Analysis on the turning point of dynamic in-plane compressive strength for a plain weave composite

Xioyu Wng , Zhixing Li ,*, Licheng Guo ,**, Zhenxin Wng , Jiuzhou Zho

a Department of Astronautic Science and Mechanics, Harbin Institute of Technology, Harbin,150001, China

b AECC Commercial Aircraft Engine Co, LTD, Shanghai, 201108, China

Keywords:Plain weave composite Dynamic strength Quantitative criterion Turning point Failure mechanism

ABSTRACT Experimental investigations on dynamic in-plane compressive behavior of a plain weave composite were performed using the split Hopkinson pressure bar.A quantitative criterion for calculating the constant strain rate of composites was established.Then the upper limit of strain rate, restricted by stress equilibrium and constant loading rate, was rationally estimated and confirmed by tests.Within the achievable range of 0.001/s-895/s,it was found that the strength increased first and subsequently decreased as the strain rate increased.This feature was also reflected by the turning point(579/s)of the bilinear model for strength prediction.The transition in failure mechanism, from local opening damage to completely splitting destruction, was mainly responsible for such strain rate effects.And three major failure modes were summarized under microscopic observations: fiber fracture, inter-fiber fracture, and interface delamination.Finally, by introducing a nonlinear damage variable, a simplified ZWT model was developed to characterize the dynamic mechanical response.Excellent agreement was shown between the experimental and simulated results.

1.Introduction

Fiber reinforced composites have been increasingly employed as protective structures in military, aviation, and marine industries because of their improved mechanical performance and superior damage tolerance[1-3].They are inevitably exposed to shock loads during long-term services, such as hail, bird strikes, and ballistic impacts.To ensure safety and reliability, the rate-dependent behavior of materials must be correctly understood.

There have been numerous publications on dynamic properties of different kinds of woven composites.Such as plain weave carbon/epoxy composites [4,5], plain weave E-glass fiber composites[6,7],plain weave aramid polyamide composites[8,9],plain weave basalt fiber composites [10], satin weave composites [11], satin weave carbon/Kevlar hybrid composites [12], twill weave composites[13],and many other 3D woven composites[14-17].Most of the above studies adopted the maximum or average rate to estimate the strain rate level of specimens in SHPB tests.These methods are commonly used for metallic materials, and their applicability to composite materials needs to be further discussed and validated systematically.In addition,it is inherently difficult for brittle composites to attain a constant strain rate,that is,a plateau.Even if it exists, the start/end range for computing the average strain rate is unclear.Inconsistent strain rates might be derived.As far as the authors are aware, a unified standard for calculating the strain rate of composites is still missing.

Plain weave carbon fiber reinforced composite is one of the most widely used in practice.Hosur[18]studied the strain rate effects of stitched and unstitched plain weave composites.Experimental results showed that both in-plane compressive strengths were enhanced as the strain rate increased from 320/s to 1149/s.Naik et al.[19]conducted SHPB tests on a typical plain weave composite along the warp, fill, and thickness directions in a range of 680/s-2890/s.It was found that there existed no evident rate dependency on in-plane properties.A stress wave attenuation within the specimen was used to account for the differences in peak force between the loading bars.Li et al.[20] investigated the ratedependent behavior of warp-knitted and plain weave composites over a range from 221/s to 1337/s.The in-plane compressive strength of warp-knitted composite was significantly improved,whereas that of plain weave composite changed little.Zhang et al.[21] discussed the dynamic mechanical response of plain weave laminates(CFS-400/CFSR-AB).Strain rate ranged from 4.0×10-5/s to 160/s.A critical strain rate, 30/s, was obtained after a series of high strain rate tests, beyond which there was an obvious rising trend of tensile strength and elastic modulus.

