This is a repository copy of Initiation and Propagation of Transverse Cracking in Composite Laminates.

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1 This is a repository copy of Initiation and Propagation of Transverse racking in oposite Lainates. White Rose Research Online URL for this paper: Article: Ye, J., La, D. and Zhang, D. 1 Initiation and Propagation of Transverse racking in oposite Lainates. oputational Materials Science, ISSN Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by eailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. eprints@whiterose.ac.uk

2 AEPTED FOR PUBLIATION IN OMPUTATIONAL MATERIALS SIENE Initiation and Propagation of Transverse racking in oposite Lainates Jianqiao Ye 1 *, Dennis La 1 and Dau Zhang 1 Institute for Resilient Infrastructure iri, School of ivil Engineering, University of Leeds, Leeds, LS 9JT, UK School of Mechanical, Aerospace and ivil Engineering, University of Manchester, PO Bo 88, Manchester, M6 1QD, UK. Abstract The atri cracking transverse to loading direction is usually the one of ost coon observations of daages in coposite lainates. The initiation and propagation of transverse cracks have been a longstanding issue in the last few decades. In this paper, a three-diensional stress analysis ethod based on the state space approach is used to copute the stresses, including the inter-lainar stresses near transverse cracks in lainated coposites. The stress field is then used to estiate the energy release rate, fro which the initiation and propagation of transverse cracking are predicted. The proposed ethod is illustrated by nuerical solutions and is validated by available eperiental results. To the best knowledge of the authors, the predictions of crack behaviour for non-syetrical lainates and lainates subect to in-plane shearing are presented for the first tie in the literature. Introduction The first for of daages in lainates is usually atri icrocracks. The ost coon observation of icrocracking is cracking in 9º plies during aial loading in the º directions Narin,. Transverse cracking is therefore the ost coon daage ode in coposite aterials. An iediate effect of transverse cracking is to cause stiffness degradations of the lainate. Stress singularities near the crack tips at the ply interface ay initiate interlainar delaination. Delaination is not necessarily the ultiate structural failure, but it ay result in fibre-atri debonding and fibre rupture, which will eventually lead to the loss of structural integrity. The * orresponding author: E-ail:.ye@leeds.ac.uk, 1

3 ultiate failure of a coposite lainate follows the occurrence of transverse cracking, longitudinal cracking, delaination and fibre breaking. The initiation and propagation of transverse cracks in coposite lainates have been the focus of failure investigation in the last few decades. Etensive investigations have been carried out both eperientally and analytically. Garrett and Bailey 1977, Parvii et al and Bailey and Parvii 1981 are aongst the earliest researchers who carried out etensive eperients to observe transverse cracks. They found that cracks fored in a direction parallel to the transverse reinforceent and the thickness of the 9 plies had significant effect on the cracking process. Flaggs and Kural 198 presented the results of an eperiental study confiring that the constrained transverse cracking phenoena observed in the 9 ply of uniaially loaded [ /9 ] s coposite lainates was also ehibited by the ore general [θ/9 ] s class of coposite lainates. Nairn and Hu 199, Liu and Nairn 199 and Nairn et al carried out a series of eperients on crack density as a function of applied load. For all the lainates tested, the characteristic cracking curve had no cracks until an onset stress was reached. After the initial crack, the crack density typically increases very rapidly. The onset stress decreases as the thickness of the 9 plies increases. Yokoeki et al. 5 investigated crack accuulation in ultiple plies of [/θ /9 ] s lainates θ=3, 45 and 6. Most of the eperiental investigations showed that the first daage ode was usually transverse cracking. Both the thickness of 9 layers and the stiffness of constrain layers affected the initiation and propagation of transverse cracks. The aority of earlier analytical work on transverse cracking assued that cracks fored when the stress or strain reached the transverse strength of a ply aterial. Garrett and Bailey 1977 assued that a transverse ply had a unique breaking strain, ε tu, and strength tu. If a stress is applied in a direction parallel to the longitudinal plies, the transverse ply will fail at a stress tu. Using the sae strength criteria, Parvii, et al 1978 reported ore detailed studies for a glass fibre reinforced epo coposite. Leblond et al studied ultiplication of transverse cracks as a function of applied stress in cross-ply lainates. The crack developent was assued to be controlled by the fracture stress in the 9 plies. However the strength based theory usually can not provide a good prediction of transverse cracking because

