A new expression for crack opening stress determined based on maximum crack opening displacement under tension compression cyclic loading

Transcription

1 ORIGINAL CONTRIBUTION doi: /ffe A new expression for crack opening stress determined based on maximum crack opening displacement under tension compression cyclic loading J. J. CHEN, M. YOU and Y. HUANG School of Naval Architecture, Dalian University of Technology, Dalian , Liaoning China Received Date: 3 January 2017; Accepted Date: 29 April 2017; Published Online: 10 July 2017 ABSTRACT The maximum crack opening displacement is introduced to investigate the effect of compressive loads on crack opening stress in tension compression loading cycles. Based on elastic plastic finite element analysis of centre cracked finite plate and accounting for the effects of crack geometry size, Young s modulus, yield stress and strain hardening, the explicit expression of crack opening stress versus maximum crack opening displacement is presented. This model considers the effect of compressive loads on crack opening stress and avoids adopting fracture parameters around crack tip. Besides, it could be applied in a wide range of materials and load conditions. Further studies show that experimental results of da/dn ΔK curves with negative stress ratios could be condensed to a single curve using this crack opening stress model. Keywords compressive load effect; crack opening stress; finite element analysis; maximum crack opening displacement; tension compression loading. NOMENCLATURE a = initial half crack length E = Young s modulus k = slope = maximum stress intensity factor K op = crack opening stress intensity factor L = half-length of plate L e = the smallest element size in crack tip n = strain-hardening exponent of material R = stress ratio, R = σ min /σ max r y = crack-tip plastic zone size r p = crack-tip forward plastic zone size = maximum crack opening displacement corresponding to σ max (MCOD) = maxmum crack opening displacement corresponding to σ min W = half-width of plate = effective stress intensity factor range ΔK = stress intensity factor range Δu = variable quantity of maximum crack opening displacement (VMCOD) σ op = crack opening stress (COS) σ s = yield stress = maximum stress in the cyclic tensile load, which is positive = minimum stress in the cyclic tensile load, which is negative = crack opening stress when the remote applied minimum stress is zero ν = Poisson s ratio K max u max u min ΔK eff σ max σ min σ op_0 Correspondence: Y. Huang. huangyi@dlut.edu.cn 29

2 30 J. CHEN et al. INTRODUCTION Compressive loads play important roles in fatigue crack behaviour. Crack would be closed within the compressive loading stage, but compressive loads still contribute to fatigue crack propagation as confirmed by a great amount of significant studies. Iranpour 1 recommended that the compressive stress cycles should be considered when establishing the fatigue crack growth rate of materials. Zhang 2 4 confirmed that the compression part of the load cycle had a significant effect on the near crack-tip parameters. Besides, Benz 5 revealed that the crack closure behaviour was in good correlation with cyclic plastic deformations at the crack tip at negative stress ratios. Based on plasticity induced crack closure (PICC) as proposed by Elber, 6 the crack opening stress (COS), corresponding to the load at which the contact between the crack surfaces is broken, is closely correlated with fatigue crack propagation rate. Thus, in order to more accurately predict crack propagation rate, it is necessary to identify the magnitude of the effect of compressive loads on COS in tension compression loading cycles. Previously, a great amount of studies focused on COS in cyclic tensile loads have been developed. For example, Nakagaki 7 proposed a finite element (FE) procedure to analysis crack closure effect, while Peeker 8 introduced a numerical model and Savaidis 9 used an analytical procedure. Besides analytical or numerical approaches, Vormwald 10 used an experimental method to study COS. Recently, Correia 11 and Blason 12 ameliorated the fatigue crack growth model with the consideration of COS. Thereafter, a theoretical model that considered crack closure effect was applied to the P355NL1 steel by Correia. 