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Computation of stress intensity factor in cracked plates under bending in static and fatigue by a hybrid method
International Journalof Fatigue
International Journal of Fatigue 29 (2007) 1904–1912
www.elsevier.com/locate/ijfatigue
Computation of stress intensity factor in cracked plates under bending in static and fatigue by a hybrid method B.K. Hachi a, S. Rechak b, M. Haboussi
c,*
, M. Taghite c, G. Maurice
c
a
c
Department of Electromechanical Engineering, C.U. de Djelfa, BP 3117 Ain-Cheih, 17000 Djelfa, Algeria b LGMD, Department of Mechanical Engineering, E.N.P., BP 182 Harrach 16200, Algiers, Algeria LEMTA, Nancy University, CNRS, 2 avenue de la Foreˆt de Haye BP 160 F-54504, Vandoeuvre Cedex, France Received 29 August 2006; received in revised form 17 April 2007; accepted 22 April 2007 Available online 3 May 2007
Abstract A hybrid weight-function technique is presented. It consists of dividing an elliptical crack into two zones, then using the appropriate weight function in the area where it is more efficient. The proportion between zones is determined by optimizing two crack parameters (axis ratio and curvature radius). Stress intensity factors for plates containing elliptical and semi-elliptical cracks are hence computed by a self developed computer code. Static and fatigue loadings of bending are considered. The results found by the present approach are in good correlation with the analytical solutions (when available) as well as with those of other researchers. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Hybridization; Weight-function; Stress intensity factor; Elliptical crack; Fatigue-crack-growth
1. Introduction In designing component structures, one must take into account the pre-existence of crack like flaws. The knowledge of the intrinsic parameter ‘‘stress intensity factor (SIF)’’ is required for the prediction of not only the fracture strengths but also for the crack-growth rates. The determination of SIF is a complex problem where the analytical solutions are not always available. Thus, to solve this kind of problem researchers attempt to use numerical approaches such as weight-function methods which are now mostly adopted because of their simplicity and their effectiveness. The development of weight-functions in fracture mechanics started with the work of Bueckner [1], based on the formulation by the Green’s function, for a semi-infinite crack in an infinite medium. The investigation of the weight-functions on the one hand and the evaluation of *
Corresponding author. Tel.: +33 3 83595530; fax: +33 3 83595551. E-mail address: [email protected] (M. Haboussi). 0142-1123/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2007.04.005
the energy balance formula of Rice [2] on the other hand, allowed the extension of the use of the weight-functions by several authors such as Oore and Burns [3] and Bortmann and Banks-Sills [4]. In 1986, Gao and Rice [5] introduced the study of the stability of the rectilinear form of a semi-infinite crack front during its coplanar propagation from which result the values of stress intensity factor (SIF) along the crack front. Recently, Sun and Wang [6] gave in-depth interpretations of the energy release rate of the crack front. Other investigations followed related especially to the crack shape (ellipse, half of ellipse, quarter of ellipse, rectangle, . . .) as well as to the mode of rupture (modes I, II, III or mixed) and to the large domain of application (elastoplastic, elastodynamic, thermoelastic . . .). Among those works, one can chronologically mention Fett et al. [7], Vainshtok [8], Dominguez and Gallego [9], Rooke and Aliabadi [10], Orynyak and Borodii [11], Zheng et al. [12], Kiciak et al. [13], Pommier et al. [14], Krasowsky et al. [15], Hachi et al. [16] and Christopher et al. [17]. The principle of the weight-function technique consists of employing one or more known solutions (known as reference solutions) of a particular case in order to find the
B.K. Hachi et al. / International Journal of Fatigue 29 (2007) 1904–1912
solution for the general case. The reference solution generally comes from the analytical results (exact). But in some cases, the absence of such results obliges researchers to use approximate solutions which could be already existing weight-functions. In this paper, a method improving the calculation of SIF in mode I for elliptical and semi-elliptical cracks is developed by means of hybridization of two weight-functions and coupling with the point weight function method (PWFM). This method is applied to plate under static and fatigue bending loads. In the first section, a detailed presentation of the hybridization approach is proposed followed by the numerical implementation. In this latter, a special emphasis is made on the treatment of singularities. The coupling of the proposed approach of hybridization with PWFM is then presented in the following section. The resulting formalism is applied for the modelling of elliptical and semi-elliptical cracks in plates. Concluding remarks will end this paper. This work is an extension of an already published study of crack modelling by hybridization performed for infinite body and pressurized cylinders [18]. 2. Presentation of the hybridization technique The solution of the SIF in mode I using the weight-function technique is given by the general form [3]: Z K IQ0 ¼ W QQ0 qðQÞ: dS ð1Þ S
where KIQ 0 is the stress intensity factor in mode I at the Q 0 point of the crack front. WQQ 0 is the weight-function related to the problem, it corresponds to K IQ0 when a unit concentrated and symmetrical force q(Q) is applied to the arbitrary Q point of the crack. S is the crack area. Our study is based on the hybridization of two weightfunctions deduced from a Green’s function formulation. The first one is developed by Oore and Burns [3] to model any closed shape of a crack in an infinite body, including elliptical cracks. Its expression is as follows: pffiffiffi 2 W QQ0 ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ R p:lQQ0 : C ðqdCÞ2
1905
such as PðhÞ ¼ ðsin2 h þ a4 cos2 hÞ=ðsin2 h þ a2 cos2 hÞ and a = a/b. The principle of hybridization is to divide, as shown in Fig. 1, the elliptical crack into two zones, an internal zone I (ellipse in grey) and an external zone II (in white), then to use each of the two weight functions in the area where it is more efficient. The two zones are defined by the following relations: ( 2 y 2 x þ a0 6 1 ðzone IÞ b0 ð4Þ x 2 y 2 þ > 1 ðzone IIÞ 0 0 a b where a 0 and b 0 are such as a 0 /a = b 0 /b = b and b 2 [0,1], b being the proportion between the two zones and a, b are the axes of ellipse (Fig. 1). The weight-function of Eq. (3) is intended exclusively for cracks of elliptical form. Nevertheless, it presents an additional singularity (1 r/R)1/2 compared to Eq. (2). This makes Eq. (3) less efficient in the vicinity of the crack front (r ! R). This argument leads us to make the following choice: W QQ0 ¼ W QQ0 of Eq: ð3Þ if Q 2 zone I W QQ0 ¼ W QQ0 of Eq: ð2Þ if Q 2 zone II
ð5aÞ ð5bÞ
The hybridization verifies well the two geometrical limits of an elliptical crack, a ! 0 and a ! 1, as long as the two functions (2) and (3) satisfy the two following conditions: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 r 2 0 W QQ0 ! W QQ ðpenny-shapedÞ ¼ pffiffiffiffiffiffi 2 ð6Þ p pRlQQ0 a!1 pffiffiffiffiffiffi 2d 0 0 ð7Þ W QQ ! W QQ ðstraightÞ ¼ pffiffiffi 2 p plQQ0 a!0 where d is the shortest distance between the crack front and the arbitrary Q point. Eqs. (6) and (7) present the weight-functions of, respectively, a penny-shaped crack and a straight line crack in an infinite and a semi-infinite body [15]. It remains to determine the appropriate proportion b between two zones I and II. It is important to point out that the function (2) of Oore and Burns has a form which is very similar to the functions (6) and (7) for, respectively, a straight line crack and a penny-shaped crack. Moreover,
Q
The second one is developed by Krasowsky et al. [15] to model elliptical cracks in an infinite body. Its expression is as follows: 2:P1=4 ðhÞ W QQ0 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 pa 1 Rr 2ðuÞ :l2QQ0 : C ðuÞ
ð3Þ dC ðqQ Þ2
In expressions (2) and (3), r and u are the polar coordinates of an arbitrary point Q. lQQ 0 is the distance between the Q 0 point and the arbitrary Q point. C is the curve of the ellipse (the crack front), and qQ is the distance between the Q point and the elementary segment dC; P is a function
Fig. 1. Subdivision of the elliptical crack in two zones and its geometrical parameters.
