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Elastic T-stress for cracks in test specimens subjected to non-uniform stress distributions
Engineering Fracture Mechanics 69 (2002) 1339–1352 www.elsevier.com/locate/engfracmech
Elastic T-stress for cracks in test specimens subjected to non-uniform stress distributions X. Wang
*
Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, Ont., Canada K1S 5B6 Received 23 August 2001; received in revised form 13 November 2001; accepted 19 November 2001
Abstract Finite element analyses have been conducted to calculate elastic T-stress solutions for cracked test specimens. The Tstress solutions are presented for single edge cracked plates, double edge crack plates and centre cracked plates. Uniform, linear, parabolic or cubic stress distributions were applied to the crack face. The results for uniform and linear stress distributions were used to derive weight functions for T-stress for the corresponding specimens. The weight functions for T-stress are then verified against several linear and non-linear stress distributions. The derived weight functions are suitable for the T-stress calculation for cracked specimens under any given stress field. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Elastic T-stress; Weight function; Crack in test specimen
1. Introduction The elastic T-stress, or the second term of the Williams [1] series expansion for linear elastic crack tip fields, represents the stress acting parallel to the crack plane. Larsson and Carlsson [2] and Rice [3] showed that the sign and magnitude of the T-stress substantially change the size and shape of the plane strain crack tip plastic zone at finite load levels. Bilby et al. [4] showed that the T-stress can strongly affect the magnitude of hydrostatic triaxiality in the near crack tip elastic–plastic fields. The important features emerging from these works is that the sign and magnitude of T-stress can substantially alter the level of crack tip stress triaxiality, hence influence crack tip constraint. Positive T-stress strengthens the level of crack tip stress triaxiality and leads to high crack tip constraint; while negative T-stress reduces the level of crack tip stress triaxiality and leads to the loss the crack tip constraint. The later works by Betegon and Hancock [5], Du and Hancock [6], O’Dowd and Shih [7], and Wang [8] indicated that the T-stress, in addition to the J-integral, provides an effective two-parameter characterisation of plane strain elastic–plastic crack tip fields in a variety of crack configurations and loading conditions. Recently, the application of two-parameter fracture mechanics to include the constraint effect in the failure assessment procedure is becoming more and more established [9]. In order to apply the *
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0013-7944/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 1 ) 0 0 1 4 9 - 7
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Fig. 1. Specimen geometries, co-ordinates and notation: (a) SECP, (b) DECP and (c) CCP.
two-parameter fracture mechanics methodology, it is important to provide T-stress solutions for the crack configuration under consideration. Especially accurate T-stress solutions for test specimens are important for the collection of appropriate constraint-dependent fracture toughness data. Similar to the stress intensity factors (SIFs), the values of the T-stress strongly depend on the type of loading as well as on the relative crack length and the overall geometry. Several numerical/analytical methods were developed to calculate the elastic T-stresses for test specimens. Many researchers (Leevers and Radon [10], Cardew et al. [11], Kfouri [12] and Sham [13]) have provided T-stress solutions for uniform tension or bending loading conditions. Non-linear stress distributions can occur as consequence of thermal stress, residual stress and stress concentration. The T-stress solutions under these non-linear stress distributions are required. There are some calculations for specific non-linear loading/crack geometries (Fett [14] and Hooton et al. [15]). However, detailed parametric solutions of T-stress for cracked specimens under non-uniform stress distributions are not available. Weight function method is a very powerful well-known method for calculating SIFs under arbitrary stress distributions. In Refs. [13–15], analogy to the method for the SIF, the weight function method for the calculation of T-stress is developed. It is a very powerful technique since it removes restrictions on the stress distributions. Sham [13] provided methods for the determinations of weight functions for T-stress numerically; while Fett [14] and Hooton et al. [15] demonstrated methods for the derivation of weight functions from reference T-stress solutions. In the present paper, detailed finite element analyses were conducted to calculate the T-stress solutions for single edge cracked plate specimen (SECP), double edge cracked plate specimen (DECP) and centre cracked plate specimen (CCP). Fig. 1 demonstrated specimens analysed. The T-stress solutions are presented with a=t values of 0.2, 0.4, 0.6 and 0.8. Uniform, linear, parabolic or cubic stress distributions are applied to the crack face. The T-stress results for uniform and linear stress distributions in the specimens were used to derive weight functions for T-stress for the corresponding specimens. The weight functions were then verified using the results for several linear and non-linear stress distributions.
2. Finite element analysis for cracks in test specimens 2.1. Finite element model Two-dimensional finite elements were used to model the symmetric half or quarter of the SECP, DECP and CCP specimen. Fig. 1 shows the geometry, notation and co-ordinates system used. The finite element
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Fig. 2. Typical finite element mesh.
analyses were made using A B A Q U S version 5.8 [16] with eight-noded isoparametric solid elements. In order to model the square root singularity at the crack tip, elements with two mid-side nodes at the quarter points (a degenerate quadrilateral element with one line collapsed) were used and the separate crack tip nodal points were constrained to have the same displacement [17]. The loads were applied directly to the crack face. Four types of loading were applied to each crack geometry, with the following stress distributions: x n rðxÞ ¼ r0 1 ð1Þ a with n ¼ 0–3, r0 is the nominal stress and a is the crack depth (for edge crack specimen) or half crack length (for centre crack specimen). The h=t ratio is fixed to 3 for all analysis in the current calculation. A typical finite element mesh is plotted in Fig. 2. 2.2. Extraction of the T-stress The interaction integral method introduced by Cardew et al. [11], Kfouri [12] and Nakamura and Parks [18] is used to extract the elastic T-stress in the current analysis. For a plane strain problem, assume a load
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Fig. 3. A point force f applied at the crack tip, the auxiliary field for the interaction integral.
