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A bi-directional damage model for matrix cracking evolution in composite laminates under fatigue loadings
Journal Pre-proofs A Bi-directional damage model for matrix cracking evolution in composite laminates under fatigue loadings Wenxuan Qi, Weixing Yao, Haojie Shen PII: DOI: Reference:
S0142-1123(19)30521-3 https://doi.org/10.1016/j.ijfatigue.2019.105417 JIJF 105417
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
23 August 2019 2 December 2019 3 December 2019
Please cite this article as: Qi, W., Yao, W., Shen, H., A Bi-directional damage model for matrix cracking evolution in composite laminates under fatigue loadings, International Journal of Fatigue (2019), doi: https:// doi.org/10.1016/j.ijfatigue.2019.105417
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© 2019 Published by Elsevier Ltd.
A Bi-directional damage model for matrix cracking evolution in composite laminates under fatigue loadings Wenxuan Qia, Weixing Yaoa, b, 1, Haojie Shenc a
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing
University of Aeronautics and Astronautics, Nanjing 210016, China b
Key Laboratory of Fundamental Science for National Defense-Advanced Design
Technology of Flight Vehicle, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China c
Nanjing Research Institute on Simulation Technique, Nanjing 210016, China.
Abstract An analytical model for prediction of transverse matrix cracks in composite laminates under fatigue loading was proposed. The micromechanics method was utilized to characterize the damaged behavior of composite laminates. The initial matrix crack initiation life was defined and deduced based on micromechanics method. Then the relation between initial matrix crack initiation life and critical energy release rate was established to obtain the distribution of initial matrix crack initiation life in cracked plies, and the evolution of matrix crack density and stiffness degradation were predicted. The prediction results were compared with experimental results and the comparison
1 Corresponding author: State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China E-mail address: [email protected] 1
shows good agreement. Key words: A laminates, B fatigue, B transverse cracking 1. Introduction Similar to damage mechanisms under quasi static loading, main damage mechanisms of composite materials under fatigue loading include transverse matrix cracking, local delamination, interfacial debonding and fiber breakage and so on [1]. As the main damage mechanism in the early and middle stage of damage evolution process, transverse matrix cracks could result in the significant degradation of material mechanical properties and affect the load bearing capacity. Also, local delamination would initiate at the tips of matrix cracks, which could lead to the further stiffness degradation and result in the catastrophic failure of composite structures. Hence it is necessary to accurately research the initiation and evolution of matrix cracks in composite laminates under fatigue loading. In the past decades, several researchers have presented much works on the matrix cracks in composite laminates under fatigue loading. Payan et al. [2] defined several damage variables and the associated thermodynamic forces based on the theory of damage mechanics to model the behavior of damaged composite laminates under fatigue loading. The evolution laws of damage variables were established in the model and the influence of stress ratio was considered. Besides, the evolution model of inelastic strain was proposed to take account of the inelastic strain during the damage evolution process. Carraro et al. [3, 4] modelled the mechanical behavior of composite materials under multiaxial cyclic loading. In their model, a criterion for the non-fiber 2
controlled fatigue behavior was proposed, and two parameters, i.e. the local maximum principal stress and local hydrostatic stress, were defined to represent the driving forces of damage initiation. An equivalent energy release rate was also defined for the propagation of the initiated cracks in the model. Glud et al. [5] established an off-axial matrix cracks initiation and propagation model for composite laminates under fatigue loading. The influence of stress ratio on the off-axial matrix cracks initiation was considered in the model, and the Weibull distribution was adopted to predict the evolution of matrix crack density. Kawai et al. [6] established a ply basis fatigue life prediction model, the inelastic deformation and in situ strength of the ply were considered in the model. A first ply fatigue failure criterion was used to predict the fatigue life of composite laminates. Hosoi et al. [7] investigated the behavior of the transverse matrix cracks growth under high cycle fatigue loading. The free-edge effect was taken into consideration and the micromechanics method was utilized to modify the energy release rate for matrix cracking. Wu et al. [8] investigated the damage mechanisms of woven composites under very high cycle fatigue loading and proposed a prediction model. In the model, surface crack density was adopted to characterize the damage state of composites, and the relation between surface crack density and fatigue life was established to predict the fatigue life of composites under very high cycle fatigue loading. Shen et al. [9] investigated the damage evolution in cross-ply glass fiber reinforced plastic laminates under fatigue loading experimentally. Four configurations of lay-up with different thickness of cracking plies were chosen to study the in situ behavior of damage 3
initiation and evolution. In the experiment, light transmission method was adopted to observe and record damage state and eight stress levels were chosen for each configuration of cross-ply laminates. Also, the degradation of longitudinal modulus of laminates under fatigue loading were obtained. In the present paper, the influence of transverse matrix cracks on the bi-directional stiffness properties of composite laminates were studied and the micromechanics-based method was adopted to characterize the influence, the initiation and evolution of transverse matrix cracks were also investigated. The initial matrix crack initiation life was considered as a fatigue property of composite materials which reflects the resistance ability of composite materials on cracking, and the initial matrix crack initiation life curves of cracked plies were obtained according to the experimental results in this paper. The relation between the initial matrix crack initiation life and the maximum stress in cracked plies were obtained to predict the initiation of initial matrix cracks. Besides, the relation between the critical energy release rate and initial matrix cracks initiation life were established, and the distribution of local matrix cracks initiation life under fatigue loading in cracked plies was obtained based on the distribution of critical energy release rate under static loading, then the evolution of matrix crack density was obtained based on the distribution. 2. Damaged mechanical behavior under fatigue loading According to the crack opening displacement theory in the micromechanics, the stiffness degradation of composite materials was related to the matrix crack density and surface displacements of matrix cracks. With the plane stress assumption, the relation 4
