Anuncio

Journal of Constructional Steel Research 146 (2018) 76–83 Contents lists available at ScienceDirect Journal of Constructional Steel Research A continuum model for damage evolution simulation of the high strength bridge wires due to corrosion fatigue Bin Sun Department of Engineering Mechanics, Jiangsu Key Laboratory of Engineering Mechanics, Southeast University, Nanjing 210096, China a r t i c l e i n f o Article history: Received 18 December 2017 Received in revised form 15 March 2018 Accepted 26 March 2018 Available online 4 April 2018 Keywords: Damage Corrosion fatigue High strength steel Bridge wires Continuum damage mechanics a b s t r a c t Continuum damage model and simulation algorithm are developed to simulate the corrosion fatigue process of high strength bridge cable steel wires. The developed model can be used to predict the damage curves during the corrosion fatigue process based on the concept of continuum damage mechanics (CDM). The algorithm can be used to simulate the corrosion fatigue damage evolution process of bridge wires from local damage to failure. As case study, the developed model and algorithm have been applied to simulate corrosion fatigue damage evolution of bridge wires under cyclic tensile in 3.5 wt% NaCl solution at 6 Hz, and the numerical prediction results are compared with experimental results. It shows that the developed model and algorithm are reasonable and can be used to study and describe corrosion fatigue damage evolution of bridge wires. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction Many long-span cable-stayed bridges have been built in the world, which are in the order of kilometers. Since stay cables are the main bearing component of long-span cable-stayed bridges [1], their service lives play an important role in the safety of the whole bridge. However, bridge cables are subjected with aggressive corrosion environments and cyclic loading during their service time, and prone to corrosion fatigue failure. Many old cable-stayed bridges all over the world have deteriorated cables, in which some of the steel wires for the main bridge cables are heavily corroded and fractured [2,3]. Rehabilitation work of deteriorated wires due to corrosion fatigue was carried out on the Brooklyn Bridge and the other bridges [4,5]. Since many researches pointed that fatigue life in corrosive environment is signiﬁcantly lower than it in dry air [6,7], it shows that aggressive corrosive environment has an important effect on the fatigue performance of metallic structures, which is because corrosion pits initiate and grow on the material surface in corrosive environments and then continue to cause crack initiation and growth even at very low stress levels [8,9]. Therefore, in order to better evaluate fatigue damage and life of bridge cable steel wires in service corrosive environment, it is necessary to consider the combined action of stress and corrosion. Although many researches have been done, the corrosion fatigue phenomenon of bridge cable steel wires are still necessary to be studied [10]. E-mail address: [email protected] https://doi.org/10.1016/j.jcsr.2018.03.031 0143-974X/© 2018 Elsevier Ltd. All rights reserved. The objective of this paper is to develop a model within continuum damage mechanics (CDM) framework which can be used to simulate the damage evolution of bridge cable steel wires due to the combined action of stress and corrosion. The concept of CDM is ﬁrst presented by Kachanov [11] and widely used in engineering application due to its simplicity, in which a damage variable is used to describe the degradation of material mechanical properties. Although CDM has been widely used to develop cumulative fatigue damage models due to cyclic stress, e.g. Manson [12], Chaboche [13], Fatemi [14], Franke [15], very few continuum models are developed to describe the damage of the high strength bridge cable steel wires due to combined action of cyclic stress and corrosion. In this study, a continuum corrosion fatigue damage model of bridge wires is developed based on an evolution law for damage accumulation due to cyclic stress [13], pit and crack growth rates [16], and pit-to-crack transition criteria [17]. Using the developed model, corrosion fatigue damage simulation algorithm is also developed. Finally, as a case study, corrosion fatigue damage evolution of a pre-split high strength bridge cable steel wire is simulated under cyclic loading within 3.5 wt % NaCl solution at 6 Hz. 2. Corrosion fatigue damage model of bridge wires based on CDM In