deformation, yield and fracture of unidirectional composites in transverse loading

$: Deformation, yield and fracture of unidirectional composites in transverse loading$
Deformation, yield and fracture of unidirectional composites in transverseloading

1. Influence of fibre volume fraction and test-temperature

J.M.M. de Kok* , H.E.H. Meijer

Centre for Polymers and Composites, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands

Received 9 April 1994; accepted 30 July 1998

Abstract

The influence of the fibre volume fraction and test-temperature on the transverse tensile properties of glass fibre reinforced epoxy is studiedusing experimental and numerical techniques. The numerical analyses are based on micromechanical models with square and hexagonal fibrepackings. Special attention has been directed towards the identification of the necessary failure criteria. Using a von Mises failure criterion,an increase in transverse tensile strength is predicted at higher fibre volume fractions with both models. This is in good quantitativeagreement with experimentally determined transverse flexural strengths. With decreasing test-temperatures, higher transverse strengthsare obtained. This is primarily caused by the temperature dependence of the yield stress of the matrix. The counteracting influence of theresidual thermal stresses and the temperature dependent matrix ductility consequently proved to be less significant for the transverse strength.q 1999 Published by Elsevier Science Ltd. All rights reserved.

Keywords:C. Micromechanics; B. Strength; A. Glass fibres; Epoxy

1. Introduction

High performance composites are generally composed ofunidirectional fibre reinforced laminates. Such laminatespossess a pronounced anisotropy given the large differencesin fibre and matrix properties, and especially the perfor-mance in transverse direction are poor. Therefore in struc-tural applications usually stacked plies with different fibreorientations are used, allowing for a considerable stiffnessand strength in more than one direction. However, even inthese laminates low off-axis strains can lead to prematurefailure in the individual layers. Therefore, the low transversefailure strain of unidirectional composites can be regardedas one of the major limitations in the application of compo-site materials.

To improve the transverse failure strain, an intensiveexperimental study would be required in order to determinethe influence of parameters such as the fibre volume frac-tion, fibre–matrix bonding, fibre coating properties andmatrix ductility. Considering the large number of para-meters involved, it is useful to combine experiments onwell defined (model) composite systems with micromechan-ical analyses. Since in such analyses a parameter variation is

easily accomplished, the amount of experimental work cansignificantly be reduced.

The transverse deformation and fracture of metal–matrix[1–5] and polymer–matrix [6–16] composites have beenstudied using finite difference or finite element methods(FEM) on micromechanical models. These numericalanalyses offer the opportunity to reveal the deformation(and fracture) on a microscale and may lead to a consider-able improvement in the fundamental insight of the influ-ence of the distinct parameters. However, in most studieshypothetical strength criteria are used, such as a maximumprinciple stress criterion for polymer matrices [8–13], andmany papers do not contain sufficient experimental data toverify the numerical results [12–16]. As a result of thecomplex three-dimensional stress situations in compositesthe choice of a failure criterion like the maximum principalstress might lead to the wrong conclusions. To avoid this,attention must be focused on the determination of thematerials failure criteria [17,18].

This investigation is part of a detailed study that focuseson the transverse tensile properties of unidirectional compo-sites and that combines experiments and finite elementanalyses on various composite systems. In this first papera system is studied based on glass fibres in a relatively brittleepoxy matrix. The objective is to develop an appropriate

Composites: Part A 30 (1999) 905–916

1359-835X/99/$ - see front matterq 1999 Published by Elsevier Science Ltd. All rights reserved.PII: S1359-835X(98)00170-5

* Corresponding author.

micromechanical model that can be used for transversestrength predictions in subsequent studies on fibre rein-forced epoxies. By using isotropic fibres like glass and anisotropic matrix material micromechanical modelling iseasier but, more importantly, more reliable and comprehen-sible. Mechanical properties and failure criteria, necessaryas input parameters for the micromechanical analyses, havebeen determined experimentally. Experimental results arecompared with the results of numerical analyses that arebased on square and hexagonal fibre packing arrays. Theinfluence of parameters like fibre volume fraction andtest-temperature will be presented to validate the modellingin wider applications. In three subsequent papers, we willdeal with fibre–matrix adhesion, matrix ductility and inter-phase properties, respectively.