13 COS is widely investigated under cyclic tensile loads but less studied under compressive loading conditions. For compressive loads, previously, Schijve, 14 Koning, 15 Newman 16 and Lang 17 proposed empirical equations to determine COS, which included negative stress ratios. Some researchers found that maximum stress intensity factor (SIF) rather than the stress ratio was the controlling parameter of COS. Antunes 18 presented empirical models to calculate COS based on maximum and total range of SIF; Lei 19 suggested that COS depended on maximum SIF and crack-tip constraint conditions. Correia 13 also used the maximum SIF to estimate the COS at different stress ratio levels. Other parameters were also used to investigate the fatigue crack propagation behaviour under tension compression cyclic loading. For example, Cauthen 20 computed crack growth rate from crack-tip opening displacement and used compression pre-cracking method to measure the fatigue crack threshold. Hu 21 established a crack closure model for GH2036 superalloy in terms of residual crack-tip opening displacement and stress ratio. Besides, other researchers used numerical analysis to obtain the residual stresses in crack tip to characterize the crack closure effects. Glinka calculated the residual stresses to assess the actual crack driving force based on local strain-based approaches 25 ; Correia and his collaborators 29,30 also made several important studies on this residual stress-based crack propagation model. However, Silva 31 reported that COS became negative for some materials such as CK45, Al7175 and crack closure concept even could not well explain fatigue crack growth at negative stress ratios. Crack opening stress is difficult to obtain in engineering practice, because it depends a plenty of parameters such as crack geometry size, stress ratio, material property and strain hardening. Up to now, researchers have not reached a unified agreement on the adequate models to determine COS in tension compression loading cycles. Most empirical equations earlier are limited to the experimental range of material types and stress ratios. The accuracy of experimental approach results depends highly on experimental measurement location and the technique employed. 32,33 Furthermore, most numerical analysis of COS earlier is associated with the calculations of crack-tip stress field and displacement field, but relevant fracture parameters in crack-tip field could not be conveniently obtained in practice. Hence, to accurately obtain COS in tension compression loading cycles in practice, more convenient fracture parameters that quantify the damage processes occurring at the crack tip are in desperate need. In this work, the effect of compressive loads on COS based on the maximum crack opening displacement (MCOD) is quantitatively investigated. Based on elastic plastic FE analysis of centre cracked finite plate and accounting for the effects of applied load, crack geometry size, Young s modulus, yield stress and strain hardening, the explicit expression of the COS versus MCOD is presented (COS MCOD model). In addition, the expression is applicable under small-scale and large-scale yielding. Afterwards, the COS MCOD model is applied to calculate the effective SIF range. And the results show that da/dn ΔK curves of different materials under tension compression cyclic loading are condensed to a single curve. This COS MCOD model could be applicable to centre cracked finite plates with different materials under tension compression cyclic loading. THEORETICAL CONSIDERATIONS Researchers have made several attempts on the understanding of crack closure mechanisms under tension compression cyclic loading, namely, PICC and roughness induced crack closure (RICC). Pommier 34