Failure mechanism also changes at different strain rates apart from the mechanical property [22,23].Chen et al.[24] conducted SHPB tests on a carbon fiber woven composite from 400/s to 1400/s.The three-phase failure mechanism of on-impact compression,crack-induced unloading, and crack deviation-caused further condensation was mainly responsible for the greatly increased strength and toughness.Reis et al.[25] investigated the rate sensitivity on off-axis compression for plain weave composites ranging from 87/s to 1294/s.The viscoelastic characteristic of epoxy resins would cause a brittle fracture at a higher strain rate.It promoted a transition from longitudinal cracking failures to delamination buckling failures.According to the study of plain weave composites by Lu et al.[26], the in-plane properties and failure patterns are influenced by strain rate.Strength and failure strain were both enhanced under compressive loads within 200/s-1000/s.The specimen was inclined to be broken up and divided into little pieces at a high strain rate.

Although many works on dynamic behaviors of composites have been done, there still exists something worth noting [27-29].On the one hand,the lack of a strain rate criterion prohibits researchers from reaching a consensus on strain rates and,in turn,on strain rate effects, even for the same SHPB tests.On the other hand, the strength of most composites is reported to be improved with the increasing strain rate.However, this enhancement cannot be infinite.It is quite likely that the dynamic strength has a turning point or critical value.For woven composites [30,31], relevant investigations are very limited, and much fewer on the dynamic viscoelastic constitutive model that can precisely characterize the properties under a certain strain rate.

This paper aimed to investigate the dynamic in-plane compressive behavior of a plain weave composite ranging from 0.001/s to 895/s.A quantitative criterion for calculating the constant strain rate of composites was established to represent the strain rate level of specimens in SHPB tests.Then the achievable strain rate limit of the studied composite materials was theoretically and experimentally estimated.Moreover,the turning point of the dynamic strength was determined through the proposed bilinear relation.Its deformation process and failure mechanism were recorded and analyzed based on high-speed camera.Finally,a simplified viscoelastic constitutive model with damage effects was developed to characterize the corresponding dynamic compressive behavior under different strain rates.

2.Material and experimental procedures

2.1.Specimen design

The 6 mm thick plain weave composite was fabricated using the hot pressing method.Carbon fiber T300/3 K and 601 epoxy resin are utilized in this process.A series of optical micrographs were taken to obtain the representative volume element (RVE) construction.The dimension of RVE in the red box is around 4 mm×4 mm (see Fig.1).Furthermore, the specimen design deserves careful consideration in high strain rate tests.The main principles can be listed as follows:

(1) The recommended specimen diameter is generally within 80% of the bar to prevent excessive expansion during the deformation.

(2) The two loading surfaces must keep parallel and smooth to prevent premature failure due to stress concentration.

(3) The specimen size should be large enough to ensure the integrity of the fabric structure.

On account of these, the specimens were processed to 12 mm square by a water jet cutting machine and polished with 240 grit sandpaper.

2.2.Quasi-static compression test

The quasi-static compression tests were carried out on the Zwick Roell 100 test machine.The upper grip was controlled at a speed of 0.72 mm/min; correspondingly, the strain rate of the specimen was about 0.001/s.The axial force was recorded by the force sensor, while the axial displacement was acquired using the Digital Imaging Correlation (DIC) technique.The axial force and displacement information were processed in the form of stressstrain curves.At least five specimens were tested to ensure the effectiveness.

2.3.High strain rate compression test

The high strain rate compression tests were carried out using the SHPB apparatus,as shown in Fig.2.The loading bars are made of high-strength maraging steel.Its density and elastic modulus are 7930 kg/m3and 206 GPa.The length of the 20 mm diameter striker,incident, and transmission bars are 150 mm, 2000 mm, and 1500 mm.A rubber disk with a diameter of 8 mm and a thickness of 1 mm was selected as the pulse shaper to control the first loading pulse to achieve the desired constant strain rate.And for experiments under different strain rates,we used the same configuration of shapers to minimize the impact of environmental factors.A Phantom V-1212 camera with 80,000 fps was positioned 0.5 m away to record the failure process during tests.Since the dynamic tests often show variability, a large number of specimens were tested.At least three specimens were tested for each strain rate level within the range of 0.001/s-650/s,and at least two specimens for each strain rate level within the range of 650/s-895/s.

The strain rate ˙ε, strain ε, and stress σ of the specimen can be calculated on the basis of one-dimensional stress wave theory and stress equilibrium assumption, which can be given as

Fig.1.Schematic illustration of the RVE construction: (a) Microscopic observation; (b) Weave pattern; (c) Compression specimen.