2. Experimental

2.1. Neat materials

The composites studied consisted of E-glass fibres fromPPG Industries Fibre Glass bv (084-M28) and an Aralditeepoxy system from Ciba Geigy, based on diglycidyl ether ofbisphenol-A (LY556) with tetra-hydro-methyl-phthalic-anhydride (HY917) as a curing agent and methyl-imidazoleaccelerator (DY070) in a weight ratio of 100:90:1. To obtainthe necessary input parameters for the numerical analyses,the neat materials were characterized using tension tests atroom temperature at a strain rate of 1023 s21. To identify anappropriate failure criterion for the epoxy matrix, besidesuniaxial tension other tests were used like simple shear,uniaxial compression and plane strain compression. ThePoisson ratio of the epoxy was determined using both alongitudinal extensometer and a transverse extensometer.The thermal expansion coefficient was determined by moni-toring the length of a sample with an extensometer fromroom temperature to 1008C. The data obtained are shownin Table 1.

The ultimate stress of the epoxy is determined as a func-tion of the test-temperature with tension tests applying aninitial strain rate of 1023 s21. Some representative stress–strain curves are shown in Fig. 1. It is clear that the yieldstress decreases with increasing temperature and the plasticstrain significantly increases. However at room temperaturethe epoxy is rather brittle and when the yield stress is

reached abrupt failure follows. As can be expected for poly-mer materials [19–21], the yield stress of the epoxy matrixdecreases linearly with temperature, as shown in Fig. 2. Thiscan be described by Eyring’s flow equation, for the highstress region given as [22]

_1 � Ae2

DU2synkT

� �or sy � DU

n1

kTn

ln_1

A

� ��1�

whereA is a constant,n is the shear activation volume,k isBoltzmann’s constant andT is the absolute temperature.DUis the activation energy for the flow process.

Simple shear tests were performed on cast moulded platesof 100× 60 × 2 mm. For these tests a grip-to-grip distancewas used of 10 mm and specimen length of 100 mm. Castmoulded plates of thickness 6 mm were used for the uniaxial

J.M.M. de Kok, H.E.H. Meijer / Composites: Part A 30 (1999) 905–916906

Table 1(Thermal) mechanical properties of fibre and matrix

Material Youngsmodulus(GPa)

Poissonratio(–)

Thermalexpansioncoefficient(1026/8C)

E-glass 70 0.22a 7a

Epoxy 3.2 0.37 67.5

Fig. 1. Stress–strain curves of the epoxy matrix at different test-tempera-tures.

Fig. 2. Temperature dependence of yield stress of the epoxy matrix.

compression tests. Tests were performed on polished cubicspecimens with a gauge length of 25× 6 × 6 mm. With the25 mm gauge length both grip effects and kink wereavoided. Planar compression tests were performed onepoxy specimens with a gauge length of 40× 12 × 2 mm.The specimens were supported by two Teflon covered steelplates to create a plane strain condition [23]. The yieldbehaviour in these test conditions can be described by amodified form of Eyring’s flow equation [21] given as

_1eq� _10e2

DU2seqn2pVkT

� ��2�

where _10 is a pre-exponential factor,p is the hydrostaticcomponent of stress (mean stress),V is the pressure activa-tion volume and_1eq the equivalent strain rate [21,24]:

_1eq��2p3

�� _11 2 _12�2 1 � _12 2 _13�2 1 � _13 2 _11�2

q�3�

where _11, _12 and _13 are the principal strain rates, while,finally, seq is the equivalent stress, also known as the vonMises stress [21,24]:

seq� 1��2p

��s1 2 s2�2 1 �s2 2 s3�2 1 �s3 2 s1�2

q�4�

wheres1, s2 ands3 are the true principal stresses.

