Hu Junshn,Zhng Kifu,*,Cheng Hui,Liu Ping,Zou Peng,Song Dnlong

aThe Ministry of Education Key Lab of Contemporary Design and Integrated Manufacturing Technology,Northwestern Polytechnical University,Xi’an 710072,China

bSchool of Mechanical Engineering,Northwestern Polytechnical University,Xi’an 710072,China

Stress analysis and damage evolution in individual plies of notched composite laminates subjected to in-plane loads

Hu Junshana,Zhang Kaifua,*,Cheng Huib,Liu Pinga,Zou Penga,Song Danlonga

aThe Ministry of Education Key Lab of Contemporary Design and Integrated Manufacturing Technology,Northwestern Polytechnical University,Xi’an 710072,China

bSchool of Mechanical Engineering,Northwestern Polytechnical University,Xi’an 710072,China

Damage evolution;Failure mode;Failure strength;Layer-by-layer stresses;User-defined subroutine

This work aims to investigate local stress distribution,damage evolution and failure of notched composite laminates under in-plane loads.An analytic method containing uniformed boundary equations using a complex variable approach is developed to present layer-by-layer stresses around the notch.The uniformed boundary equations established in series together with conformal mapping functions are capable of dealing with irregular boundary issues around the notch and at infinity.Stress results are employed to evaluate the damage initiation and propagation of notched composites by progressive damage analysis(PDA).A user-de fined subroutine is developed in the finite element(FE)model based on coupling theories for mixed failure criteria and damage mechanics to ef ficiently investigate damage evolution as well as failure modes.Carbon/epoxy laminates with a stacking sequence of[45°/0°/-60°/90°]sare used to investigate surface strains,in-plane load capacity and microstructure of failure zones to provide analytic and FE methods with strong validation.Good agreement is observed between the analytic method,the FE model and experiments in terms of the stress(strain)distributions,damage evaluation and ultimate strength,and the layerby-layer stress components vary according to a combination effect of fiber orientation and loading type,causing diverse failure modes in individuals.

1.Introduction

Carbon fiber reinforced polymer(CFRP)composites are widely used in all fields of aerospace,automobile,electronic power,and mechanical engineering for several advantages offered over metals,ceramics,and plastics.These include low density,robust specific strength,low thermal expansion,corrosion resistance and designable characteristics for lightweight,efficient structures.Cutouts,especially circular and elliptical shapes,commonly appear in composite structural components due to requirements for stability,maneuverability,low weight optimization and accessibility of other systems.This is often the case in aircraft where composite structures such as wing spars and thin walls are drilled for electronic wires and hydraulic pipes1or to facilitate assembly operations.2Other instances occur not as part of the initial design of essential structures but due to material defects or unexpected damage during a service cycle.3These are serious and dangerous cases since they are not supposed to be there.Whatever their reasons for being,the integrity and continuity of fiber and matrix in the composite are destroyed;this makes the notch region the weakest part of the structure and causes serious local stress concentrations in the vicinity.As a consequence,structural capacity is reduced and uncertain reactions to external loads may occur and lead to unexpected fractures in service.In order to avoid potential safety hazards,accurate analysis of local stress levels and damage evolution of notched laminates are of great significance to the utilization of the material and lay the foundation for engineering applications.

