolutions of fatigue crack front shapes obtained from these iterative numerical simulations were validated successfully against some experimental results. Among the latest developments we also point out a simplified numerical-analytical technique [43], which is capable to capture the general tendencies of the crack front shape evaluation and provide quite accurate qualitative and quantitative assessments of fatigue crack growth.
3. Variation of stress intensity factor across plate thickness
By using the finite element method, the variation of local (3D) stress intensity factors along the plate thickness direction for cracks under different mode (mode I, II, III or mixed mode) loadings was thoroughly investigated and reported in a number of papers [58-67]. Some representative examples will be given in the succeeding text.
In 2007, She and Guo [37] conducted a set of finite element studies of through-the-thickness cracks stressed in mode I and mode II. The local stress intensity factors were then evaluated by using the quarter point displacement extrapolation method [68]. Figure 5 schematically shows the variation of the normalized local stress intensity factor Kl( z)/K° along the crack front for models stressed in mode I for different Poisson's ratio of the material (v = = 0.15, 0.30, 0.50), where K° is the remote applied stress intensity factor, Kl(z) is the local stress intensity factor along the crack front direction (or z-direction). The numerical results presented in Fig. 5 show an increase above 2D solution at the centre line (z/h = 0), while the values fall towards the expected value of zero as the free surface is approached (z/h = 1). Similarly, Fig. 6 shows the normalized local stress intensity factor Kn (z)/K° along the thickness direction for model stressed in mode II. The dependence of Kjj( z) has an opposite tendency to mode I loading near the intersection of the crack front with the free surfaces, or at z/h ~ 1. It increases rapidly near the vertex. Noting that the influence of Poisson's ratio on the stress intensity factor for mode II is quite weak, and the results at different values of Poisson's ratio are not shown for the sake of the clarity of the figure.
The numerical results presented above (Figs. 5 and 6) as well as in many other papers have many implications in the evaluation of fatigue and fracture. These results, in particular, can explain that the initiation of fracture for symmetric loading (mode I) has a tendency to occur in the central part of the specimen. In the case of the antisymmetric loading (or mode II) the fracture initiation is normally shifted
Fig. 5. Normalized local stress intensity factor along the crack front for models stressed in mode I with different Poisson's ratio v = 0.15 (1), 0.30 (2), 0.50 (3)
Fig. 6. The dependence of the intensity of the coupled mode KO(z) (1-3) and the primary mode Kn(z) (4) across the plate thickness in the area near the crack front. v = 0.1 (1), 0.3 (2), 0.5 (3, 4)
Compression
Free surface
Tension
Fig. 7. Illustration of coupled fracture modes due to Poisson's effect and the redistribution of stresses close to the free surfaces for a crack subjected to shear (a) and anti-plane loading (b)
to the free surfaces, as reported in several experimental studies [69-71]. However, there were no systematic studies focusing on crack initiation spots for brittle materials.
4. Coupled modes
It was known for a long time that shear and anti-plane fracture modes are coupled [24]. It means that shear or antiplane loading also generates coupled 3D antiplane and shear singular stress states, respectively. These singular stress states (or coupled fracture modes) cannot be identified or investigated within the framework of plane (2D) theory of elasticity and are currently ignored or considered to be negligible in theoretical and experimental investigations as well as in standards and failure assessment codes of structural components [29].
The coupled mode generated due to shear loading was called the out-of-plane mode, or mode O (as suggested by Kotousov [72]), to distinguish it from the conventional (or global) fracture mode III. It was demonstrated that the out-of-plane mode is generated due to the 3D effects related to Poisson's ratio of the material. Figure 7, a [73, 74] illustrates the mechanism of formation of this singular mode for a particular configuration: sharp corner with a zero opening angle (crack) subjected to shear loading. Intuitively, such a loading will create compressive and tensile zones along two opposite free edges, and Poisson's effect will lead to a scissoring motion of the faces generating conditions similar to the tearing mode (mode III) in classical fracture mechanics but symmetric with respect to the midplane of the plate. It can be realized that the dependence of the strength of the singularity of the out-of-plane mode is exactly the same as for fracture mode III [72]. However, there are essential differences between these two singular modes. The out-of-plane singular mode is a localized fracture mode, which rapidly decays with the radial distance from the crack front. It is significantly affected by Poisson's ratio in contrast to fracture mode III. A different coupled mode is generated in the case of the presence of mode III loading. In this case the characteristic equation describing the degree of
singularity of the coupled mode is the same as for mode II. Several numerical studies have demonstrated that the intensity of this coupled mode is not significantly affected by Poisson's ratio. It is generated due to a redistribution of the transverse shear stresses at the free plate surfaces as illustrated in Fig. 7, b, which has the similar pattern regardless of the value of Poisson's ratio.