Fig.2.Schematic diagram of SHPB apparatus.

3.Experimental results

Detailed data are summarized in Table 1 and Fig.3.Each data for a different strain rate is a representative one and is not the average for each case.The strain rate is estimated using the proposed quantitative criterion Eq.(4).The peak point of the stress-strain curve represents the failure strength, and its corresponding strain and time are the failure strain and failure time.To ensure repeatability, the standard deviations (SD) of compressive strength are also calculated.The absorbed energy of the specimen is obtained by integrating the stress-strain curve numerically up to the peak point.

3.1.Quantitative criterion for calculating the constant strain rate

The aim of conducting high strain rate tests is to obtain stressstrain curves at different strain rate levels and investigate the rate-dependent behavior of composites accordingly.The strain rate for each curve is expected to be constant, especially for those materials sensitive to the loading rate.Fig.4 shows typical signals on the loading bars.It also should be mentioned that the strain gauges on the incident and transmission bars have the same configuration,and they were measured at the same frequency (10 MHz) using a dynamic strainmeter.According to Eq.(1), the plateau of the reflected signal can indicate a nearly constant strain rate state.

Table 1 Results of quasi-static and high strain rate tests.

Fig.3.Stress-strain curve at different strain rate level.

Fig.4.Typical signals on the loading bars.

Quantitative criterion for calculating the constant strain rate of composites is established in this section.Since the failure of composites is a property degradation process rather than an instantaneous state,the average of a period in the strain rate history curve,instead of an instant strain rate at a specific time, is much more appropriate to represent the strain rate level.Moreover, strain is more accessible and intuitive than stress for determining material failure.The deformation of the studied composites, within the range of 70%-100%failure strain,is sufficient to cover all the typical failure processes.On this basis,the average of the strain rate curve segment,corresponding to the above strain range,is quite enough to be taken as an indicator to characterize the strain rate level of the woven composites.Its expression can be expressed as

Fig.5.Strain rate history curves.

where ˙εcons(70%)is the proposed constant strain rate in this paper.t1and t2are the moments when the specimen strain reaches 70%and 100% of its failure strain, respectively.Fig.5 shows the strain rate history curves of specimen Nos.1-8.The start point corresponds to t1, while the endpoint corresponds to t2.

The reasonableness of the start point of the proposed quantitative criterion is discussed below, as is shown in Table 2.Results show that the composite specimen cannot maintain absolute stability when the strain rate is higher than 808/s.It continues to increase, and the specimen can break before reaching an expected constant state.It would make no sense to have the maximum strain rate ˙εmaxand the strain rate at the failure time ˙εfailin such cases.Moreover,considering that a longer stable target segment means a more representative and desirable loading condition, ˙εcons(60%)and ˙εcons(70%) are better than the others, including ˙εmax, ˙εfail,˙εcons(80%), and ˙εcons(90%).

Then the differences between the remaining two are compared.A low-level slope is important to reach a constant strain rate since it reflects how fast the strain rate changes.The secant slope is used in this paper.Figs.6-8 show the typical strain rate history curve and secant slope of specimen No.1, No.5, and No.7, respectively.It should be noted that a little variation of 1/s in strain rate can make a slope of 1.0×107/s2because the interval between two data points is extremely small, only 1.0 × 10-7s.Although the secant slope of specimen No.1 at 60%failure strain is higher than that at 70%failure strain, 80% failure strain, and 90% failure strain, they are all at a lower-level slope,i.e.a nearly constant strain rate.As for specimen No.5 and specimen No.7,it is easy to find that the secant slope at 60%failurestrain is much greater than that at 70%failurestrain,80%failure strain,and 90%failure strain.Taking the above into account,˙εcons(70%) is the best in the proposed quantitative criterion for calculating the constant strain rate of composites under different strain rate levels.