between the damaged stiffness matrix of the elementary ply and the matrix crack density and normalized surface displacements of matrix cracks could be obtained [10], 1
q Qd = I + 0 Q0U Q0 E1
(1)
where Qd denotes the damaged stiffness matrix, ρ denotes the density of transverse crack, I is the unit matrix, q denotes the ratio of the extension length to the total length, which considers the condition that the transverse crack has not run through the width direction of specimen, U denotes the normalized displacement matrix of matrix crack surface. E10 and Q0 are the undamaged longitudinal modulus and stiffness matrix respectively. According to the equation (1), there is no variables related to the external load, which indicates that the influence of matrix cracks on the material properties is just determined by the damage state and material properties and independent of the loading form, hence equation (1) holds both under quasi static loading and fatigue loading. Then the elastic modulus of the damaged ply can be obtained by the above equation,
Q11d Q22d Q12d 2 E1 Q22d E2
Q11d Q22d Q12d 2 Q11d
(2)
G12 Q66d where E1(ρ), E2(ρ) and G12(ρ) are damaged longitudinal modulus, damaged transverse modulus and damaged shear modulus respectively. The above equations characterize the influence of transverse matrix cracks on bidirectional material properties of damaged plies. Then the constitutive model of 5
damaged composite laminates could be obtained based on the classic laminate theory. 3. Damage evolution model under fatigue loading 3.1 Initial matrix cracks initiation life curve In order to study the initiation and evolution of matrix cracks in composite materials under fatigue loading, a relation between the stress level in cracked plies and number of cycles when cracks initiate should be established, and then the initiation of matrix cracks could be predicted based on the relation. With the increase of cycle number, transverse matrix crack density increases in cracked plies, which results in the change of material properties and stress state in cracked plies. However, the initiation of the first matrix crack only depends on the undamaged material properties and external load, so it could be assumed that under fatigue loading, the first matrix crack initiates in the cracked plies after a known constant amplitude load spectrum, and then the following matrix cracks initiate under variable amplitude loadings whose stress level decreases, as shown in Fig.1. Hence, the relation between the stress level in cracked plies and the cycle number when the first crack initiates, which is named as the initial matrix crack initiation life Nini, could reflect the basic fatigue property of cracked plies in composite laminates, and the curve of this relation could be called as the S-Nini curve of the cracked plies. Then the curve could be considered as a fatigue property of composite material and adopted to predict the initiation of matrix cracks in cracked plies. In the experiment, the S-Nini curve could be obtained by recording the cycle numbers when the first transverse matrix crack initiates under different stress levels.
6
Fig. 1 The decrease of stress in cracked plies with the increase of cycle number. 3.2 Initial matrix crack initiation equation As a fatigue property of composite materials, the S-Nini curve could be obtained by experimental results, but also could be obtained by theoretical derivation. Plenty of studies show that the evolution of transverse matrix cracks of composite materials under fatigue loading could be determined by a Paris-law like equation [11-13], G d k( ) Gti dN
(3)
where ρ denotes transverse matrix cracks, N denotes cycle number, ∆G=Gmax-Gmin denotes the range of energy release rate, Gti denotes the energy release rate when the first matrix crack initiates in cracked plies under quasi static loading, k and α are material parameters. With regard to the initiation of initial matrix crack in cracked plies, we could have
1 L dN N ini d
(4)
where L denotes the length of cracked plies, Nini denotes the initial matrix crack 7
initiation life. According to equation (3) and equation (4), the equation of Nini and ∆G could be obtained, N ini
1 G - ) ( kL Gti
(5)
Based on the micromechanics method, the range of energy release rate under tension-tension fatigue loading could be determined by maximum and minimum stress in the cracked plies [14], 90 2 90 2 G ( max ) ( min ) = 90 2 Gti ( ti )
(6)
where σ90max and σ90min are maximum stress and minimum stress in cracked plie respectively, σ90ti denotes the initiation stress of initial matrix crack in cracked plies under quasi static loading. According to equation (5) and equation (6), we could obtain the following equation, 90 2 90 2 90 2 90 2 ) ) ( min ( max ) ) ( min 1 ( max N ini = ( ) ) C( 90 2 90 2 kL ( ti ) ( ti )
(7)
where C=1/kL is a material parameter. When the stress ratio of the fatigue loading is smaller enough, σ90min would be much smaller than σ90max, so the above equation could be simplified as
N ini =C (
90 2 ( max ) ) 90 2 ( ti )
(8)
Then we could obtain the logarithmic form of the above equation, 90 lg N ini = lg C 2 lg ti90 2 lg max
(9)
When the maximum stress in composite laminates under fatigue loading is known, the maximum stress in cracked plies could be calculated based on the classic theory 8
laminate theory, and then the initiation life of initial matrix crack in cracked plies could be determined according to the above equation. 3.3 Random distribution of material property Initial defects in composite materials results in the random distribution of material mechanical properties, which causes the different resistance ability on cracking in materials and leads to the evolution process of matrix cracking. In many quasi static models, the critical energy release rate was assumed to obey two-parameter Weibull distribution to obtain the evolution process of matrix cracking [15, 16], and the prediction results were compared with experimental results to verify the distribution assumption. For fatigue model, the initial matrix crack initiation life was considered as a fatigue property of composite materials, hence it is also random distributed due to the initial defects in composite materials. In order to be consistent with the quasi static models, the distribution of the initial matrix crack initiation life is deduced based on the two-parameter Weibull distribution of critical energy release rate in quasi static models. Under quasi static loading, the relation between critical energy release rate and initiation stress of initial matrix crack in cracked plies could be obtained, GC =