force of magnitude f applied at the crack tip in the direction parallel to the crack as illustrated in Fig. 3. Using superscript ‘L’ to designate the stress and displacement fields, the analytical solution gives [19] f f cos3 h rL22 ¼ cos h sin2 h pr pr f rL12 ¼ cos2 h sin h pr
ð2aÞ
1 t2 f r sin2 h ln ¼ þ d 2ð1 tÞ E p 1þt f uL2 ¼ fð1 2tÞh cos h sin hg 2E p
ð2bÞ
rL11 ¼
and uL1
where the subscripts 1, 2 suggests a local Cartesian co-ordinate system; r and h are the local polar coordinates; d is a reference distance; E and t are the Young’s modulus and Poisson’s ratio of the material, respectively. Employing the above solution as an auxiliary field, Cardew et al. [11] and Kfouri [12] extracted the T-stress for 2D crack problems by introducing an interaction J-integral based on Eshelby’s theorem. Nakamura and Parks [18] extended this method to 3D crack problems and provided the domain integral formulations. Combined with finite element method, it was shown to be a simple and effective method for the determination of the T-stress of 2D and 3D crack specimens. For 2D crack problems, the interaction integral can be written in the domain integral form: Z 1 ouL oui oqk oqk I¼ rij i þ rLij rLij eij dS ð3Þ Da S oxk oxk oxj oxk where Da is the virtual crack advance, S is an area which encloses the crack tip, qk defines the virtual extension of the crack tip; rij , eij and ui are the stress, strain and displacement components of the 2D crack problem; rLij , eLij and uLi are the corresponding components in the point load auxiliary solution given by Eqs. (2a) and (2b). The crack tip T-stress is related to I by (Kfouri [12]) T ¼
E I 1 t2 f
ð4Þ
The computation of the domain integral, Eq. (3), is readily compatible with finite element formulation [18,20]. Once I is determined, the T-stress can be obtained from Eq. (4) directly.
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A B A Q U S (version 5.8) was used to solve the finite element model. A post processing F O R T R A N program was developed to interface with A B A Q U S to calculate the domain-integral, Eq. (3), from Gauss integration scheme. Eight-noded two-dimensional quadrilateral elements with 3 3 Gauss integration were used in the finite element models. A ‘pyramid’ qk function [21] was adopted to simulate the virtual crack extension. The point-load auxiliary field was derived from Eqs. (2a) and (2b) by taking f ¼ 1. Three contours are used to calculate I. Good independence of the I value on the choice of contour was obtained. In the current calculation, the average value of I results from outer two contours are used to calculate crack tip T-stress.
2.3. Verification of the finite element model In order to verify the finite element models and the method of extracting T-stress values, T-stress values for cracks in SECP, DECP and CCP under far-end uniform tension loading were calculated. Comparisons were made with results from Sham [13] for SECP and Kfouri [12] for DECP and CCP specimens. The results for SECP specimen are presented in Table 1, the maximum difference was 3.62% and most are within 2%. The results for DECP and CCP specimens are presented in Table 2, the maximum difference is within 2.21% for DECP specimens and 5.04% for CCP specimens. Through these verifications, the present method is considered suitable for analyses of T-stress for cracks in test specimens. 2.4. Finite element results The T-stress values for cracks in SECP, DECP and CCP specimens with a=t of 0.2, 0.4, 0.6 or 0.8 subjected to constant, linear, quadratic or cubic stress distributions as expressed in Eq. (1) have been determined. The results of normalised T-stress, V ¼ T =r0 , are summarised in Tables 3–5. Table 1 Comparison of normalised T-stress solutions, T =r0 , from present FEM calculation and solution from Sham [13] for single edge cracked plate (SECP) specimen under far-end tension a=t
Solution from [13] V
Present calculation V
Difference (%) 100 jV V j=V
0.2 0.4 0.6 0.8
0.5918 0.5853 0.0278 5.9755
0.5949 0.5782 0.0268 5.8298
0.53 1.20 3.62 2.44
Table 2 Comparison of normalised T-stress solutions T =r0 , from present FEM calculation and solution from Kfouri [12] for DECP and CCP specimen under far-end tension a=t 0.2 0.4 0.6
Solution from [12] V
Present calculation V
Difference (%) 100 jV V j=V
DECP
CCP
DECP
CCP
DECP
CCP
0.5276 0.5541 0.5788
1.0277 1.1363 1.3803
0.5312 0.5429 0.5659
1.0289 1.1394 1.4484
0.69 2.01 2.21
0.11 0.27 5.04
Table 3 Normalised T-stress, V ¼ T =r0 , for SECP specimen under crack face pressure rðxÞ
a=t ¼ 0:2
a=t ¼ 0:4
a=t ¼ 0:6
a=t ¼ 0:8
Constant Linear Parabolic Cubic
0.4050 0.2831 0.2205 0.1813
0.4217 0.2953 0.2300 0.1889
1.0405 0.7019 0.5328 0.4300
6.8297 4.5876 3.4559 2.7714
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Table 4 Normalised T-stress, V ¼ T =r0 , for DECP specimen under crack face pressure rðxÞ
a=t ¼ 0:2
a=t ¼ 0:4
a=t ¼ 0:6
a=t ¼ 0:8
Constant Linear Parabolic Cubic
0.4687 0.3216 0.2482 0.2028
0.4570 0.3167 0.2454 0.2010
0.4340 0.3065 0.2393 0.1969
0.3999 0.2955 0.2353 0.1957
Table 5 Normalised T-stress, V ¼ T =r0 , for CCP specimen under crack face pressure rðxÞ
a=t ¼ 0:2
a=t ¼ 0:4
a=t ¼ 0:6
a=t ¼ 0:8
Constant Linear Parabolic Cubic
0.02890 0.01713 0.01163 0.00871
0.1394 0.0802 0.0554 0.0421
0.4484 0.2557 0.1771 0.1350
1.5689 0.8652 0.5911 0.4474