3 A NEW EXPRESSION FOR CRACK OPENING STRESS 31 found that the COS decreased with the increase of maximum stress at negative stress ratios; he proved that the COS depended strongly on the plasticity at the crack tip. Peeker 8 proposed crack closure model that contained any distribution of manufacture-introduced residual stress. Recently, Antunes 35 investigated the main numerical parameters affecting PICC. The importance of RICC on fatigue crack propagation has also been investigated; for example, Akhtar 36 proposed that the superior fatigue crack growth resistance of the weldment could be attributed to the occurrence of RICC mechanism. Silva 31 made roughness measurement experiments and concluded that at negative stress ratios, RICC was not a relevant mechanism. From the perspective of PICC, the crack closure is linked closely to the monotonic and reversed plastic deformation occurring at the crack tip; therefore, COS has a certain relationship with crack-tip plastic zone that is simply expressed as σ op e r y (1) where σ op is COS, r y is crack-tip plastic zone. In our previous work, 37 we proposed that monotonic plastic zone size ahead of crack could be determined by MCOD during loading; the crack-tip reverse plastic zone size could be determined by variable quantity of MCOD (VMCOD). Hence, the crack-tip plastic zone is linked to VMCOD under tension compression cyclic loading; the relationship can be simply and distinctly expressed as r y e Δu (2) According to Eqs (1) and (2), it is clear that COS is related to VMCOD during the crack propagation and the relationship between COS and VMCOD could be simply and markedly written as σ op e Δu (3) In Eq. (3), Δu represents the VMCOD and can be written as Δu ¼ u max u min (4) u max and u min represent the resulting MCOD perpendicular to crack surfaces corresponding to the maximum stress σ max and minimum stress σ min of the applied load in Fig. 1a. Fig. 1 (a) Definition of maximum crack opening displacement. (b) Geometry of investigated specimen. FINITE ELEMENT ANALYSIS Establishment of finite element models A two-dimensional FE model is used. The geometric dimensions of a centre cracked specimen under uniaxial uniform loads are shown in Fig. 1b, half-width of the plate W = 40 mm, half-length of the plate L = 40 mm, half initial crack length a = 4 mm. The material is assumed to be elastic-perfectly plastic, and the yield stress σ s = 480 MPa, Young s modulus E = 72GPa, Poisson s ratio ν = 0.3. The two-dimensional elastic plastic FE analysis of the specimen is performed in ANSYS Eight-node quadrilateral elements, target elements and contact elements are used for this model. Plane stress condition is assumed. In view of the symmetry, the FE model only consists of half of the geometry. Moreover, node-release scheme at the maximum load of a loading cycle is used to model crack propagation. When the stresses in crack tip change from compressive to tensile, the crack is taken to be fully open. And the remote applied stress is defined as the COS when the stresses at the crack tip pass through zero. The load is applied on the plate in the direction of perpendicular to the crack surface. A mesh convergence analysis is performed with the FE model. The load level σ max /σ 0 = 0.3; R = 0 is employed to the model. The smallest element size in crack tip is characterized by L e /r p, where L e is the minimum element size and r p is the crack tip forward plastic zone. L e /r p ranges from 1/3 to 1/20. When the smallest element size L e /r p in crack tip is less than 1/10 (Fig. 3), the stable value of COS converges. In this work, the smallest element size in crack tip is 1/10 of the forward plastic zone. McClung 38 also suggested the criteria that the crack-tip forward plastic zone had 20 triangle elements or 10 four-node quadrilateral elements. The FE model has elements and nodes.

4 32 J. CHEN et al. Validation of the finite element model To confirm the analysis, the stress levels σ max /σ s ranging from 0.1 to 0.6 are employed to the model shown in Fig. 2. Figure 4a illustrates the typical transient performance of COS during crack propagation. The COS increases as crack grows and then gradually attains a stable value. According to McClung s 39 numerical results, the crack must be advanced a distance on the order of the original crack-tip plastic zone size before opening levels are valid in FE simulations. Figure 4b shows the obtained stable COS at different load levels. Compared with the previous research, 40 the data show a good consistency, which give more confidence to the FE model used in this work. Correlation between crack opening stress and maximum crack opening displacement In our previous work, the relationship between COS and VMCOD under cyclic tensile loading (stress ratio R 0) could be expressed as Eq. (5). σ op σ s ¼ 0:024 þ0:172! 