1 ( ti90 ) 2 2E 0
(10)
where E0 is the initial elastic modulus, GC denotes the critical energy release rate for matrix cracking, σ90ti denotes the initiation stress of initial matrix crack. According to the above equation, the distribution of σ90ti could be determined based on the two-parameter Weibull distribution of GC. Then in equation (9), α, C and σ90max are constants, hence the distribution of Nini could be determined based on the 9
distribution of σ90ti in cracked plies. 3.4 Damage accumulation model for matrix cracking evolution In order to establish a damage accumulation model for matrix cracking evolution, the following three aspects should be considered, (1) the definition of damage caused by a single cycle on the materials, (2) how does the damage accumulate after several cycles, (3) the definition of critical damage when material fails. From the perspective of macroscopic phenomenology, the Miner fatigue damage is adopted to define the damage for matrix cracking in composite materials under fatigue loading, Di =
1 N ini
(11)
According to the above definition and the S-Nini curve of cracked plies, damage caused by a single cycle on the materials could be obtained, which is D1= 1/Nini. With regard to the damage accumulation, it is assumed that every single cycle is independent of each other, hence the fatigue damage could accumulate linearly, which indicates that the linear Miner cumulative damage theory is adopted. Then, damage caused by n cycles under constant amplitude loading could be obtained, n
n
i 1
i 1
D n Di =
1 ( N ini ) i
(12)
where (Nini)i denotes the crack initiation life corresponding to the maximum stress Si. According to the definition of critical damage, when Dn=DCR, matrix crack initiates, and the critical damage is assumed that DCR=1 under the constant amplitude loading. 10
4. Results and discussion In order to verify the model, the experimental results of E-glass/YPX-3300 composite laminates with two configurations [02/904]s and [0/904]s are compared to the prediction results, including the evolution of matrix crack density and the degradation of stiffness. 4.1 Fatigue experiment introduction Experiments of several cross-ply laminates of glass-fiber reinforced composite under fatigue loading with four stress levels were carried out [9]. In the experiments, glass-fiber reinforced composites made from E-glass/YPX-3300 were employed, and the geometry and dimension of specimen are shown in Fig.2. In the experiments, glassfiber reinforced composites made from Henderson composite materials co., LTD was adopted. The high strength glass fiber and 3300 epoxy resin from Kunshan Yubo composite materials co., LTD were employed. The composite laminates were manufactured by hand lay-up of pre-preg followed autoclave curing, and the volume fraction of the fibers is 70%. Several material properties of E-glass/YPX-3300 are shown in Table 1. Configurations of cross-ply laminates include [02/904]s and [0/904]s, and the thickness of each ply is 0.125mm.
Fig. 2 Specimen configuration (dimensions in mm) 11
Table 1 Material properties of E-glass/YPX-3300 E1 (MPa) 43157
E2 (MPa) G12 (MPa) 10810
4215
ν12 0.306
The experiment was carried out according to the ASTM D3479 test standard [17], continuous tests were conducted with a MTS 370.10 test machine, which is a servohydraulic test machine with hydraulic grips. Tests were carried out under load control with a frequency of 8 Hz and a load ratio of R=0.1. The sinusoidal waveform of applied load was selected. And for each configuration of laminates, four stress levels were selected, whose maximum stresses are 60%, 70%, 80% and 90% of the initiation stress of initial transverse matrix crack under quasi static loading respectively. Due to the transparency of E-glass/epoxy material system and the convenience of light transmission method, this method was used to record the damage state of composite laminates simultaneously in the experiment. For the matrix cracking mechanism, the crack density was used as damage variables, and it was defined by dividing the numbers of cracks in the damaged plies by the length of the ply. In this study, the average line crack density was adopted to characterize the matrix crack damage states. Then the damaged stiffness properties could be deduced from each hysteresis loop. In order to study the stiffness degradation of composite laminates with different configurations intuitively, the normalized longitudinal modulus was calculated, which was defined by dividing the damaged longitudinal modulus by initial value.
12
4.2 Determination of material parameters According to the experimental results, the S-Nini curves of two cross-ply laminates could be obtained. In the fatigue experiment, the initiation lives of initial matrix crack in cracked plies in laminates under four stress levels were recorded, and the maximum stress in cracked plies could be determined with classic laminate theory (CLT). The two S-Nini curves are shown in Fig.3.
Fig. 3 S-lgNini curves of two laminates. Then according to the S-Nini curves, two material parameters C and α could be determined, which are shown in Table 2. Table 2 Material parameters in S-Nini curve of two cross-ply laminates Configuration
C
α
[0/904]s
160.63
-4.63
[02/904]s
72.88
-5.51
In cross-ply laminates, 90ºplies are supported by the neighboring 0ºplies, with the 13
thicker 0ºplies, the ply thickness ratio(defined as the thickness of the 0ºplies divided by the thickness of the 90ºplies) is larger and hence the constraint becomes stronger, which is considered to result in the in-situ effect. The in-situ effect causes the difference between the S-Nini curves, hence the model is demonstrated as lay-up dependent. 4.3 Model verification and discussions The comparison between the prediction results and experiment results of matrix crack evolution under four stress levels is shown in Fig.4.
(a)
14
(b) Fig. 4 Comparing the prediction results of matrix crack density in two laminates with experimental results. (a) [0/904]s, (b) [02/904]s. The comparison between the prediction results and experiment results of normalized longitudinal modulus under four stress levels is shown in Fig.5 and Fig.6.
15
(a)
(b)
(c)
16
(d) Fig. 5 Comparing the prediction results of normalized longitudinal modulus of [0/904]s laminates with experimental results. (a) 60%, (b) 70%, (c) 80%, (d) 90%.