The determination of stress fields in notched anisotropic plates has been the focus of many scholars for a long time.Muskhelishvili4first introduced complex potential theory to the isotropic elastic plates and successfully obtained an accurate solution of stress distribution.Analytical solutions for the stress distribution around holes of different shapes in anisotropic plates were given by Lekhnitskii5using series methods.Savin6presented a much simpler approach by conformal mapping of Cauchy integrals.Gao7used a biaxial loading factor together with an arbitrary orientation angle previously used to solve the problem of a plate with biaxial loading at infinity,avoiding the superposition of a solution with two uniaxial loading problems.The stress analysis research above established the base for later study of notched composite laminates.Ukadgaonker et al.developed a general solution for stress around oval holes,8triangular holes9with rounded corners10and irregular shaped holes11in cross-ply and angleply orthotropic plates under in-plane loading by superposition of two stage solutions for boundary conditions.The stresses were employed together with a series of failure criteria,which were an extension of the Von Mises criterion for quadratic interaction,to calculate the first ply failure(FPF)strength.Similar solutions around a rectangular hole in an infinite isotropic and anisotropic plate were given by Pan et al.12and Rao et al.,13respectively,using the complex variable method.Sharma14suggested general solutions for determining the stress distribution around polygonal holes and investigated the effect of hole geometry and loading pattern on the stress concentration factor.Batista15used a modified solution to solve problems of stress distribution around polygonal holes of complex geometry in an infinite plate subjected to uniform loading at infinity.Remeepazhand and Jafari16also studied the central polygonal hole problem in composite plates using two simple equations with parameters λ,c,nandwcontrolling the size,shape and bluntness of corners.Toubal et al.17experimentally investigated the tensile strain field of composite plates in the presence of stress concentration caused by geometrical cutouts consisting of circular holes by Electronic Speckle Pattern Interferometer(ESPI).The stress obtained in experiments is consistently lower than the analytical and numerical models.

This literature mostly concentrated on stress distribution in anisotropic plates as well as laminated composites,which are equivalent anisotropic plates,thus the stresses around cutouts were simplified to be uniform throughout,regardless of thickness.It should be noted that composite laminates contain several plies which possess dissimilar properties due to different fiber orientations,so the stress distribution in individual plies should be thoroughly investigated.

The above stress research methodologies and conclusions establish a theoretical basis for damage and failure of anisotropic plates.As for failure analysis,the damage propagation process is often depicted based on progressive damage analysis(PDA)utilizing numerical and experimental methods.18Lapczyk and Hurtado19proposed an anisotropic damage model suitable for predicating failure and post-failure behavior in fiber-reinforced materials.The plane stress formulation is used and the response of the undamaged material is assumed to be linearly elastic.The evaluation law is based on fracture energy dissipation and implemented in a finite element code.Zahari and EI-Zafrany20developed a progressive analysis algorithm based on Tsai-Hill failure to model the non-linear material behavior and capture the compressive response of woven glass/epoxy composite plates via non-linear finite element analysis.Rakesh et al.21introduced a generic finite element model to investigate the failure of unidirectional glass fiber reinforced plastic(UD-GFRP)composite laminates with drilled holes under tensile testing and compared the results with experimental work done earlier.Effects of joint geometry and stacking sequence on the bearing strength and damage mode were investigated by Ondurucu et al.22and specimens were examined for failure modes using a scanning electron microscope(SEM).Kim et al.23employed rosette strain gauges to measure the strain around the joint hole during insertion of stainless steel pins into glass fiber reinforced plastic(GFRP)specimens,and results were compared with the finite element method.Satapathy et al.24presented a modified fiber failure fatigue model to characterize the behavior of laminated composites with a central circular hole under in-plane fatigue loading.Martins et al.25studied the influence of diameter and thickness on failure pressure during tube burst tests and employed the progressive failure analysis using a damage model by a user subroutine(UMAT)implemented in ABAQUS software to understand the behavior of composite tubes under internal pressure.Lee et al.26proposed an evaluation method for the progressive failure of composite laminates built upon Puck failure criterion by implanted UMAT to efficiently analyze the progressive failure phenomenon in glass/carbon fiber-reinforced composite laminates.Compression tests were performed by Aljibori et al.27on 16 fiber-glass laminated plates with and without circular cut-outs to investigate the effects of varying the centrally located circular cut-out sizes and fiber angle-ply orientations on the ultimate load.Similar experiments were also conducted by Abu et al.28to investigate the influence of cut-outs on multilayer Kevlar-29/epoxy composite laminated plates.

These remarkable works contribute a lot to the failure analysis of notched anisotropic composites whose layups are quasiisotropic,cross-ply,and angle-ply.At the same time,numerical and advanced experimental methods for progressive failure have greatly developed.However,the failure criteria employed in the literature were linear expressions;whereas composite laminatesin specialapplication are notquasi-isotropic lay-up,composite materials accumulate damage in shear,leading to nonlinear stress-strain relations.The relationship between stress distribution and damage evolution thus needs further investigation.