2 ΔuE aσ s ð1 þ nþ 2! ΔuE aσ s ð1 þ nþ 2 0:015 where 0.2 ΔuE/aσ s (1 + n) 2 3.2, Δu is VMCOD, nis the strain hardening exponent, E is Young s modulus and a is half initial crack length. (5) Fig. 3 The transient response of crack opening stress at different mesh density. The MCOD of FE model under tension compression loading (σ max = 192 MPa, σ min = 96 MPa) is calculated. Figure 5 shows that the values of MCOD virtually become zero during the compressive loading stage; therefore, only the values of VMCOD could be obtained during tensile loading stage. Consequently, the relationship between COS and VMCOD in the previous work is not appropriate for tension compression loading; hence, a quantitative investigation will be made on the effect of compressive loads on COS. In the calculation model as shown in Fig. 2, σ min varies in the range of 384 to 0 MPa, σ max = 192 MPa; the calculated results of stable COS at different Fig. 2 Finite element discretization of the model. [Colour figure can be viewed at wileyonlinelibrary.com]

5 A NEW EXPRESSION FOR CRACK OPENING STRESS 33 Fig. 4 Illustrations on the calculation of crack opening stress (COS). (a) The transient response of COS at R =0,σ max /σ s = 0.3. (b) Comparison of stable COS against normalized maximum stress. same trend is in accord with the numerical results of Antunes 41 and experimental ones of Lang 42 and Yu 43 : The crack propagation SIF of Al 7475-T7351 and the COS intensity of 2024-T351 aluminium alloy decreased linearly with increasing magnitude of the compressive peak stress. In fact, the increase of compressive peak stress results in the increase of reversed plastic deformation at the crack tip, which then leads to the decrease of residual plastic deformation and also the COS. Fig. 5 The calculated maximum crack opening displacement in tension compression loading cycles. 8 >< σ op =σ s ¼ 0:21ðσ min =σ s >: σ op 0 =σ s ¼ 0:024 Þþσ op 0 =σ s! 2 aσ s ð1 þ nþ 2 þ 0:172 aσ s ð1 þ nþ 2! 0:015 (6) loading conditions are shown in Fig. 6b. The COS decreases linearly with the increasing of compressive peak stress σ min, and the linear relationship between COS and minimum stress is depicted as Eq. (6). The where σ min is minimum stress and σ op_0 represents COS corresponding to the loading case of σ min = 0. It can be recognized that σ op_0 can be determined by MCOD by Eq. (5) for the loading case of σ min = 0, because u min =0, Fig. 6 Illustration of the relationship between crack opening stress and minimum stress. (a) Loading history of different loading conditions. (b) The liner relationship between σ op /σ s and σ min /σ s.

6 34 J. CHEN et al. Δu = u max ; therefore, COS can be easily calculated based on Eq. (6). RESULTS AND DISCUSSION Effect of maximum stress Equation (6) is established on the basis of σ max = 192 MPa; further studies are thus conducted to investigate the effect of maximum stress. The parameters of the calculation models in Fig. 2 are a = 4 mm, L = 40 mm, W = 40 mm, E = 72 GPa, σ s = 480 MPa and ν = 0.3. σ max ranges from 120 to 300 MPa, and the relationship between COS and minimum stress is calculated for each case. A linear relationship between σ op /σ s and σ min /σ s can be established at different σ max. 41 Table 1 shows the values of the parameters k and σ op_0 /σ s in Eq. (7) at different σ max. σ op =σ s ¼ kðσ min =σ s Þþσ op 0 =σ s (7) where k is the slope of liner expression. Table 1 shows that σ max is linked to slope k. The slope k increases as σ max increases. This result is in accord with the results obtained by other researchers. Antunes 41 observed that the slope of the linear trend decreases with the reduction of maximum SIF. Silva 31 also showed that COS varied with maximum SIF at the same negative R ratio. When comparing with σ op_0 /σ s and σ max, the increase of σ max results in the increase of u max, which then leads to an elevated σ op_0 /σ s as indicated by Eq. (6); σ op_0 /σ s and σ max have a close relationship with each other. As a result, the slope k can be associated with σ op_0 /σ s through σ max. Here, k and σ op_0 /σ s fit well with a curve shown in Fig. 7, and an expression of this function is given as Fig. 7 Relationship between the slope k and σ op_0 /σ s. Eq. (8) by fitting FE analysis data. And the least square method is adopted. 