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in their study, Oore and Burns particularly used the conditions (6) and (7) to deduce their final expression (2) (see pages 203 and 204 of Ref. [3]). This is not the case for the function (3) of Krasowsky et al. [15] which is deduced from the elementary variation of the Rice’s energy balance during an elementary propagation of the crack. By construction, the weight-function (2) is preferable to the function (3) in the two following cases: When the crack front is close to a circle (a ! 1); this is confirmed numerically as the graph of Fig. 2 shows it for the parameters f1 and f2 defined by R R ðW QQ0 ÞEq: ð2Þ dS ðW QQ0 ÞEq: ð3Þ dS S S and f 2 ¼ R : f1 ¼ R ðW QQ0 ÞEq: ð6Þ dS ðW QQ0 ÞEq: ð6Þ dS S S This graph shows clearly that the function (2) converges more quickly towards the weight-function of a pennyshaped crack compared to the function (3). When the crack front is close to a straight line (low values of a with value of h far from zero). In fact, these two cases correspond to situations where the variation of the curvature radius Rc of the crack front is weak, excluding a very narrow zone corresponding of very weak value of Rc (h close to zero with low values of a). In this case and for a relatively high variation of the curvature radius of ellipse, the weight-function (3) of Krasowsky et al. is more adapted. This is confirmed by the presence via the function P(h) of curvature radius Rc ¼ aa P3=2 ðhÞ in the expression of the function (3). Consequently, more the radius of curvature is relatively weak or its derivative (spatial gradient of the radius of curvature) is high, more the zone I should increase with respect to the zone II and vice versa. Taking into account all these considerations, we propose the relative parameter expressed by ðb minðRc ; aÞÞ b1 ¼ b
for the representation of the influence of the curvature radius compared to the large axis of the ellipse b, and the following one oRc oRc oh min ð9Þ b2 ¼ oRcoh c oR oh max oh min to represent the influence of the gradient of the curvature rac dius for a given a. In this relation oR is the partial derivative oh with respect to the angular position of the point Q 0 , its maximum and minimum values are calculated by ‘‘sweeping’’ completely the contour of the ellipse for a given a. The computation of the partial derivative is achieved numerically. The proportion parameter b takes the value: b ¼ maxðb1 ; b2 Þ 3. Numerical implementation, meshing and singularities According to Eqs. (1) and (2) or Eq. (3) two integrals are to be considered; a surface integral and a curvilinear integral (contour integral) of the crack front (C) involved in the weight functions (2) and (3). 3.1. The surface integral A treatment technique of the singularity 1=l2QQ0 in the integral Eq. (1) has been efficiently employed by Krasowsky et al. [15]. It consists in surrounding the (Q 0 ) point by a small half-circle of R0 radius on which the integral is analytically evaluated using Eq. (7): Z pffiffiffiffiffiffi pffiffiffiffiffi 2d 1 0 pffiffiffi 2 qðQ0 Þ dS 0 1:21703 R0 qðQ0 Þ K SI ¼ ð11Þ ðS 0 Þ p p lQQ0 In this relation, S 0 represents the surface of the half-circle and R0 is a parameter which is equal to R0 ¼ ð1=30 1=20Þ minðRi ðQ0 Þ; aÞ
ð8Þ
ð10Þ
ð12Þ
in order to minimize the error generated by the curvature in Q 0 (Ri is the radius of curvature of the crack front). In order to complete the calculation of the SIF which is 0 00 now split up in to K SI ¼ K SI þ K SI , the remaining SIF on (S00 ) is evaluated by the hybridization method. The meshing of (S00 ) is generated by drawing first, concentric half-circles centred at Q 0 of progressively increasing radii R0, r0, r1, r2, r3, . . . then, half-lines with Q 0 as origin, hence dividing the half-plane of the ellipse into n portions, approximately 60 portions in practice as shown in Fig. 3a. 3.2. The contour integral
Fig. 2. Comparison of weight-functions (2) and (3) with the penny-shaped one in the vicinity of a = 1.
R When Q2 is very close to the crack front, the integral ½dC=ðqQ Þ becomes singular and its numerical calculaC tion becomes delicate. To deal with this singularity, we first ignore a very narrow band of thickness D near the contour. This narrow band D = ca with c = 1/300 is shown in Fig. 3b. The effective ellipse surface becomes:
B.K. Hachi et al. / International Journal of Fatigue 29 (2007) 1904–1912
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Fig. 3. Discretization of the elliptical crack: (a) Discretization of the crack area. (b) Discretization of the crack front. 2
2
ðx=bÞ þ ðy=aÞ 6 ð1 D=RðhÞÞ
2
ð13Þ
In their work, Krasowsky et al. [15] consider a thickness D as variable (D(u) = c cos(T(u)) min (Ri(u),a), T is the angle between (OQ 0 ) and the normal of C at Q 0 (see Fig. 1)), making the automatic meshing of the crack very complex without gaining in the exactitude of the computation. The error is approximately estimated as R c pffiffiffi R 1 pffiffiffi ð 0 x dx= 0 x dxÞ ¼ c3=2 for a changing D whereas it is equal to (2c)3/2 for a constant D. At any rate, the shortest distance between two successive arcs of circles must not be lower than the thickness D. Therefore, we must verify at each calculation step the condition below: D ¼ ðriþ1 ri Þmin ¼ r0 R0
ð14Þ
By combining Eq. (14) with Eq. (12), one can get the progression of the ri radius as follows: m ¼ ðriþ1 ri Þ=ri ¼ ðr0 R0 Þ=R0 1=15 1=10
ð15Þ
The numerical calculation of the contour integral requires the discretization of the crack front (C) with a finite number of points delimiting straight segments of length dC (Fig. 3b). The error of linearization of dC is obtained by the use of Taylor’s series expansion of sin(dw/2) in the vicinity of zero: _
e¼
d C dC _
dC
ðdwÞ 24
2
ð16Þ
h p i wðiÞ ¼ arctan a tan i 2N
ð20Þ
To insure that the error e remains weak, Krasowsky et al. [15] imposed the following condition (Fig. 4): l1 ðuÞ=D 6 l ¼ l1 =l0 2 ½0:1; 0:2
ð21Þ
When coupling this condition with Eqs. (16), (17) and (21) we obtain the lower limit of N: pffiffiffiffiffiffiffiffi N P p=ð2 8lcÞ ð22Þ As long as the Q point is far from (C), the integral could be transformed into the summation: Z 4N dC X ðdCÞi ¼ ð23Þ 2 q q2iQ C Q i¼1 When Q is very close to the kth segment, the integral must analytically be computed on this segment in order to avoid any risk of singularity. It is equal to: Z c2 Z dC dc 2 2 2 q ðdCÞk c1 c þ d Q 1 c2 c1 ð24Þ ¼ arctan arctan d d d where the parameters c1, c2 and d are shown in Fig. 4. Eq. (23) thus becomes: Z Z k1 4N dC X ðdCÞi dC X ðdCÞi ¼ þ þ ð25Þ 2 2 q qiQ q2iQ C qQ ðdCÞk Q i¼1 i¼kþ1
dw is the angular opening of the dC. The opening dw decreases with the reduction of the curvature radius. To render uniform the error e on the entire contour (C), we take its average value (corresponding to a circular contour): 1 p 2 1 e¼ ¼ ðdwr Þ2 ð17Þ 24 2N 24 where N is the number of points on the quarter of contour (C) and wr is the reduced angle defined as (Fig. 3b): w ¼ arctan½a tanðwr Þ
ð18Þ
The reduced angle for the ith point is then given by: p ð19Þ wr ðiÞ ¼ iðdwÞ ¼ i 2N By substitution of Eq. (19) in Eq. (18), the angular position of the ith point becomes:
Fig. 4. Zoom of the elementary subdivision of the crack front.