(a)
17
(b)
(c)
18
(d) Fig. 6 Comparing the prediction results of normalized longitudinal modulus of [02/904]s laminates with experimental results. (a) 60%, (b) 70%, (c) 80%, (d) 90%. According to the comparison, the prediction results of normalized longitudinal modulus and matrix cracks evolution in the two cross-ply laminates under four stress levels are in good agreement with experimental results. In the semi-log coordinates, there are three obvious periods of evolution process in predicted matrix crack density evolution curve, including initiation, evolution and tendency to saturation, and the results agree well with the experimental results, which indicates that the proposed model could predict the matrix crack density evolution in composite laminates under fatigue loading with different stress levels. The prediction results of normalized longitudinal modulus are also in good agreement with experimental results, which indicates that the matrix crack damage characterization model under quasi static model is also suitable for the fatigue model. 19
5. Conclusions A damage model for prediction of matrix cracking initiation and evolution in composite laminates under fatigue was proposed in this paper. The influence of transverse matrix cracks on material stiffness properties was studied, the transverse matrix cracking initiation and evolution was also modelled. (1) The micromechanics based model was verified for composite laminates under fatigue loading, and mechanical behavior of composite laminates with matrix cracks was modelled based on the crack opening displacement (COD) theory in micromechanics, the damaged elastic modulus was also obtained. (2) The initial matrix crack initiation life was considered as a fatigue property of composite materials. The equation between the initial matrix crack initiation life and maximum stress in cracked plies was established, and the equation was adopted to predict the initiation of matrix cracks in cracked plies. (3) The distribution of the initial matrix crack initiation life in cracked plies was obtained based on the distribution of critical energy release rate in quasi static model, which ensures the consistency in fatigue model and quasi static model. Then the distribution was utilized to predict the evolution of matrix crack density in cracked plies. (4) The experimental results of two cross-ply laminates under fatigue loading with four stress levels were compared with the prediction results, including the degradation of normalized longitudinal modulus and evolution of matrix cracks density, and the comparison showed a good agreement, which indicates that the proposed model is effective and reliable in predicting the stiffness degradation and matrix crack evolution 20
in composite laminates under fatigue loading with different stress levels.
Acknowledgements This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References: [1] Talreja R, Singh, C V. Damage and Failure of Composite Materials. 1st ed. New York: Cambridge University Press, 2012. [2] Payan J, Hochard C. Damage modelling of laminated carbon/epoxy composites under
static
and
fatigue
loadings.
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2002;24(2):299-306.
https://doi.org/10.1016/S0142-1123(01)00085-8. [3] Carraro PA, Quaresimin M. A damage based model for crack initiation in unidirectional composites under multiaxial cyclic loading. Composites Science & Technology. 2014;99(4):154-163. https://doi.org/10.1016/j.compscitech.2014.05.012. [4] Carraro PA, Maragoni L, Quaresimin M. Prediction of the crack density evolution in multidirectional laminates under fatigue loading. Composites Science & Technology. 2017;145:24-39. https://doi.org/10.1016/j.compscitech.2017.03.013. [5] Glud JA, Dulieu-Barton JM, Thomsen OT, Overgaard LCT. A stochastic multiaxial fatigue model for off-axis cracking in FRP laminates. Int J Fatigue. 2017;103:576-590. https://doi.org/10.1016/j.ijfatigue.2017.06.012. [6] Kawai M, Honda N. Off-axis fatigue behavior of a carbon/epoxy cross-ply laminate and predictions considering inelasticity and in situ strength of embedded plies. Int J 21
Fatigue. 2008;30(10):1743-1755. https://doi.org/10.1016/j.ijfatigue.2008.02.009. [7] Hosoi A, Arao Y, Kawada H. Transverse crack growth behavior considering freeedge effect in quasi-isotropic CFRP laminates under high-cycle fatigue loading. Compos
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https://doi.org/10.1016/j.compscitech.2008.09.003. [8] Wu T, Yao W, Xu C. A VHCF Life Prediction Method Based on Surface Crack Density
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https://doi.org/10.1016/j.ijfatigue.2018.04.028. [9] Shen H, Yao W, Qi W, Zong J. Experimental investigation on damage evolution in cross-ply laminates subjected to quasi-static and fatigue loading. Composites Part B Engineering. 2017;120:10-26. https://doi.org/10.1016/j.compositesb.2017.02.033. [10] Shen H, Yao W, Wu Y. Synergistic Damage Mechanic Model for Stiffness Properties of Early Fatigue Damage in Composite Laminates. Procedia Engineering. 2014;74(2):199-209. https://doi.org/10.1016/j.proeng.2014.06.250. [11] Kobayashi S, Terada K, Ogihara S, Takeda N. Damage-mechanics analysis of matrix cracking in cross-ply CFRP laminates under thermal fatigue. Composites Science & Technology. 2001;61(12):1735-1742. https://doi.org/10.1016/S02663538(01)00077-X. [12] Llobet J, Maimí P, Essa Y, Martin DLEF. Progressive matrix cracking in carbon/epoxy cross-ply laminates under static and fatigue loading. Int J Fatigue. 2019;119:330-337. https://doi.org/10.1016/j.ijfatigue.2018.10.008. [13] Takeda N. Microscopic fatigue damage progress in CFRP cross-ply laminates. 22
Composites. 1995;26(12):859-867. https://doi.org/10.1016/0010-4361(95)90879-5. [14] Hosoi A, Sakuma S, Fujita Y, Kawada H. Prediction of initiation of transverse cracks in cross-ply CFRP laminates under fatigue loading by fatigue properties of unidirectional CFRP in 90° direction. Composites Part A: Applied Science and Manufacturing. 2015;68:398-405. https://doi.org/10.1016/j.compositesa.2014.10.022. [15] Vinogradov V, Hashin Z. Probabilistic energy based model for prediction of transverse cracking in cross-ply laminates. International Journal of Solids & Structures. 2005;42(2):365-392. https://doi.org/10.1016/j.ijsolstr.2004.06.043. [16] Varna J, Joffe R, Talreja R. A synergistic damage-mechanics analysis of transverse cracking in [±θ/904]S laminates. Composites Science & Technology. 2001;61(5):657665. https://doi.org/10.1016/S0266-3538(01)00005-7. [17] ASTM Standard. Standard Test Method for Tension-Tension Fatigue of Polymer Matrix Composite Materials. ASTM. 2007.
23
Declarations of interest: none.