In the present study,an analytical model for layer-by-layer stress around notches in individual plies of general composite laminates is investigated using a complex variable approach that contains uniformed boundary functions.The functions established in series together with conformal mapping functions are capable of dealing with irregular boundary issues both around the notch and at infinity.A stress solution is employed to evaluate the progressive damage and failure of notched laminates based on coupling theories for mixed failure criteria and damage mechanics.A user-defined subroutine is then developed in ABAQUS to simulate the damage evolution under tensile and compressive loading.Experiments for strain and failure are conducted using a 3D digital speckle measurement system and microscopy study to validate the proposed analysis method.

2.Analysis for local stress distribution

2.1.Problem configuration of notched laminates

A composite laminate with a traction free notch under inplane loading is investigated in Fig.1.The inplane loading is in the form of stress resultants in an arbitrary coordinate systemx′oy′rotated clockwise by an angle β from the global coordinate systemxoy,as shown in Fig.1(a).Dis the diameter of the circle that encloses the notch.It can be written in terms of Gao’s loading conditions7using stress loading σ∞x′，y′as follows:

The notched plate is projected to a unit circle on ζ plane under normalized loading in Fig.1(b)with an angle θ to denote hole position.The equivalent loading σ∞x，yalongxoyaxes corresponding to original loading alongx′oy′axes is obtained by transformation of axis:

By substituting Eqs.(3)and(4)into Eq.(7),a forth order differential equation is obtained:

Fig.1 Problem configuration and solution of a notched composite under in-plane loading.

The general solution of bi-harmonic Eq.(6)is determined by roots of the following characteristic equation:

In general,it can be proved that Eq.(7)has four distinct roots,which are always complex conjugates in pairs:

where α1,α2,β1,β2are real numbers and β1＞ 0,β2＞ 0.μ1,μ2and their conjugates are complex parameters of a plane stress state.They are determined by effective compliance coefficients aijand can be used to describe the degree of material anisotropy.

Through the integral operation of Eq.(6),the stress functionU(x，y)can be expressed by the real part of two complex functionsF1(z1)andF2(z2):

wherezp=x+ μpy(p=1，2)are anisotropic complex coordinates,andShrepresents the boundary conditions of the laminate in Fig.1(a).So,the problem is reduced to finding two complex analytical stress functions that satisfy the boundary conditions at the notch edge as well as infinity.

2.2.Determination of analytic stress functions

Here,in the issue of plane stress state,the uniformed boundary equations φ(z1)and ψ(z2)are developed in the form of derivatives ofF1andF2:

where parameters γ and γ′are complex variables of the notch boundary.ParametersB,B′andC′are loading condition contents.Parametersanandbnare series coefficients determined by the boundary conditionsSharound the cutouts and at infinity.

2.2.1.Remote boundary conditions

At the infinite plate edge(|z|→∞),the mean stress components reach the applied stress loading of the normalized boundary condition:

Through substituting Eqs.(9)and(11)into Eq.(4),the parametersB,B′andC′are obtained by the following equation group:

2.2.2.Boundary conditions at the notch edge

At the notch edge,which is traction free,the notch area on thezplane is projected onto the exterior of a unit circle on ζ complex plane using the conformal mapping method(Fig.1).The unit circle is defined by the Euler formula ζ=eiθin Fig.1(b).This opens up the possibility of a simple general approach to determine the stress functions of the actual notch contour independently.

The mapping function for the elliptic notch in anisotropic cases fromzplane to ζ plane is defined by:

whereR=(a+b)/2 is a constant for size of the hole,andm1=(a-b)/(a+b)is defined to control the ellipticity.Parametersaandbare major and minor axes of the elliptic hole,respectively.These are typically:(1)m1=0 ora=b:circular holes;(2)0＜m1＜1 ora≠b:elliptical holes;(3)m1→ 1 ora?b:blunted cracks.