CDM, macroscopic state variable D is used to describe the distribution, characterization and growth of micro-structural defects. As shown in Fig. 1, D is deﬁned as the gradual loss of effective cross-sectional area Ae due to material degradation processes [18]: D ¼ 1−Ae =A0 ð1Þ B. Sun / Journal of Constructional Steel Research 146 (2018) 76–83 77 A Fig. 1. Schematic diagram of physical damage as deﬁned in CDM framework. radius Re which is similar to the deﬁnition of the effective cross-sectional area Ae. where A0 is the elemental area of the representative volume element (RVE) shown in Fig. 1, in which all properties including micro-structural defects are represented by homogenized variables [19]. For the undamaged material with effective cross sectional area Ae = A0, the damage variable D equals to zero based on Eq. (1). Using the deﬁnition of D shown in Eq. (1), the effective stress σe can be deﬁned as [19]: σ e ¼ σ=ð1−DÞ DCF ¼ 1−Re =R ¼ Rl =R ð3Þ where R is the initial radius for the undamaged material, Rl is the loss of length along the radius within the cross section of the bridge wire. As shown in Eq. (3), in order to obtain the corrosion fatigue damage of bridge wires, Rl should ﬁrst be calculated. In this study, the loss of the length along the radius within cross section of bridge wires Rl were di- ð2Þ where σ is the nominal stress. According to the symmetry feature of cylindrical bridge wires, here the corrosion fatigue damage rates are assumed to be the same along the radius in their cross section. Such as shown in Fig. 2, the loss of length rate along the two arbitrary radius R1, R2 due to corrosion fatigue are same based on the above assumption. Similar to deﬁnition of damage variable due to cyclic stress shown in Eq. (1), here corrosion fatigue damage model of bridge wires DCF is deﬁned based on the effective vided into two conditions Rl 1 and Rl 2 (Rl ¼ Rl 1 þ Rl 2 ): in the ﬁrst condition, Rl 1 can be obtained based on the pit initiation and growth to cracks from outside surface in contact with environment to material inward of bridge wires due to combined action of cyclic stress and corrosion; in the second condition, Rl 2 can be obtained based on loss of effective cross-sectional area within inward of bridge wires due to the action of cyclic stress, where has not been corroded. R O R Rl Re R O R R Fig. 2. Schematic diagram of deﬁnition of the corrosion fatigue damage model of bridge wires. 78 B. Sun / Journal of Constructional Steel Research 146 (2018) 76–83 For calculating the loss of the length along the radius within cross the integration of Eq. (4) from t = 0 to current time t leads to: section of bridge wires for the ﬁrst condition Rl 1 , the pit and crack growth models in previous works [16] are used in this study: Here the used depth x of pit with time t model can be described as [16]: D F ðt Þ ¼ x ¼ α c t βc where t = N/f, f is loading frequency of the cyclic loads. Based on Eq. (1), loss of effective cross-sectional area of bridge wires due to cyclic stress Al = A0 − Ae can be expressed as: ð4Þ where αc, βc are the model parameters. Based on Eq. (4), the pit growth rate model can be written as: dx ¼ βc α c 1=βc xð1−1=βc Þ dt ð5Þ !1−α1 f Δσ β f 1−α f tf 2M 0 ð13Þ Al ¼ D F A0 ð14Þ Then, based on Eqs. (13) and (14), the effective loss of the length along the radius within cross section of the cylindrical bridge wires for the second condition Rl 2 , can be expressed as: Here the used crack growth rate model can be described as [16]: Rl 2 ¼ dx ¼ Cσ P xq dt ð6Þ where C, p, q are the model parameters, and the crack length is also denoted by x for consistency. Based on the pit-to-crack transition criteria of Kondo [17], i.e. the critical depth at which the crack growth rate exceeds that for the pit, the critical pit-to-crack transition size can be obtained based on the Eqs. (5) and (6). xc ¼ β =ð1þβc ðq−1ÞÞ βc α c 1=βc c P Cσ ð7Þ Based on Eqs. (4), (5), (6) and (7), two cases are considered in the calculation of Rl 1 : pit growth stage and crack growth stage. For the ﬁrst case, when x ≤ xc, i.e. t ≤ tp, based on the Eqs. (4) and (7), where xc αc tp ¼ 1=βc ð8Þ Rl 1 can be expressed as: Rl 1 ¼ α c t βc ; t ≤t p ð9Þ For the second case, when x N xc, i.e. t N tp, Rl 1 can be obtained based on Eq. (6): Z Rl 1 xc 1 dx ¼ Cσ P xq Z t dt ð10Þ tp Based on Eq. (10), Rl 1 for the second case can be expressed as: Rl 1 ¼ ð1−qÞ 1=ð1−qÞ xc þ Cσ P t−t p ð1−qÞ 1−q ð11Þ For calculating the loss of the length along the radius within cross section of bridge wires for the second condition Rl 2 , an evolution law for damage accumulation DF due to cyclic stress in previous works [13] are used in this study to calculate loss of effective cross-sectional area based on Eq. (1): dD F ¼ D F α f Δσ 2M0 β f dN ð12Þ where Δσ is the stress range, N is cycles number and αf, βf, M0 are the model parameters. By using the initial condition (DF(0) = 0 for t = 0), pﬃﬃﬃﬃﬃﬃ DFR ð15Þ Up to this point, combining with Eqs. (3), (9), (11), (13) and (15), the developed continuum corrosion fatigue damage model DCF(t) of the cylindrical bridge wires with current time t due to the combined action of stress and corrosion can be obtained based on two conditions: When t ≤ tp, DCF(t) can be expressed as: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D F ðt Þ þ α c t βc =R ð16 aÞ where tp can be found in Eq. (8), and DF(t) can be found in Eq. (13). When t N tp, DCF(t) can be expressed as: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D F ðt Þ þ ð1−qÞ 1=ð1−qÞ xc þ Cσ P t−t p ð1−qÞ =R 1−q ð16 bÞ 3. Model parameters determination and application of the developed model In order to verify the developed model, one group corrosion fatigue life data of pre-split bridge cable steel wires is chosen from Ref. [1], which are from the corrosion fatigue experiments of high strength steel test sample under cyclic tensile in 3.5 wt% NaCl solution at 6 Hz with constant stress amplitude. Using Eqs. (16-a), (16-b), DCF(Nf/f) = 1, where Nf is corrosion fatigue life, is chosen to ﬁt the experimental data for obtaining model parameters. The ﬁtted model parameters are summarized in Table.1, and the parameters are dimensionless. Based on the model parameters shown in Table 1, the developed model are used to predict the corrosion fatigue life Nf of the bridge wires using the no-linear equation DCF(Nf/f) = 1 by combining with Eqs. (16-a), (16-b), and compared with experimental data shown in Fig. 3. It shows that the predicted results based on the model agree well with the experiment data, which shows that the developed corrosion fatigue model and the ﬁtted model parameters are reasonable and effective. As an application of the developed model, two continuum damage evolution curves respectively with stress level Δσ = 500MPa (Condition A) and Δσ = 300MPa (Condition B) shown in Fig. 3, are predicted by the developed corrosion fatigue damage model Eqs. (16-a), (16-b) and given in Fig. 4. It shows that the developed model can be used to describe the progressive degradation of material behavior for easy engineering application using a damage variable based on the concept of CDM. Table 1 Parameters of model. Model αf parameters M0 βf βc αc C q p 0.99 702.31 9.12 0.689 1.27 × 10−5 7.29 × 10−9 0.798 2.01 B. Sun / Journal of Constructional Steel Research 146 (2018) 76–83 79 550 500 450 Stress range (MPa) 400 f 350 Numerical curve predicted by the model Experimental data 300 250 200 150 2 10 3 4 10 10 5 10 N Fig. 3. Corrosion fatigue life of high strength steel bridge wires predicted by the model and experiment. 4. FE model-based corrosion fatigue damage simulation algorithm Since cycle-by-cycle corrosion fatigue simulation is computationally prohibitive, here block cycle jump technique [20] is used to speed-up the corrosion fatigue damage simulation in the algorithm. In the block cycle jump technique, here a few seconds Δt of cyclic loading for test sample of bridge wires in corrosion solution is considered to be one standard dynamic loading block with a certain repeated number ΔT, and the choice of the Δt and ΔT is based on the balance of computational time and precision of the corrosion fatigue damage simulation process. The smaller the Δt and ΔT are, the better accuracy is achieved while the longer computational time is expected. In the corrosion fatigue damage simulation using the algorithm, one standard dynamic loading block is ﬁrstly applied on the FE model of the bridge wires, the stress time histories of all elements of the model can be obtained. Then Eqs. (16-a), (16-b) can be used to evaluate the corrosion fatigue damage increment ΔDe CF ðtÞ of all elements due to one standard Fig. 4. Continuum corrosion fatigue damage evolution curves predicted by the model. 