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4. The hybrid method coupling with the point weight-function method (PWFM) The PWFM developed by Orynyak and Borodii [11] consists in the development of the appropriate modified weight-function for semi-elliptical crack based on the elliptical crack one in order to take into account the free edge effect. The weight-function is then split up into asymptotic and corrective parts [11]: W QQ0 ¼ W
A QQ0
þW
C QQ0
ð26Þ
C where W A QQ0 is the asymptotic part and W QQ0 is the corrective one. In the following, we give compact forms of C A WA QQ0 and W QQ0 . For the first function W QQ0 , we propose: ! " # l2QQ0 y Q0 yQ ðEllipseÞ A W QQ0 ¼ W QQ0 1 ð27Þ 1þ 2 1 a a lQ Q0 x
where yQ and yQ 0 are the projections of Q and Q 0 on the (Oy) axis of the ellipse, Qx is the symmetrical point of Q with respect to (Ox) axis of the ellipse, by considering (Oxz) as the free edge plan (see Fig. 1). The weight-function ðEllipseÞ W QQ0 is given by the Eq. (5). For the second function W CQQ0 , the expression is: W
C QQ0
¼W
A QQ0
r minðlQQ0 ; a2 Þ 1 Dðh; aÞ ¼ W CQQ0 Dðh; aÞ R a2 ð28Þ 2
where D(h, a) is the only unknown function to be determined. It is related to the geometry of the crack and the angular position of the point of calculation Q 0 . In order to determine this function, we define the dimensionless SIF, G(h, a), as follows: EðkÞ Gðh; aÞ ¼ pffiffiffiffiffiffi 1=4 K I ðh; aÞ paP ðhÞ
ð29Þ
where E(k) is an elliptical integral of the second kind and pffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ 1 a2 . By using the Eq. (1) giving the integral expression of KI and Eq. (26), we obtain: Gðh; aÞ ¼ I A ðh; aÞ þ I C ðh; aÞDðh; aÞ
ð30Þ A
Based on Eq. (28), the dimensionless integrals I and ICcan be expressed as: Z EðkÞ I A ðh; aÞ ¼ pffiffiffiffiffiffi 1=4 rðx; yÞW AQQ0 dS and I C ðh; aÞ paP ðhÞ ðSÞ Z EðkÞ ¼ pffiffiffiffiffiffi 1=4 rðx; yÞW CQQ0 dS ð31Þ paP ðhÞ ðSÞ If the solution G0(h, a) is known for a particular loading r0 (this solution is generally denoted as a reference one), then for any other stress distribution r(x, y), the repeated usage of Eq. (30) allows one to deduce the D(h, a) value and to determine the general solution G(h, a):
G0 ðh; aÞ I A 0 ðh; aÞ Gðh; aÞ ¼ I ðh; aÞ þ I ðh; aÞ I C0 ðh; aÞ A
C
ð32Þ
More development details of the coupling between our hybrid technique and the PWFM method are presented in [19]. 5. Results and discussions For the modeling of the elliptical and the semi-elliptical cracks by the hybrid weight-function technique, we developed a computer operational software with graphic interface named HWFun by using the C++ object-oriented language. The following are practical applications based on the HWFun code for plate containing cracks under static and fatigue bending. Because of the complexity of the analytical developments concerning the evaluation of the FIC, the exact solutions are available only for the embedded elliptical cracks under particular loadings (see Ref. [20]). Whereas for the surface cracks, the existence of such analytical developments is quasi impossible. Consequently, the tests of validation rest on the comparison of the results obtained by the present approach with the analytical solution (in the case of embedded elliptical cracks) like with the results of other studies in the event of its unavailability (for the case of the surface cracks). It is important to point out that the present method was successfully tested for an embedded crack in infinite body with a significant reduction of the error concerning the SIF. This study is available in Refs. [18,19]. 5.1. Plate containing embedded transverse elliptical crack subjected to bending The problem of gaps and inclusions in plates subjected to bending is usually encountered in practice. Such defects influence the structure toughness by weakening their resistance and increasing their risk of cracking. In the calculation of resistance of such structures, these defects are often simulated by plane elliptical cracks, more or less flattened as shown in Fig. 5. The exact solution for this problem is given by the following relation [20]: rffiffiffiffiffiffi 1=4 pa 2 K I ¼ Cr cos h þ a2 sin2 h /
k 2 EðkÞ cos h 1 ð33Þ ð1 þ k 2 ÞEðkÞ ð1 k 2 ÞKðkÞ where r is the applied stress, / E(k)2 for small values of (r/re), re is the elastic limit of the material, K(k) is the elliptic integral of the first kind, C is a correction factor depending on geometrical parameters h, a, a/d and the material characteristics. For a Poisson’s ratio m = 0.3, Ref. [20] gives the value of C in abacuses form. For example, C 1 for the values of a/d 6 0.2, h 2 [ p/2, p/2] and a 2 [0.1,1.0]. The case of the plate subjected to pure bending, as presented in Fig. 5 with a/d 6 0.2, d 6 c and r(y) = r0(1 + (y/a)), is solved by the present approach via the developed code HWFun. The obtained results are reported
B.K. Hachi et al. / International Journal of Fatigue 29 (2007) 1904–1912
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Fig. 5. Embedded elliptical crack in an infinite plate under bending.
on the graphs of Fig. 6 where we can observepgood ffiffiffiffiffiffi correlations between the values of K I ¼ K I EðkÞ= ðr0 paP1=4 ðhÞÞ computed by our code HWFun and those of the analytical solution. Compared with Krasowsky’s weight-function results, we point out that the precision of our approach is still appreciable when varying the angle h from p/2 to p/2 with a = 0.6 especially in the interval hr 2 [ 30, + 30](hr = arctan (tan(h)/a)) (see Fig. 6a). At the dangerous point h = p/2 (maximum value of K I Þ, where the free edge effect is more accentuated, the hybrid approach is still more efficient than the Krasowsky’s weight function results for various values of a as shown in Fig. 6b. 5.2. Surface semi-elliptical crack in finite plate subjected to tension and bending The problem of surface semi-elliptical crack in finite plate subjected to tension and bending (Fig. 7) was treated
Fig. 7. Semi-elliptical surface crack in finite plate under tension and bending loads.
by Newman and Raju [21,22] based on the FEM and empirical relations, respectively. The same problem was treated also by other authors such as Isida et al. [23] based on modified body force method (MBFM) and Gonc¸alves and de-Castro [24] based on line spring model (LSM). We adopted for this example the dimensionless SIF EðkÞ ffiffiffi ffi where r0 = max(St,Sb), St and S b ¼ 3M F I ¼ Kr0Ip being, pa wt2 respectively, the tension and the bending stresses (Fig. 7). In this study, the semi-elliptical cracks are modelled by coupling the present hybrid weight-function method with the point weight-function method (PWFM) presented previously. The reference solution is the one given by Newman
Fig. 6. Dimensionless SIF for elliptical crack in an infinite plate under bending: (a) for a = 0.6 and h 2 [ p/2, + p/2], (b) for h = p/2 and a 2 [0.1, 1.0].