By the projection process and taking Eq.(13)into the uniformed boundary Eq.(10)on ζ plane,the boundary equations in ζ plane are written as:

It is seen that the boundary of the notch is free of traction,which indicates that the mean resultant forces on the notch are zero:

Parameters γ and γ′are confirmed to be zero,and the logarithmic terms drop out.The rest of the parameters in series terms are obtained by substituting Eqs.(14)and(16)into Eq.(15):

whereK1,K2,K3andK4are complex constants.They are determined by:

2.3.Local stresses around notches of individual plies

By substituting the analytic stress functions into Eq.(4),the mean stress components around the hole in global coordinates(Cartesian coordinates in Fig.1)are obtained through the following equations:

According to the classical laminate theory29,30,laminate strains can be written in terms of the midplane strains ε0x，yand midplane curvatures κx，yas follows:

where εx, εyand γxyarex-direction,y-direction and shear strain components inzthickness layer,and ε0x, ε0yand γ0xyarex-direction,y-direction and shear strain components in the midplane.κx,κyand κxyarex-direction,y-direction and torsional curvatures in midplane,respectively.

For the symmetric laminates studied here,it can be proven that the coupling stiffness coefficients Bij(i，j=1，2，6)are zero,which indicates the resultant and moment terms are uncoupled.Hence the curvature of midplane κx，yis equal to zero.So,the strain at an arbitrary point of the notch edge throughout its thickness is the same as midplane strain and the mean strain:

The stress components in thekth ply in principal direction σ(k)are calculated by the following equation modi fied from

L，TStress-Strain Equation and the stress transformation of axis:

3.Analysis for damage and failure

As mentioned before,one of the main purposes of determining local stress distribution in individual plies around the notch is to investigate the damage and failure of the perforated laminate accurately.In progressive damage analysis(PDA)theory,31composites are materials characterized by complicated failure phenomena that interact.The onset of failure in an individual ply generally does not lead to the structure’s collapse,and this may still not occur until the failure has spread to each of the multiple plies,indicating that some plies fail before others.Since the notch region is the weakest part of a composite plate,the damage initiates around the notch.

3.1.Failure criterion

Most failure criteria are extended from the von Mises criterion for quadratic interaction.32They allow the failure of a whole ply to be checked,but ignore the distinct failure modes of composites.Given this,the mixed failure criteria,which combines Hanshin criteria and maximum criteria,differs greatly from them.It not only allows distinguishing between failure in tension compression as well as fiber and matrix,but also takes nonlinear shear stress-strain behavior into consideration.The mixed failure criteria33are given by:

whereemt,emc,eft,efcandefsare failure indices that denote matrix tensile,matrix compression,fiber tensile,fiber compression and the fiber-matrix shear-out,respectively.The parameter α is a material constant that is determined experimentally according to Chang and Lessard34and commonly takes the value of 2.44 × 10-8MPa-3in the User’s Manual of Abaqus 6.11.Once an index exceeds 1.0,the corresponding failure occurs.The termsXt,Xcare tensile and compressive strengths in the fiber direction andYt,Ycare tensile and compressive strengths in the traverse direction.Scis the in-plane shear strength.

3.2.Material property degradation rules

From the phenomenological viewpoint,the defect initiation/propagation phenomenon for a material is strongly related to the degradation of the material’s capacity,and the most visible aspect is the degradation of the material’s stiffness.Therefore,it is possible to describe the progressive failure of a composite laminate by numerically representing the correlation between an increase in the material’s internal damage and a decrease in the material’s stiffness.The Camanho and Matthews degradation rules35(Table 1)are adopted because they are closest to the real conditions according to the research of Chang.34ParametersE1,E2,G12and υ12are the longitudinal Young’s modulus,transverse Young’s modulus,shear modulus and Poison’s ratios of the lamina.E′1,E′2,G′

12and υ′12are material properties after degradation.

