Jin SONG,Weidong WEN,Hito CUI

aCollege of Energy and Power Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

bDepartment of Mechanical and Biomedical Engineering,City University of Hong Kong,Hong Kong 999077,China

1.Introduction

Textile composites are being widely applied in the field of aerospace engineering due to their excellent mechanical properties,i.e.high specific stiffness/strength and outstanding fatigue resistance.2.5D Woven Composites(2.5D-WC)not only possess a superior delamination resistance capacity in comparison with 2D laminated composites,but also have a simpler structural configuration than 3D textile composites.Recently,many parts in the aero-engine field,i.e.woven fan/compressor blades and casing,have been manufactured using resin matrix composites.Nevertheless,the characteristics of long-term service and elevated temperature environment in aero-engine inevitably result in difficulty of fatigue-related theoretical research,especially for the study with respect to the fatigue life prediction model at un-ambient temperatures.1,2

Many researches have reported about the mechanical properties and prediction models of woven composites based on experimental and finite element methods.Montesano et al.3,4investigated the mechanical behavior of 2D triaxially woven composites at different temperatures by experiment,and found that fatigue behavior was not sensitive to temperature at 120°C.Selezneva et al.5experimentally investigated the failure mechanism in off-axis 2D woven laminates at ambient temperature(20 °C),105,160 and 205 °C,and demonstrated that the woven yarns began to straighten out and rotated towards the loading direction just prior to failure.Vieille and Taleb6studied the influence of temperature and matrix ductility on the behavior of notched 2D woven composites at ambient temperature(20 °C)and 120 °C,and the results revealed that the highly ductile behavior of thermoplastic laminates was quite effective to accommodate the overstresses near the hole at the temperature higher than the glass transition temperatureTm.Koumpias et al.7predicted the strength of 3D fully woven composites at ambient temperature based on a homogenized Representative Volume Element(RVE).Zhou et al.8studied the damage and failure characterization of 2D woven composites under different uniaxial and biaxial loadings at ambient temperature by adopting a two-step,multi-scale progressive damage analysis.Li et al.9developed a micromechanical finite element model to predict the effective mechanical properties of woven fabric composites at elevated temperatures.Although there have been several works in predicting mechanical properties of textile composites by simulation,the specific research pertaining to 2.5D-WC is scarce as yet.Previous works in terms of establishing and simulating the mechanical behavior of 2.5D-WC at ambient temperature have been done by us.1,10,11The geometric model,strength prediction model and damage behavior of 2.5D-WC under the warp and weft static loading at ambient temperature have been systematically analyzed.

Additionally,to the best of our knowledge,very few simulation models related to the fatigue life of woven composites have been reported.Dai and Mishnaevsky12simulated the fatigue life of hybrid fiber reinforced composites at ambient temperature based on X-FEM and unit cell models.Hao et al.13predicted the fatigue behavior of 3D 4-direction braided composites at ambient temperature based on the unit cell approach,where the prediction model takes into account the variation of stiffness and strength of components induced by cyclic loading.Qiu14proposed modified residual stiffness and residual strength models,in which the influence of fiber volume fraction was considered.Coupled with the progression damage approach,the fatigue life of 2.5D-WC was predicted at ambient temperature.

Surprisingly,there is almost no published literature about predicting the fatigue behavior of woven composites at unambient temperatures using numerical approach.However,the immense popularity of woven composites in the aeroengine generally experiences a long-term service under the un-ambient temperatures.Therefore,it is meaningful to establish a temperature-dependent fatigue life prediction model of woven composites,especially in the aero-engine field.

In this work,our principal objective is to establish the fatigue life prediction model that can evaluate the temperature dependent fatigue behavior of woven composites.Taking 2.5D-WC as a specific research object,three stress levels of warp fatigue loading at 20 and 180°C were employed to verify the rationality of fatigue life prediction model.Afterwards,the damage evolution histories at 20 and 180°C were quantitatively observed based on the simulation model.Finally,the fracture morphologies at 20 and 180°C obtained by simulation and testing were compared.This work could provide an available approach in predicting fatigue behavior at different temperatures,which will further facilitate the engineering application of 2.5D-WC.