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and Raju [21] for a reference loading of tension (St = 1 and Sb = 0). In the following, we compute FI for pure bending (St = 0 and Sb = 1), the results are summarized in the form of graphs in Fig. 8. Various geometric configurations are considered, h 2 {0,p/2} for a 2 {0.2,0.4,0.6,1.0}. In absence of the analytical solution for this problem, the good correlation of the actual results with those of other researchers [21–24] is obvious for both the depth point A and the surface point B as shown in Fig. 8. The maximum difference between the two results is observed when the minor axis a is close to the thickness t of the plate (a/t ! 1). This can primarily be explained by the effect, due to the second free edge (y = t) of the finite plate (see Fig. 7), which is not taken into account in our approach. This shift is more visible when the point of calculation is very close to the free surface (for instance, the point A). This effect is more pronounced in the case of thin plates (for instance: a/b = 0.2 and a/t = 0.8) because of its double influence (double edge effects). It should be known that the bending
stress applied on the crack is given by Sb (1 2y/t) as a function of the y coordinate of an arbitrary point Q of the crack. Moreover, an effect of the a/t ratio is introduced in the SIF evaluation via the reference solution given by Newman and Raju [21] used in the relation (32). This latter is derived from the coupling between the hybridization approach and the PWFM method to treat semi-elliptical cracks. The cases of St = 1 and Sb = 1(tension + bending) and of a polynomial as well as an exponential stress distributions applied on the crack are also treated by the present approach for which the same remarks of the previous case (pure bending) concerning the quality of results were observed. 5.3. Prediction of fatigue-crack-growth for surface crack in plate under bending In this section, we intend to treat the problem of ductile fracture which is the case of fatigue-crack-growth. By this
Fig. 8. Dimensionless SIF at depth point and surface point for various geometric configurations with a pure bending (St = 0 and Sb = 1).
B.K. Hachi et al. / International Journal of Fatigue 29 (2007) 1904–1912
study, we want to demonstrate the flexibility and the efficiency of our approach. We limit ourselves to the work of Newman and Raju [21,22] for the sake of comparison. Hence, we study the example of fatigue-crack-growth of a finite plate containing semi-elliptical surface crack subjected to pure bending (St = 0 and Sb = 1). The most popular law in the computation of crack-growth rates is the Paris relationship given by the following: da ¼ C A ðDK A Þn dN db n ¼ C B ðDK B Þ dN
ð34Þ ð35Þ
where a and bare the ellipse axes, N, the number of the stress cycles, n, an exponent usually equal to 4 (ductile fracture) for a wide range of materials (The 2014-T651 aluminium alloy is one of them). CA and CB are the crack growth coefficients for, respectively, the points A and B of the crack. DK is the stress intensity factor range, defined as: DK ¼ ðK I Þmax ðK I Þmin max
min
n
Db is obtained by dividing the surface crack extension into a large number of increments. Fig. 9b and c show the comparison between experimental and predicted fatigue-crack-growth patterns of surface cracks in plates subjected to bending (St = 0 and Sb = 1) for both Newman and Raju approach [22] and the present one, respectively. It should be mentioned that all the experimental data points are obtained for the 2014-T651 aluminium alloy [25]. It is clear from Fig. 9c that the results obtained by the actual approach are in good correlation with the experiments. The reason for showing the two Fig. 9b and c is to demonstrate the total concordance between the predicted results and those of Newman and Raju [22]. The predicted results show that the cracks tend to approach a common propagation (see Fig. 9b and c) as pointed out by Corn [25]. This observation was also made by Newman and Raju [22]. 6. Conclusion
ð36Þ
ðK I Þ and ðK I Þ are the SIF corresponding, respectively, to the max-stress and the min-stress in stress cycles. In our case the max-stress is Sb and the min-stress is 0 (Fig. 9a). The relationship between CA and CB is assumed equal to [22]: C B ¼ ð0:9Þ C A
1911
ð37Þ
An increment Db at the surface point B induces an increment in-depth Da which we compute using the equation below: n n C A DK A DK A Da ¼ Db ¼ Db ð38Þ C B DK B 0:9DK B
In this work, a hybrid weight-function-based on Green’s function formulation is developed for modelling elliptical and semi-elliptical cracks in elastic and homogeneous bodies. In this modelling, a computation of stress intensity factors in mode I is performed. The idea of hybridization leads us to an optimization problem of two geometrical parameters (the ratio axes and curvature radius of ellipse). The development of the numerical procedures, the treatment of the various singularities, and the meshing implementation have been presented in full details. A computer code named HWFun has been developed and tested on various practical applications concerning plates. The results obtained show a clear reduction in the error
Fig. 9. Predicted fatigue-crack-growth patterns for plate under cyclic pure bending (St = 0 and Sb = 1): (a) The stress cycle applied to the plate. (b) Newman-Raju’s results compared with experimental data. (c) HWFun results compared with experimental data.
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for an embedded elliptical crack in plate under bending compared to classical weight-function-based approach. The semi-elliptical surface cracks in plates under bending were also analyzed by the present approach of hybridization. Compared to other results obtained by using various techniques, it shows a good accuracy. The present approach was also tested on the problem of fatiguecrack-growth in plate under bending. The obtained results confirm its efficiency and accuracy. The idea of hybridization undoubtedly opens horizons for the treatment of other complex problems in fracture mechanics such as mixedmode, interaction among cracks. References [1] Bueckner HF. A novel principle for the computation of stress intensity factors. Z Angew Meth 1970;50:529–46. [2] Rice JR. Some remarks on elastic crack-tip stress fields. Int J Solids Struct 1972;8:751–8. [3] Oore M, Burns DJ. Estimation of stress intensity factors for embedded irregular cracks subjected to arbitrary normal stress fields. Trans ASME J Pres Vess Technol 1980;102(2):202–11. [4] Bortmann Y, Banks-Sills L. An extended weight function method for 2D and 3D-problems. J Appl Mech 1983;50:907–9. [5] Gao H, Rice JR. Shear stress intensity factors for planar crack with slightly curved front. J Appl Mech 1986;53:774–8. [6] Sun CT, Wang CY. A new look at energy release rate in fracture mechanics. Int J Fract 2002;113:295–307. [7] Fett T, Mattheck C, Munz D. Approximate weight function for 2D and 3D-problems. Eng Anal Boundary Elements 1989;6:37–48. [8] Vainshtok VA. Application of the weight function method to solving multi-parametric three-dimensional fracture mechanics problems. Int J Fract 1991;47:201–12. [9] Dominguez J, Gallego F. Time domain boundary element method for dynamic stress intensity factor computations. Int J Numer Meth Eng 1992;33:635–47. [10] Rooke DP, Aliabadi MH. The use of fundamental fields to obtain weight functions for mixed-mode cracks. Int J Eng Sci 1994;32: 155–66. [11] Orynyak IV, Borodii MV. Point weight function method application for semi-elliptical mode I cracks. Int J Fract 1995;70:117–24.