In our experiments, strain rate effects are circumventedby determining the yield stress at equal values of the equiva-lent strain rate [21] (1023s21). The yield stresses measuredare shown in Table 2, combined with the decomposed prin-cipal stressess1 ands2 [24,25]. The true yield stresss t intension or compression has been determined from the engi-neering yield stressse by correcting the effect of elasticdeformation prior to yielding [19] by

st � se�1 1 1y� �5�

where1 y is the strain at yield. With the principal stresses,s1

ands2, a yield envelope can be generated, as illustrated inFig. 3a. This envelope can reasonably be approached by thevon Mises criterion with a little sensitivity of the yield stressto hydrostatic pressure [25–27]. Since the data were notsufficient to determine the pressure dependence of theyield stress, this influence will be disregarded in the numer-ical analyses.

2.2. Composite manufacturing

Composite specimens were manufactured using the fila-ment winding technique. Glass fibre strands were impreg-nated in an epoxy bath and wound on a framework. Theresin rich fibres were degassed by placing the frameworkwith fibres in a vacuum oven at 400 mmHg and 608C for30 min. After degassing, the composite is moulded usingteflon sheets to ensure demoulding after cure. The resultingthickness of the laminates, and thus fibre volume fraction, iscontrolled by spacers. After applying 5 bar pressure, theresin is cured in the press at 808C for 4 h. Postcuring isperformed at 1408C for 8 h. Unidirectional compositeswere obtained varying in fibre volume fraction from 43%to 73%. High-quality (transparent) composites resultedwhich were basically free of voids.

The manufactured laminates were cut with a diamondsaw into rectangular specimens with a width of 20 mm.To avoid grip effects, tensile specimens with a typicaldumbbell shape were machined using a special grindingmachine. In order to avoid catastrophic influence of surfaceflaws, the specimen edges were carefully finished using asequence of 300, 1200 and 4000 SiC sandpaper [28].

J.M.M. de Kok, H.E.H. Meijer / Composites: Part A 30 (1999) 905–916 907

Table 2Yield stresses in various tests

Test method Uniaxial tension Simple shear Uniaxial compression Planar compression

Strain at yield (%) 5.9 – 2 7.0 2 7.0True yield stress (MPa) 99 53 2 111 2 114Ratios y:s1:s2 1:1:0 1:1:2 1 2 1: 2 1:0 2 1: 2 1:1/2First Princ. stress (MPa) 99 53 2 111 2 114Second Princ. stress (MPa) 0 2 53 0 2 57Von Mises stress (MPa) 99 92 111 99Hydrostatic stress (MPa) 33 0 2 37 2 57

Fig. 3. Failure envelope of the epoxy matrix.

2.3. Testing

Tension tests were performed on a Frank type 81565tensile machine with an initial strain rate of 1023 s21. Card-board and grinding paper is used between specimens andspecimen grips to prevent slip and premature failure in thegrips. Three-point bending tests were performed accordingto the ASTM D790-81 standard with an initial strain rate of1023 s21. Span-to-depth ratios between 30 and 46 wereused. Tests at room temperature were also performed on aFranck type 81565 tensile machine. A Zwick Rel SB 3122hydraulic tensile machine with a thermostatically controlledoven was used for testing at2 308C and 1 708C.

2.4. Numerical analyses

The numerical analyses are based on a two-dimensionalgeneralized plane strain model using the FEM code MARC(version k4.2). A square and hexagonal array model wereused. In Fig. 4 the meshes used are shown, both created withthe preprocessor MENTAT 5.4.3. The particular curve onthe right side of the hexagonal mesh allows for calculationson composites with high fibre volume fractions. Quadraticeight-node generalized plane strain elements were used toachieve accurate results with a constant strain in the fibredirection. In the vicinity of the interface the element sizewas reduced because of the expected high stress gradients[2]. Perfect fibre–matrix bonding was modelled by attach-ing fibre and matrix with coinciding nodes. The fibrevolume fraction could be varied by changing the radius ofthe fibre. This was easily achieved by changing the materialparameters of the circular layers at the fibre–matrix inter-face from fibre to matrix parameters or vice versa. In thisway, 10 different fibre volume fractions could be computedwith both meshes: with the square model from 38% to 68%and with the hexagonal model from 39% to 72%. For bothfibre and matrix, linear elastic behaviour is assumed. Theinput parameters used are shown in Table 1.