The T-stress for an application can be obtained by superposition of these solutions provided the stress distribution has been expressed as a polynomial of order three. However, the weight function for T-stress derived in the following will remove any restrictions on the stress distributions. 3. Weight functions for T-stresses Finite element calculations of T-stress for cracks in test specimens with four kinds of stress distribution have been presented in Section 2. However, it is impossible to foresee the variety of loading occurring in engineering applications, especially when complicated transient or residual stresses are involved. The weight function method is one of the most efficient methods to derive SIFs for complex stress distributions [22,23]. This method was also applied by several researchers [13–15] to derive T-stress solutions for complex stress distributions. In this section, the theoretical background is reviewed and weight functions for T-stress in cracked specimens are then derived and validated. 3.1. Theoretical background For mode I crack problems, using the superposition method, it can be demonstrated (Chen [24]) that for a cracked body as shown in Fig. 4(a), loaded by a stress field Q, the T-stress is the superposition of Tstresses for two cases. The first is the T-stress for the same cracked body loaded by a crack face pressure, rðxÞ, induced by the remote stress field Q in the uncracked body, as shown in Fig. 4(b), the second case is the T-stress in the uncracked body under the remote stress field Q (Fig. 4(c)). Therefore, the T-stress for the problem shown in Fig. 4(a) can be calculated from: T ¼ Tcrack pressure þ Tuncrack
ð5Þ
The T-stress for a cracked body with loading applied to the crack surface can be calculated by integrating the product of the weight function tðx; aÞ and the stress distribution, rðxÞ, on the crack plane: Z a Tcrack pressure ¼ rðxÞtðx; aÞ dx ð6Þ 0
where rðxÞ is the stress distribution on the crack face and tðx; aÞ is the weight function for T-stress (it is alternatively called second order weight function by Sham [13], or Green’s function for T-stress by Fett
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Fig. 4. Weight function for T-stress: (a)–(d).
[14]). The weight function for T-stress depends on the crack geometry and is independent to the loading conditions. Mathematically, the weight function, tðx; aÞ, is the Green’s function for the T-stress. It represents the T-stress at the crack tip for pair of unit point loads acting on the crack face at the location x as shown in Fig. 4(d). The T-stress for an uncracked body with applied load system Q, as shown in Fig. 4(c), can be found from the stresses at the crack tip location in the uncracked body [24]: Tuncrack ¼ ðrx ry Þx¼a ð7Þ
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Note that the Tuncrack here represents the T-stress at the crack tip under load system Q when the crack is completely closed (Fig. 4(c)), it is different from the T-stress limit value when a approaches zero. It depends on the crack half-length a as demonstrated in Eq. (7). Substituting Eqs. (6) and (7) into Eq. (5), the T-stress for cracked body as shown in Fig. 4(a), loaded by a stress field Q, is obtained: Z a T ¼ rðxÞtðx; aÞ dx þ ðrx ry Þx¼a ð8Þ 0
Eq. (8) provides very efficient way for calculating T-stress. From alternative derivations, Eq. (8) was also obtained by Sham [13], Fett [14] and Hooten et al. [15]. Once the weight function for T-stress tðx; aÞ is determined, and a stress analysis of uncracked body is conducted to obtain rðxÞ and ðrx ry Þx¼a , the corresponding T-stress can be calculated for any loading condition. There are different ways to obtain the weight function for T-stress, tðx; aÞ. Sham [13] extended Bueckner’s [22] theory of fundamental fields to the determination of T-stress. And it was demonstrated that tðx; aÞ represents a combination of displacement from fundamental field and displacement from regular field [13]. Using a unified finite element method, Sham [13] obtained the numerical solution for tðx; aÞ. Recently, Fett [14] and Hooten et al. [15] have proposed closed form approximations for tðx; aÞ, and the tðx; aÞ can be obtained from reference T-stress solutions. The approach of Fett [14] is used here. 3.2. General forms for weight function for T-stress Based on the analogy to the weight function for SIFs, Fett [14] proposed to use a form, which can be re-written in the following format, to approximate the weight function tðx; aÞ:
2 x 1=2 x 3=2 x 5=2 x n1=2 tðx; aÞ ¼ D1 1 þ D2 1 þ D3 1 þ þ Dn 1 ð9Þ pa a a a a where crack tip is at x ¼ a, and D1 ; D2 ; . . . and Dn are constants for given crack geometry. In Ref. [14], the following two-terms expression was then used to approximate the weight functions for T-stress for edge cracks:
2 x 1=2 x 3=2 D1 1 tðx; aÞ ¼ þ D2 1 ð10Þ pa a a where D1 and D2 can be decided from two reference T-stress solutions. This weight function form of Eq. (10) is used in the current research to derive weight functions for cracked specimens. Note that Eqs. (9) and (10) were proposed in Ref. [14] for edge cracks, a different functional form was proposed for internal cracks in Ref. [25]. In the current research, Eq. (10) is used for both edge (SECP, DECP) and internal (CCP) cracks, and it will be shown later that excellent accuracy is achieved for both edge and internal cracks. 3.3. Weight function for test specimens Two reference solutions for T-stress are used to decide D1 and D2 in Eq. (10): uniform crack face pressure and linear decreasing stress distribution corresponding to n ¼ 0, and n ¼ 1 in Eq. (1), respectively. 3.3.1. Reference T-stress solutions The numerical T-stress results for cracks in SECP, DECP and CCP specimens presented in Section 2 were approximated by empirical formulae fitted with an accuracy of 1% or better for SECP and DECP specimens and 2% or better for CCP specimens. The results for a=t ¼ 0 were obtained by smooth ex-
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trapolation. The choice of equations for the extrapolation was based on engineering judgement. The range of applicability for these equations is 0 6 a=t 6 0:8. The results for the uniform crack face pressure: rðxÞ ¼ r0