24
Highlights: (1) The micromechanics based model was verified for composite laminates under fatigue loading in this paper, and mechanical behavior of composite laminates with transverse matrix cracks was modelled based on micromechanics theory. (2) The initiation of matrix cracks in cracked plies was predicted with the definition of the initial matrix crack initiation life, which was considered as a fatigue property of composite materials in this paper, and the equation between the initial matrix crack initiation life and maximum load stress in cracked plies was deduced. (3) The evolution of matrix crack density in cracked plies was predicted with the distribution of the initial matrix crack initiation life in cracked plies, and the distribution was obtained based on the distribution of critical energy release rate in quasi static model, which ensures the consistency in fatigue model and quasi static model. (4) The experiment of cross-ply laminates under fatigue loading with four stress levels was conducted, several material parameters were determined based on the experimental results, and the prediction results were compared with the experimental results, including the degradation of normalized longitudinal modulus and evolution of matrix cracks density, and the comparison shows a good agreement
25
S0142-1123(19)30521-3 https://doi.org/10.1016/j.ijfatigue.2019.105417 JIJF 105417
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
23 August 2019 2 December 2019 3 December 2019
Please cite this article as: Qi, W., Yao, W., Shen, H., A Bi-directional damage model for matrix cracking evolution in composite laminates under fatigue loadings, International Journal of Fatigue (2019), doi: https:// doi.org/10.1016/j.ijfatigue.2019.105417
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier Ltd.
A Bi-directional damage model for matrix cracking evolution in composite laminates under fatigue loadings Wenxuan Qia, Weixing Yaoa, b, 1, Haojie Shenc a
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing
University of Aeronautics and Astronautics, Nanjing 210016, China b
Key Laboratory of Fundamental Science for National Defense-Advanced Design
Technology of Flight Vehicle, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China c
Nanjing Research Institute on Simulation Technique, Nanjing 210016, China.
Abstract An analytical model for prediction of transverse matrix cracks in composite laminates under fatigue loading was proposed. The micromechanics method was utilized to characterize the damaged behavior of composite laminates. The initial matrix crack initiation life was defined and deduced based on micromechanics method. Then the relation between initial matrix crack initiation life and critical energy release rate was established to obtain the distribution of initial matrix crack initiation life in cracked plies, and the evolution of matrix crack density and stiffness degradation were predicted. The prediction results were compared with experimental results and the comparison
1 Corresponding author: State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China E-mail address: [email protected] 1
shows good agreement. Key words: A laminates, B fatigue, B transverse cracking 1. Introduction Similar to damage mechanisms under quasi static loading, main damage mechanisms of composite materials under fatigue loading include transverse matrix cracking, local delamination, interfacial debonding and fiber breakage and so on [1]. As the main damage mechanism in the early and middle stage of damage evolution process, transverse matrix cracks could result in the significant degradation of material mechanical properties and affect the load bearing capacity. Also, local delamination would initiate at the tips of matrix cracks, which could lead to the further stiffness degradation and result in the catastrophic failure of composite structures. Hence it is necessary to accurately research the initiation and evolution of matrix cracks in composite laminates under fatigue loading. In the past decades, several researchers have presented much works on the matrix cracks in composite laminates under fatigue loading. Payan et al. [2] defined several damage variables and the associated thermodynamic forces based on the theory of damage mechanics to model the behavior of damaged composite laminates under fatigue loading. The evolution laws of damage variables were established in the model and the influence of stress ratio was considered. Besides, the evolution model of inelastic strain was proposed to take account of the inelastic strain during the damage evolution process. Carraro et al. [3, 4] modelled the mechanical behavior of composite materials under multiaxial cyclic loading. In their model, a criterion for the non-fiber 2
controlled fatigue behavior was proposed, and two parameters, i.e. the local maximum principal stress and local hydrostatic stress, were defined to represent the driving forces of damage initiation. An equivalent energy release rate was also defined for the propagation of the initiated cracks in the model. Glud et al. [5] established an off-axial matrix cracks initiation and propagation model for composite laminates under fatigue loading. The influence of stress ratio on the off-axial matrix cracks initiation was considered in the model, and the Weibull distribution was adopted to predict the evolution of matrix crack density. Kawai et al. [6] established a ply basis fatigue life prediction model, the inelastic deformation and in situ strength of the ply were considered in the model. A first ply fatigue failure criterion was used to predict the fatigue life of composite laminates. Hosoi et al. [7] investigated the behavior of the transverse matrix cracks growth under high cycle fatigue loading. The free-edge effect was taken into consideration and the micromechanics method was utilized to modify the energy release rate for matrix cracking. Wu et al. [8] investigated the damage mechanisms of woven composites under very high cycle fatigue loading and proposed a prediction model. In the model, surface crack density was adopted to characterize the damage state of composites, and the relation between surface crack density and fatigue life was established to predict the fatigue life of composites under very high cycle fatigue loading. Shen et al. [9] investigated the damage evolution in cross-ply glass fiber reinforced plastic laminates under fatigue loading experimentally. Four configurations of lay-up with different thickness of cracking plies were chosen to study the in situ behavior of damage 3