The boundary conditions of the models are given as zerodisplacement in thex andy directions for all nodes at thelines AD and AB, respectively. The nodes at the line CD aretied to point C with the displacements in they direction of

these nodes equal to they displacement of point C, as givenby

vP � vC �6�wherev is the displacement in they direction, the subscriptsare node labels and P is a node at CD. With the square arraymodel the nodes at the line BC are tied to point C with thedisplacements in thex direction of these nodes equal to the xdisplacement of point C, as given by

uQ � uC �7�whereu is the displacement in thex direction and Q is anode at BC. For the hexagonal model another boundarycondition is used for side BC, where the displacements ofnodes at BS are linked to side SC and the midpoint S [2,12],as given by

uR � 2uS 2 u�R �8�

vR � 2vS 2 v�R � vC 2 v�R; with vS � 12

vC �9�

where R is a node at BS and at SC equidistant from point S.Transverse loading, with the residual thermal stresses

being ignored, is realized by subscribing a fixed displace-ment of 1023 × AD in the y direction for node-point C,which results in an average applied transverse tensile strainof 0.1%. Transverse tensile moduli are determined by divid-ing the average transverse stress by the applied strain. Topredict the transverse strength only the stresses in the matrixhave been considered, assuming perfect adhesion and thusmatrix dominated failure. Furthermore it is assumed thatreaching the yield stress at room temperature will result instrain-localization and abrupt failure, as observed fromtensile testing at 228C; see Fig. 1. Stress concentrationsare determined by dividing the maximum local stress bythe average stress. By dividing the ultimate stress of thematrix by the stress magnification factor the transversestrength can be predicted. The transverse failure strain iscalculated by dividing the predicted strength by the calcu-lated transverse modulus.

The test-temperature is varied by applying a thermal loadon the model geometry to simulate the temperature jump


Fig. 4. Finite element meshes of (a) the square and (b) hexagonal fibre packing model, respectively.

from cure temperature to test-temperature. A negative ther-mal load on the model geometry results in a temperaturedecrease and subsequently in thermal shrinkage. Residualthermal stresses are introduced as a result of the differencein thermal expansion coefficient of fibre and matrix, seeTable 1. It is assumed that no relaxation of thermallyinduced stresses occurs below the glass transition tempera-ture (Tg) of the epoxy matrix, which is 1408C. Cooling fromcure temperature (1408C) to test-temperatures of2 30, 22and 708C is realized, using thermal loads of2 170, 2 118and 2 708C, respectively. By applying a transverse loadafter the thermal load, on each position the matrix stressesincrease linearly with the applied load. However, the localmaxima do not increase linearly, since the position of themaxima shifts to another position and, moreover, because ofthe nonlinear formulation of the von Mises stress (Eq. (4)),even at every single position this stress usually does notincrease linearly with the applied transverse stress. An addi-tional transverse loading is generated by applying an incre-mental pointload up to 150 MPa at node-point C in steps of10 MPa. By determining the maximum local von Misesstress with increasing load the transverse strength ispredicted. Since the critical failure stress of the epoxymatrix is temperature dependent, for each test-temperaturea different value is used; see Table 3.

3. Results and discussion

3.1. Influence of fibre volume fraction

3.1.1. ModulusAll composites experimentally tested showed perfect

linear elastic behaviour up to failure. The transverse modu-lus strongly increases with increasing fibre volume fraction,as could be expected; see Fig. 5. The modulus determinedby three-point bending is systematically lower than themodulus determined by tensile testing, despite the largespan-to-depth ratios used in the bending tests. Numericalpredictions were made using both types of packing. Mostexperimental data are found between the two calculatedlines, but the best results are obtained with the squaremodel, which is in good agreement with previous studies[6].