A large computational effort was recently directed to the characterization of these coupled modes for various structural components [38, 75]. Figure 6 shows the numerical results of the intensities of the out-of-plane mode at different Poisson's ratio (v = 0.1, 0.3, 0.5). These results have been obtained by Berto et al. [27] and reproduced for the validation purposes by He et al. [47] using a standard numerical procedure as described in Nakamura and Parks [24, 25] and She and Guo [37] for the analysis of 3D effects near the crack tip. From Fig. 6, it is clearly seen that the intensities of the out-of-plane mode are comparable with the intensities of the primary mode (mode II). The maximum values of this coupled mode are located in the vicinity of the free plate surfaces [76].
A comprehensive review devoted to coupling between the primary and local modes was recently published by Kotousov et al. [29]. The authors demonstrated that the coupled local modes have many interesting features, which are capable of advancing our understanding of scale effects, mixed-mode fracture, crack initiation and fatigue growth phenomena [4, 59-61, 69, 70, 77-79]. Among these features is the generation of the singular coupled modes by nonsingular (in 2D sense) shear or antiplane loading [27, 28], see, for example, Fig. 8 (which shows the numerical results of the intensities of the out-of-plane mode across the plate thickness generated by the nonsingular term b3). From the point of view of classical (2D) linear elastic fracture mechanics, the brittle or quasi-brittle crack propagation is impossible in these cases as the energy release rate is zero. However, from the 3D considerations, which are more realistic and accurate, the nonsingular terms of William's solution are capable to generate the singular coupled fracture modes, which, in turn, can initiate brittle fracture. In this situation a strong plate thickness effect on the intensity
Fig. 8. The dependence of the singular coupled mode KO(z) across the plate thickness for h = 1 mm. v = 0.1 (1), 0.3 (2), 0.5 (3)
of the coupled modes can be explicitly derived from dimensional considerations. For example, such considerations predict an increase of the intensity of the mode O with an increase of the plate thickness (Fig. 8). An opposite dependence can be derived for the coupled mode generated by the antiplane loading [4]. However, it is quite difficult to find the experimental confirmations of the above mentioned dependences and effects from the test results described in the literature, primary, because the previous experimental studies were largely based on the 2D framework [16] and all 3D effects were largely disregarded. It is, therefore, recognized that a future work is needed in order to verify the above identified 3D effects as well as their influence on brittle fracture.
5. 3D corner (vertex) singularity
Another characteristic feature of 3D solutions of elastic problems with cracks is presence of the three-dimensional (3D) corner singularity, sometimes called vertex singularity. The 3D corner singularity was first discovered in the late 1970s and early 1980s by Benthem and a number of other researchers, who employed a finite difference scheme and the eigenfunction expansion method to demonstrate that at the intersection/vertex of the crack front and a free surface, the square root singularity disappears, and at such a point, one has to deal with a 3D corner singularity [80-82]. As opposite to the in-plane singularities, whose strength is described by universal square root behavior, the strength of the corner singularity depends on Poisson's ratio as well as the intersection angle between the crack front and the free surface. For 3D corner singularity, the polar coordinates in Fig. 1 are replaced by spherical coordinates (r, 0, with origin at the corner point. The angle ^ is measured from the crack front. The stress intensity measure KX is utilized to characterize the corner point singularity, where X is a parameter defining the 3D corner singularity. Stresses are pro-
/X i 1
r and displacements to KX / r , where r is the distance from the corner point. The problem of a corner singularity is well documented in a number of articles in the last thirty years [83-85].