3.2.Upper limit of strain rate

It is meaningful to figure out the upper limit of strain rate in SHPB tests,which contributes to performing the reliability analysis and studying the strain rate effects.Most researchers agree that stress equilibrium and constant strain rate are essential prerequisites.However,the ideal loading condition cannot be satisfied during the whole test.It takes time to bring the specimen close to a state of stress equilibrium and constant strain rate.The strainaccumulated during the rising phase of the strain rate may have caused damage before reaching the expected very high strain rate level.Thus, there exists an upper limit of strain rate, which is constrained by the following two aspects: stress equilibrium state and achievable constant strain rate level.

Table 2 Effect of key parameters on strain rate estimations.

Fig.6.Strain rate history curve and secant slope of specimen No.1 (55/s).

Fig.7.Strain rate history curve and secant slope of specimen No.5 (585/s).

According to Chen[32],the stress equilibrium requires multiple reflections of the loading pulse.The required number of reflections is related to different materials and loading conditions.Assume that n represents the number of reflections.Then the time for specimens to reach dynamic equilibrium is

Fig.8.Strain rate history curve and secant slope of specimen No.7 (808/s).

The other limitation is to achieve a constant strain rate, which proves to be a more powerful constraint.The specimen takes time to arrive at a target strain rate level.If this level is set too high, it might fail first.For brittle composites, a linear load is needed to attain a constant state

Considering that the failure strain of plain weave composites changed little at a high strain rate, the average of the last five in Table 1 is taken as the failure strain.Besides, it has been reported that n ≥6 and ξ = 0.2 are acceptable for analysis [34].Relevant parameters used for calculation can be seen in Table 3.According to Eq.(12),the final result is 779/s,which indicates that the limitation of constant strain rate state(779/s)is more restrictive than that of stress equilibrium condition (1286/s) when evaluating the upper limit of strain rate.In addition,as illustrated in Fig.5,specimen No.7 has a pretty short plateau, while specimen No.8 has no plateau due to its continuous increase in strain rate.The achievable upper limit is probably around 808/s.A close match between the theoretical calculation and the experimental estimation proves its validity.

3.3.Strain rate effect on strength

This paper studies the rate dependency of in-plane compressive strength.All the tests should ensure the condition of stress equilibrium and constant strain rate as much as possible.Parameter R is employed to quantify the stress equilibrium.It represents the ratio of stress differences between the two sides of the specimen [35].Fig.9 shows a typical stress-strain curve at a constant strain rate(387/s) and a detailed equilibrium history curve.

According to Table 1,the compressive strength is enhanced from 401 to 702 MPa when the strain rate is increased from 0.001/s to 587/s, about 95% higher than that under static loading.Then, it is reduced to 515 MPa when the strain rate continues to increase up to 895/s,but still 28%higher than the static strength.Once the strain rate exceeds 808/s,a constant strain rate can hardly be maintained.However,this does not change the fact that the failure strength falls as the strain rate increases.More test results are shown in Fig.10.An empirical bilinear model is adopted to describe such ratedependent behavior, given as Eq.(13).The turning point of the strain rate effects is also determined.

Fig.9.Stress-strain curve at a constant strain rate (387/s) and a stress equilibrium history curve.

Fig.10.Strain rate sensitivity analysis of failure strength.

where σfailand ˙ε are the failure strength and the strain rate.The coefficient of determination (R2) measures the accuracy of the linear fit; the closer R2(0.94, 0.92) is to 1, the better the fitting result.This bilinear model for strength prediction would be helpful in finite element analysis of the studied composites.

Moreover,a lower limit for rate-dependent behavior can also be determined,as is shown in Fig.10.It is acceptable in engineering for strength of composites to fluctuate within 10%.Then a threshold of 435 MPa, about 1.1 times the 395 MPa, was obtained.The lower limit can be calculated as 60/s by Eq.(13), beyond which obvious rate dependency exists.

Table 3 Parameters for estimating the upper limit of strain rate.

3.4.Failure analysis and damage morphology

Although most researchers believe that the compressive strength of composites will increase monotonically, in this paper,the dynamic compressive strength is increased first and then decreased with the rising strain rate.It is considered that the properties of the matrix and the transformation of the failure mechanism are responsible for the rate-dependent behavior of the studied plain weave composite.