initiation and evolution. In the experiment, light transmission method was adopted to observe and record damage state and eight stress levels were chosen for each configuration of cross-ply laminates. Also, the degradation of longitudinal modulus of laminates under fatigue loading were obtained. In the present paper, the influence of transverse matrix cracks on the bi-directional stiffness properties of composite laminates were studied and the micromechanics-based method was adopted to characterize the influence, the initiation and evolution of transverse matrix cracks were also investigated. The initial matrix crack initiation life was considered as a fatigue property of composite materials which reflects the resistance ability of composite materials on cracking, and the initial matrix crack initiation life curves of cracked plies were obtained according to the experimental results in this paper. The relation between the initial matrix crack initiation life and the maximum stress in cracked plies were obtained to predict the initiation of initial matrix cracks. Besides, the relation between the critical energy release rate and initial matrix cracks initiation life were established, and the distribution of local matrix cracks initiation life under fatigue loading in cracked plies was obtained based on the distribution of critical energy release rate under static loading, then the evolution of matrix crack density was obtained based on the distribution. 2. Damaged mechanical behavior under fatigue loading According to the crack opening displacement theory in the micromechanics, the stiffness degradation of composite materials was related to the matrix crack density and surface displacements of matrix cracks. With the plane stress assumption, the relation 4
between the damaged stiffness matrix of the elementary ply and the matrix crack density and normalized surface displacements of matrix cracks could be obtained [10], 1
q Qd = I + 0 Q0U Q0 E1
(1)
where Qd denotes the damaged stiffness matrix, ρ denotes the density of transverse crack, I is the unit matrix, q denotes the ratio of the extension length to the total length, which considers the condition that the transverse crack has not run through the width direction of specimen, U denotes the normalized displacement matrix of matrix crack surface. E10 and Q0 are the undamaged longitudinal modulus and stiffness matrix respectively. According to the equation (1), there is no variables related to the external load, which indicates that the influence of matrix cracks on the material properties is just determined by the damage state and material properties and independent of the loading form, hence equation (1) holds both under quasi static loading and fatigue loading. Then the elastic modulus of the damaged ply can be obtained by the above equation,
Q11d Q22d Q12d 2 E1 Q22d E2
Q11d Q22d Q12d 2 Q11d
(2)
G12 Q66d where E1(ρ), E2(ρ) and G12(ρ) are damaged longitudinal modulus, damaged transverse modulus and damaged shear modulus respectively. The above equations characterize the influence of transverse matrix cracks on bidirectional material properties of damaged plies. Then the constitutive model of 5
damaged composite laminates could be obtained based on the classic laminate theory. 3. Damage evolution model under fatigue loading 3.1 Initial matrix cracks initiation life curve In order to study the initiation and evolution of matrix cracks in composite materials under fatigue loading, a relation between the stress level in cracked plies and number of cycles when cracks initiate should be established, and then the initiation of matrix cracks could be predicted based on the relation. With the increase of cycle number, transverse matrix crack density increases in cracked plies, which results in the change of material properties and stress state in cracked plies. However, the initiation of the first matrix crack only depends on the undamaged material properties and external load, so it could be assumed that under fatigue loading, the first matrix crack initiates in the cracked plies after a known constant amplitude load spectrum, and then the following matrix cracks initiate under variable amplitude loadings whose stress level decreases, as shown in Fig.1. Hence, the relation between the stress level in cracked plies and the cycle number when the first crack initiates, which is named as the initial matrix crack initiation life Nini, could reflect the basic fatigue property of cracked plies in composite laminates, and the curve of this relation could be called as the S-Nini curve of the cracked plies. Then the curve could be considered as a fatigue property of composite material and adopted to predict the initiation of matrix cracks in cracked plies. In the experiment, the S-Nini curve could be obtained by recording the cycle numbers when the first transverse matrix crack initiates under different stress levels.
6
Fig. 1 The decrease of stress in cracked plies with the increase of cycle number. 3.2 Initial matrix crack initiation equation As a fatigue property of composite materials, the S-Nini curve could be obtained by experimental results, but also could be obtained by theoretical derivation. Plenty of studies show that the evolution of transverse matrix cracks of composite materials under fatigue loading could be determined by a Paris-law like equation [11-13], G d k( ) Gti dN
(3)
where ρ denotes transverse matrix cracks, N denotes cycle number, ∆G=Gmax-Gmin denotes the range of energy release rate, Gti denotes the energy release rate when the first matrix crack initiates in cracked plies under quasi static loading, k and α are material parameters. With regard to the initiation of initial matrix crack in cracked plies, we could have
1 L dN N ini d
(4)
where L denotes the length of cracked plies, Nini denotes the initial matrix crack 7
initiation life. According to equation (3) and equation (4), the equation of Nini and ∆G could be obtained, N ini
1 G - ) ( kL Gti
(5)
Based on the micromechanics method, the range of energy release rate under tension-tension fatigue loading could be determined by maximum and minimum stress in the cracked plies [14], 90 2 90 2 G ( max ) ( min ) = 90 2 Gti ( ti )
(6)
where σ90max and σ90min are maximum stress and minimum stress in cracked plie respectively, σ90ti denotes the initiation stress of initial matrix crack in cracked plies under quasi static loading. According to equation (5) and equation (6), we could obtain the following equation, 90 2 90 2 90 2 90 2 ) ) ( min ( max ) ) ( min 1 ( max N ini = ( ) ) C( 90 2 90 2 kL ( ti ) ( ti )
(7)
where C=1/kL is a material parameter. When the stress ratio of the fatigue loading is smaller enough, σ90min would be much smaller than σ90max, so the above equation could be simplified as
N ini =C (
90 2 ( max ) ) 90 2 ( ti )
(8)
Then we could obtain the logarithmic form of the above equation, 90 lg N ini = lg C 2 lg ti90 2 lg max
(9)
When the maximum stress in composite laminates under fatigue loading is known, the maximum stress in cracked plies could be calculated based on the classic theory 8