ð11Þ
T1r ¼ r0 V0
ð12Þ
are where V0 represents the normalised T-stress, and it is a function of a=t. The results for a linearly decreasing crack face pressure: x rðxÞ ¼ r0 1 ð13Þ a are T2r ¼ r0 V1
ð14Þ
where V1 represents the normalised T-stress for this loading condition. The expressions for V0 and V1 for SECP, DECP and CCP specimens are presented as follows. For SECP specimen a 8:0252 V0 ¼ 0:4012 þ 38:5340 t a 8:0877 V1 ¼ 0:2813 þ 26:1749 t for DECP specimen V0 ¼ 0:4729 0:09814 V1 ¼ 0:3239 0:05101
a 2 t a 2 t
0:02501 þ 0:01011
a 4 t a 4 t
for CCP specimen V0 ¼
V1 ¼
1 a a2 2:7206 0:4774 exp 1:4690 t t 1 a a2 4:5984 0:8062 exp 2:5343 t t
3.3.2. Weight functions for T-stress By substituting Eqs. (11)–(14) into Eq. (8), two equations with two unknowns were established. Note that the uncracked T-stresses are zero in Eq. (8) for those crack pressure cases (no external load Q is applied and therefore rx and ry are zeros in Eq. (7)). The parameters in the weight function expression were solved and are: pð5V0 7V1 Þ ð15Þ D1 ¼ 15 16 D2 ¼ 165 pð35V1 21V0 Þ The weight function for T-stress can then be determined from Eq. (10).
ð16Þ
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3.4. Validation of weight functions In this section, the weight functions were verified using T-stress solutions for several linear and nonlinear stress fields. 3.4.1. Two non-linear stress distributions First, the weight functions derived above for SECP, DECP and CCP specimens were validated using finite element results for two non-linear stress fields. Using Eq. (8), T-stress values were calculated for the following crack face stress distributions: x 2 rðxÞ ¼ r0 1 a
ð17Þ
x 3 rðxÞ ¼ r0 1 a
ð18Þ
Note that under those crack face pressure, the uncracked T-stress is zero, therefore Eq. (8) becomes Eq. (6). The T-stress results calculated from the derived weight functions and from the present finite element calculations for the above stress distributions for all the test specimens are shown in Figs. 5–7. The differences between the normalised T-stress values were less than 2.4%, 2.1% and 3.3% for SECP, DECP and CCP specimens, respectively. As a comparison, the T-stress results for constant and linear stress distributions Eqs. (11) and (13) are also plotted in Figs. 5–7.
Fig. 5. Comparison of weight function based T-stress and FEM data, SECP specimen.
X. Wang / Engineering Fracture Mechanics 69 (2002) 1339–1352
Fig. 6. Comparison of weight function based T-stress and FEM data, DECP specimen.
Fig. 7. Comparison of weight function based T-stress and FEM data, CCP specimen.
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3.4.2. Far-field tension and bending For SECP specimens, the weight functions were then used to calculate the T-stress values under far-field tension and bending cases. Note that the uncracked T-stresses are not zeros in those cases. For far-field tension, Tuncrack ¼ ðrx ry Þ ¼ r0 ð19Þ x¼a
and crack face pressure rðxÞ ¼ r0 where r0 is the nominal stress for tension. For far-field bending, a Tuncrack ¼ ðrx ry Þx¼a ¼ r0 1 2 t and crack face pressure a a x rðxÞ ¼ r0 1 2 þ r0 2 1 t t a
ð20Þ
ð21Þ
ð22Þ
where r0 is the nominal stress for bending. The T-stress results calculated from the derived weight functions using Eq. (8) and from finite element calculations from [13] for SECP are shown in Fig. 8. Excellent agreements are achieved. 3.4.3. Comparison with Fett’s results In Ref. [14], the T-stress for single edge cracked plates under the following thermal stress field were calculated:
Fig. 8. Comparison of weight function based T-stress and results from Sham [13] for far-end tension and bending loads, SECP specimen.
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Fig. 9. Comparison of weight function based T-stress and results from Fett [14] for non-linear thermal stress distribution, SECP specimen.
rðxÞ ¼ r0
x x 2 2 4 þ4 3 t t
ð23Þ
The currently derived weight functions for SECP specimens are used to calculate the T-stress for this stress distribution using Eq. (8). Fig. 9 shows the comparison between the current weight function prediction and the results from [14]. Good agreement is achieved.
4. Conclusions Finite element analyses have been conducted to calculate the T-stress for cracks in single edge cracked, double edge cracked, and centre cracked plates. Constant, linear, quadratic and cubic stress distributions were applied to the crack face. The analysis procedures and results were verified with existing solutions in the literature. Based on the present finite element calculations for T-stress in the test specimens, weight functions for Tstress were derived. They are valid for 0 6 a=t 6 0:8 for all the test specimens. The weight functions were verified using T-stress solutions for several linear and non-linear stress fields. Excellent agreements (maximum difference less than 3.3%) are achieved. The weight functions can be used to calculate T-stress for any given stress field rðxÞ. These closed form weight functions for T-stress derived here make them suitable for fracture analysis, for example, to account for crack tip constraint under local non-linear residual stresses, using the method demonstrated in Refs. [9,15].
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Acknowledgements The author gratefully acknowledges the financial supports from the Natural Science and Engineering Research Council (NSERC) of Canada and Materials and Manufacturing Ontario (MMO). He would also want to thank Hibbitt, Karlsson & Sorenson, Inc. for making A B A Q U S available under an academic license to the Carleton University.