Generally,epoxy resin tends to be tougher when the strain rate increases.It provides a stronger constraint on the lateral displacement of the fiber bundles and restricts the buckling deformation of the specimen.Since the fiber bundles are the main load-bearing structure of the composite, the strength is increased.On the other hand,the epoxy resin also becomes more brittle[23,36]and hence absorbs less energy at a higher strain rate.The severe matrix crack and interface delamination are more likely to occur under dynamic compression loads, which explains the decrease in strength.These opinions are supported by Table 1 and the displayed damage morphology.

More in-depth analyses on failure through a high speed-camera are as follows.Typical images are shown in Figs.11-13, which record the deformation process of the specimen at 387/s, 585/s, and 751/s,respectively.It is evident that the failure process comes more rapidly,and the specimen is destroyed much more seriously.All the damage initiates on the right side of the specimen, near the end face of the transmission bar.Fig.11 displays typical failure process of specimen at 387/s.With the growth of the stress level,the right side of the compressive specimen has a clear tendency to open up,presented in a broom shape.Several micro-cracks have appeared at its foot, and gradually expand to the left.The initiation and expansion of these micro-cracks reduce the mechanical properties of such material, indicating that the damage has occurred and the specimen is about to lose its load-bearing capacity.

Fig.12 displays typical failure process of specimen at 585/s.The broom still forms, accompanied by some visible micro-cracks,when it comes to 62.5 μs.Then a main crack, parallel to the central axis,can be observed in the middle area.It has almost expanded to the other side.However,the integrity in volume is still preserved,without missing fragments or splitting into two pieces.Taking the one at 387/s as a reference,the cracks of the specimen at 585/s are denser in number,deeper in extent,and broader in distribution.It suggests an increase in energy absorption during the failure process.The toughness and strength are enhanced accordingly.

Fig.13 displays typical failure process of specimen at 751/s.When damage evolves,the delamination is manifested by the main crack that cuts through the whole specimen.The specimen fails catastrophically and breaks into two pieces directly under the excessive compressive load.It is worth noting that the one at 751/s has fewer cracks than that at 585/s.The strength decreases as well.

Together with the above observations, it is found that the increased strain rate causes a transition in failure mechanism,from local opening damage to completely splitting destruction.The energy absorption capacity usually determines the degree of damage.As a consequence, the rate dependency of failure is thought to be related to the absorbed energy, which is further confirmed by Figs.14 and 15 using an optical microscope.When the strain rate is lower than 585/s,the specimen can maintain relative integrity,e.g.,specimen Nos.1-5.More cracks are observed on the specimen at a higher strain rate than those at a lower strain rate.The toughness and the impact resistance of such material are enhanced since crack initiation and propagation dissipate much more absorbed energy.Compressive strength is increased as well.However,once the strain rate is higher than 585/s,the specimen is immediately crushed and completely split into several fragments when subjected to shock compression.In such circumstances,the crack growth is unstable.A small driving force can make the crack propagate rapidly and cause a catastrophic failure[37,38].The absorbed energy and compressive strength are reduced instead with the increased strain rate.

The view that the failure mechanism changes is somewhat supported by the fact that a higher strain rate will make the epoxy resin tougher,but a further increase in loading rate may let it more brittle.These findings reasonably explain the turning point of the dynamic strength of the studied composites.Furthermore, as is shown in Figs.14 and 15,three major failure modes are identified:fiber fracture, inter fiber fracture, and interface delamination.The fiber fracture is caused by the buckling and kinking of the yarns,whilst the inter fiber fracture is due to the matrix cracking within the fiber bundles.Both fracture behaviors are raised by the dynamic compression load.According to high-speed photography, the opening process of the specimen is regarded as a motivation to induce the formulation and propagation of interface delamination.

3.5.Dynamic compressive constitutive model

Studying the constitutive models has always been an important but difficult task in material science, especially for high strain rate tests.Based on stress-strain curves in Fig.3, the nonlinear viscoelastic ZWT model is selected to reflect the dynamic compressive behavior.Nevertheless, the original ZWT model [39-41] requires so many parameters and thus makes it impractical in engineering.Furthermore,the damage that accumulates during the deformation until failure can not be ignored either.It would reduce the residual strength and stiffness.A damage variable is therefore needed.Similar phenomena have been observed in other dynamic studies of pure epoxy resin [39], and glass/epoxy composites [42].