3.1.2. StressesTransverse tensile loading results in calculated matrix

stresses as illustrated in Fig. 6. The applied transverse stressis mainly transferred through the fibres and the matrix inseries with the fibres. This can be seen in Fig. 6(a) and (e),where the dark grey zones with high stresses continue verti-cally in the direction of the applied load. Due to the presenceof the rigid fibres considerable strain magnifications arefound in the matrix, which can reasonably be approximatedusing the analytical model of Kies, given by

SMF� 1M

1C� 1

1 1EM

EF2 1

� � ��2Kp

VF

r �10�

with K � 2 for the square array model [29] andK � √3 for

the hexagonal array model. Furthermore1 is the strain,E isthe modulus andVF is the volume fraction of the fibres. Thesubscripts M, F and C are for matrix, fibre and composite,respectively. This strain magnification factor varies from 3to 7 for the fibre volume fractions considered. The matrixparallel (or next) to the fibres is hardly loaded (stresses orstrained). Because the contraction of the matrix in thex andz directions is constrained by the rigid fibres, considerabletensile stresses are generated perpendicular to the appliedload, see Fig. 6(b), (c), (f) and (g). Despite that in theseblack-and-white pictures the contrast is low, stresses ofmore than half times the applied stress are observed. Thisresults in a von Mises stress lower than the axial stresssYY,see Fig. 6(d) and (h).

For each fibre volume fraction the maximum in principalstress, von Mises stress and hydrostatic stress has beendetermined. Fig. 7 shows the local maxima normalized tothe average stress for both model geometries. As could beexpected, the principal stress increases with increasing fibrevolume fraction [8,10,12]. Also the hydrostatic stressincreases with fibre volume fraction and appears to bequite high, especially when compared to the hydrostaticstress in uniaxial tensile loadings. In the composite matrix


Table 3Yield stress of the epoxy matrix as a function of temperature

Temperature (8C) 2 30 22 70

Yield stress (MPa) 123.9 94.2 67.2

Fig. 5. Transverse modulus as a function of fibre volume fraction, asmeasured in tension (W) and three point bending (X) and predicted withthe square (———) and hexagonal fibre packing model (- - -).

stresses are close to triaxial, which results in von Misesstresses that are considerably lower than the maximum prin-cipal stress and almost equal to the average applied stress.Although in general the matrix stresses increase withincreasing fibre volume fraction, the von Mises stressdecreases, because the matrix stresses are closer to triaxialwith increasing fibre volume fraction.

3.1.3. StrengthIn Fig. 8 the predicted strengths are shown using a von

Mises and a principal stress failure criterion. Both criteriayield substantial different results, quantitative as well asqualitative. The curves obtained from the principal stress

criterion are in agreement with the existing literature[8,10,12] and show a decreasing transverse strength withhigher fibre volume fraction. Comparing the experimentallydetermined transverse strengths (stress-to-failure), a largedifference is observed between the results of the tensiontests and the results of the three-point bending tests, seealso Fig. 8.

If only one fibre volume fraction had been studied ofapproximately 60 vol%, like the case in many other paperson this subject [8–11], a reasonable good agreement wouldhave been found between the results from the tensile testsand the predictions using the principal stress criterion.


Fig. 6. Stress contour plots of the square (50.3 vol% fibre, (a), (b), (c), (d)) and hexagonal (49.1 vol% fibre, (e), (f), (g), (h)) fibre packing model, respectively,in transverse loading. Results are plotted for the stresses normalized to the applied stress iny-direction in y-direction ((a), (e)),x-direction ((b), (f)) andz-direction ((c), (g)) and, finally, the von Mises stress ((d), (h)).

Fig. 7. Maximum principal stress, von Mises stress and hydrostatic stress inthe matrix as a function of fibre volume fraction, as calculated with thesquare (———) and hexagonal fibre packing model (- - -).

Fig. 8. Transverse strengths as a function of fibre volume fraction, asmeasured in tension (W) and three point bending (X), and predicted withthe square (———) and hexagonal fibre packing model (- - -), using a vonMises criterion and an ultimate stress criterion.