4 the strength of 9 o plies of a lainate is usually not the sae as that of a different lainate. Due to the drawbacks and liitations of the strength based ethods, the aority of recent work was based fracture echanics using the energy ethod to predict transverse cracking. Most energy odels used a representative volue eleent RVE to predict net crack foration when the energy released due to crack foration reached the critical strain energy release rate G c. It has been widely recognised that for the sae aterial ply lainates with different laination profiles, the value of G c alost keeps constant Nairn,. onsequently, the ethod applies for a wide variety of lainates fro a single value of G c. Parvii et al deonstrated that a siple shear lag analysis used in conunction with the Griffith energy criterion can be used to accurately predict atri cracking. Flaggs 1985 ade use of a strain energy release rate fracture criteria in conunction with an approiate two-diensional shear-lag odel to predict tensile atri failure. Wang et al 1985 eployed the energy release rate ethod of classical fracture echanics to odel various atri crack interactions. Dvorak and Laws 1987 investigated the first ply failure using a critical energy release rate criteria and later Laws and Dvorak 1988 presented a odel for progressive transverse cracking based on statistical fracture echanics. Nairn 1989;, Liu and Nairn, 199 and Nairn and Hu 199 carried out a series of study on atri cracking by finite fracture echanics. Zhang et al. 199 and Fan and Zhang 1993 proposed the equivalent constraint odel EM, in which the energy release rate due to transverse ply cracking, incorporating residual theral stresses, was derived. Mcartney 1998; ; 4; 5 investigated ply crack developent for various laination profiles; fro cross-ply to general syetric lainates, subected to aial etension or ied ode loading. Sith and Ogin 1999 calculated the critical bending oent at transverse cracking under fleural loads using a fracture echanics approach. Joffe et al. 1 used a crack-closure technique to calculate the energy release caused by cracking. A Monte-arlo siulation in increental strain-controlled loading was used to odel the transverse cracking process. The 9 layer was divided into a large nuber of eleents and a critical energy release rate G c was assigned to each eleent according to Weibull distribution. Yokoeki el al. eployed energy release rate to investigate crack initiation and propagation across specien width. Energy 3

5 release rates associated with crack propagation in the width direction were calculated using a three-diensional FEA. Subsequently Yokoeki et al. 5 used the sae ethod to study icro-cracking behaviour induced by atri cracks in adacent plies. Li and Li 5 calculated energy release rates for transverse cracking and delaination under the generalised plane strain condition. By introducing the iniu strain energy density criterion to a non-linear FE analysis, Sirivedin et al. 6 predicted atri crack propagation in continuous-carbon fibre/epo coposites. It is obvious that for an energy based ethod, an accurate prediction of the stress distribution within a RVE is essential to an accurate estiate of the potential energy within the eleent. This is particularly difficult for a lainated RVE with transverse cracks. Traditional analysis of lainated coposites used classic or higher order plate theories that usually provided unsatisfactory predictions to interfacial stresses and the stress singularities at the tips of transverse cracks. Zhang and Ye 7a recently developed an analytical odel that can provide accurate predictions to the stress fields, including all the interfacial stresses, and a satisfactory approiation to the stress singularities near ply cracks. The odel was based on a state space approach that has been successfully used to solve a variety of stress probles Soldatos and Ye, 1994; Ye and Soldatos, 1994a, b, 1995; Ye and Sheng, 3; Ye et al., 4; Zhang and Ye, 7b. opared with other analytical odels, this new odel takes full three-diensional consideration of lainar properties, displaceents and interfacial stress continuities at all aterial interfaces. The odel can also deal with both syetric and non-syetric lainates with a universal approach. A coprehensive account of the ethodology can be found in Ye. In cobination with the state space forulation Ye, this paper presents a odel to predict crack propagation by using an energy based approach. An accurate stress distribution within a RVE with cracks is obtained fro the state space solution. The stresses are then used to copute the energy release rate in the crack propagation analysis. Nuerical results are obtained and copared with the tests results available in the literature. Results are also presented for non-syetric lainates and lainates subected to in-plane shearing. Fro the authors best knowledge, these results are new and are not available in the literature. Stresses of angle-ply lainates with transverse cracks 4

6 Solution of an angle-ply laina onsider an off-ais laina Fig. 1 with principal aterial directions 1--3 in the global -y- coordinate syste. The laina has constant thickness h, width L and infinite length. The displaceents in the, y and directions are denoted by u, v and w, respectively. Suppose that the laina is subected to a unifor tension by the application of a constant longitudinal strain in the y direction, ε,which represents a lainate that is long and relatively unifor in one direction, such as aircraft wing panels. The constant strain in the y direction also represents the state of stress at a point in a aterial subect to a generalied plane strain, where the stresses and strains in other two directions are ore doinating. The laina is ade of a hoogeneous, onoclinic and linearly elastic aterial whose principal aterial direction 1, i.e., the fiber direction, has an angle of θ to the ais. a Stress-strain relations The basic constitutive equation for thero-elastic stress analysis is Herakovich, 1998 T { } = [ ]{} ε { ε }. 1 Here, the atrices [ ], {} T ε and { } ε are stiffness atri, total strains and theral strains, respectively. For a linearly elastic onoclinic aterial, [ ] =, where the i are stiffness coefficients that can be epressed in ters of Young s oduli, Poisson s ratios and shear oduli. {} ε = [ ε ε ε ε ε ε ] T, 3 T { } = {α }ΔT yy y ε, 4 where Δ T denotes teperature change, { α } = [ α α α α, 5 T yy ] where α, α yy, α and α are the coefficients of aial theral epansion relative to the, y, directions and shear theral epansion, respectively. b Equilibriu equations 5