Because of the difficulties associated with the analytical modeling of the corner geometries, the main results in this area have been obtained using various numerical techniques, in particular, the finite element method [80-82, 86, 87]. The dependences of X as a function of Poisson's ratio v (when intersection angle P = 90°) are given in Fig. 9. Similar to the in-plane singularities, the stress states due to the 3D corner singularity can be uncoupled in the case of symmetric (mode I) and antisymmetric (mode II) loading. Another interesting observation is that Poisson's ratio affects the strength of the 3D singularity for symmetric and antisymmetric loading in a different way. In mode I, a higher Poisson's ratio leads to a lower strength of singular behavior, and the tendency is opposite for mode II.
Mode II -- —+ Crack front / Free surface
+ Bazant and Estenssoro [80] ^ o Benthem [82]
Fig. 9. Effect of Poisson's ratio on X, intersection angle P = 90°
The problem of 3D corner singularity was recently reexamined by He et al. [47]. A detailed numerical study of the 3D corner singularity was also conducted recently by Pook [39]. In particular, all these studies verified the earlier findings of Nakamura and Parks [25], i.e. the size of the 3D corner singularity field in a thin plate appears to be about 3-5 % of the thickness in spherical radius and depends weakly on the Poisson's ratio. Despite on the limited volume of the cracked plate affected by the 3D corner singularity this 3D effect can have a significant influence on fracture and fatigue phenomena. For example, Bazant and Estenssoro [80], Heyder et al. [88] and other researchers [89, 90] suggested that the presence of the corner singularity might lead to the deviation of the fatigue crack front from the orthogonal direction in the vicinity of the free surface. This phenomenon was observed in many fatigue crack growth studies for various materials [70, 71]. A comprehensive review of some of the early investigation on the 3D corner singularity was recently published by Pook [60].
5.1. Implications to fatigue crack growth and crack path
As is well known, a typical front of a fatigue crack propagating in a plate is usually curved and it tends to intersect a free surface at some angle [61]. From energy consideration this angle can be linked to a critical angle, at which the 3D corner singularity disappears and, as a result, the stress state at the corner point has the same singular behavior as the rest of the crack front, i.e. the usual square root singular behavior [60]. This critical angle depends on the mode of loading and Poisson's ratio of the material and is not affected by other elastic constants [80]. However, in the absence of exact analytical solutions, only limited data for the critical intersection angle appear to be available. These are based on either experimental data [91, 92] or numerical calculations [80-82]. Figure 10 shows the dependences of the critical intersection angle Pc as a function of Poisson's ratio v (when X = 0.5). It is clear from the figure that the critical intersection angle increases with Poisson's ratio for mode I case and decreases for mode II case.
The critical angle Pc in the case of mode II can be approximated by the following equation [60]:
Fig. 10. Effect of Poisson's ratio on intersection angle, X = 0.5
ßc =tan
-1
1 -v
(5)
From Fig. 10 it can be see that this expression provides a reasonably close fit to the numerical data presented in [80]. A similar expression
ßc =tan
-1
(6)
provides a fit to the mode I data plotted in Fig. 10. Experimental evidences of the effect of the corner singular stress on the crack front shape are shown in Fig. 11 [70, 71]. It is clear from this figure that the critical angle is greater than 90° for specimens subjected to fatigue loading in mode I, and less than 90° in the case of mode II.