2.Fatigue life prediction model

The temperature-dependent fatigue life prediction model of woven composites subjected to uniaxial tension-tension loading mainly includes:fatigue damage criteria,damage propagation model,geometry/finite element model and periodic boundary conditions.

2.1.Fatigue damage criteria

Several damage criteria in terms of composite materials,such as Misses,Tsai-Wu and Hashin criteria,have been proposed to solve different engineering issues.As the 3D Hashin criterion has been successfully applied in estimating the strength of woven composites at ambient temperature previously15–17,a modified 3D Hashin criterion taking into account temperature and cycle number will be proposed in this work.Furthermore,based on the previous studies,1failure mechanisms of woven composites can be hypothetically related to two failure modes(two directions)for anisotropic fiber yarns:yarn breaking and matrix cracking.Nevertheless,in addition to temperature,the mechanical properties of fiber yarns are generally sensitive to the volume fraction of fiber in fiber yarns(or called fiber aggregation density).14Therefore,the corresponding failure criteria can be given as follows:

Yarn longitudinal damage(breakage in axial direction,or 1-axis direction):

where σij(i,j=1,2,3)are the stress components;X11is the longitudinal tensile strength of fiber yarn;S12andS13are the shear strength of fiber yarn;β is the shear contribution factor;nis the cycle number;Vfis the fiber aggregation density;Tis the temperature.

Yarn transversal damage(Interior matrix cracking or fiber matrix shear-out failure in in-plane direction,or 2/3-axis direction):

whereY22is the transversal tensile strength of fiber yarn;S23is the shear strength of fiber yarn.

Pure resin matrix failure:

whereXmis the strength of pure polymer.

In order to obtain the corresponding strengths mentioned in Eqs.(1)–(3),the relevant models were proposed in our previous work,18and listed in Tables 1 and 2,in whichEf1andEf2are the longitudinal modulus and transversal modulus of carbon fiber;Emis the modulus of matrix;Vmis the volume fraction of matrix;μf12and μmare the Poisson ratio of carbon fiber and matrix;E11（T,Vf）,E22（T,Vf）andE45（T,Vf）are the longitudinal modulus,transversal modulus and 45°direction modulus of yarns,related to longitudinal direction of carbon fiber at a given temperatureTand fiber volume fractionVf;μ12and μ21are the Poisson ratio of yarns;G12（T,Vf）is the shear modulus of yarns;αx（Vf,ΔT）and αy（Vf,ΔT）are the coefficient of thermal expansion of yarns in longitudinal and transversal directions;αm（ΔT）is the coefficient of thermal expansion of matrix;Tmand ΔTare the glass transition temperature of matrix and temperature difference;T0is the ambient temperature.

2.2.Damage propagation model

Owing to the effect of cyclic loading,localized damages inside different constituents will inevitably take place.Here,the related damage theory will be firstly established and damage propagation models based on different constituents will be proposed in the following sections.

2.2.1.Basic theory

Based on Murakami’s damage theory,the damage model is attributed to the reduction of effective bearing area.The damage tensor ω was adopted as19

where ωiand miare the principal value and principle unit vector of damage tensor;?is the Cartesian product operation.

In addition,the constitutive equation for the damaged composites can be derived by the effective stress vector σ*,which can be defined by the actual stress vector σ as

where I is the unit matrix;M（ω）is the Murakami’s damage tensor.

Based on the hypothesis of equivalent strain energy,the damage variable can be introduced into the constitutive equation with respect to temperature in each cycle,namely

where C（ω,n,T）is the damaged stiffness tensor of fiber yarn;C（n,Vf,T）is the undamaged stiffness tensor.

Table 1 Mechanical models of fiber yarns and pure resin with various temperatures and fiber volume fractions.

Table 2 Mechanical properties of component materials at ambient temperature.

Since the engineering constants are easy to obtain,the damaged stiffness coefficients can be transferred by engineering constants as

whereE,Gand μ are the elastic modulus,shear modulus and Poisson’s ratio at a given temperature in the analyzed cyclic increment,respectively.In order to obtain the above mechanical properties,viz.E,Gand μ,the temperature-dependent models were established in Tables 1 and 2.8

Since the resin is considered as an isotropic material,the damaged stiffness matrix for the pure resin can be defined as20d1=d2=d3.