3.3.Progressive failures of notched laminates

After using Eq.(23)to obtain the stress distribution around the notch in each individual ply,the mixed failure criteria is employed to check the failure state around the notch edge.On the occurrence of some sort of failure,the failing ply should be degraded according to Table 1,which causes redistribution of stresses around the notch.Thus the failure state of the notch edge will be rechecked.The procedure of stress redistribution and failure recheck will continue until all plies fail or fiber failure occurs,causing sharp decline of carrying capacity.So,the location of damage initiation and final failure strength can be obtained during the analysis.Failure strength here refers to the maximum permissible remote stress loading applied.30

4.Numerical implementation

The stress distribution,damage and strength of notched specimens were also investigated using FE analysis implemented in the commercial software ABAQUS 6.11/standard.A composite laminate with a general stacking sequence of[45°/0°/-60°/90°]swas chosen as the investigated object,and the material properties are given in Table 2.The specimen configuration for the open hole tensile(OHT)test is illustrated in Fig.2 according to ASTM D5766M-02a Standard.36The open hole compressive(OHC)test followed ASTM D 6484/6484M-04.37The specimen geometry was similar to that used for the OHC test.The aspect ratio met the conditionW/D≥4 so that the external boundary effect was negligible.38

In the FE model,the 3D eight-node layered continuum shell element was employed for the laminate.The mesh was structured using linear elements with reduced-integration(SC8R),and the notch region was highly refined by partition to capture the high near-hole stress gradients.A planar mesh size of 0.188 mm×0.275 mm in(inx-yplane)in the vicinity of the hole was used while a coarser mesh was used away from the zone of the interest to computation cycles.In the thickness direction,a single layer of elements was used to model each ply.The approximate element size was 0.206 mm×0.188 mm×0.375 mm.Successive space discretization was compared,varying the element size surrounding the hole to verify that the final mesh size was enough to provide accurate results without expensive computational cost.A total of 92,345 eight-node brick elements were employed to model the specimen.The specimen was encastre on one side and applied tensile or compressive stress loading on the other side.Two lines,between which the length wasL-2S,were marked to define the elongation corresponding to the part of the specimen that was not griped.The mesh and boundary conditions(BCs)are shown in Fig.3.

The progressive model is implemented in ABAQUS by applying a separate subroutine called user-define field(USDFLD).39In the analysis,the material properties depend on six field variables(FVs).The first five FVs from FV1 to FV5 represent failure indexes corresponding toemt,emc,eft,efcandefs.The last one,FV6,denotes the shear damage parameter.The algorithm is illustrated in Fig.4.In the beginning,the FVs and the solution-dependent state variables(SDVs)are set to 0 and the material properties are equal to their initial values.Then the load is increased gradually;For each load increment,several iterations are necessary before the analysis converges to an equilibrium state.At the end of each increment,stresses,failure indices and shear damage parameter are computed at the integration points.The value of the shear damage parameter is directly assigned to the corresponding FVs and used to degrade the shear modulus.The values of the failure indices are stored as SDVs.Once an SDV exceeds 1.0,the corresponding FV is set to 1 and maintains this value until the end of the analysis.Meanwhile,the material properties are automatically reduced according to the degradation rules in Table 1.With each increasing applied load,the procedure is repeated until the program terminates due to excessive element distortion.Since equilibrium will not be reestablished after degradation of properties,small load increments should be set to obtain an accurate solution.40

Table1 Property degradation rulesofCamanho and Matthews.

5.Experimental investigation

The damage and failure behavior of notched composite laminates is also investigated on a Bairoe electronic universal testing machine with a load capacity of 100 kN.The tests were run in displacement control at a rate of 2 mm/min.Loading stopped after dropping 30%from the peak load.A non-contact measurement method,3D digital speckle measurement system(DSMS),was used to investigate the strain field in the vicinity of the hole in the notched composite plate.The system is based on a 3D digital image correlation(DIC)system with binocular stereo vision technique that together perform fullfield 3D measurements of surface strain and out-of-plane deformation of the joints,as shown in Fig.5(a).The principle is to track an applied surface pattern during loading bycontinuously taking digital images of the surface.41Two charge coupled device(CCD)cameras and two light emitting diode(LED)sources are required for measurement.To ensure data synchronization,DSMS was connected with the testing machine by a synchronous data transmission device.