A number of careful and sophisticated numerical studies (using the finite element or boundary element methods) in the past were devoted to the evaluation of fatigue crack front shapes attracting the critical angle consideration, which have had some limited success [77, 78, 93, 94]. For example, Sevcik et al. [48] estimated the fatigue crack front shape in a semi-infinite plane media by assuming constant stress singularity exponent (k = 0.5) throughout the curved
crack front. The numerical predictions demonstrated a good agreement with the experimental observations [92]. However, some further experimental studies showed that the critical angle concept is valid for rather brittle materials only, such as glasses or brittle plastics (e.g. PMMA) or in the case of small scale plasticity. Generally speaking, this concept is not supported by experimental evidences when the material behavior and loading conditions result into large (in comparison with the plate thickness) plastic deformations [95, 96]. In these situations the stress state in the vicinity of the vertex point is controlled by plasticity phenomena [97-100]. The latter can explain the differences in the crack front shapes in brittle and plastic materials subjected to the same loading conditions. For brittle materials, the effect of the 3D corner singularity is significant, thus, the crack front is controlled by the stress intensity factor range and the critical angle identifies the crack front shape near the free surfaces. While, for sufficiently plastic materials, the influence of the 3D corner singularity disappears, instead, it is the plastic crack closure phenomena and the effective stress intensity factor range control the fatigue crack growth rates and crack front shape evolution [52-55, 101].
6. Crack-tip displacement field near crack tip
The 3D displacement field near straight front of a through-the-thickness crack subjected to in-plane (mode I/II) loading was recently re-examined by He et al. [44-46], on the basis of Williams' solution in conjunction with a detailed 3D finite element analysis. Figure 12 presents the numerical results [45, 102] of the out-of-plane displacement uz (r, 0, z) along the crack bisector line (0 = 0) at the free surface (z = h) of a plate, together with outcomes of an experimental study [103] and an analytical solution obtained
Fig. 11. Fatigue crack growth of aluminum alloy 2024 loaded in mode I [71] (a), maraging steel loaded in mode II [70] (b). Reproduced with permission
Fig. 12. Normalized out-of-plane surface displacements along the crack direction as a function of x/h. In this figure E is Young's modulus, v is Poisson's ratio, K™ is the remotely applied stress intensity factor in mode I, and h is the half plate thickness
based on the first order plate theory [104]. The cracks in these studies were subjected to mode I loading and the material was considered to be linear-elastic and isotropic.
As it can be observed in Fig. 12, relatively far from the crack tip (r > h) all studies are in a good agreement and predict that the out-of-plane displacements follow the classical plane stress solution of the plane theory of elasticity. However, in the close vicinity of the crack tip the displacement field is different to the predictions by either the classical plane stress (uz ~ r~12) or plane strain (uz = 0) solutions. One important finding demonstrated by He et al. [45] is that the out-of-plane displacement very near the crack tip does not depend on the angular position 0 and is almost constant along the radial direction confined by five percent of the plate thickness (or r/h < 0.1, see Fig. 12). Therefore, it can be stated that the out-of-plane displacement field is a function of the out-of-plane coordinate only or, mathematically uz (r, 0, z) ~ uz (z). The present authors linked the near crack-tip field (r/h < 0.1) to the 3D corner singularity and suggested that, in the near crack-tip area, the out-of-plane displacement follows the vertex singular solutions, or uz ~ £XuI. Here £ (= 1 - z/h) is the through-the-thick-ness coordinate with the origin at the free surface (£ = 0 corresponds to the vertex point and £ = 1 located at the intersection of the midplane and crack front), and XuI is the strength of the corner singularity (or power of the displacement at the free surface), which is associated with mode I loading. For the whole range of Poisson's ratios, the normalized out-of-plane displacement Uz very near the crack front, as obtained from the numerical simulations, can be approximated by the following equation [45]:
U „ =
uz (r, 0, z )E
*-1.34V (1 -£X *).
(7)
KfVh
At the free surface (z = ±h), above equation becomes: uz (r, 0, ± h)E
U z =-
krVh
~ -1.34 v.
(8)
For mode II case, the normalized out-of-plane surface displacement Uz in the close vicinity of the crack front can be expressed in the form of [46]:
U,
uz (r, 0, ± h)E sin(0/ 2) K
A( v)
h
+ B (V)
h
(9)
where K^ is the remotely applied stress intensity factor in mode II, XO is the strength of the out-of-plane singularity (XO = 0.5 for sharp cracks), XuII is the strength of corner singularity (see Fig. 9), A and B are fitting coefficients, which can be found from the following equations:
XuII(v) = 0.4v2 - 0.46V + 0.497, a(v ) = -3.125V2 + 0.874V -1.355, B (v ) = 2.75V2 + 0.529V +1.34.