2.2.2.Sudden degeneration strategy

According to the failure models given in Section 2.1 and the damage theory shown in Section 2.2.1,the direct discount strategy by directly discounting the variablesdiwas used to assess the performance of the damaged elements during fatigue cycling.However,due to the typical difference in the damage severity between warps and wefts,it causes inconsistencies in the discount degree of warps and wefts.Referring to the previous work on this matter,1the corresponding sudden discount strategy is listed in Table 3.

2.2.3.Gradual degeneration strategy

Under fatigue loading,materials with high bear-capacity usually follow a relatively slow degradation induced by softening and micro-cracking inside the yarns prior to failure.The temperature-dependent residual stiffness and strength models were therefore proposed to estimate the gradual degeneration response of materials components.As mentioned earlier,the mechanical properties of fiber yarns are in general sensitive to the fiber aggregation density as well.Hence,the residual stiffness and residual strength models taking into account temperature and fiber aggregation density were proposed as follows(Derivation process can be found in Sections A1–A2 of Appendix A):

Residual stiffness model

whereE（n,Vf,T）andE（0,Vf,T）are the residual stiffness and initial stiffness at the given temperatureTand fiber aggregation densityVfrespectively;X（n,Vf,T）andX（0,Vf,T）represent the residual strength and initial strength respectively;p= σmax/X（0,Vf,T）is the ratio of applied maximum stress and initial strength at the corresponding temperature;nandNfcorrespond to the cycle number and fatigue life at the related temperature respectively;c1,c2,b,Trandk1–k5are the correlative parameters of residual stiffness and strength models.Notice that these parameters can be found in Table 4.

Table 3 Sudden degradation strategy.

Table 4 Gradual degradation model parameters of fiber yarns.

2.3.Geometric and finite element models of 2.5D-WC

Taking into consideration the discrepancy in configuration between the out most layer and internal layers resulting from Resin Transfer Molding(RTM)technology(see Fig.1(a)and(b)),two types of geometric model,called inner-cell model and full-cell model,will be established.In accordance with the micrograph and previous work,10,11the crosssectional configuration of weft yarns is assumed to be a quadratic curve(Fig.1(c)):

A rectangular shape is adopted to describe the crosssectional configuration of warp yarns,while the extended outlines of weft and warp yarns are represented as a straight line and piecewise function,viz.quadratic line and straight line:

In the following process,the inner-cell model can be firstly introduced(see Fig.1(c)).

(1)The boundary dimensions of the inner-cell model:

whereNkandNjare the number of weft yarns at the same height and the number of warp yarns in each cell;MwandMjare the weft arranged density and warp arranged density.

(2)The cross-sectional sizes of warp yarn:

whereAjis the cross-sectional area of warp yarn;Teis the linear density of yarns;ρ is the material density;Pjis the fiber aggregation density of warp yarns;W1jandW2jare the width and height of warp yarns,andW2j+W2w=Lhis the thickness of the inner-cell model.

(3)The cross-sectional sizes of weft yarn and inclination angle:

whereAwis the cross-sectional area of weft yarns;Pwis the fiber aggregation density of weft yarns;Lzis the height in the thickness direction;Nhis the layer number of weft yarns;W2wis the height of weft yarns.

In addition,a condition of the first-order continuity in pointCmust be satisfied to ensure the smooth transition in that point:

where θ is the inclination angle of warp yarns.

Meanwhile,the inclination angle can also be described by using the following relationship:

By Eq.(16),the inclination angle can be calculated by the bisection method.Furthermore,according to the continuity condition in pointC,the following equation can be obtained:

The configuration of weft is obtained by adjustingW1wto make sure that the area of weft is equal toAw:

Therefore,the unknown parametersW1w,~a,~cand θ can be calculated according to simultaneous Eqs.(15)–(18).Ultimately,based on the woven parameters listed in Table 5,the inner-cell model can be established by the above parameters(see Fig.1(d)).