However, by varying the fibre volume fraction thisagreement is found to be a coincidence. Given thedifferences in slope there is no qualitative agreement,because the experimental transverse tensile strengthincreases with fibre volume fraction. It is clear thatthe uniaxial tension tests result in relatively lowstrengths due to unwanted size effects, where thestrength is determined by defects and flaws [30]. Inthree-point bending, failure is less controlled by flawsbecause only a small volume of material is highlystressed. Since the flexural transverse strengths areclose to the matrix strength, they are expected to be areliable estimation of the real “intrinsic” material

strength. By using the von Mises criterion, which isthe criterion experimentally found (see Fig. 3 and thediscussion above), a good agreement is found betweenthe predicted strengths and the flexural transversestrengths, both quantitatively and qualitatively; seeFig. 8.

Due to the triaxial stress state, the use of the vonMises criterion yields a higher predicted strength thanobtained with the principal stress criterion. It can beexpected, however, that the accompanying high hydro-static stresses might cause cavitation of the matrix andsubsequently exceptional brittle behaviour [17,18].Considering the flexural strength measured, the matrixis still able to sustain these triaxial loads. When theyield stress is reached, however, localized plastic flowmay induce an increased hydrostatic stress and subse-quent cavitation [23]. It is important to notice that boththe experimental and predicted transverse stress to fail-ure increases with increasing fibre volume fraction. Thisis in contrast with the often quoted hypothesis that thetransverse strength is low as a result of stress concen-trations in the matrix and that the transverse strengthwill decrease with increasing fibre volume fraction as aresult of increasing stress concentrations.

Since the transverse modulus increases considerably withincreasing fibre volume fraction, the transverse failure straindecreases with increasing fibre volume fraction; see Fig. 9.Obviously, also for the measured transverse failure-strains alarge difference is found between the results of the tensiontests and the three-point bending tests. Using a von Misesfailure criterion, a good agreement is found between theresults from the hexagonal array and the experiments.Because the square array model generally overestimatesthe transverse flexural modulus, lower failure strains arepredicted.


Fig. 9. Transverse failure strain as a function of fibre volume fraction, asmeasured in tension (W) and three point bending (X) and predicted with thesquare (———) and hexagonal fibre packing model (- - -), using a vonMises criterion and an ultimate stress criterion.

Fig. 10. Stress contour plots of the square (50.3 vol% fibre, (a), (b), (c), (d)) and hexagonal (49.1 vol% fibre, (e), (f), (g), (h)) fibre packing model, respectively,in thermal loading. Results are plotted for an applied thermal load of 18C for the radial stress ((a), (e)), hoop stress ((b), (f)) and axial stress ((c), (g)) and, finally,the von Mises stress ((d), (h)).

3.2. Influence of test-temperature

3.2.1. StressesThermal loading of the composites results in matrix stres-

ses as illustrated in Fig. 10. For understanding the stresssituations, the deformation can be considered as a resultof a relative expansion of the fibres. This results in tensilehoop stresses in the matrix and compressive stresses normalto the fibre–matrix interface, with the highest compressivestresses where the fibres are adjacent, as shown in Fig. 10(a),(b), (e) and (f). Because matrix shrinkage is constrained inthe fibre direction, tensile axial stresses are generated; seeFig. 10(c) and (g). The combined tensile stresses andcompressive stresses result in von Mises stresses higherthan the distinct principal stresses; see Fig. 10(d) and (h).

As a result of cooling down from cure temperature to test-temperature, significant residual stresses can be generated.However, since stress relaxation is ignored, in reality ther-mal residual stresses will be somewhat lower than thosepredicted here. Higher fibre volume fractions yield higherthermal stresses, as shown in Fig. 11. Of course, the residualthermal stresses increase proportionally with increasingtemperature jump, given the linear thermal-elasticity ofthe systems under investigation. By normalizing the thermalstresses to the applied thermal load, all data are reduced totwo curves; see Fig. 11(b). The local maximum in the vonMises stress proves to be much higher with the square modelthan with the hexagonal model. This is attributed to the


Fig. 11. (a) Maximum von Mises stress due to cooling, and (b) maximumvon Mises stress normalized to the applied thermal load as a function offibre volume fraction, predicted with the square (———) and hexagonalfibre packing model (- - -), with the test-temperature as parameter.