Table 2 Material properties of carbon-epoxy T300/3526.

Fig.2 Specimen configuration of composite for open hole test.

Fig.3 Mesh and BCs of the FE model.

Fig.4 Algorithm for ABAQUS user-defined subroutine.

Fig.5 Test machine with digital speckle measurement system and specimen processing.

The CFRP laminate was fabricated through hand lay-up method using resin prepregs,and specimens were processed with a water jet cutter to avoid defects caused by cutting heat.The material properties and specimen geometry are the same as the FE model in Section 5.Holes were drilled and reamed with carbide-tipped tools to attain fine surface quality,.then surfaces of the specimens were polished with sandpaper(first using a grain size of 800 then a grain size of 2000)to ensure a smooth surface.All the specimens were cleaned with alcohol to remove grease and dust before the test.A white background was painted on the testing surface of the specimens using spray paint,and later small black spray paint droplets were subtly applied with an appropriate density to make a high contrastpattern,as shown in Fig.5(b).Images of speckle pattern were recorded at a frequency of 5 Hz.The VIC 3D software calculated the accumulated strain by comparing the subsequent images with the first image,which was taken as a reference.

6.Results and discussion

6.1.Experimental veri fication of the strain distribution around the notch

The strain distributions around the notch on the surface of the specimen under tensile and compressive loading are presented in Figs.6 and 7,respectively.The test was repeated three times,and the strains in the vicinity of the open hole were extracted by DSMS(Fig.5(b))to verify the validation of the theory and FE model.Since the strains around the notch are center symmetric,only those whose angular position θ ranges from 0°to 180°are revealed.

The experimental results show relatively good trending alongside the proposed analytical solution and FE method with acceptable amplitude errors.There are two possible reasons for the errors:Either the open hole is a singularity at the surface of the specimen,or the fragmentary grid on the hole boundary cannot be used to calculate the strain field.42In fact,the strains obtained by DSMS are the same as for the ring belt,which is a small offset from the hole edge.The strains fade much more sharply along the radial direction at the points where the strain concentrations are the most serious at the edge of the hole,so the obtained strains from ring belt near these points are apparently smaller than those from the hole edge.

The small error between analytical and FE results is due to the algorithm in the FE method,which is a numerical and approximate solution as compared to an analytical solution.Actually,the strains under tension and compression possess the opposite distribution law due to the tensile and compressive deformation state of specimen in either thexoryaxial direction.For strains in thexaxial direction,maximum strain occurs at the 90°point where by pass load goes through net tension plane and causes larger strain than any other part of the specimen.As for strains in theyaxial direction,extreme points occur at regions around 0°and 90°due to transverse contracting under tension(Fig.6(b))or transverse stretching under compression(Fig.7(b)).Extreme points at 45°and 135°are caused by shear effect.

6.2.Stress distribution around the hole in individual layers

The layer-by-layer stress distributions under tensile and compressive loading are provided separately in Figs.8 and 9 to give clear insight into the mechanical behavior of notched composite laminates.Good agreement is observed between theory and FEA,thus providing confidence in the accuracy of the present results.Stress components in individual plies are shown to be highly non-uniform throughout the laminate thickness.The normalized longitudinal stresses in an individual layer have a much larger magnitude than the transverse and tangential stresses,reflecting the specific characteristic of anisotropic material in which carbon fibers bear most of the load.

It is seen that the stress concentration level in a layer depends on the relative angle between the fiber orientation and loading direction.The same dependence exists for concentration regions.For longitudinal stress,the smaller the angle is,the higher concentration level found near the region where the fiber direction is tangential to the hole perimeter.This can explain why the 0°layer exhibit the highest stress peaks of±8.65 near 91.8°and is the most effective in carrying the axial load while the 90°layer has the smallest concentration level of±2.25 at 0°.The transverse stresses in each layer exhibit peaks at locations where the fiber direction deviates most from the loading direction or where the fiber direction is normal to the notch boundary.So the 90°layer shows the largest concentration of±0.62 near 90°while the 0°layer has a level of±0.18 near 0°.Actually,there is adverse distribution rule between longitudinal and transverse stresses.For tangential stress influenced by the shear effect,higher degrees of concentration are found in layers whose fiber orientation possesses an angle in the loading direction.