80 B. Sun / Journal of Constructional Steel Research 146 (2018) 76–83 dynamic loading block with a certain repeated number ΔT for speed-up the damage simulation based on the block cycle jump technique: ΔDe CF ðt Þ ¼ De CF ðt Þ−De CF ðt−Δt ΔT Þ ¼ ∂DCF ðt−Δt ΔT Þ Δt ΔT ð17Þ ∂t After corrosion fatigue damage increment of all elements can be calculated due to one standard dynamic loading block, the stiffness matrices of the FE model can be updated based on the following equations of motion for bridge wires coupled with the developed corrosion fatigue damage model: n o n o ½Mb d€b þ ½Cb d_b þ ½Kb fdb g ¼ f Fb g ð18Þ where Mb is the mass matrices, Cb is damping matrices, d€b, d_b, and db are respectively the acceleration, velocity and displacement vector of the bridge wires, Fb is the vector of cyclic loading acting on the bridge wires within corrosion solution, and the Kb is stiffness matrices of FE model of the bridge wires coupled with the developed corrosion fatigue damage model, and can be expressed as: Kb ¼ nelem X Ke ð19Þ e¼1 where the nelem is the total number of elements of the FE model, Ke is the stiffness matrices of the element and can be expressed as: corrosion fatigue damage model Eqs. (16-a), (16-b) introduced in the Section 2. Finally, repeat to apply another one standard dynamic loading block and implement the above computational procedure again until the entire corrosion failure. The implementation procedure and ﬂow chart for the FE model-based corrosion fatigue damage simulation algorithm of bridge wires is given in Fig. 5. 5. Numerical analyses on corrosion fatigue damage evolution of a bridge wire 5.1. Numerical example of corrosion fatigue damage simulation of a bridge wire In order to study the corrosion fatigue damage mechanisms of bridge wires, many corrosion fatigue experiments of pre-split high strength steel bridge wires under constant stress amplitude cyclic tensile in 3.5 wt% NaCl solution with different stress levels and loading frequencies were carried out in Ref. [1]. In order to verify the developed corrosion fatigue damage model and simulation algorithm, one corrosion fatigue experiment of a pre-split bridge cable steel wire under cyclic tensile with Δσ = 500MPa and Δσ = 300MPa in 3.5 wt% NaCl solution at 6 Hz is chosen as a numerical case study, which respectively are the condition A and B shown in Fig. 3. And the schematic illustration and FE model of the numerical example is shown in Fig. 6. 5.2. The simulated results by the developed model and simulation algorithm Z BT ðI−DÞEBdV Ke ¼ ð20Þ e where B is strain–displacement relationship matrix, E is the elastic matrix describing the elastic stress–strain relationship, and I is the unit matrix, and D is damage matrix and can be calculated based on the developed As a case study of the developed algorithm, the corrosion fatigue damage evolution process of the bridge wire under the condition A and B described in Section 5.1 are respectively simulated and given in Figs. 7 and 8. Using the algorithm described in Section 4, the corrosion fatigue damage increment of the bridge wire for condition A is evaluated Fig. 5. Flow chart of the corrosion fatigue damage simulation algorithm. B. Sun / Journal of Constructional Steel Research 146 (2018) 76–83 81 Fig. 6. Numerical example of a bridge wire for corrosion fatigue damage simulation. every 10 s with a repeated number ﬁve until the bridge wire is entire failure, i.e. Δt = 10, ΔT = 5, and the stiffness matrices of the FE model is also updated once the new corrosion fatigue damage increment can be calculated. For condition B, Δt = 10, ΔT = 20 are chosen to simulate the fatigue damage evolution of bridge wire. The choice of the Δt and ΔT for the numerical example here is based on the balance of Fig. 7. Corrosion fatigue damage evolution simulation of the bridge wire for condition A. 82 B. Sun / Journal of Constructional Steel Research 146 (2018) 76–83 Fig. 8. Corrosion fatigue damage evolution simulation of the bridge wire for condition B. computational time and precision of the corrosion fatigue damage simulation process, and can also be chosen as the other numerical value. It can be seen from Figs. 7 and 8 that fatigue damage distribution of the bridge wire can be predicted with different loading times. For condition A, corrosion fatigue damage ﬁrst initiate after N = 900 shown in Fig. 7-a. Then corrosion fatigue damage accumulates gradually up to the bridge wire corrosion fatigue failure shown in Fig. 7-d, it shows that the bridge wire loses its bearing capacity and to be failure after N = 1800. In other words, the corrosion fatigue life of the bridge wire under condition A is predicted to be 1800 using the developed algorithm, while the corrosion fatigue life predicted by experiment for the same condition is 2189, and the error is 17.8%. For condition B, the corrosion fatigue life of the bridge wire is predicted to be 9600 using the developed algorithm, while the corrosion fatigue life predicted by experiment for the same condition is 12,545, and the error is 23.5%. It shows that the simulated results predicted by the developed algorithm is reasonable. 