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Elastic T-stress for cracks in test specimens subjected to non-uniform stress distributions X. Wang
*
Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, Ont., Canada K1S 5B6 Received 23 August 2001; received in revised form 13 November 2001; accepted 19 November 2001
Abstract Finite element analyses have been conducted to calculate elastic T-stress solutions for cracked test specimens. The Tstress solutions are presented for single edge cracked plates, double edge crack plates and centre cracked plates. Uniform, linear, parabolic or cubic stress distributions were applied to the crack face. The results for uniform and linear stress distributions were used to derive weight functions for T-stress for the corresponding specimens. The weight functions for T-stress are then verified against several linear and non-linear stress distributions. The derived weight functions are suitable for the T-stress calculation for cracked specimens under any given stress field. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Elastic T-stress; Weight function; Crack in test specimen
1. Introduction The elastic T-stress, or the second term of the Williams [1] series expansion for linear elastic crack tip fields, represents the stress acting parallel to the crack plane. Larsson and Carlsson [2] and Rice [3] showed that the sign and magnitude of the T-stress substantially change the size and shape of the plane strain crack tip plastic zone at finite load levels. Bilby et al. [4] showed that the T-stress can strongly affect the magnitude of hydrostatic triaxiality in the near crack tip elastic–plastic fields. The important features emerging from these works is that the sign and magnitude of T-stress can substantially alter the level of crack tip stress triaxiality, hence influence crack tip constraint. Positive T-stress strengthens the level of crack tip stress triaxiality and leads to high crack tip constraint; while negative T-stress reduces the level of crack tip stress triaxiality and leads to the loss the crack tip constraint. The later works by Betegon and Hancock [5], Du and Hancock [6], O’Dowd and Shih [7], and Wang [8] indicated that the T-stress, in addition to the J-integral, provides an effective two-parameter characterisation of plane strain elastic–plastic crack tip fields in a variety of crack configurations and loading conditions. Recently, the application of two-parameter fracture mechanics to include the constraint effect in the failure assessment procedure is becoming more and more established [9]. In order to apply the *
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0013-7944/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 1 ) 0 0 1 4 9 - 7
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Fig. 1. Specimen geometries, co-ordinates and notation: (a) SECP, (b) DECP and (c) CCP.
two-parameter fracture mechanics methodology, it is important to provide T-stress solutions for the crack configuration under consideration. Especially accurate T-stress solutions for test specimens are important for the collection of appropriate constraint-dependent fracture toughness data. Similar to the stress intensity factors (SIFs), the values of the T-stress strongly depend on the type of loading as well as on the relative crack length and the overall geometry. Several numerical/analytical methods were developed to calculate the elastic T-stresses for test specimens. Many researchers (Leevers and Radon [10], Cardew et al. [11], Kfouri [12] and Sham [13]) have provided T-stress solutions for uniform tension or bending loading conditions. Non-linear stress distributions can occur as consequence of thermal stress, residual stress and stress concentration. The T-stress solutions under these non-linear stress distributions are required. There are some calculations for specific non-linear loading/crack geometries (Fett [14] and Hooton et al. [15]). However, detailed parametric solutions of T-stress for cracked specimens under non-uniform stress distributions are not available. Weight function method is a very powerful well-known method for calculating SIFs under arbitrary stress distributions. In Refs. [13–15], analogy to the method for the SIF, the weight function method for the calculation of T-stress is developed. It is a very powerful technique since it removes restrictions on the stress distributions. Sham [13] provided methods for the determinations of weight functions for T-stress numerically; while Fett [14] and Hooton et al. [15] demonstrated methods for the derivation of weight functions from reference T-stress solutions. In the present paper, detailed finite element analyses were conducted to calculate the T-stress solutions for single edge cracked plate specimen (SECP), double edge cracked plate specimen (DECP) and centre cracked plate specimen (CCP). Fig. 1 demonstrated specimens analysed. The T-stress solutions are presented with a=t values of 0.2, 0.4, 0.6 and 0.8. Uniform, linear, parabolic or cubic stress distributions are applied to the crack face. The T-stress results for uniform and linear stress distributions in the specimens were used to derive weight functions for T-stress for the corresponding specimens. The weight functions were then verified using the results for several linear and non-linear stress distributions.
2. Finite element analysis for cracks in test specimens 2.1. Finite element model Two-dimensional finite elements were used to model the symmetric half or quarter of the SECP, DECP and CCP specimen. Fig. 1 shows the geometry, notation and co-ordinates system used. The finite element
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Fig. 2. Typical finite element mesh.
analyses were made using A B A Q U S version 5.8 [16] with eight-noded isoparametric solid elements. In order to model the square root singularity at the crack tip, elements with two mid-side nodes at the quarter points (a degenerate quadrilateral element with one line collapsed) were used and the separate crack tip nodal points were constrained to have the same displacement [17]. The loads were applied directly to the crack face. Four types of loading were applied to each crack geometry, with the following stress distributions: x n rðxÞ ¼ r0 1 ð1Þ a with n ¼ 0–3, r0 is the nominal stress and a is the crack depth (for edge crack specimen) or half crack length (for centre crack specimen). The h=t ratio is fixed to 3 for all analysis in the current calculation. A typical finite element mesh is plotted in Fig. 2. 2.2. Extraction of the T-stress The interaction integral method introduced by Cardew et al. [11], Kfouri [12] and Nakamura and Parks [18] is used to extract the elastic T-stress in the current analysis. For a plane strain problem, assume a load
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Fig. 3. A point force f applied at the crack tip, the auxiliary field for the interaction integral.