Subsequently,the full-cell model can be established based on the inner-cell model.Fig.1(e)illustrates the forming process of 2.5D-WC based on RTM technology.Based on the assumptions,the processing characteristic as mentioned above and the inner-cell model,the full-cell model considering the outermost layer structure can be established as shown in Fig.1(d).

Based on the full-cell model,a bottom-to-up approach21was used to establish the finite element model.Considering the balance between calculation efficiency and simulation precision,a free meshing mode with the capability of automatically refining certain domains was adopted.Six surfaces were initially meshed,with identical mesh configuration on each of the opposite surface(see Fig.2(a)).Fig.2(b)exhibits the finite element models of each components.The total number of mesh elements reached up to 92924.The inner-cell and full-cell finite element models were exhibited in Fig.2(c).Here,the effect of mesh size on the mechanical properties will be discussed in Section 3.2.

2.4.Periodic boundary conditions

The stress-strain field in the unit cell should have the same properties of translational symmetry as the composite itself.In order to ensure displacement compatibility and force continuity of opposite surfaces in the Representative Volume Cell(RVC),periodic boundary conditions should be imposed in the simulation.For the 3D problem shown in Fig.3,the set of nodes for each boundary are represented bya-f,respectively.The macroscopic displacement constraints for the corresponding nodes on surfaces can be defined as10

whereUi(i=x,y,z)andFi(i=x,y,z)are the displacements and loads in thex,yandzdirections.

Based on the discrete method,it is convenient to establish the boundary conditions in software.Assume that the average mechanical properties of RVC are equal to the average properties of 2.5D-WC,and the equivalent average stress can be defined as10

2.5.Stress analysis process

In the case of fatigue loading,since the applied load in the form of sine wave constantly varies,analyzing any states in the process will inevitably lead to a large amount of calculation.Based on the fatigue theory,the applied maximum load plays a critical role in crack growth rate and fatigue life.Therefore,the stress analysis in the proposed fatigue life prediction model will be established based on the following assumptions:

(1)Stress analysis is only carried out at the peak of applied load in each discrete cycle.

(2)Thermal equilibrium state has been reached ahead of fatigue analysis.

(3)Continuous fatigue loading process is artificially divided into several discrete cycles,namely n0,n0+Δn,...,n·Δn.

Table 5 Woven parameters of 2.5D-WC.

Fig.2 Finite element meshing process and corresponding mesh model.

Fig.3 Periodic boundary conditions for 3D unit with a cuboid-shaped configuration.

Consequently,the constitutive relationship taking into account temperature in each discrete cycle can be derived in matrix form as follows(Derivation process can be found in Section A.3 of Appendix A):

where δ（Δu）and δ（Δε）are the virtual deformation increment vector and virtual strain increment vector in matrix form respectively when the cycle number increases fromnth to(n+Δn)th;α（Vf,ΔT）is the coefficient vector of thermal expansion;ˉTn+Δnis the force vector loaded on the surfaceSσat the(n+ Δn)th cycle;Superscript ‘T” is the transposition operation;ΔTis the temperature variation.

According to the results obtained by solving the constitutive function Eq.(21)using Finite Element Method(FEM)at the point of maximum stress in each given cycle,damage gradually accumulate in the elements under the cyclic process defined by the degradation strategies in Section 2.2.If the element stress surpasses the fatigue damage criteria defined in Section 2.1,the relevant element will be considered as failure.The FEM calculation is carried out until the ultimate failure conditions are satisfied,where complete failure of the material occurs.Here,ultimate failure conditions are defined as:when the amount of damaged elements in the warps and wefts with respect to longitudinal damage in the yarn exceeds 50%.

The process of simulation can be shown as the flowchart in Fig.4.

Fig.4 Flowchart of simulation analysis based on RVC using FEM.

3.Results and discussion

In order to verify the rationality of the fatigue life prediction model proposed in this work,2.5D-WC were used to be an object.A set of sixteen specimens(T300 carbon fiber/QY8911-IV resin)was tested under fatigue loading in the warp direction at ambient temperature(eight samples,20°C)and un-ambient temperature(eight samples,180°C).A sinusoidal tension-tension cyclic loading was applied along the warp direction at constant maximum stress amplitude and frequency of 10 Hz using a hydraulic MTS 809.The minimum to maximum stress ratio was set to 0.1.In accordance with the static testing results,1the virgin strengths of 2.5D-WC at 20 and 180°C were found to be 515.09 and 431.89 MPa,respectively.Three stress levels of 90%,87%,80%at 20°C and 80%,75%,73%at 180°C were applied to investigate the fatigue life and the damage mechanisms by simulation.More details on the material and experimental method can be found in Section A.4 of Appendix A.