Fig.6 Strain distribution around circular notch of composite laminates under tensile loading.

Fig.7 Strain distribution around circular notch of composite laminates under compressive loading.

Fig.8 Stress distribution around circular notch in individual layers of composite laminates under tensile loading.

Fig.9 Stress distribution around circular notch in individual layers of composite laminates under compressive loading.

6.3.Damage initiation and propagation of notched composite laminates

Damage initiation and propagation was evaluated by comparing screenshots of various failure modes in different increment steps(IS).Some of the important damage initiation locations were also predicted using the analytical method in section 3.3.A microscopy study that investigated micro-scale failures occurring in the vicinity of the hole using scanning electron microscope(SEM)is shown in Fig.12 to validate the damage mode simulated in the FE model.

Fig.10 Damage initiation and propagation of notched composite laminates under tensile loading.

Fig.11 Damage initiation and propagation of notched composite laminates under compressive loading.

Fig.10 shows the failure mode(i.e.,matrix tensile,matrix compress,fiber tensile,fiber compress,and fiber-matrix shear)in different ply orientations under tensile loading.Matrix tensile failure,which onset first at 92.72°in the 90°layer,was the dominant failure mode in early increment steps.This damage,though possibly not affecting tensile strength much,was caused by transverse stress concentration and spread to other layers with increasing loading.Fiber tensile failure occurred at 88.71°in the 0°layer and was accompanied by matrix compressive failure in later increment steps.As loading increased,fiber damage propagated across the net tension plane causing dramatic decrease in load capacity.Matrix- fiber shear failure was observed due to excessive transverse shrinkage in late loading increments.The failure mode under compressive loading where matrix compressive failure initiation was observed at 88.4°in the 90°layer and propagates in early increment steps is presented in Fig.11.Fiber compressive failure was found along with matrix- fiber shear in the 0°layer before specimen buckling deformation.The initiation location of fiber compressive damage occurred where fiber direction was normal to the hole perimeter.Tensile failure rarely appeared in the process.

Microscopy investigation of damaged specimen form tests was conducted to back up the progressive failure results from FE model.In the microscopic morphology,matrix crack(Fig.12(a))and fiber breakage(Fig.12(b)),which corresponded to matrix tensile failure and fiber tensile failure from FE results in Fig.10,were observed as common micro-scale failure modes in OHT specimens.Matrix crush(Fig.12(c)),fiber buckling(Fig.12(d))and matrix- fiber shear(Fig.12(e))were dominant failure modes in compressive specimens,just as the FE model simulated in Fig.11.Slight delamination is also found in the experiment because excessive load destroyed the compatibility of deformation among layers.This is beyond the elastic range and not considered in the analytical solution.

To further support the progressive failure results from the FE model,strain concentration and crack growth path on the specimen surface were captured by DSMS to compare with the damage of the 45°layer(surface layer)in the composites,and the results are listed in Figs.13 and 14.For OHT testing,the initial damage points induced by matrix tensile failure(Fig.13(c))matched very well with the zones of strain concentration(Fig.13(a)),and the damage propagation path follows the crack profile.Fiber tensile failure(fiber tensile rupture)is also observed in both test results(Fig.13(b))and the FE results(Fig.13(d)).As for OHC testing,the initial damage points of failure(Fig.14(c))were in accordance with the zones of strain concentration around the hole(Fig.14(a)).The fibermatrix shear-out failure in the 45°layer(Fig.14(d))resulted in surface cracking of the specimen(Fig.14(b))because they possessed the same growth path.The above analysis provided sufficient credibility to the validation of the FE model for progressive failure analysis.

Fig.12 Scanning electron microscope images of damage types in the vicinity of the notch.

Fig.13 Comparison of failure results between experimental test and FE model under tensile loading.

Fig.14 Comparison of failure results between experimental test and FE model under compressive loading.