[12] Zheng XJ, Kiciak A, Glinka G. Weight functions and stress intensity factors for internal surface semi-elliptical crack in thick-walled cylinder. Eng Fract Mech 1997;58(3):207–21. [13] Kiciak A, Glinka G, Eman M. Weight functions and stress intensity factors for corner quarter-elliptical crack in finite thickness plate subjected to in-plane loading. Eng Fract Mech 1998;60(2):221–38. [14] Pommier S, Sakae C, Murakami Y. An empirical stress intensity factor set of equations for a semi-elliptical crack in semi-infinite body subjected to a polynomial stress distribution. Int J Fatigue 1999;21:243–51. [15] Krasowsky AJ, Orynyak IV, Yu Gienko A. Approximate closed from weight function for an elliptical crack in an infinite body. Int J Fract 1999;99(1–2):117–30. [16] Hachi BK, Belkacemi Y, Rechak S. Computation of the stress intensity factor and stress analysis in 3D by using the Green’s function in displacement discontinuity method. Revue de me´canique applique´e et the´orique 2003;1(5):289–310 [in French]. [17] Christopher T, Sankarnarayanasamy K, Rao BN. Fracture behaviour of maraging steel tensile specimens and pressurized cylindrical vessels. Fatigue Fract Eng Mater Struct 2004;27:177–86. [18] Hachi BK, Rechak S, Belkacemi Y, Maurice G. Modelling of elliptical cracks in infinite body and in pressurized cylinder by a hybrid weight function approach. Int J Press Vess Piping 2005;82:917–24. [19] Hachi BK, Rechak S, Ouadjaout M, Haboussi M. Semi-elliptical cracks modelling by coupling of the hybrid weight function method with the point weight function method. In: Proceedings of CFM’2005, No. 887, U.T. Troyes-France. [20] Barthelemy B. Notions pratiques de la me´canique de la rupture. Paris: Eyrolles; 1980 [in French]. [21] Newman Jr JC, Raju IS. Analyses of surface cracks in finites plates under tension or bending loads. NASA TP-1578, Washington (DC): National Aeronautic and Space Administration; 1979. [22] Newman Jr JC, Raju IS. An empirical stress-intensity factor equation for the surface crack. Eng Fract Mech 1981;15(1–2):185–92. [23] Isida M, Noguchi H, Yoshida T. Tension and bending of finite plates with a semi-elliptical surface crack. Int J Fract 1984;26:157–88. [24] Gonc¸alves JPM, de-Castro PMST. Application of the line spring model to some geometries, and comparison with three-dimensional results. Int J Press Vess Piping 1999;76:551–60. [25] Corn DL. A study of cracking techniques for obtaining partial thickness cracks of pre-selected depths and shapes. Eng Fract Mech 1971;3(1):45–52.
International Journal of Fatigue 29 (2007) 1904–1912
www.elsevier.com/locate/ijfatigue
Computation of stress intensity factor in cracked plates under bending in static and fatigue by a hybrid method B.K. Hachi a, S. Rechak b, M. Haboussi
c,*
, M. Taghite c, G. Maurice
c
a
c
Department of Electromechanical Engineering, C.U. de Djelfa, BP 3117 Ain-Cheih, 17000 Djelfa, Algeria b LGMD, Department of Mechanical Engineering, E.N.P., BP 182 Harrach 16200, Algiers, Algeria LEMTA, Nancy University, CNRS, 2 avenue de la Foreˆt de Haye BP 160 F-54504, Vandoeuvre Cedex, France Received 29 August 2006; received in revised form 17 April 2007; accepted 22 April 2007 Available online 3 May 2007
Abstract A hybrid weight-function technique is presented. It consists of dividing an elliptical crack into two zones, then using the appropriate weight function in the area where it is more efficient. The proportion between zones is determined by optimizing two crack parameters (axis ratio and curvature radius). Stress intensity factors for plates containing elliptical and semi-elliptical cracks are hence computed by a self developed computer code. Static and fatigue loadings of bending are considered. The results found by the present approach are in good correlation with the analytical solutions (when available) as well as with those of other researchers. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Hybridization; Weight-function; Stress intensity factor; Elliptical crack; Fatigue-crack-growth
1. Introduction In designing component structures, one must take into account the pre-existence of crack like flaws. The knowledge of the intrinsic parameter ‘‘stress intensity factor (SIF)’’ is required for the prediction of not only the fracture strengths but also for the crack-growth rates. The determination of SIF is a complex problem where the analytical solutions are not always available. Thus, to solve this kind of problem researchers attempt to use numerical approaches such as weight-function methods which are now mostly adopted because of their simplicity and their effectiveness. The development of weight-functions in fracture mechanics started with the work of Bueckner [1], based on the formulation by the Green’s function, for a semi-infinite crack in an infinite medium. The investigation of the weight-functions on the one hand and the evaluation of *
Corresponding author. Tel.: +33 3 83595530; fax: +33 3 83595551. E-mail address: [email protected] (M. Haboussi). 0142-1123/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2007.04.005
the energy balance formula of Rice [2] on the other hand, allowed the extension of the use of the weight-functions by several authors such as Oore and Burns [3] and Bortmann and Banks-Sills [4]. In 1986, Gao and Rice [5] introduced the study of the stability of the rectilinear form of a semi-infinite crack front during its coplanar propagation from which result the values of stress intensity factor (SIF) along the crack front. Recently, Sun and Wang [6] gave in-depth interpretations of the energy release rate of the crack front. Other investigations followed related especially to the crack shape (ellipse, half of ellipse, quarter of ellipse, rectangle, . . .) as well as to the mode of rupture (modes I, II, III or mixed) and to the large domain of application (elastoplastic, elastodynamic, thermoelastic . . .). Among those works, one can chronologically mention Fett et al. [7], Vainshtok [8], Dominguez and Gallego [9], Rooke and Aliabadi [10], Orynyak and Borodii [11], Zheng et al. [12], Kiciak et al. [13], Pommier et al. [14], Krasowsky et al. [15], Hachi et al. [16] and Christopher et al. [17]. The principle of the weight-function technique consists of employing one or more known solutions (known as reference solutions) of a particular case in order to find the
B.K. Hachi et al. / International Journal of Fatigue 29 (2007) 1904–1912
solution for the general case. The reference solution generally comes from the analytical results (exact). But in some cases, the absence of such results obliges researchers to use approximate solutions which could be already existing weight-functions. In this paper, a method improving the calculation of SIF in mode I for elliptical and semi-elliptical cracks is developed by means of hybridization of two weight-functions and coupling with the point weight function method (PWFM). This method is applied to plate under static and fatigue bending loads. In the first section, a detailed presentation of the hybridization approach is proposed followed by the numerical implementation. In this latter, a special emphasis is made on the treatment of singularities. The coupling of the proposed approach of hybridization with PWFM is then presented in the following section. The resulting formalism is applied for the modelling of elliptical and semi-elliptical cracks in plates. Concluding remarks will end this paper. This work is an extension of an already published study of crack modelling by hybridization performed for infinite body and pressurized cylinders [18]. 2. Presentation of the hybridization technique The solution of the SIF in mode I using the weight-function technique is given by the general form [3]: Z K IQ0 ¼ W QQ0 qðQÞ: dS ð1Þ S
where KIQ 0 is the stress intensity factor in mode I at the Q 0 point of the crack front. WQQ 0 is the weight-function related to the problem, it corresponds to K IQ0 when a unit concentrated and symmetrical force q(Q) is applied to the arbitrary Q point of the crack. S is the crack area. Our study is based on the hybridization of two weightfunctions deduced from a Green’s function formulation. The first one is developed by Oore and Burns [3] to model any closed shape of a crack in an infinite body, including elliptical cracks. Its expression is as follows: pffiffiffi 2 W QQ0 ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ R p:lQQ0 : C ðqdCÞ2
1905
such as PðhÞ ¼ ðsin2 h þ a4 cos2 hÞ=ðsin2 h þ a2 cos2 hÞ and a = a/b. The principle of hybridization is to divide, as shown in Fig. 1, the elliptical crack into two zones, an internal zone I (ellipse in grey) and an external zone II (in white), then to use each of the two weight functions in the area where it is more efficient. The two zones are defined by the following relations: ( 2 y 2 x þ a0 6 1 ðzone IÞ b0 ð4Þ x 2 y 2 þ > 1 ðzone IIÞ 0 0 a b where a 0 and b 0 are such as a 0 /a = b 0 /b = b and b 2 [0,1], b being the proportion between the two zones and a, b are the axes of ellipse (Fig. 1). The weight-function of Eq. (3) is intended exclusively for cracks of elliptical form. Nevertheless, it presents an additional singularity (1 r/R)1/2 compared to Eq. (2). This makes Eq. (3) less efficient in the vicinity of the crack front (r ! R). This argument leads us to make the following choice: W QQ0 ¼ W QQ0 of Eq: ð3Þ if Q 2 zone I W QQ0 ¼ W QQ0 of Eq: ð2Þ if Q 2 zone II
ð5aÞ ð5bÞ
The hybridization verifies well the two geometrical limits of an elliptical crack, a ! 0 and a ! 1, as long as the two functions (2) and (3) satisfy the two following conditions: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 r 2 0 W QQ0 ! W QQ ðpenny-shapedÞ ¼ pffiffiffiffiffiffi 2 ð6Þ p pRlQQ0 a!1 pffiffiffiffiffiffi 2d 0 0 ð7Þ W QQ ! W QQ ðstraightÞ ¼ pffiffiffi 2 p plQQ0 a!0 where d is the shortest distance between the crack front and the arbitrary Q point. Eqs. (6) and (7) present the weight-functions of, respectively, a penny-shaped crack and a straight line crack in an infinite and a semi-infinite body [15]. It remains to determine the appropriate proportion b between two zones I and II. It is important to point out that the function (2) of Oore and Burns has a form which is very similar to the functions (6) and (7) for, respectively, a straight line crack and a penny-shaped crack. Moreover,
Q
The second one is developed by Krasowsky et al. [15] to model elliptical cracks in an infinite body. Its expression is as follows: 2:P1=4 ðhÞ W QQ0 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 pa 1 Rr 2ðuÞ :l2QQ0 : C ðuÞ
ð3Þ dC ðqQ Þ2
In expressions (2) and (3), r and u are the polar coordinates of an arbitrary point Q. lQQ 0 is the distance between the Q 0 point and the arbitrary Q point. C is the curve of the ellipse (the crack front), and qQ is the distance between the Q point and the elementary segment dC; P is a function
Fig. 1. Subdivision of the elliptical crack in two zones and its geometrical parameters.