3.1.Parameters and temperature correlation analysis

Many studies have evidenced that the mechanical properties of resin matrix composites generally decease with increasing temperature,especially at elevated temperatures,which can be attributed to the obvious decline of the resin’s mechanical properties inside the yarns or interlamination.Therefore,a series of T300/QY8911-IV unidirectional composites,viz.[0]8(Vf= 47.20%),[0]12(Vf=51.89%,62.97%),[0]16(Vf=64.32%),[±45]2s(Vf=44.40%),[±45]3s(Vf=52.10%,56.59%)and[±45]4s(Vf=65.00%),were manufactured by RTM technology.Fatigue life,residual stiffness and residual strength tests were conducted at 20,160 and 200°C,and the correlation parameters in the residual stiffness and strength models(Eqs.(8)and(9))were obtained using genetic algorithm,and are given in Table 4.Fig.5 exhibits the compared results of predicted curves based on fitting parameters and test data.It can be seen that the fatigue-dependent models are well coherent with the corresponding testing results at specific fiber volume fraction and temperature.Additionally,the fatigue life,residual stiffness and strength models were also verified in several other situations,and the related results can be found in Section A5 of Appendix A.

Fig.5 Fatigue-related mechanical behaviors with different fiber volume fraction and temperature.

3.2.Stress distributions of 2.5D-WC at ambient and un-ambient temperatures

Fig.6 exhibits the deformed contour images of stress distribution subjected to warp tension-tension fatigue loading at the maximum stress level of the 10th cycle.Based on the stress contour at 20°C,warps carry the majority of the applied load during the loading cycle(see Fig.6(a)),and stress intensity is obviously enhanced inside the weft yarns.Stress concentration occurs in the intersection regions between the warps and wefts at 180°C(see Fig.6(b)).The highly stressed domains in wefts can be attributed to the elevated temperature which results in the resin softening and disaccord deformation of the warps and wefts induced by different thermal expansion coefficients.Therefore,the influence of temperature on the initial stress distribution of 2.5D-WC subjected to fatigue loading in the warp direction is significant.

3.3.Mesh dependency

In FEM,mesh size plays a crucial role in analytical precision,especially for the complex geometric model.Hence,it is worth to discuss the mesh size before fatigue analysis.Here,we carried out a series of simulation to predict the initial warp stress in the case of 0.1%strain,where the mesh sizes are 0.06,0.08,0.10,0.15,0.20,0.25,0.30 and 0.40,corresponding to the entire mesh numbers of 954674,481972,247519,139902,92924,60499,57347 and 53292,respectively.

Fig.7 illustrates the warp stress and solving time curves as the function of mesh number.It can be clearly seen that the warp stress decreases with the increase of mesh size,but later it generally trends to be horizontal,indicating that the influence of mesh size on the mechanical properties dependency is slowly weakening.Noticeably,although the mesh size has some effect on the predicted stress,the maximum deviation and error are merely 1.2 MPa and 2.8%respectively,which means that the mechanical properties are not susceptible to mesh size.However,the solving time has a sharp upward trend during this process.Specifically,when the mesh size is equal to 0.05,the mesh number in the aggregate will exceed 106,which is quite hard to converge in ANSYS.Since the fatigue life prediction is a relatively complex process that takes too much calculating time,we therefore choose 0.20 as the mesh size in this work.

3.4.Fatigue lives of 2.5D-WC at ambient and un-ambient temperatures

Based on the full-cell model along with fatigue life prediction model,the fatigue lives of 2.5D-WC at 20 and 180°C subjected to warp tension-tension cyclic loading were predicted.The corresponding results compared with the testing results are exhibited in Table 6.

Fig.6 Stress distribution and stress concentration regions in deformed 2.5D-WC at the 10th cycle.