(10)
7. Application to experimental evaluation of stress intensity factor
Almost all current experimental methods of determining stress intensity factors are based on the assumption that the state of stress near the crack tip is plane stress or plane strain [105-108]. Therefore, these methods rely on strain or displacement measurements made outside the near crack tip region affected by the 3D effects. In this section, we present an advanced experimental technique for the evaluation of stress intensity factors from the measurements of the out-of-plane (transverse) displacements in the surface area controlled by 3D effects. The evaluation of stress intensity factors is possible when the process zone is sufficiently small, and the displacement field generated by the 3D effects is fully encapsulated by the K-dominance region.
The proposed theoretical equations (8)-(10) established a link between the applied (or remote) stress intensity factors and the out-of-plane surface displacement in the area very close to the vertex point. This link forms a conceptual basis for the new experimental technique for the evaluation of the stress intensity factors using the out-of-plane displacement very near the vertex point, specifically within the surface area encapsulating the crack tip with radius less than five percent of the total plate thickness.
To validate this technique, the authors of the current paper conducted a careful experimental study, which utilized an edge cracked semicircular specimen (made of PMMA) subjected to symmetric/antisymmetric loading (Fig. 13). The digital image correlation technique [109] was employed to extract the out-of-plane surface displacement around the crack tip region of the tested samples. The details of this procedure are not provided here but can found in [45, 46].
Figures 14, a and 14, b show the contour plot of the out-of-plane surface displacement around the crack tip for specimens loaded in mode I and mode II, respectively. It can be observed from Fig. 14, a that there indeed exists a small
Fig. 13. Loading set up for specimens in pure mode I (a) and pure mode II conditions (b)
circular area around the crack tip, where the measured transverse surface displacements are almost constant as predicted by theoretical equation (8).
The experimental results were demonstrated a good agreement with the theoretically derived equations (8)-(10). The outcomes of the experimental study confirmed that the developed method has a sufficient accuracy suitable for many engineering applications. Moreover, it was found that the out-of-plane displacement field near the crack tip is not affected by the higher order nonsingular terms of the crack tip asymptotic expansion (1). This implies that the accuracy of this new developed method could be much better in comparison with the traditional experimental approaches, which always rely on a fitting of the 2D series expansion of displacements or strain near the crack tip to the experimental data.
8. Plate thickness effect
Numerous numerical [110-112], analytical [22, 66, 113] and experimental [88, 103] studies have suggested that the region of the 3D effects is confined within a close proximity to the crack tip and propagate in the plane directions to approximately one-half of the plate thickness. In this section we consider a plane problem of a sharp crack in a plate of thickness 2h when the zone of the 3D stress state is fully
encapsulated by K-dominance region, in which the stress state can be represented by the leading term of classical asymptotic expansion (William's solution). We summarize some recent investigations on the influence of the plate thickness on the intensity of various 3D singular modes. Let us first consider the in-plane 3D singular modes.
8.1. In-plane singularities
Similar to the common definition of the stress intensity factors for plane problems of elasticity, see Eq. (2), we introduce the local stress intensity factors for in-plane singularities as
KI(z) = lim av (x, 0, z)(2nx)1-kl (11)
for mode I loading, and
Kjj(z) = lim t (x, 0, z)(2nx)1-kl1 (12)
for mode II loading;
ki=kn=2
for crack geometry, and these can be different for sharp notch being functions of the notch opening angle.
Due to the linearity of the elastic problems, the distribution of the local stress intensity factors must be proportional to the remotely applied stress intensity factors Kapp and K5pp. Because the asymptotic behavior of the applied
98
and local in-plane stress fields is the same as described above, the only possible form of the function describing the intensity of the local in-plane mode is
Kj( z) = Kapp F ^ Z-,v] and (13)
Kn(z) = KTF,, [z, v), (14)
where FI and Fn are dimensionless functions, which depend on the position z/h and Poisson's ratio v only. It is important to highlight that there is no effect of the plate thickness on the shape of the stress intensity factor distribution along the crack front for this local singular mode. A detailed description of FI and Fn has been given in Sect. 3 of the current paper (see Figs. 5 and 6) and will not be repeated here.