force of magnitude f applied at the crack tip in the direction parallel to the crack as illustrated in Fig. 3. Using superscript ‘L’ to designate the stress and displacement fields, the analytical solution gives [19] f f cos3 h rL22 ¼ cos h sin2 h pr pr f rL12 ¼ cos2 h sin h pr
ð2aÞ
1 t2 f r sin2 h ln ¼ þ d 2ð1 tÞ E p 1þt f uL2 ¼ fð1 2tÞh cos h sin hg 2E p
ð2bÞ
rL11 ¼
and uL1
where the subscripts 1, 2 suggests a local Cartesian co-ordinate system; r and h are the local polar coordinates; d is a reference distance; E and t are the Young’s modulus and Poisson’s ratio of the material, respectively. Employing the above solution as an auxiliary field, Cardew et al. [11] and Kfouri [12] extracted the T-stress for 2D crack problems by introducing an interaction J-integral based on Eshelby’s theorem. Nakamura and Parks [18] extended this method to 3D crack problems and provided the domain integral formulations. Combined with finite element method, it was shown to be a simple and effective method for the determination of the T-stress of 2D and 3D crack specimens. For 2D crack problems, the interaction integral can be written in the domain integral form: Z 1 ouL oui oqk oqk I¼ rij i þ rLij rLij eij dS ð3Þ Da S oxk oxk oxj oxk where Da is the virtual crack advance, S is an area which encloses the crack tip, qk defines the virtual extension of the crack tip; rij , eij and ui are the stress, strain and displacement components of the 2D crack problem; rLij , eLij and uLi are the corresponding components in the point load auxiliary solution given by Eqs. (2a) and (2b). The crack tip T-stress is related to I by (Kfouri [12]) T ¼
E I 1 t2 f
ð4Þ
The computation of the domain integral, Eq. (3), is readily compatible with finite element formulation [18,20]. Once I is determined, the T-stress can be obtained from Eq. (4) directly.
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A B A Q U S (version 5.8) was used to solve the finite element model. A post processing F O R T R A N program was developed to interface with A B A Q U S to calculate the domain-integral, Eq. (3), from Gauss integration scheme. Eight-noded two-dimensional quadrilateral elements with 3 3 Gauss integration were used in the finite element models. A ‘pyramid’ qk function [21] was adopted to simulate the virtual crack extension. The point-load auxiliary field was derived from Eqs. (2a) and (2b) by taking f ¼ 1. Three contours are used to calculate I. Good independence of the I value on the choice of contour was obtained. In the current calculation, the average value of I results from outer two contours are used to calculate crack tip T-stress.
2.3. Verification of the finite element model In order to verify the finite element models and the method of extracting T-stress values, T-stress values for cracks in SECP, DECP and CCP under far-end uniform tension loading were calculated. Comparisons were made with results from Sham [13] for SECP and Kfouri [12] for DECP and CCP specimens. The results for SECP specimen are presented in Table 1, the maximum difference was 3.62% and most are within 2%. The results for DECP and CCP specimens are presented in Table 2, the maximum difference is within 2.21% for DECP specimens and 5.04% for CCP specimens. Through these verifications, the present method is considered suitable for analyses of T-stress for cracks in test specimens. 2.4. Finite element results The T-stress values for cracks in SECP, DECP and CCP specimens with a=t of 0.2, 0.4, 0.6 or 0.8 subjected to constant, linear, quadratic or cubic stress distributions as expressed in Eq. (1) have been determined. The results of normalised T-stress, V ¼ T =r0 , are summarised in Tables 3–5. Table 1 Comparison of normalised T-stress solutions, T =r0 , from present FEM calculation and solution from Sham [13] for single edge cracked plate (SECP) specimen under far-end tension a=t
Solution from [13] V
Present calculation V
Difference (%) 100 jV V j=V
0.2 0.4 0.6 0.8
0.5918 0.5853 0.0278 5.9755
0.5949 0.5782 0.0268 5.8298
0.53 1.20 3.62 2.44
Table 2 Comparison of normalised T-stress solutions T =r0 , from present FEM calculation and solution from Kfouri [12] for DECP and CCP specimen under far-end tension a=t 0.2 0.4 0.6
Solution from [12] V
Present calculation V
Difference (%) 100 jV V j=V
DECP
CCP
DECP
CCP
DECP
CCP
0.5276 0.5541 0.5788
1.0277 1.1363 1.3803
0.5312 0.5429 0.5659
1.0289 1.1394 1.4484
0.69 2.01 2.21
0.11 0.27 5.04
Table 3 Normalised T-stress, V ¼ T =r0 , for SECP specimen under crack face pressure rðxÞ
a=t ¼ 0:2
a=t ¼ 0:4
a=t ¼ 0:6
a=t ¼ 0:8
Constant Linear Parabolic Cubic
0.4050 0.2831 0.2205 0.1813
0.4217 0.2953 0.2300 0.1889
1.0405 0.7019 0.5328 0.4300
6.8297 4.5876 3.4559 2.7714
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Table 4 Normalised T-stress, V ¼ T =r0 , for DECP specimen under crack face pressure rðxÞ
a=t ¼ 0:2
a=t ¼ 0:4
a=t ¼ 0:6
a=t ¼ 0:8
Constant Linear Parabolic Cubic
0.4687 0.3216 0.2482 0.2028
0.4570 0.3167 0.2454 0.2010
0.4340 0.3065 0.2393 0.1969
0.3999 0.2955 0.2353 0.1957
Table 5 Normalised T-stress, V ¼ T =r0 , for CCP specimen under crack face pressure rðxÞ
a=t ¼ 0:2
a=t ¼ 0:4
a=t ¼ 0:6
a=t ¼ 0:8
Constant Linear Parabolic Cubic
0.02890 0.01713 0.01163 0.00871
0.1394 0.0802 0.0554 0.0421
0.4484 0.2557 0.1771 0.1350
1.5689 0.8652 0.5911 0.4474