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in their study, Oore and Burns particularly used the conditions (6) and (7) to deduce their final expression (2) (see pages 203 and 204 of Ref. [3]). This is not the case for the function (3) of Krasowsky et al. [15] which is deduced from the elementary variation of the Rice’s energy balance during an elementary propagation of the crack. By construction, the weight-function (2) is preferable to the function (3) in the two following cases: When the crack front is close to a circle (a ! 1); this is confirmed numerically as the graph of Fig. 2 shows it for the parameters f1 and f2 defined by R R ðW QQ0 ÞEq: ð2Þ dS ðW QQ0 ÞEq: ð3Þ dS S S and f 2 ¼ R : f1 ¼ R ðW QQ0 ÞEq: ð6Þ dS ðW QQ0 ÞEq: ð6Þ dS S S This graph shows clearly that the function (2) converges more quickly towards the weight-function of a pennyshaped crack compared to the function (3). When the crack front is close to a straight line (low values of a with value of h far from zero). In fact, these two cases correspond to situations where the variation of the curvature radius Rc of the crack front is weak, excluding a very narrow zone corresponding of very weak value of Rc (h close to zero with low values of a). In this case and for a relatively high variation of the curvature radius of ellipse, the weight-function (3) of Krasowsky et al. is more adapted. This is confirmed by the presence via the function P(h) of curvature radius Rc ¼ aa P3=2 ðhÞ in the expression of the function (3). Consequently, more the radius of curvature is relatively weak or its derivative (spatial gradient of the radius of curvature) is high, more the zone I should increase with respect to the zone II and vice versa. Taking into account all these considerations, we propose the relative parameter expressed by ðb minðRc ; aÞÞ b1 ¼ b
for the representation of the influence of the curvature radius compared to the large axis of the ellipse b, and the following one oRc oRc oh min ð9Þ b2 ¼ oRcoh c oR oh max oh min to represent the influence of the gradient of the curvature rac dius for a given a. In this relation oR is the partial derivative oh with respect to the angular position of the point Q 0 , its maximum and minimum values are calculated by ‘‘sweeping’’ completely the contour of the ellipse for a given a. The computation of the partial derivative is achieved numerically. The proportion parameter b takes the value: b ¼ maxðb1 ; b2 Þ 3. Numerical implementation, meshing and singularities According to Eqs. (1) and (2) or Eq. (3) two integrals are to be considered; a surface integral and a curvilinear integral (contour integral) of the crack front (C) involved in the weight functions (2) and (3). 3.1. The surface integral A treatment technique of the singularity 1=l2QQ0 in the integral Eq. (1) has been efficiently employed by Krasowsky et al. [15]. It consists in surrounding the (Q 0 ) point by a small half-circle of R0 radius on which the integral is analytically evaluated using Eq. (7): Z pffiffiffiffiffiffi pffiffiffiffiffi 2d 1 0 pffiffiffi 2 qðQ0 Þ dS 0 1:21703 R0 qðQ0 Þ K SI ¼ ð11Þ ðS 0 Þ p p lQQ0 In this relation, S 0 represents the surface of the half-circle and R0 is a parameter which is equal to R0 ¼ ð1=30 1=20Þ minðRi ðQ0 Þ; aÞ
ð8Þ
ð10Þ
ð12Þ
in order to minimize the error generated by the curvature in Q 0 (Ri is the radius of curvature of the crack front). In order to complete the calculation of the SIF which is 0 00 now split up in to K SI ¼ K SI þ K SI , the remaining SIF on (S00 ) is evaluated by the hybridization method. The meshing of (S00 ) is generated by drawing first, concentric half-circles centred at Q 0 of progressively increasing radii R0, r0, r1, r2, r3, . . . then, half-lines with Q 0 as origin, hence dividing the half-plane of the ellipse into n portions, approximately 60 portions in practice as shown in Fig. 3a. 3.2. The contour integral
Fig. 2. Comparison of weight-functions (2) and (3) with the penny-shaped one in the vicinity of a = 1.
R When Q2 is very close to the crack front, the integral ½dC=ðqQ Þ becomes singular and its numerical calculaC tion becomes delicate. To deal with this singularity, we first ignore a very narrow band of thickness D near the contour. This narrow band D = ca with c = 1/300 is shown in Fig. 3b. The effective ellipse surface becomes:
B.K. Hachi et al. / International Journal of Fatigue 29 (2007) 1904–1912
1907
Fig. 3. Discretization of the elliptical crack: (a) Discretization of the crack area. (b) Discretization of the crack front. 2
2
ðx=bÞ þ ðy=aÞ 6 ð1 D=RðhÞÞ
2
ð13Þ
In their work, Krasowsky et al. [15] consider a thickness D as variable (D(u) = c cos(T(u)) min (Ri(u),a), T is the angle between (OQ 0 ) and the normal of C at Q 0 (see Fig. 1)), making the automatic meshing of the crack very complex without gaining in the exactitude of the computation. The error is approximately estimated as R c pffiffiffi R 1 pffiffiffi ð 0 x dx= 0 x dxÞ ¼ c3=2 for a changing D whereas it is equal to (2c)3/2 for a constant D. At any rate, the shortest distance between two successive arcs of circles must not be lower than the thickness D. Therefore, we must verify at each calculation step the condition below: D ¼ ðriþ1 ri Þmin ¼ r0 R0
ð14Þ
By combining Eq. (14) with Eq. (12), one can get the progression of the ri radius as follows: m ¼ ðriþ1 ri Þ=ri ¼ ðr0 R0 Þ=R0 1=15 1=10
ð15Þ
The numerical calculation of the contour integral requires the discretization of the crack front (C) with a finite number of points delimiting straight segments of length dC (Fig. 3b). The error of linearization of dC is obtained by the use of Taylor’s series expansion of sin(dw/2) in the vicinity of zero: _
e¼
d C dC _
dC
ðdwÞ 24
2
ð16Þ
h p i wðiÞ ¼ arctan a tan i 2N
ð20Þ
To insure that the error e remains weak, Krasowsky et al. [15] imposed the following condition (Fig. 4): l1 ðuÞ=D 6 l ¼ l1 =l0 2 ½0:1; 0:2
ð21Þ
When coupling this condition with Eqs. (16), (17) and (21) we obtain the lower limit of N: pffiffiffiffiffiffiffiffi N P p=ð2 8lcÞ ð22Þ As long as the Q point is far from (C), the integral could be transformed into the summation: Z 4N dC X ðdCÞi ¼ ð23Þ 2 q q2iQ C Q i¼1 When Q is very close to the kth segment, the integral must analytically be computed on this segment in order to avoid any risk of singularity. It is equal to: Z c2 Z dC dc 2 2 2 q ðdCÞk c1 c þ d Q 1 c2 c1 ð24Þ ¼ arctan arctan d d d where the parameters c1, c2 and d are shown in Fig. 4. Eq. (23) thus becomes: Z Z k1 4N dC X ðdCÞi dC X ðdCÞi ¼ þ þ ð25Þ 2 2 q qiQ q2iQ C qQ ðdCÞk Q i¼1 i¼kþ1
dw is the angular opening of the dC. The opening dw decreases with the reduction of the curvature radius. To render uniform the error e on the entire contour (C), we take its average value (corresponding to a circular contour): 1 p 2 1 e¼ ¼ ðdwr Þ2 ð17Þ 24 2N 24 where N is the number of points on the quarter of contour (C) and wr is the reduced angle defined as (Fig. 3b): w ¼ arctan½a tanðwr Þ
ð18Þ
The reduced angle for the ith point is then given by: p ð19Þ wr ðiÞ ¼ iðdwÞ ¼ i 2N By substitution of Eq. (19) in Eq. (18), the angular position of the ith point becomes:
Fig. 4. Zoom of the elementary subdivision of the crack front.