laminate theory, and then the initiation life of initial matrix crack in cracked plies could be determined according to the above equation. 3.3 Random distribution of material property Initial defects in composite materials results in the random distribution of material mechanical properties, which causes the different resistance ability on cracking in materials and leads to the evolution process of matrix cracking. In many quasi static models, the critical energy release rate was assumed to obey two-parameter Weibull distribution to obtain the evolution process of matrix cracking [15, 16], and the prediction results were compared with experimental results to verify the distribution assumption. For fatigue model, the initial matrix crack initiation life was considered as a fatigue property of composite materials, hence it is also random distributed due to the initial defects in composite materials. In order to be consistent with the quasi static models, the distribution of the initial matrix crack initiation life is deduced based on the two-parameter Weibull distribution of critical energy release rate in quasi static models. Under quasi static loading, the relation between critical energy release rate and initiation stress of initial matrix crack in cracked plies could be obtained, GC =
1 ( ti90 ) 2 2E 0
(10)
where E0 is the initial elastic modulus, GC denotes the critical energy release rate for matrix cracking, σ90ti denotes the initiation stress of initial matrix crack. According to the above equation, the distribution of σ90ti could be determined based on the two-parameter Weibull distribution of GC. Then in equation (9), α, C and σ90max are constants, hence the distribution of Nini could be determined based on the 9
distribution of σ90ti in cracked plies. 3.4 Damage accumulation model for matrix cracking evolution In order to establish a damage accumulation model for matrix cracking evolution, the following three aspects should be considered, (1) the definition of damage caused by a single cycle on the materials, (2) how does the damage accumulate after several cycles, (3) the definition of critical damage when material fails. From the perspective of macroscopic phenomenology, the Miner fatigue damage is adopted to define the damage for matrix cracking in composite materials under fatigue loading, Di =
1 N ini
(11)
According to the above definition and the S-Nini curve of cracked plies, damage caused by a single cycle on the materials could be obtained, which is D1= 1/Nini. With regard to the damage accumulation, it is assumed that every single cycle is independent of each other, hence the fatigue damage could accumulate linearly, which indicates that the linear Miner cumulative damage theory is adopted. Then, damage caused by n cycles under constant amplitude loading could be obtained, n
n
i 1
i 1
D n Di =
1 ( N ini ) i
(12)
where (Nini)i denotes the crack initiation life corresponding to the maximum stress Si. According to the definition of critical damage, when Dn=DCR, matrix crack initiates, and the critical damage is assumed that DCR=1 under the constant amplitude loading. 10
4. Results and discussion In order to verify the model, the experimental results of E-glass/YPX-3300 composite laminates with two configurations [02/904]s and [0/904]s are compared to the prediction results, including the evolution of matrix crack density and the degradation of stiffness. 4.1 Fatigue experiment introduction Experiments of several cross-ply laminates of glass-fiber reinforced composite under fatigue loading with four stress levels were carried out [9]. In the experiments, glass-fiber reinforced composites made from E-glass/YPX-3300 were employed, and the geometry and dimension of specimen are shown in Fig.2. In the experiments, glassfiber reinforced composites made from Henderson composite materials co., LTD was adopted. The high strength glass fiber and 3300 epoxy resin from Kunshan Yubo composite materials co., LTD were employed. The composite laminates were manufactured by hand lay-up of pre-preg followed autoclave curing, and the volume fraction of the fibers is 70%. Several material properties of E-glass/YPX-3300 are shown in Table 1. Configurations of cross-ply laminates include [02/904]s and [0/904]s, and the thickness of each ply is 0.125mm.
Fig. 2 Specimen configuration (dimensions in mm) 11
Table 1 Material properties of E-glass/YPX-3300 E1 (MPa) 43157
E2 (MPa) G12 (MPa) 10810
4215
ν12 0.306
The experiment was carried out according to the ASTM D3479 test standard [17], continuous tests were conducted with a MTS 370.10 test machine, which is a servohydraulic test machine with hydraulic grips. Tests were carried out under load control with a frequency of 8 Hz and a load ratio of R=0.1. The sinusoidal waveform of applied load was selected. And for each configuration of laminates, four stress levels were selected, whose maximum stresses are 60%, 70%, 80% and 90% of the initiation stress of initial transverse matrix crack under quasi static loading respectively. Due to the transparency of E-glass/epoxy material system and the convenience of light transmission method, this method was used to record the damage state of composite laminates simultaneously in the experiment. For the matrix cracking mechanism, the crack density was used as damage variables, and it was defined by dividing the numbers of cracks in the damaged plies by the length of the ply. In this study, the average line crack density was adopted to characterize the matrix crack damage states. Then the damaged stiffness properties could be deduced from each hysteresis loop. In order to study the stiffness degradation of composite laminates with different configurations intuitively, the normalized longitudinal modulus was calculated, which was defined by dividing the damaged longitudinal modulus by initial value.
12
4.2 Determination of material parameters According to the experimental results, the S-Nini curves of two cross-ply laminates could be obtained. In the fatigue experiment, the initiation lives of initial matrix crack in cracked plies in laminates under four stress levels were recorded, and the maximum stress in cracked plies could be determined with classic laminate theory (CLT). The two S-Nini curves are shown in Fig.3.
Fig. 3 S-lgNini curves of two laminates. Then according to the S-Nini curves, two material parameters C and α could be determined, which are shown in Table 2. Table 2 Material parameters in S-Nini curve of two cross-ply laminates Configuration
C
α
[0/904]s
160.63
-4.63
[02/904]s
72.88
-5.51
In cross-ply laminates, 90ºplies are supported by the neighboring 0ºplies, with the 13
thicker 0ºplies, the ply thickness ratio(defined as the thickness of the 0ºplies divided by the thickness of the 90ºplies) is larger and hence the constraint becomes stronger, which is considered to result in the in-situ effect. The in-situ effect causes the difference between the S-Nini curves, hence the model is demonstrated as lay-up dependent. 4.3 Model verification and discussions The comparison between the prediction results and experiment results of matrix crack evolution under four stress levels is shown in Fig.4.
(a)
14
(b) Fig. 4 Comparing the prediction results of matrix crack density in two laminates with experimental results. (a) [0/904]s, (b) [02/904]s. The comparison between the prediction results and experiment results of normalized longitudinal modulus under four stress levels is shown in Fig.5 and Fig.6.
15
(a)
(b)
(c)
16
(d) Fig. 5 Comparing the prediction results of normalized longitudinal modulus of [0/904]s laminates with experimental results. (a) 60%, (b) 70%, (c) 80%, (d) 90%.