6.4.Failure strength

The load-displacement plots of experimental and FE simulation results for the notched composite laminates are shown in Figs.15 and 16.The slope of the linearity in the experimental tests(i.e.elastic specimen stiffness)is seen to be very well captured by the numerical models,indicating that the composite can be regarded as linear-elastic material despite its brittleness.The stiffness of the specimen(slope of the curve)from the test is smaller than that from FE results due to the defects of the composite laminates and the degradation rules for the material properties.The OHC test shows slight nonlinear behavior at the start of the curve because the specimen slightly buckled under compressive loading.Since the OHC specimens were not ruptured,compressive failure is much more moderate than in the OHT test.The nonlinear behavior observed near the peak load is associated with severe damage in the specimens leading to fracture.

Fig.15 Load-displacement curves of notched composite laminates under tensile test.

Fig.16 Load-displacement curves of notched composite laminates under compressive test.

In the tensile test,there is a protuberant point at a load of 247 MPa,which is mainly caused by the initiation of fiber tensile damage around the notch in the 0°layer along with serious matrix tensile failure in all plies.With each load increment,the fiber tensile damage spreads across the tension plane at 0°through thickness into 90°,-60°,+45°layers in the vicinity of the hole,resulting in drastic fluctuations in the curve.The experimental ultimate strength was 484.6 MPa with an error of 7.7%compared with the FE result of 524.9 MPa.In the compressive test,the main damage mode in an earlier state was matrix compressive failure.Fiber buckling failure occurs,especially in the 0°layer in the net tension plan,causing a quick shock in the load capacity curve.As the load increased,buckling failure extended beyond the surroundings of the notch and caused bending of the specimen.Since the specimens were not ruptured,the compressive failure is much more moderate than in the tensile test.There was a difference of 5.3%between the test strength of 206.1 MPa and FE result of 217.5 MPa.

7.Conclusions

This research investigates the layer-by layer stress components,and the initiation and propagation of damage and failure in notched composite laminates by analytic method and FE model,both of which are validated by experimental tests.There is good agreement between the analytic method and the FE model in terms of the stress(strain)distributions,damage initiation and location of failure modes,and experiments on strain by DSMS and microscopy study by SEM provide them with strong suitability and validation.The proposed analytic fundamentals present a dependable and manageable alternative to evaluate stress level and damage initiation quickly,while the FE model simulates the damage evolution process thoroughly.Several conclusions can be drawn from the research:

(1)The stress components are all non-uniform and periodic in different plies and caused by the anisotropy of the laminates.The magnitude of longitudinal stress is much larger than that of transverse stress and tangential stress due to the high elastic modulus of the fiber contrasting with the matrix.

(2)The layer-by-layer stress distributions around cutouts are affected by fiber orientation and loading direction.The high stress level of σLmainly occurs where the fiber direction is tangential to the hole perimeter.The concentration of σTis observed where the fiber direction is normal to the notch boundary.The τLTis more influenced by shear effect and possesses more extreme points than σLand σT.

(3)Matrix dominant failure is common in early states of inplane loading.It initiates at the hole edge in the ply whose fiber orientation is perpendicular to the load direction and then propagates to other plies.Fiber damage occurs and builds until matrix damage causes ruptures.While fiber buckling occurs in most plies,fiber breakage is found in the ply where the fiber orientation is parallel with loading direction.

(4)Load-displacement plots exhibit a long linear state,indicating that the composites can be regarded as linearelastic material despite of their brittleness.Matrix damage evolution does not affect tensile strength much while fiber damage initiation may cause sudden shocks in plots.CFRP laminates possess better tensile load capacity than compressive ones.

Acknowledgements

This work was sponsored by the National Natural Science Foundation of China,with three different programs(No.51275410,No.51305349 and No.51305352)that supports the present work financially.The authors would like to acknowledge the editors and the anonymous reviewers for their insightful comments.

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23 February 2016;revised 5 April 2016;accepted 5 August 2016

Available online 21 December 2016

?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.

E-mail address:zhangkf@nwpu.edu.cn(K.Zhang).

Peer review under responsibility of Editorial Committee of CJA.