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4. The hybrid method coupling with the point weight-function method (PWFM) The PWFM developed by Orynyak and Borodii [11] consists in the development of the appropriate modified weight-function for semi-elliptical crack based on the elliptical crack one in order to take into account the free edge effect. The weight-function is then split up into asymptotic and corrective parts [11]: W QQ0 ¼ W
A QQ0
þW
C QQ0
ð26Þ
C where W A QQ0 is the asymptotic part and W QQ0 is the corrective one. In the following, we give compact forms of C A WA QQ0 and W QQ0 . For the first function W QQ0 , we propose: ! " # l2QQ0 y Q0 yQ ðEllipseÞ A W QQ0 ¼ W QQ0 1 ð27Þ 1þ 2 1 a a lQ Q0 x
where yQ and yQ 0 are the projections of Q and Q 0 on the (Oy) axis of the ellipse, Qx is the symmetrical point of Q with respect to (Ox) axis of the ellipse, by considering (Oxz) as the free edge plan (see Fig. 1). The weight-function ðEllipseÞ W QQ0 is given by the Eq. (5). For the second function W CQQ0 , the expression is: W
C QQ0
¼W
A QQ0
r minðlQQ0 ; a2 Þ 1 Dðh; aÞ ¼ W CQQ0 Dðh; aÞ R a2 ð28Þ 2
where D(h, a) is the only unknown function to be determined. It is related to the geometry of the crack and the angular position of the point of calculation Q 0 . In order to determine this function, we define the dimensionless SIF, G(h, a), as follows: EðkÞ Gðh; aÞ ¼ pffiffiffiffiffiffi 1=4 K I ðh; aÞ paP ðhÞ
ð29Þ
where E(k) is an elliptical integral of the second kind and pffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ 1 a2 . By using the Eq. (1) giving the integral expression of KI and Eq. (26), we obtain: Gðh; aÞ ¼ I A ðh; aÞ þ I C ðh; aÞDðh; aÞ
ð30Þ A
Based on Eq. (28), the dimensionless integrals I and ICcan be expressed as: Z EðkÞ I A ðh; aÞ ¼ pffiffiffiffiffiffi 1=4 rðx; yÞW AQQ0 dS and I C ðh; aÞ paP ðhÞ ðSÞ Z EðkÞ ¼ pffiffiffiffiffiffi 1=4 rðx; yÞW CQQ0 dS ð31Þ paP ðhÞ ðSÞ If the solution G0(h, a) is known for a particular loading r0 (this solution is generally denoted as a reference one), then for any other stress distribution r(x, y), the repeated usage of Eq. (30) allows one to deduce the D(h, a) value and to determine the general solution G(h, a):
G0 ðh; aÞ I A 0 ðh; aÞ Gðh; aÞ ¼ I ðh; aÞ þ I ðh; aÞ I C0 ðh; aÞ A
C
ð32Þ
More development details of the coupling between our hybrid technique and the PWFM method are presented in [19]. 5. Results and discussions For the modeling of the elliptical and the semi-elliptical cracks by the hybrid weight-function technique, we developed a computer operational software with graphic interface named HWFun by using the C++ object-oriented language. The following are practical applications based on the HWFun code for plate containing cracks under static and fatigue bending. Because of the complexity of the analytical developments concerning the evaluation of the FIC, the exact solutions are available only for the embedded elliptical cracks under particular loadings (see Ref. [20]). Whereas for the surface cracks, the existence of such analytical developments is quasi impossible. Consequently, the tests of validation rest on the comparison of the results obtained by the present approach with the analytical solution (in the case of embedded elliptical cracks) like with the results of other studies in the event of its unavailability (for the case of the surface cracks). It is important to point out that the present method was successfully tested for an embedded crack in infinite body with a significant reduction of the error concerning the SIF. This study is available in Refs. [18,19]. 5.1. Plate containing embedded transverse elliptical crack subjected to bending The problem of gaps and inclusions in plates subjected to bending is usually encountered in practice. Such defects influence the structure toughness by weakening their resistance and increasing their risk of cracking. In the calculation of resistance of such structures, these defects are often simulated by plane elliptical cracks, more or less flattened as shown in Fig. 5. The exact solution for this problem is given by the following relation [20]: rffiffiffiffiffiffi 1=4 pa 2 K I ¼ Cr cos h þ a2 sin2 h /
k 2 EðkÞ cos h 1 ð33Þ ð1 þ k 2 ÞEðkÞ ð1 k 2 ÞKðkÞ where r is the applied stress, / E(k)2 for small values of (r/re), re is the elastic limit of the material, K(k) is the elliptic integral of the first kind, C is a correction factor depending on geometrical parameters h, a, a/d and the material characteristics. For a Poisson’s ratio m = 0.3, Ref. [20] gives the value of C in abacuses form. For example, C 1 for the values of a/d 6 0.2, h 2 [ p/2, p/2] and a 2 [0.1,1.0]. The case of the plate subjected to pure bending, as presented in Fig. 5 with a/d 6 0.2, d 6 c and r(y) = r0(1 + (y/a)), is solved by the present approach via the developed code HWFun. The obtained results are reported
B.K. Hachi et al. / International Journal of Fatigue 29 (2007) 1904–1912
1909
Fig. 5. Embedded elliptical crack in an infinite plate under bending.
on the graphs of Fig. 6 where we can observepgood ffiffiffiffiffiffi correlations between the values of K I ¼ K I EðkÞ= ðr0 paP1=4 ðhÞÞ computed by our code HWFun and those of the analytical solution. Compared with Krasowsky’s weight-function results, we point out that the precision of our approach is still appreciable when varying the angle h from p/2 to p/2 with a = 0.6 especially in the interval hr 2 [ 30, + 30](hr = arctan (tan(h)/a)) (see Fig. 6a). At the dangerous point h = p/2 (maximum value of K I Þ, where the free edge effect is more accentuated, the hybrid approach is still more efficient than the Krasowsky’s weight function results for various values of a as shown in Fig. 6b. 5.2. Surface semi-elliptical crack in finite plate subjected to tension and bending The problem of surface semi-elliptical crack in finite plate subjected to tension and bending (Fig. 7) was treated
Fig. 7. Semi-elliptical surface crack in finite plate under tension and bending loads.
by Newman and Raju [21,22] based on the FEM and empirical relations, respectively. The same problem was treated also by other authors such as Isida et al. [23] based on modified body force method (MBFM) and Gonc¸alves and de-Castro [24] based on line spring model (LSM). We adopted for this example the dimensionless SIF EðkÞ ffiffiffi ffi where r0 = max(St,Sb), St and S b ¼ 3M F I ¼ Kr0Ip being, pa wt2 respectively, the tension and the bending stresses (Fig. 7). In this study, the semi-elliptical cracks are modelled by coupling the present hybrid weight-function method with the point weight-function method (PWFM) presented previously. The reference solution is the one given by Newman
Fig. 6. Dimensionless SIF for elliptical crack in an infinite plate under bending: (a) for a = 0.6 and h 2 [ p/2, + p/2], (b) for h = p/2 and a 2 [0.1, 1.0].