6. Conclusions Major conclusions from this study can be summarized as follows: (1) A physical continuum corrosion fatigue damage model is developed to describe the damage process of high strength bridge cable steel wires. (2) The FE model-based algorithm is also developed to simulate the damage evolution of the bridge wires due to corrosion fatigue using the developed model. (3) As case study, the developed model and algorithm have been applied to predict continuum damage evolution of bridge wires under cyclic tensile in 3.5 wt% NaCl solution at 6 Hz with constant stress amplitude. By comparison of the numerical prediction results with the experimental results, it shows that developed model and the algorithm can be used to describe the corrosion fatigue evolution process of bridge wires and obtain reasonable results. Acknowledgements The works described in this paper are ﬁnancially supported by Jiangsu Province Natural Sciences Fund subsidization project (BK20170655, BK20170677), and the Fundamental Research Funds for the Central Universities (3205007817), to which the authors are most grateful. References [1] J.H. Jiang, A.B. Ma, W.F. Weng, G.H. Fu, Y.F. Zhang, G.G. Liu, F.M. Lu, Corrosion fatigue performance of pre-split steel wires for high strength bridge cables, Fatigue Fract. Eng. Mater. Struct. 32 (9) (2009) 769–779. [2] S. Nakamura, K. Suzumura, Experimental study on fatigue strength of corroded bridge wires, J. Bridg. Eng. 18 (3) (2012) 200–209. [3] R. Betti, A.C. West, G. Vermaas, Y. Cao, Corrosion and embrittlement in high-strength wires of suspension bridge cables, J. Bridg. Eng. 10 (2) (2005) 151–162. [4] R. Betti, B. Yanev, Conditions of suspension bridge cables, Proc., The New York city Workshop on Safety Appraisal of Suspension Bridge Main Cables, NCHRP, New Jersey, 1998. [5] F.L. Stahl, C.P. Gagnon, Cable Corrosion, ASCE Press, New York, 1996. [6] T. Palin-Luc, R. Pérez-Mora, C. Bathias, G. Dominguez, P.C. Paris, J.L. Arana, Fatigue crack initiation and growth on a steel in the very high cycle regime with sea water corrosion, Eng. Fract. Mech. 77 (11) (2010) 1953–1962. [7] S. Yang, H.Q. Yang, G. Liu, Y. Huang, L.D. Wang, Approach for fatigue damage assessment of welded structure considering coupling effect between stress and corrosion, Int. J. Fatigue 88 (2016) 88–95. [8] S. Ishihara, S. Saka, Z.Y. Nan, T. Goshima, H. Shibata, B.L. Ding, Study on the pit growth during corrosion fatigue of aluminum alloy, Int. J. Mod. Phys. B 20 (2006) 3975–3980. [9] W.M. Tian, S.M. Li, B. Wang, J.H. Liu, M. Yu, Pitting corrosion of naturally aged AA 7075 aluminum alloys with bimodal grain size, Corros. Sci. 113 (2016) 1–16. [10] S.I. Nakamura, K. Suzumura, Hydrogen embrittlement and corrosion fatigue of corroded bridge wires, J. Constr. Steel Res. 65 (2) (2009) 269–277. [11] L.M. Kachanov, Time of the rupture process under creep condition, TVZ Akad Nauk SSR Otd Tech Nauk, 8, 1958, pp. 26–31. [12] S.S. Manson, G.R. Halford, Practical implementation of the double linear damage rule and damage curve approach for treating cumulative fatigue damage, Int. J. Fract. 17 (2) (1981) 169–192. [13] J.L. Chaboche, P.M. Lesne, A non-linear continuous fatigue damage model, Fatigue Fract. Eng. Mater. Struct. 11 (1) (1988) 1–17. [14] A. Fatemi, L. Yang, Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials, Int. J. Fatigue 20 (1) (1998) 9–31. B. Sun / Journal of Constructional Steel Research 146 (2018) 76–83 [15] L. Franke, G. Dierkes, A non-linear fatigue damage rule with an exponent based on a crack growth boundary condition, Int. J. Fatigue 21 (8) (1999) 761–767. [16] A. Turnbull, L.N. McCartney, S. Zhou, A model to predict the evolution of pitting corrosion and the pit-to-crack transition incorporating statistically distributed input parameters, Corros. Sci. 48 (8) (2006) 2084–2105. [17] Y. Kondo, Prediction of fatigue crack initiation life based on pit growth, Corrosion 45 (1989) 7–11. 83 [18] J. Lemaitre, J.L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge, 1990. [19] J. Lemaitre, R. Desmorat, Engineering Fracture Mechanics: Ductile, Creep, Fatigue and Brittle Fracture, Springer-Verlag, Berlin Heidelberg, 2005. [20] J. Fish, M. Bailakanavar, L. Powers, T. Cook, Multiscale fatigue life prediction model for heterogeneous materials, Int. J. Numer. Methods Eng. 91 (10) (2012) 1087–1104.