2:46 k ¼ 10:2 σ op 0 =σ s (8) The application conditions of Eq. (8) are σ s = 480 MPa and 0.25 σ max /σ s 0.625; thus, Eq. (8) is applicable under small-scale and large-scale yielding. Here, the σ op_0 /σ s can be determined by MCOD using Eq. (5). Then, the slope k could be calculated based on the obtained value of σ op_0 /σ s using Eq. (8). The relationship between σ op /σ s and σ min /σ s has also been established by Eq. (7). Depending on the previous analysis, the COS in tension compression loading cycles could be calculated by Eq. (9). 8 σ op =σ s ¼ kðσ min =σ s Þþσ op 0 =σ s >< 2:46 k ¼ 10:2 σ op 0 =σ s! 2 >: σ op 0 =σ s ¼ 0:024 aσ s ð1 þ nþ 2 þ 0:172! aσ s ð1 þ nþ 2 0:015 (9) Table 1 Results of k and σ op_0 at different σ max σ max /MPa k σ op_0 /σ s However, COS is influenced by yield stresses, Young s modulus, strain hardening and crack geometry. These effects on σ op_0 have been considered in previous work. It is note that the relationship between the slope k and σ op_0 /σ s is crucial to determine COS in tension compression loading cycles. Hence, in order to expand the applied range of Eq. (9), particular attention is paid to the effects of these parameters on the relationship between k and σ op_0 /σ s. Effect of yield stress To investigate the effect of yield stress, the parameters of this calculation model are a = 4 mm, L = 40 mm, W = 40 mm and E = 72 GPa. σ max ranges from 60 to

7 A NEW EXPRESSION FOR CRACK OPENING STRESS MPa, and the yield stress σ s = 100, 200, 250, 300, 350, 400, 480 and 600 MPa. Slope k and σ op_0 /σ s are calculated at different yield stress. Figure 8 shows the values of k and σ op_0 /σ s at different yield stress. It could be seen that slope k and normalized σ op_0 /σ s fit well with a curve, which can be expressed as Eq. (8). Effect of Young s modulus For the model shown in Fig. 2, σ max = 192 MPa, σ min = 192 MPa, a = 4 mm, L = 40 mm, W =40mm and σ s = 480 MPa; Young s modulus varies in the range of GPa. Figure 9 shows the transient response of COS at different Young s modulus. It is evident that COS is nearly not influenced by Young s modulus. The same trend could be noted in the literature, 44 which proposed that the ratio of yield stress to elastic modulus had essentially no effect on COS. Effect of initial crack length The crack geometry may be characterized by the nondimensional form a/w. In the finite plate model, E = 72 GPa, σ s = 480 MPa, ν = 0.3, σ max = 192 MPa, σ min = 192 MPa and W = 40 mm; a/w ranges from 0.05 to 0.3. The COS at different a/w is calculated by FE method and Eq. (9) presented earlier. Table 2 shows both the results of COS estimated on the basis of MCOD by Eq. (9) and those calculated with FE simulations. When a/w 0.3, the MCOD increases with the increasing of initial crack length; nevertheless, the stable value of COS hardly changes. The size of the plastic zone relative to initial crack length changes very little. 39 And the results of COS by finite element method analysis are compared with by MCOD through Eq. (9); the relative error is within 3%, indicating that the proposed Eq. (9) has a high precision in the case of a/w 0.3. Effect of strain hardening The materials presented in the previous work are assumed to be elastic-perfectly plastic. In previous research, 31,45 the effect of compressive loads on crack closure behaviour was linked to the strain hardening. To study the straining hardening effect on COS, kinematic power-law model 37 of relationship between the stress and strain [Eq. (10)] has been adopted. σ ¼ ( σ n s 1 þ ε p =ε s σ > σ s Eε s σ σ s (10) Fig. 8 Relationship between the slope k and σ op_0 /σ s at different σ s. Fig. 9 The transient response of crack opening stress at different Young s modulus. where σ is the flow stress, σ s is the yield stress, ε p is the plastic strain, ε s = σ s /E is yield strain and n is the strain hardening exponent. The calculation model has a = 4 mm, L = 40 mm, W = 40 mm, E = 72 GPa and σ s = 480 MPa. The applied load σ max ranges from 60 to 240 MPa. The model has a series of strain hardening exponent n = 0.1, 0.2 and 0.3. The relationship between k and σ op_0 /σ s is calculated corresponding to different strain hardening exponent as shown in Fig. 10a. An obvious difference could be seen between the elastic-perfectly plastic material and strain hardening material. With the increase of strain hardening exponent, the slope k decreases at same σ op_0 / σ s. And the strain hardening contributes to increased COS. The Tzamtzis 46 research is available to support this conclusion. Accounting for the strain hardening effects, 1/(1 + n) 2.46 is introduced to Eq. (8). Figure 10b shows the new normalized results of slope k/(1 + n) 2.46 and σ op_0 /σ s (1 + n) Here we found that the results could be plotted by the same curve, namely, the relationship