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and Raju [21] for a reference loading of tension (St = 1 and Sb = 0). In the following, we compute FI for pure bending (St = 0 and Sb = 1), the results are summarized in the form of graphs in Fig. 8. Various geometric configurations are considered, h 2 {0,p/2} for a 2 {0.2,0.4,0.6,1.0}. In absence of the analytical solution for this problem, the good correlation of the actual results with those of other researchers [21–24] is obvious for both the depth point A and the surface point B as shown in Fig. 8. The maximum difference between the two results is observed when the minor axis a is close to the thickness t of the plate (a/t ! 1). This can primarily be explained by the effect, due to the second free edge (y = t) of the finite plate (see Fig. 7), which is not taken into account in our approach. This shift is more visible when the point of calculation is very close to the free surface (for instance, the point A). This effect is more pronounced in the case of thin plates (for instance: a/b = 0.2 and a/t = 0.8) because of its double influence (double edge effects). It should be known that the bending
stress applied on the crack is given by Sb (1 2y/t) as a function of the y coordinate of an arbitrary point Q of the crack. Moreover, an effect of the a/t ratio is introduced in the SIF evaluation via the reference solution given by Newman and Raju [21] used in the relation (32). This latter is derived from the coupling between the hybridization approach and the PWFM method to treat semi-elliptical cracks. The cases of St = 1 and Sb = 1(tension + bending) and of a polynomial as well as an exponential stress distributions applied on the crack are also treated by the present approach for which the same remarks of the previous case (pure bending) concerning the quality of results were observed. 5.3. Prediction of fatigue-crack-growth for surface crack in plate under bending In this section, we intend to treat the problem of ductile fracture which is the case of fatigue-crack-growth. By this
Fig. 8. Dimensionless SIF at depth point and surface point for various geometric configurations with a pure bending (St = 0 and Sb = 1).
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study, we want to demonstrate the flexibility and the efficiency of our approach. We limit ourselves to the work of Newman and Raju [21,22] for the sake of comparison. Hence, we study the example of fatigue-crack-growth of a finite plate containing semi-elliptical surface crack subjected to pure bending (St = 0 and Sb = 1). The most popular law in the computation of crack-growth rates is the Paris relationship given by the following: da ¼ C A ðDK A Þn dN db n ¼ C B ðDK B Þ dN
ð34Þ ð35Þ
where a and bare the ellipse axes, N, the number of the stress cycles, n, an exponent usually equal to 4 (ductile fracture) for a wide range of materials (The 2014-T651 aluminium alloy is one of them). CA and CB are the crack growth coefficients for, respectively, the points A and B of the crack. DK is the stress intensity factor range, defined as: DK ¼ ðK I Þmax ðK I Þmin max
min
n
Db is obtained by dividing the surface crack extension into a large number of increments. Fig. 9b and c show the comparison between experimental and predicted fatigue-crack-growth patterns of surface cracks in plates subjected to bending (St = 0 and Sb = 1) for both Newman and Raju approach [22] and the present one, respectively. It should be mentioned that all the experimental data points are obtained for the 2014-T651 aluminium alloy [25]. It is clear from Fig. 9c that the results obtained by the actual approach are in good correlation with the experiments. The reason for showing the two Fig. 9b and c is to demonstrate the total concordance between the predicted results and those of Newman and Raju [22]. The predicted results show that the cracks tend to approach a common propagation (see Fig. 9b and c) as pointed out by Corn [25]. This observation was also made by Newman and Raju [22]. 6. Conclusion
ð36Þ
ðK I Þ and ðK I Þ are the SIF corresponding, respectively, to the max-stress and the min-stress in stress cycles. In our case the max-stress is Sb and the min-stress is 0 (Fig. 9a). The relationship between CA and CB is assumed equal to [22]: C B ¼ ð0:9Þ C A
1911
ð37Þ
An increment Db at the surface point B induces an increment in-depth Da which we compute using the equation below: n n C A DK A DK A Da ¼ Db ¼ Db ð38Þ C B DK B 0:9DK B
In this work, a hybrid weight-function-based on Green’s function formulation is developed for modelling elliptical and semi-elliptical cracks in elastic and homogeneous bodies. In this modelling, a computation of stress intensity factors in mode I is performed. The idea of hybridization leads us to an optimization problem of two geometrical parameters (the ratio axes and curvature radius of ellipse). The development of the numerical procedures, the treatment of the various singularities, and the meshing implementation have been presented in full details. A computer code named HWFun has been developed and tested on various practical applications concerning plates. The results obtained show a clear reduction in the error
Fig. 9. Predicted fatigue-crack-growth patterns for plate under cyclic pure bending (St = 0 and Sb = 1): (a) The stress cycle applied to the plate. (b) Newman-Raju’s results compared with experimental data. (c) HWFun results compared with experimental data.
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for an embedded elliptical crack in plate under bending compared to classical weight-function-based approach. The semi-elliptical surface cracks in plates under bending were also analyzed by the present approach of hybridization. Compared to other results obtained by using various techniques, it shows a good accuracy. The present approach was also tested on the problem of fatiguecrack-growth in plate under bending. The obtained results confirm its efficiency and accuracy. The idea of hybridization undoubtedly opens horizons for the treatment of other complex problems in fracture mechanics such as mixedmode, interaction among cracks. References [1] Bueckner HF. A novel principle for the computation of stress intensity factors. Z Angew Meth 1970;50:529–46. [2] Rice JR. Some remarks on elastic crack-tip stress fields. Int J Solids Struct 1972;8:751–8. [3] Oore M, Burns DJ. Estimation of stress intensity factors for embedded irregular cracks subjected to arbitrary normal stress fields. Trans ASME J Pres Vess Technol 1980;102(2):202–11. [4] Bortmann Y, Banks-Sills L. An extended weight function method for 2D and 3D-problems. J Appl Mech 1983;50:907–9. [5] Gao H, Rice JR. Shear stress intensity factors for planar crack with slightly curved front. J Appl Mech 1986;53:774–8. [6] Sun CT, Wang CY. A new look at energy release rate in fracture mechanics. Int J Fract 2002;113:295–307. [7] Fett T, Mattheck C, Munz D. Approximate weight function for 2D and 3D-problems. Eng Anal Boundary Elements 1989;6:37–48. [8] Vainshtok VA. Application of the weight function method to solving multi-parametric three-dimensional fracture mechanics problems. Int J Fract 1991;47:201–12. [9] Dominguez J, Gallego F. Time domain boundary element method for dynamic stress intensity factor computations. Int J Numer Meth Eng 1992;33:635–47. [10] Rooke DP, Aliabadi MH. The use of fundamental fields to obtain weight functions for mixed-mode cracks. Int J Eng Sci 1994;32: 155–66. [11] Orynyak IV, Borodii MV. Point weight function method application for semi-elliptical mode I cracks. Int J Fract 1995;70:117–24.
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