The T-stress for an application can be obtained by superposition of these solutions provided the stress distribution has been expressed as a polynomial of order three. However, the weight function for T-stress derived in the following will remove any restrictions on the stress distributions. 3. Weight functions for T-stresses Finite element calculations of T-stress for cracks in test specimens with four kinds of stress distribution have been presented in Section 2. However, it is impossible to foresee the variety of loading occurring in engineering applications, especially when complicated transient or residual stresses are involved. The weight function method is one of the most efficient methods to derive SIFs for complex stress distributions [22,23]. This method was also applied by several researchers [13–15] to derive T-stress solutions for complex stress distributions. In this section, the theoretical background is reviewed and weight functions for T-stress in cracked specimens are then derived and validated. 3.1. Theoretical background For mode I crack problems, using the superposition method, it can be demonstrated (Chen [24]) that for a cracked body as shown in Fig. 4(a), loaded by a stress field Q, the T-stress is the superposition of Tstresses for two cases. The first is the T-stress for the same cracked body loaded by a crack face pressure, rðxÞ, induced by the remote stress field Q in the uncracked body, as shown in Fig. 4(b), the second case is the T-stress in the uncracked body under the remote stress field Q (Fig. 4(c)). Therefore, the T-stress for the problem shown in Fig. 4(a) can be calculated from: T ¼ Tcrack pressure þ Tuncrack
ð5Þ
The T-stress for a cracked body with loading applied to the crack surface can be calculated by integrating the product of the weight function tðx; aÞ and the stress distribution, rðxÞ, on the crack plane: Z a Tcrack pressure ¼ rðxÞtðx; aÞ dx ð6Þ 0
where rðxÞ is the stress distribution on the crack face and tðx; aÞ is the weight function for T-stress (it is alternatively called second order weight function by Sham [13], or Green’s function for T-stress by Fett
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Fig. 4. Weight function for T-stress: (a)–(d).
[14]). The weight function for T-stress depends on the crack geometry and is independent to the loading conditions. Mathematically, the weight function, tðx; aÞ, is the Green’s function for the T-stress. It represents the T-stress at the crack tip for pair of unit point loads acting on the crack face at the location x as shown in Fig. 4(d). The T-stress for an uncracked body with applied load system Q, as shown in Fig. 4(c), can be found from the stresses at the crack tip location in the uncracked body [24]: Tuncrack ¼ ðrx ry Þx¼a ð7Þ
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Note that the Tuncrack here represents the T-stress at the crack tip under load system Q when the crack is completely closed (Fig. 4(c)), it is different from the T-stress limit value when a approaches zero. It depends on the crack half-length a as demonstrated in Eq. (7). Substituting Eqs. (6) and (7) into Eq. (5), the T-stress for cracked body as shown in Fig. 4(a), loaded by a stress field Q, is obtained: Z a T ¼ rðxÞtðx; aÞ dx þ ðrx ry Þx¼a ð8Þ 0
Eq. (8) provides very efficient way for calculating T-stress. From alternative derivations, Eq. (8) was also obtained by Sham [13], Fett [14] and Hooten et al. [15]. Once the weight function for T-stress tðx; aÞ is determined, and a stress analysis of uncracked body is conducted to obtain rðxÞ and ðrx ry Þx¼a , the corresponding T-stress can be calculated for any loading condition. There are different ways to obtain the weight function for T-stress, tðx; aÞ. Sham [13] extended Bueckner’s [22] theory of fundamental fields to the determination of T-stress. And it was demonstrated that tðx; aÞ represents a combination of displacement from fundamental field and displacement from regular field [13]. Using a unified finite element method, Sham [13] obtained the numerical solution for tðx; aÞ. Recently, Fett [14] and Hooten et al. [15] have proposed closed form approximations for tðx; aÞ, and the tðx; aÞ can be obtained from reference T-stress solutions. The approach of Fett [14] is used here. 3.2. General forms for weight function for T-stress Based on the analogy to the weight function for SIFs, Fett [14] proposed to use a form, which can be re-written in the following format, to approximate the weight function tðx; aÞ:
2 x 1=2 x 3=2 x 5=2 x n1=2 tðx; aÞ ¼ D1 1 þ D2 1 þ D3 1 þ þ Dn 1 ð9Þ pa a a a a where crack tip is at x ¼ a, and D1 ; D2 ; . . . and Dn are constants for given crack geometry. In Ref. [14], the following two-terms expression was then used to approximate the weight functions for T-stress for edge cracks:
2 x 1=2 x 3=2 D1 1 tðx; aÞ ¼ þ D2 1 ð10Þ pa a a where D1 and D2 can be decided from two reference T-stress solutions. This weight function form of Eq. (10) is used in the current research to derive weight functions for cracked specimens. Note that Eqs. (9) and (10) were proposed in Ref. [14] for edge cracks, a different functional form was proposed for internal cracks in Ref. [25]. In the current research, Eq. (10) is used for both edge (SECP, DECP) and internal (CCP) cracks, and it will be shown later that excellent accuracy is achieved for both edge and internal cracks. 3.3. Weight function for test specimens Two reference solutions for T-stress are used to decide D1 and D2 in Eq. (10): uniform crack face pressure and linear decreasing stress distribution corresponding to n ¼ 0, and n ¼ 1 in Eq. (1), respectively. 3.3.1. Reference T-stress solutions The numerical T-stress results for cracks in SECP, DECP and CCP specimens presented in Section 2 were approximated by empirical formulae fitted with an accuracy of 1% or better for SECP and DECP specimens and 2% or better for CCP specimens. The results for a=t ¼ 0 were obtained by smooth ex-
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trapolation. The choice of equations for the extrapolation was based on engineering judgement. The range of applicability for these equations is 0 6 a=t 6 0:8. The results for the uniform crack face pressure: rðxÞ ¼ r0