(a)
17
(b)
(c)
18
(d) Fig. 6 Comparing the prediction results of normalized longitudinal modulus of [02/904]s laminates with experimental results. (a) 60%, (b) 70%, (c) 80%, (d) 90%. According to the comparison, the prediction results of normalized longitudinal modulus and matrix cracks evolution in the two cross-ply laminates under four stress levels are in good agreement with experimental results. In the semi-log coordinates, there are three obvious periods of evolution process in predicted matrix crack density evolution curve, including initiation, evolution and tendency to saturation, and the results agree well with the experimental results, which indicates that the proposed model could predict the matrix crack density evolution in composite laminates under fatigue loading with different stress levels. The prediction results of normalized longitudinal modulus are also in good agreement with experimental results, which indicates that the matrix crack damage characterization model under quasi static model is also suitable for the fatigue model. 19
5. Conclusions A damage model for prediction of matrix cracking initiation and evolution in composite laminates under fatigue was proposed in this paper. The influence of transverse matrix cracks on material stiffness properties was studied, the transverse matrix cracking initiation and evolution was also modelled. (1) The micromechanics based model was verified for composite laminates under fatigue loading, and mechanical behavior of composite laminates with matrix cracks was modelled based on the crack opening displacement (COD) theory in micromechanics, the damaged elastic modulus was also obtained. (2) The initial matrix crack initiation life was considered as a fatigue property of composite materials. The equation between the initial matrix crack initiation life and maximum stress in cracked plies was established, and the equation was adopted to predict the initiation of matrix cracks in cracked plies. (3) The distribution of the initial matrix crack initiation life in cracked plies was obtained based on the distribution of critical energy release rate in quasi static model, which ensures the consistency in fatigue model and quasi static model. Then the distribution was utilized to predict the evolution of matrix crack density in cracked plies. (4) The experimental results of two cross-ply laminates under fatigue loading with four stress levels were compared with the prediction results, including the degradation of normalized longitudinal modulus and evolution of matrix cracks density, and the comparison showed a good agreement, which indicates that the proposed model is effective and reliable in predicting the stiffness degradation and matrix crack evolution 20
in composite laminates under fatigue loading with different stress levels.
Acknowledgements This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References: [1] Talreja R, Singh, C V. Damage and Failure of Composite Materials. 1st ed. New York: Cambridge University Press, 2012. [2] Payan J, Hochard C. Damage modelling of laminated carbon/epoxy composites under
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Fatigue. 2008;30(10):1743-1755. https://doi.org/10.1016/j.ijfatigue.2008.02.009. [7] Hosoi A, Arao Y, Kawada H. Transverse crack growth behavior considering freeedge effect in quasi-isotropic CFRP laminates under high-cycle fatigue loading. Compos
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https://doi.org/10.1016/j.ijfatigue.2018.04.028. [9] Shen H, Yao W, Qi W, Zong J. Experimental investigation on damage evolution in cross-ply laminates subjected to quasi-static and fatigue loading. Composites Part B Engineering. 2017;120:10-26. https://doi.org/10.1016/j.compositesb.2017.02.033. [10] Shen H, Yao W, Wu Y. Synergistic Damage Mechanic Model for Stiffness Properties of Early Fatigue Damage in Composite Laminates. Procedia Engineering. 2014;74(2):199-209. https://doi.org/10.1016/j.proeng.2014.06.250. [11] Kobayashi S, Terada K, Ogihara S, Takeda N. Damage-mechanics analysis of matrix cracking in cross-ply CFRP laminates under thermal fatigue. Composites Science & Technology. 2001;61(12):1735-1742. https://doi.org/10.1016/S02663538(01)00077-X. [12] Llobet J, Maimí P, Essa Y, Martin DLEF. Progressive matrix cracking in carbon/epoxy cross-ply laminates under static and fatigue loading. Int J Fatigue. 2019;119:330-337. https://doi.org/10.1016/j.ijfatigue.2018.10.008. [13] Takeda N. Microscopic fatigue damage progress in CFRP cross-ply laminates. 22
Composites. 1995;26(12):859-867. https://doi.org/10.1016/0010-4361(95)90879-5. [14] Hosoi A, Sakuma S, Fujita Y, Kawada H. Prediction of initiation of transverse cracks in cross-ply CFRP laminates under fatigue loading by fatigue properties of unidirectional CFRP in 90° direction. Composites Part A: Applied Science and Manufacturing. 2015;68:398-405. https://doi.org/10.1016/j.compositesa.2014.10.022. [15] Vinogradov V, Hashin Z. Probabilistic energy based model for prediction of transverse cracking in cross-ply laminates. International Journal of Solids & Structures. 2005;42(2):365-392. https://doi.org/10.1016/j.ijsolstr.2004.06.043. [16] Varna J, Joffe R, Talreja R. A synergistic damage-mechanics analysis of transverse cracking in [±θ/904]S laminates. Composites Science & Technology. 2001;61(5):657665. https://doi.org/10.1016/S0266-3538(01)00005-7. [17] ASTM Standard. Standard Test Method for Tension-Tension Fatigue of Polymer Matrix Composite Materials. ASTM. 2007.
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Declarations of interest: none.
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Highlights: (1) The micromechanics based model was verified for composite laminates under fatigue loading in this paper, and mechanical behavior of composite laminates with transverse matrix cracks was modelled based on micromechanics theory. (2) The initiation of matrix cracks in cracked plies was predicted with the definition of the initial matrix crack initiation life, which was considered as a fatigue property of composite materials in this paper, and the equation between the initial matrix crack initiation life and maximum load stress in cracked plies was deduced. (3) The evolution of matrix crack density in cracked plies was predicted with the distribution of the initial matrix crack initiation life in cracked plies, and the distribution was obtained based on the distribution of critical energy release rate in quasi static model, which ensures the consistency in fatigue model and quasi static model. (4) The experiment of cross-ply laminates under fatigue loading with four stress levels was conducted, several material parameters were determined based on the experimental results, and the prediction results were compared with the experimental results, including the degradation of normalized longitudinal modulus and evolution of matrix cracks density, and the comparison shows a good agreement
25