Fig. 12. Maximum von Mises stress normalized to the applied thermal loadas a function of applied stress at2 308C (X), 228C (W) and 708C ( 1 ), aspredicted with (a) the square fibre packing and (b) the hexagonal fibrepacking, with the fibre volume fraction as parameter.

interfibre distance, which is considerably smaller in thesquare fibre packing than in the hexagonal fibre packing.

Following the position of the maximum von Mises stressand its value as a function of the applied transverse stress atdifferent temperatures, a general master curve can be gener-ated by normalizing both the von Mises stress and theapplied transverse stress to the applied thermal load; seeFigs. 11 and 12. As mentioned previously, with higherfibre volume fractions initially, i.e. before transverse load-ing, the residual stresses are higher. However, as a result ofthe lower stress magnifications at higher fibre volume frac-tions, the stress will increase slower with increasing load.This results in intersection points of the lines with differentfibre volume fractions, where the influence of the mechan-ical loading overrules the influence of the thermal loading.At higher mechanical loads, the von Mises stress decreaseswith increasing fibre volume fraction, which is similar to the

results with the thermal stresses ignored. The intersectionpoints are found at higher mechanical loads with higherthermal loads, because higher thermal stresses have to beovercome.

3.2.2. StrengthThe transverse strength can be predicted by determining

the transverse stress necessary to reach the ultimate matrixstress. By combining the master curves from Fig. 12 withthe temperature dependence of the yield stress (Fig. 2) thetransverse strength can be calculated as a function of test-temperature; see Fig. 13. In general, a strongly increasingstrength is predicted with decreasing temperature, becausethe yield stress of the matrix dominates the transversestrength. The effects of thermal stresses (although lesssignificant) is most pronounced around 708C, at the distinctkink in the curves of Fig. 13. The residual thermal stressesyield an enhanced transverse strength since compressivethermal stresses have to be overcome; see Fig. 10(a) and(e). An exception in this general trend is found with thesquare array model at a high fibre volume fraction. Here,with further decreasing test-temperature the transversestrength decreases since the thermally induced tensilehoop stresses (see Fig. 10) give rise to reduced transversestrengths. However, since this decrease in transversestrength is experimentally not observed, these effects ofthe thermal stresses must have been overestimated.

To compare the numerical results with experimental data,the transverse strengths determined at three different test-temperatures are shown as a function of fibre volume frac-tion in Fig. 14. Only three-point bending tests have beenused, since tensile tests showed to yield inappropriate data.At room temperature (228C) for both models, a good agree-ment is found between the calculated and the experimentalstrengths. However, a more pronounced temperature influ-ence is predicted than experimentally observed. Basically, itis expected that the transverse strength decreases withdecreasing temperature, as a result of residual thermal stres-ses caused by matrix shrinkage. However, the influence ofparameters, such as residual thermal stresses, is obscured bythe strong effects of the increase in matrix yield stress withdecreasing temperature, as was shown in Fig. 2.

The influence of the residual thermal stresses can berevealed by normalizing the transverse strengths to theyield stress of the matrix (Table 3). The normalized flexuralstrengths increase significantly with increasing temperatureto values higher than the matrix strength (1); see Fig. 14(b).Surprisingly, the influence of temperature on the experimen-tally obtained normalized strength is smaller at high fibrevolume fractions than at low fibre volume fractions,although residual thermal stresses are likely to be higherat high fibre volume fractions. Since at these high volumefractions the measured flexural strengths appear to beinbetween the predicted strengths of the square and hexa-gonal array model, the influence of residual thermal stresses


Fig. 13. Predicted transverse strength as a function of temperature, for (a)the square array model and (b) the hexagonal array model, with the fibrevolume fraction as parameter.

would be overestimated with the square array model andunderestimated with the hexagonal array model.