8 36 J. CHEN et al. Table 2 COS determined by Eq. (9) based on MCOD and FEM a/mm a/w u max /mm Results by FEM Results by Eq. (9) Relative error (%) COS, crack opening stress; FEM, finite element method; MCOD, maximum crack opening displacement. Fig. 10 (a) Relationship between the slope k and σ op_0 /σ s at different n. (b) Relationship between the slope k/(1 + n) 2.46 and σ op_0 /σ s (1 + n) 2.46 at different n. between k/(1 + n) 2.46 and σ op_0 /σ s (1 + n) 2.46 has taken into account the effect of strain hardening.! 2:46 k ð1 þ nþ 2:46 ¼ 10:2 σ op 0 σ s ð1 þ nþ 2:46 (11) Depending on the previous results, COS of strain hardening material under tension compression loading can be determined based on MCOD by Eq. (12). The definite expression of Eq. (12) is only applicable to kinematic power-law model. However, other constitutive models, such as Swift model, Voce model and the influence of Bauschinger effect, are needed to be investigated in further study. APPLICABILITY OF THE CRACK OPENING STRESS MAXIMUM CRACK OPENING DISPLACEMENT MODEL In this section, fatigue crack growth data of three materials in previous literatures are used for analysing the applicability of the COS MCOD model. The materials and geometric properties are given in Table 3. The Paris fatigue crack propagation expression 51 could be written as Eqs (13) and (14) by using Elber s crack closure concept. 6 da dn ¼ C ΔK m eff (13) K eff ¼ K max K op (14) 8 σ op =σ s ¼ kðσ min =σ s Þþσ op 0 =σ s! 2:46 k ð1 þ nþ 2:46 ¼ 10:2 σ op 0 >< σ s ð1 þ nþ 2:46! 2 >: σ op 0 =σ s ¼ 0:024 aσ s ð1 þ nþ 2 þ 0:172! aσ s ð1 þ nþ 2 0:015 (12) where ΔK eff is the effective SIF range, K max is the maximum SIF, K op is crack opening SIF, and C and m are Paris constant. The crack opening SIF could be expressed as pffiffiffiffiffi K op ¼ YðÞσ a op πa (15) where Y(a) is geometric correction coefficient.

9 A NEW EXPRESSION FOR CRACK OPENING STRESS 37 Table 3 Materials and geometric properties And the Eq. (12) gives the relationship between MCOD and COS; the relationship has already considered the effects of crack geometry size, hence, the geometric correction coefficient Y(a) = 1 in Eq. (15). According to Eqs (12) and (15), the K op could be written as 8 pffiffiffiffiffi K op ¼ σ op πa σ op =σ s ¼ kðσ min =σ s Þþσ op 0 =σ s! >< 2:46 k ð1 þ nþ 2:46 ¼ 10:2 σ op 0 σ s ð1 þ nþ 2:46! 2 >: σ op 0 =σ s ¼ 0:024 aσ s ð1 þ nþ 2 þ 0:172 E/GPa σ s /MPa a/mm σ max /MPa AM60B AL T6 alloy 49, ! aσ s ð1 þ nþ 2 0:015 (16) In our previous work, 52 for a central through-crack in an infinite plate under tensile loads, the mathematical n function relationship between the normal MCOD and SIF was deduced as Eq. (17) based on the Westergaard stress function. The Eq. (17) is able to eliminate the influence of plate width in the case of 0.05 a/w 0.5. u max ¼ 4 p ffiffiffiffiffiffiffi a=π E K max ¼ 4 p ffiffiffiffiffiffiffi a=π ΔK Eð1 RÞ (17) where u max is the resulting MCOD corresponding to the maximum stress σ max. Figures 11a 13a illustrate the experimental fatigue crack growth data of different materials that are found in previous literature. Figures of da/dn ΔK are the original ones, and based on the ΔK presented in Figs 11a 13a, the MCOD are obtained by Eq. (17). Crack opening SIF values are determined on the basis of MCOD using the model given by Eq. (16). Figures 11b 13b show da/dn ΔK eff curves. It can be seen from Figs 11b 13b that all the data are condensed to a single curve using the COS model respectively. The result implies that COS MCOD model is applicable to centre cracked finite plates with different Fig. 11 Fatigue crack growth data of AM60B magnesium alloy. (a) Experimental da/dn ΔK curves. 47 (b) da/dn ΔK eff curves. Fig. 12 Fatigue crack growth data of AL (a) Experimental da/dn ΔK curves. 48 (b) da/dn ΔK eff curves.