8.2. Out-of-plane singularities
Next, we consider the out-of-plane singular mode, or mode O, which is a local singular mode, as described previously. In the beginning, we formally introduce the local stress intensity factor for this mode. Taking into account the similarity in the stress state between the KO mode and the conventional mode III, we define this mode in a similar fashion to fracture mode III as
KO(z) = lim tvz(x, 0, z)(2nx)1-Xo. (15)
Based on the same dimensionless considerations, the only possible form of the function describing the intensity of the out-of-plane mode for the problem under consideration is
Ko( z) = Kjjpp hXjI -Xo Fo ^ z, vj. (16)
Noting that for the considered case of the through-the-thickness crack, Xn = XO = 1/2, the dependence from the plate thickness disappears. However, this is obviously not the case for the problems involving sharp notches, as Xjj > XO for all opening angles greater than zero. In these cases, the intensity of the KO mode increases boundlessly with an increase of the plate thickness leading to a strong scale effect as described in [1, 4].
8.3. 3D corner singularity
Finally, we consider scaling of the stress state associated with the 3D corner singularity, which is also a local singular mode. Similar to the previous section, we start with the definition of the stress intensity factor for the corner singular mode. This singular mode is unique and different from the above considered in-plane and out-of-plane modes because the corner singularity is concentrated in a single (vertex) point. Therefore, we define the intensity of the corner singularity as follows:
Kcj = lim ctv (x, 0, ± h)(2nx)1-Xcj (17)
for mode I loading, and
Kcjj = lim txv (x, 0, ± h)(2rcx)1-Xcn (18)
for mode II loading.
Analogous to the previously considered cases, the di-mensionless considerations lead to the following dependences as:
KCI = Kjapp h12-Xcj FCI(v) and (19)
Ken = KjIpp h12-Xcn Fcn(v). (20)
From Fig. 9, it follows that 1/2 < XCI and 1/2 > XCII, which implies that the intensity of the corner singularity KCI increases boundlessly with a decrease of the plate thickness. The intensity of the corner singularity in mode II KCII has the same asymptotic behavior if the plate thickness increases. Equations (13), (14), (16), (19) and (20) can be considered as a new scaling law for the intensities of 3D corner singularities. These stress states and plate thickness can influence the initiation conditions of brittle fracture or fatigue crack growth rates. However, these effects are yet to be confirmed experimentally.
9. Conclusion
In this paper we attempted to provide a brief review of the latest outcomes of an intensive research directed on the investigation of 3D crack problems. This review is largely focused on brittle fracture, and in a less extent, on fatigue. The aspects of 3D crack problems considered in this paper include the variation of stress intensity factor along the crack front, coupled and corner singularities, scaling law. The latest developments in this area also involve a detailed investigation of the displacement field near the crack front of a through-the-thickness crack in an elastic plate. It is demonstrated that the out-of-plane displacements at the vertex point are finite and cannot be described within the plane stress of plane strain theories of elasticity.
Based on this investigation we suggested a new experimental technique for the evaluation of the applied stress intensity factor from the measurements of the out-of-plane (transverse) displacements in the surface area controlled by 3D effects. A special attention was given to 3D corner singularity effect. Despite that the volume controlled by the corner singularity is quite small; the effect on fatigue crack growth can be quite significant. Finally, it is believed that further investigations of 3D crack problems will lead to more accurate procedures of the evaluation of fracture and fatigue, which currently still represents a significant challenge for designers and engineers.
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Zhuang He, Prof., The University of Adelaide, Australia, [email protected] Andrei G. Kotousov, The University of Adelaide, Australia, [email protected] Filippo Berto, Prof., University of Padova, Italy, [email protected] R. Branco, Polytechnic Institute of Coimbra, Portugal, [email protected]