The numerical analyses, as shown in Fig. 14(b), indicatethat at low fibre volume fractions the influence of residualthermal stresses is insignificantly low. However, experi-mentally a strong increase in normalized flexural strengthis found at these volume fractions (,50%). If the tempera-ture dependence of the normalized strength cannot be attrib-uted to the occurrence of thermal stresses, it is probablycaused by the increase in matrix ductility (plastic strain)with increasing temperature, which also may affect thetransverse strength. This speculation is confirmed by theobserved temperature influence on the transverse failurestrain; see Fig. 15. The failure strain is expected to decreasewith increasing test-temperature, similarly to the transversestrength. However, no significant influence of the test-

temperature on the failure strain is found, as a result ofincreasing matrix ductility (see Fig. 1), compensating theeffects of the matrix yield stress.

At room temperature a good agreement is found betweenthe calculated and experimental strengths, since theassumed failure mechanism of abrupt fracture at the yieldstress is valid (see Fig. 1). Ignoring the temperature depen-dence of the failure mechanism, with increased plastic strainat higher temperatures, gives rise to larger differences in thepredicted failure properties and the experimentally deter-mined values.

4. Conclusion

Three-point bending tests are essential to obtain goodexperimental data of the transverse stress and strain to fail-ure of unidirectional composites. Transverse testing of glassfibre reinforced epoxy in uniaxial tension yields too lowstrength values as a result of premature failure caused byflaws and defects. With three-point bending tests, strengthsare obtained, which are close to the matrix strength indicat-ing that the “intrinsic” material performance is measured.

The transverse stress to failure increases with increasingfibre volume fraction. By using a principal stress failurecriterion for the matrix no agreement could be obtainedbetween the experimental flexural strengths and thestrengths predicted with a square packing array or hexago-nal packing array. However, by using a von Mises failurecriterion a good agreement, both quantitatively and qualita-tively, is found between experiments and numericalanalyses. By incorporating a pressure dependence and strainrate dependence of the yield stress according to the modifiedform of Eyring’s flow equation [21], this agreement mighteven be improved.

By decreasing the test-temperature an increase in


Fig. 14. (a) Transverse strength and (b) normalized transverse strength as afunction of fibre volume fraction, at2 308C (X), 228C (O) and 708C (B), asmeasured in three point bending and predicted with the square (———)and hexagonal fibre packing model (- - -).

Fig. 15. Transverse failure strain as a function of fibre volume fraction, asmeasured in three point bending at2 308C (X), 228C (O) and 708C (B).

transverse strength is found. This is primarily caused by thetemperature dependence of the yield stress of the epoxymatrix. The numerical analyses predicted a strongertemperature dependence of the transverse strength thanexperimentally found. The difference in strengths at highfibre volume fractions might be caused by the predictedinfluence of the thermal stresses, which is overestimatedwith the square array model and underestimated with thehexagonal array model. Both models show that the presenceof residual thermal stresses generally results in an increaseof transverse strength, since matrix shrinkage yieldscompressive stresses between the fibres.

The difference between the predicted strengths and theexperimental strengths at lower fibre volume fractionscannot be attributed to the occurrence of thermal stresses,since at these volume fractions the influence of thermalstresses is found to be insignificant. The decrease of thetransverse strength with increasing test-temperature as aresult of the decreasing matrix yield stress is probablyreduced by the increase of the matrix ductility.

At room temperature a good agreement is found betweenthe calculated and experimental strength, using relativelysimple linear elastic analyses. However, by ignoring thetemperature dependence of the matrix ductility, withincreased plastic strain at higher temperatures, at othertemperatures larger differences between predicted strengthsand experimental strength are found.

Acknowledgements

The authors gratefully acknowledge H.J. Schellens for theidentification of the failure criterion of the epoxy matrix andA.E. de Jong for his large contribution in the both experi-mental and numerical work.

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deformation, yield and fracture of unidirectional composites in transverse loading

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