It can be seen from Table 6 that the fatigue life of 2.5D-WC is noticeably dependent on temperature,and the deviation between prediction values and testing values in logarithmic scale for each stress level is approximately 1,indicating that the temperature-dependent fatigue model agrees well with the test data.Additionally,the model proposed in this work was compared with the one proposed by Qiu,14which predicted the fatigue life of 2.5D T300-3 k/HCGP-1 woven composites at 20°C,using the material properties of T300-3 k/HCGP-1.The corresponding fatigue life obtained is given in Table 7.Compared with Qiu’s results,the maximum errors(the ratio of prediction value and testing value)calculated by our model at stress level of 76%,72.5%and 70%are 1.04,1.03 and 1.01 respectively,which are lower than those obtained by Qiu’s model.14

Fig.7 Influence of mesh size on warp stress of 2.5D-WC.

Table 6 Verification of predicted fatigue life and experimental results of 2.5D-WC.

3.5.Damage evolution processes and mechanisms at ambient and un-ambient temperatures

In addition to predicting the fatigue life,damage propagation curves of 2.5D-WC subjected to warp direction,along with related evolution photographs,were obtained by simulation at 20 and 180°C based on the proposed fatigue model(Eqs.(8)and(9)).The amount of damage elements as a function of the number of cycles at 20 and 180°C is illustrated in Fig.8,and the corresponding damage propagation photographs are shown in Figs.9 and 10.

Table 7 Comparison of prediction values based on presented model and Qiu’s model.

For the damage evolution process at 20°C,the damage propagation rate remains essentially constant as the number of cycles is increased,up to the occurrence of damaged elements in the traversal direction of the yarn and pure resin matrix(see Fig.8(a)).These damaged elements mostly exist in the intersection areas of warps and wefts.This is due to the resin damage inside or outside yarns(see Fig.8),which also reflects the stress concentration of woven composites located at yarn contact zones rather than interlayers or resinrich regions(see Fig.6).When the number of cycle reaches to 10010,the 2.5D-WC reaches to a saturation state,followed by a rapid increase in the percentage of damaged elements until the ultimate failure(see Fig.8(a)).It can be obviously seen from Fig.8(a)that during this damage propagation process,a large number of damaged elements in the longitudinal direction of the warp yarns emerge in the inclined area of bear-loading warps,further facilitating the failure of surrounding resin(see Fig.9).This saturation phenomenon was also observed in the static process of warp loading.1Ultimately,longitudinal damage elements in the inclination warp yarns propagate to the edge of the full-cell along the weft direction,causing the failure of 2.5D-WC.Likewise,a large amount of damaged elements in the traversal direction and failure in the resin-rich region around the damaged warps on the fracture surface were also predicted.Qiu also studied the fatigueinduced damage extension behavior of 2.5D-WC under ambient temperature condition,and consistent results were obtained by experiment.14

For the damage evolution process at 180°C as shown in Fig.10,initial damaged elements in the transversal direction and resin failure still pertain.The corresponding damage locations are also consistent with that under the ambient temperature condition.However,the amount of them is much higher,indicating that the components’properties are sensitive to the environmental temperature.18Based on the damage extension curves at 20 and 180°C,the fatigue-induced transversal damage and resin failure as the number of cycles increased.This result could give rise to a gradual decrease in the residual mechanical properties of2.5D-WC,22and non-existent saturation state(see Fig.8(b)).It was found that the ultimate failure of 2.5D-WC occurs only a few hundred cycles upon the onset of longitudinal damage in warps.These damaged elements in the longitudinal direction of warps rapidly extend from the contact areas of yarns to the edge of material along the weft direction,concentrated at the root of inclined warps(see Fig.10).A certain extent of transversal damage and resin failure still exist around the damaged warps,which is caused by the damage to principal load-bearing warps.

In addition,due to the perpendicular relationship between the weft direction and the loading direction,there is no longitudinal damage in weft yarns.These findings coincide with the experimental fracture surface as shown in Fig.11.Therefore,prior to the occurrence of longitudinal damage in warps,the transversal damage and resin rich failure are the principal damage modes.After the onset of longitudinal damage in warps,the damage extension rates in the corresponding damage locations are remarkably affected by the damaged warps.Moreover,although the fatigue life is noticeably shortened at 180°C,which is induced by the decrease in mechanical properties of resin,the damage extension behavior of 2.5D-WC is still dominated by the warp yarns owing to the 3D network configuration.