ð11Þ
T1r ¼ r0 V0
ð12Þ
are where V0 represents the normalised T-stress, and it is a function of a=t. The results for a linearly decreasing crack face pressure: x rðxÞ ¼ r0 1 ð13Þ a are T2r ¼ r0 V1
ð14Þ
where V1 represents the normalised T-stress for this loading condition. The expressions for V0 and V1 for SECP, DECP and CCP specimens are presented as follows. For SECP specimen a 8:0252 V0 ¼ 0:4012 þ 38:5340 t a 8:0877 V1 ¼ 0:2813 þ 26:1749 t for DECP specimen V0 ¼ 0:4729 0:09814 V1 ¼ 0:3239 0:05101
a 2 t a 2 t
0:02501 þ 0:01011
a 4 t a 4 t
for CCP specimen V0 ¼
V1 ¼
1 a a2 2:7206 0:4774 exp 1:4690 t t 1 a a2 4:5984 0:8062 exp 2:5343 t t
3.3.2. Weight functions for T-stress By substituting Eqs. (11)–(14) into Eq. (8), two equations with two unknowns were established. Note that the uncracked T-stresses are zero in Eq. (8) for those crack pressure cases (no external load Q is applied and therefore rx and ry are zeros in Eq. (7)). The parameters in the weight function expression were solved and are: pð5V0 7V1 Þ ð15Þ D1 ¼ 15 16 D2 ¼ 165 pð35V1 21V0 Þ The weight function for T-stress can then be determined from Eq. (10).
ð16Þ
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3.4. Validation of weight functions In this section, the weight functions were verified using T-stress solutions for several linear and nonlinear stress fields. 3.4.1. Two non-linear stress distributions First, the weight functions derived above for SECP, DECP and CCP specimens were validated using finite element results for two non-linear stress fields. Using Eq. (8), T-stress values were calculated for the following crack face stress distributions: x 2 rðxÞ ¼ r0 1 a
ð17Þ
x 3 rðxÞ ¼ r0 1 a
ð18Þ
Note that under those crack face pressure, the uncracked T-stress is zero, therefore Eq. (8) becomes Eq. (6). The T-stress results calculated from the derived weight functions and from the present finite element calculations for the above stress distributions for all the test specimens are shown in Figs. 5–7. The differences between the normalised T-stress values were less than 2.4%, 2.1% and 3.3% for SECP, DECP and CCP specimens, respectively. As a comparison, the T-stress results for constant and linear stress distributions Eqs. (11) and (13) are also plotted in Figs. 5–7.
Fig. 5. Comparison of weight function based T-stress and FEM data, SECP specimen.
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Fig. 6. Comparison of weight function based T-stress and FEM data, DECP specimen.
Fig. 7. Comparison of weight function based T-stress and FEM data, CCP specimen.
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3.4.2. Far-field tension and bending For SECP specimens, the weight functions were then used to calculate the T-stress values under far-field tension and bending cases. Note that the uncracked T-stresses are not zeros in those cases. For far-field tension, Tuncrack ¼ ðrx ry Þ ¼ r0 ð19Þ x¼a
and crack face pressure rðxÞ ¼ r0 where r0 is the nominal stress for tension. For far-field bending, a Tuncrack ¼ ðrx ry Þx¼a ¼ r0 1 2 t and crack face pressure a a x rðxÞ ¼ r0 1 2 þ r0 2 1 t t a
ð20Þ
ð21Þ
ð22Þ
where r0 is the nominal stress for bending. The T-stress results calculated from the derived weight functions using Eq. (8) and from finite element calculations from [13] for SECP are shown in Fig. 8. Excellent agreements are achieved. 3.4.3. Comparison with Fett’s results In Ref. [14], the T-stress for single edge cracked plates under the following thermal stress field were calculated:
Fig. 8. Comparison of weight function based T-stress and results from Sham [13] for far-end tension and bending loads, SECP specimen.
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Fig. 9. Comparison of weight function based T-stress and results from Fett [14] for non-linear thermal stress distribution, SECP specimen.
rðxÞ ¼ r0
x x 2 2 4 þ4 3 t t
ð23Þ
The currently derived weight functions for SECP specimens are used to calculate the T-stress for this stress distribution using Eq. (8). Fig. 9 shows the comparison between the current weight function prediction and the results from [14]. Good agreement is achieved.
4. Conclusions Finite element analyses have been conducted to calculate the T-stress for cracks in single edge cracked, double edge cracked, and centre cracked plates. Constant, linear, quadratic and cubic stress distributions were applied to the crack face. The analysis procedures and results were verified with existing solutions in the literature. Based on the present finite element calculations for T-stress in the test specimens, weight functions for Tstress were derived. They are valid for 0 6 a=t 6 0:8 for all the test specimens. The weight functions were verified using T-stress solutions for several linear and non-linear stress fields. Excellent agreements (maximum difference less than 3.3%) are achieved. The weight functions can be used to calculate T-stress for any given stress field rðxÞ. These closed form weight functions for T-stress derived here make them suitable for fracture analysis, for example, to account for crack tip constraint under local non-linear residual stresses, using the method demonstrated in Refs. [9,15].
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Acknowledgements The author gratefully acknowledges the financial supports from the Natural Science and Engineering Research Council (NSERC) of Canada and Materials and Manufacturing Ontario (MMO). He would also want to thank Hibbitt, Karlsson & Sorenson, Inc. for making A B A Q U S available under an academic license to the Carleton University.
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