10 38 J. CHEN et al. Fig. 13 Fatigue crack growth data of 7075-T6 alloy. (a) Experimental da/dn ΔK curves. 49,50 (b) da/dn ΔK eff curves. materials under tension compression cyclic loading. This work offers an alternative method to accurately determine COS in practice respectively. The result implies that COS MCOD model is applicable to centre cracked finite plates with different materials under tension compression cyclic loading. This work offers an alternative method to accurately determine COS in practice. CONCLUSION The paper focuses on investigating the magnitude of the effect of compressive loading part of tension compression loading cycles on COS in terms of MCOD. Based on the results of elastic and plastic FE analysis, the following conclusions can be noted: 1. An explicit expression [Eq. (12)] of COS versus MCOD is presented to calculate COS in tension compression loading cycles. 2. This COS model has the advantages of avoiding adopting fracture parameters around crack tip and taking into account the effects of crack geometry, applied load, yield stress, Young s modulus and strain hardening. 3. This COS model could be applied to centre cracked finite plates with different materials under tension compression cyclic loading. 4. This COS model could condense da/dn ΔK curves with different materials under tension compression cyclic loading. Acknowledgements The work is supported by the National Natural Science Foundation of China, grant No The authors would like to acknowledge National Natural Science Foundation Committee for the financial support. REFERENCES 1. Iranpour, M. and Taheri, F. (2014) Analytical and computational investigation into the influence of the compressive stress cycles on crack growth under variable amplitude loading using CTOD. Fatigue Fract. Engng. Mater. Struct., 37, Zhang, J., He, X. D., Sha, Y. and Du, S. Y. (2010) The compressive stress effect on fatigue crack growth under tension-compression loading. Int. J. Fatigue, 32, Zhang, J., He, X. D., Suo, B. and Du, S. Y. (2008) Elastic-plastic finite element analysis of the effect of compressive loading on crack tip parameters and its impact on fatigue crack propagation rate. Eng. Fract. Mech., 75, Zhang, J., He, X. and Du, S. (2007) Analysis of the effects of compressive stresses on fatigue crack propagation rate. Int. J. Fatigue, 29, Benz, C. and Sander, M. (2015) Reconsiderations of fatigue crack growth at negative stress ratios: finite element analyses. Eng. Fract. Mech., 145, Elber, W. (1970) Fatigue crack closure under cyclic tension. Eng. Fract. Mech., 2, Nakagaki, M. and Atluri, S. N. (1979) Fatigue crack closure and delay effects under mode I spectrum loading: an efficient elastic-plastic analysis procedure. Fatigue Fract. Engng. Mater. Struct., 1, Peeker, E. and Niemi, E. (1999) Fatigue crack propagation model based on a local strain approach. J. Constr. Steel Res., 49, Savaidis, G., Dankert, M. and Seeger, T. (1995) An analytical procedure for predicting opening loads of cracks at notches. Fatigue Fract. Engng. Mater. Struct., 4, Vormwald, M. and Seeger, T. (1991) The consequences of short crack closure on fatigue crack growth under variable amplitude loading. Fatigue Fract. Engng. Mater. Struct., 2-3, Correia, J. A. F. O., Blasón, S., Arcari, A. et al. (2016) Modified CCS fatigue crack growth model for the AA2019-T851 based on plasticity-induced crack-closure. Theor. Appl. Fract. Mech., 85,

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