Fig.11.Typical failure process of specimen at 387/s: (a) High speed photography; (b) Strain rate history curve.

Fig.12.Typical failure process of specimen at 585/s: (a) High speed photography; (b) Strain rate history curve.

Fig.13.Typical failure process of specimen at 751/s: (a) High speed photography; (b) Strain rate history curve.

Fig.14.Failure modes of specimen Nos.1-4 under an optical microscope (mag.× 20).

Fig.15.Failure modes of specimen Nos.5-8 under an optical microscope (mag.× 20).

To solve the above problems,a simplified ZWT model with damage effects is developed.In this work, the low-frequency component is removed due to its minor contribution.Hence, the simplified ZWT viscoelastic model (see Fig.16) only consists of a nonlinear elastic spring and a high-frequency Maxwell element,which can be expressed as

where the previous three terms correspond to the nonlinear spring.E1,α,β are its elastic parameters.The integral term corresponds to the high-frequency Maxwell element, which can be used to describe the viscoelastic response at a high strain rate.E2and θ are its elastic constant and relaxation time.

As aforementioned, the failure strain of this composite is extremely small,only about 0.01-0.02.The cubic terms βε3can also be deleted without affecting its accuracy.Assuming that the specimen is loaded at a constant strain rate, Eq.(14) can be further simplified as

The strain-induced damage will reduce the properties of composites,as is shown in Fig.3,Figs.11-15.For simplicity,an idealized scalar parameter D is adopted to quantify the damage throughout the deformation process[42].It evolves with the increase of strain,given as

Fig.16.Nonlinear viscoelastic model.

The stress-strain curves at 55/s and 74/s are disregarded because their nonlinear behaviors and strain rate effects are not evident.Considering the bilinear relation in Fig.10,the remaining six curves are divided into two parts, i.e., the rising part (200/s, 387/s, and 585/s)and the falling part(751/s,808/s,and 895/s).A python script that determines the optimal parameters is created.Root Mean Square Error(RMSE)is used to measure the deviation between the predicted and true values.In Fig.17, it is clear that the simplified ZWT model, which considers the damage effects, can accurately characterize the dynamic compressive behavior under different strain rates.The required simulation parameters are listed in Table 4.

4.Conclusions

Experimental investigations on the rate-dependent behavior of a plain weave composite were carried out using an SHPB apparatus.Results showed that there existed obvious strain rate effects in dynamic in-plane compressive strength.Meanwhile, the failure mechanism and failure modes of the studied composite were observed and summarized.The main conclusions were as follows:

1.A quantitative criterion for calculating the constant strain rate of composites was established.The average of the strain rate history curve that corresponds to the range of 70%-100% failure strain was taken as the constant strain rate.

Fig.17.Experimental and fitted stress-strain curves: (a) Specimen Nos.3-5; (b) Specimen Nos.6-8.

Table 4 Parameters for ZWT model.

2.The compressive strength of the studied composite increased first and then decreased with the rising strain rate.A turning point was found to illustrate this tendency in the proposed bilinear model.Within the range of 0.001/s-585/s,the strength was enhanced from 401 to 702 MPa.When the strain rate was further raised to 895/s, the strength was gradually reduced to 515 MPa instead.

3.The increased strain rate promoted a transition in failure mechanism from local opening damage to completely splitting destruction.Fiber fracture, inter fiber fracture, and interface delamination were identified as three major failure modes when subjected to shock compression.

4.The upper limit of strain rate was obtained by the theoretical estimation (779/s),quite close to that(808/s) inferred from the strain rate history curves.The lower limit (60/s) for ratedependent behavior was determined by reference to the quasi-static tests.

5.By introducing a nonlinear damage variable,the simplified ZWT model was developed to characterize the dynamic mechanical response of plain weave composites.Its fitting curves were highly consistent with the experimental results.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This project is financially supported by the National Science and Technology Major Project (Grant No.2017-VII-0011-0106) and Natural Science Foundation of Heilongjiang Province (Grant No.ZD2019A001).