3.6.Fatigued fracture morphologies at ambient and un-ambient temperatures

Fig.11 exhibits the comparison between the simulated fatigue fracture morphologies and corresponding testing results at 20 and 180°C.The simulation results exhibit more longitudinal fracture occurring inside the inclination warp yarns at 20°C,whereas the ones at 180°C are mainly located at the root.These are consistent with the corresponding fatigued fracture morphologies obtained through experiments.In addition,a more severe delamination phenomenon can be seen surrounding the longitudinal fractured elements in warps at 20°C than that at 180°C,which was testified by simulation and experiment.Although the stiffness and strength of QY8911-IV resin experience a gradual decrease with the increase of temperature,18temperature gives rise to the enhancement of resin’s ductility, which has been revealed by our previous work.22Therefore,the heat-resistant QY8911-IV resin(Tm～ 256 °C18)at 180 °C facilitates a more uniform bearloading status in warp yarns,resulting in an even fatigue fracture morphology at elevated temperatures,along with a few damage elements corresponding to delamination.Finally,the fracture morphologies obtained by simulation and experiment consistently demonstrate that there is no yarn longitudinal damage in weft yarns.

Fig.8 Percentages of damage elements corresponding to different failure modes during warp fatigue process at 20 and 180°C.

Fig.10 Damage propagation of 2.5D-WC during warp fatigue process at 180°C.

Fig.11 Comparison of predicted fracture morphology and experimental results of 2.5D-WC at 20 and 180°C.

As stated above,the temperature-dependent fatigue life prediction model is reasonable for predicting the fatigue life and damage evolution behavior of 2.5D-WC at different temperatures.

3.7.Temperature dependency formation

The formation of temperature dependency on the fatigue behavior can be attributed to the balance between mechanical properties of resin and ductility of resin.More specifically,the typical stress-strain curves of pure resin(QY8911-IV)at 20,160 and 200°C are shown in Fig.12(a).The modulus and strength of pure resin decrease as the temperature increases(see Fig.12(a)),resulting in decline in mechanical properties of 2.5D-WC at elevated temperatures.However,the ductility of 2.5D-WC improves at elevated temperatures,due to the softening of resin,causing the stress concentration to be effectively accommodated.6,22Meanwhile,the inclination angle of load-bearing warps decreases to some degree in the fatigue process,strengthening the load-bearing capacity in warp direction(see Fig.12(b)).22Therefore,at 20°C,a severe delamination fracture mode with the brittle failure resin was obtained,but there was a much more uniform fracture morphology at 180°C(see Fig.11).Nevertheless,the fatigue life is obviously reduced,owing to the decrease of mechanical properties of resin induced by elevated temperature.

Fig.12 Thermo-mechanical behaviors of pure resin and damage mechanism of 2.5D-WC.

4.Conclusions

In this work,we proposed a temperature-dependent fatigue life prediction model of woven composites,along with the corresponding damage propagation behavior,at ambient and un-ambient temperatures.Taking 2.5D-WC as an example,the fatigue life,damage evolution regulation and fatigue fracture morphology at 20 and 180°C were successfully predicted based on this model.Furthermore,the stress in warps is higher than that in other components,indicating that warps are the primary load-carrying elements under tension-tension fatigue loading in warp direction.At 20°C,there is an obvious saturation point in the damage propagation curves whereas damages in the non-longitudinal direction experience a gradually rising trend from the initial step to failure.Additionally,a more uniform fracture surface was obtained at 180°C than that at 20°C.This result can be attributed to release of stress concentration caused by the increase in ductility at 180°C,although the mechanical properties of components are inevitably reduced.

Acknowledgement

This work was supported by Jiangsu Innovation Program for Graduate Education(No.KYLX_0237).

