A continuum model for damage evolution simulation of the high strength bridge wires due to corrosion fatigue

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.