Appendix A

A.1.Derivation process of residual stiffness model

Based on the damage mechanics,Mao and Mahadevan23proposed a damage model as

whereDis the accumulated damage;r,m1andm2are material dependent parameters;nandNfare the cycle number and maximum cycle number(fatigue life)respectively.The characteristics of rapid damage accumulation during the first few cycles can be captured with the first term,withm1＜1;the second term shows the fast damage growth at the end of fatigue life withm2＞1;rcan be defined as the related coefficient of stress levels and temperature.Therefore,in this work,the parameters in Eq.(A1)are defined in terms of fatigue life as

where the meaning of parameters refers to Eqs.(8)and(9).Moreover,the damage variableDcan also be expressed by

Substituting Eqs.(A2)and(A3)into Eq.(A1)yields

As it is difficult to measure the residual stiffnessE（Nf,Vf,T）in the failure moment,a relationship between residual stiffness and stress level proposed by Lee24is introduced:

Substitute Eq.(A5)into Eq.(A4),and the theoretical residual stiffness model of 2.5D-WC at various temperatures can be established:

Notice that when the temperatureTis equal to ambient temperatureT0,Eq.(A6)will be degenerated into ambient condition.

A.2.Derivation process of residual strength model

The damage variableDcan also be expressed as a function of residual strength:

Substitute Eqs.(A5)and(A7)into Eq.(A1),and the theoretical residual strength model of 2.5D-WC at various temperatures can be established:

In addition,the fatigue lifeNfin Eqs.(A6)and(A8)can be obtained by

wherea,b,m,nandTrare the relevant fitting parameters,which can be found in Table 4.

A.3.Derivation process of stress-strain relationship

When cycle number increases fromnth to(n+Δn)th,the equilibrium equation at a given temperature can be depicted as

The boundary condition of force is given as

Substituting Eq.(A10)into Eq.(A11),based on the incremental form of principle of virtual work,yields

where δ（Δui）and δ（Δεij）are the virtual deformation increment and virtual strain increment respectively when the cycle number increases fromnth to(n+Δn)th.

Substituting Eq.(A13)into Eq.(A12)yields

Eq.(A14)can be expressed in matrix form:

where σn,Δσ andTσ are stress vector,stress increment vector and thermal stress vector respectively.

It is assumed that the incremental load is enough small so that the stress-strain relations at thenth cycle can be treated as linear,namely

Additionally,as for the stress induced by temperature,it can be defined as

Substituting Eqs.(A16)and(A17)into Eq.(A15),we can obtain

Eq.(A18)is the stress-strain relationship taking into account temperature during the fatigue process,which can be solved by a finite element method.

A.4.Experimental section

The material is a 2.5D woven carbon fiber-reinforced thermosetting bismaleimide resin(QY8911-IV).The 2.5D-WC with six plies of weft yarns were fabricated by a RTM technique.18Ultimately,the fatigue-related specimens with a dimension of 300 mm×25 mm were fabricated as shown in Fig.A1.22

All the tests were carried out by a MTS 810 hydraulic servodynamic material test machine with a 25.4 mm MTS-634-25 extensometer(Fig.A2).Additionally,a MTS809 furnace was used to provide un-ambient temperatures.For the ambient temperature tests,the fatigue tests were run at a frequency of 10 Hz and a stress ration of 0.1 in accordance with ASTM D3479.25For the un-ambient temperature tests,temperature was ramped up at a rate of 10°C per minute from ambient temperature to the target temperature in force control,followed by a heat preservation process for at least 30 min.Afterward,the thermos-mechanically fatigue tests were performed.22

Fig.A1 Schematic illustration of fatigue-related specimens,magnification image of tested specimens.22

Fig.A2 Photograph of MTS-810 test machine,inset 634-25 high-temperature extensometer.

A.5.Residual stiffness/strength models considering temperature and fiber volume fraction

The normalized stiffness and strength curves of yarns as the function of fiber volume fraction and temperature were numerically calculated based on the correlated parameters listed in Table 4,and are exhibited in Figs.A3 and A4.

Fig.A3 Normalized stiffness/strength vs normalized fatigue life curves of unidirectional composites at different temperatures and fiber volume fractions in longitudinal direction.

Fig.A4 Normalized stiffness/strength vs normalized fatigue life curves of[±45]ncomposites at different temperatures and fiber volume fractions.

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