7 c ε ε y y yy y y y y = = = Strain-displaceent relations u =, w v =, y ε ε yy. 6 v =, ε y u w =, ε w = u v = y. 7 Since the laina is subected to a unifor etension ε in the y direction, it follows that v ε yy = = ε. 8 y Then the generalied plane strain deforation is assued such that all coponents of stress and strain do not depend upon y. To carry out the following deductions, let α = /, 1 = 13 / 33, = / 33, 3 = / 33, 4 = 3 / 33, 5 = 3 / 33, 6 = 36 / 33, 7 = 1/ 33, 8 = 1/ 55, = 1 = / / 33, = , = / = / Δ, = / Δ, 1 44 = / Δ, 13 55, Δ. 9 Fro the third equation of Eq. 1 and Eq. 7, one has w = 1α u 6αv 7 5ε 1α 5α yy 6α α ΔT. 1 By substituting Eq. 7 into the first, second and sith equations of Eq. 1, the inplane stresses can be epressed as α yy = 3α 9α α α α 1 u 3 α 3α yy 9α 5 v 4 ε 3α 4α yy 1α ΔT α 1α yy 14α Inserting Eq. 11 into Eq. 6 and considering Eq. 1 as well as the fourth and fifth equations of Eq. 1, the following first order partial differential equation can be obtained 6

8 7 = T w v u w v u 6 yy y y Δ α α α α ε α α α α α α α α α α 1 Assuing that displaceents u, v, and w can be epressed, respectively, as = = =,,, 1,,, 1,,, w y w y L V v y v L U u y u ε, 13 where U and V are unknown boundary displaceents that can be deterined by iposing traction free conditions along the stress free surfaces see the boundary condition section. In Eq.13, the following Fourier series epansions are assued = = cos sin sin cos sin sin y Z Y X W V U w v u ξ ξ ξ ξ ξ ξ, 14 where L / π ξ =. In the case of a unifor etension in the -direction, the aial displaceent u is ero at =L/. Hence, the integer in Eq. 14 and the equations below takes only even nubers, i.e. =,, 4,. By introducing Eqs. 13 and 14 into 1 and epanding the and 1 in Eq. 13 into also Fourier series, the following non-hoogenous state space equation for an arbitrary value of is obtained { } [ ]{ } { } d d B F G F =, 15a where { } [ ] T Z Y X W V U = F, 15b

9 ξ ξ 9ξ ξ [ G ] =, 15c { B } ξ 9 14 ξ ξ 11 1 ξ ξ 6ξ L L 1 6 =,, 5ε 1α 5α yy 6α α ΔT U V,,, T, 15d { B } = 1 cos π, 1 cos π,,,, =, 4, 6. π du d π dv d The solution of the non-hoogenous state space Eq. 15 is { } [ G ] { } [ G] τ F = e F e { B τ } dτ [ D ]{ F } { }, [, h]. 16 = H In particular, at =h, { F h } [ D h ]{ F } { H h } =, 17 T 15e where [ D h ] is called transfer atri. The calculation of the two constant atrices, [ D h] and { H h} Ye. Solution of an angle-ply lainate, in Eq. 17 can be found either analytically or nuerically fro onsider an infinite long ulti-layered general angle-ply lainate of thickness H and width L. Again the lainate is subected to a constant longitudinal strain, ε.we ay iagine that it is coposed of N fictitious sub-layers, each of which ay have different thickness. However, it is assued that the thickness of all the fictitious sublayers approach ero uniforly as N approaches infinity. Assuing, in addition, that different sub-layers ay be coposed of different onoclinic aterials, two types of aterials interfaces are distinguished in the plate; the fictitious interfaces which separate sub-layers with the sae aterial properties and the real ones that separate sub-layers coposed of different aterials. Upon choosing a suitably large value of N, each individual sub-layer becoes thin. For each of the sub-layers, are the solutions. The state space equation and the for of solution of an arbitrary sub- 8

10 layer, e.g., the th one whose thickness is h, can easily be obtained by replacing h with h in Eqs The state space equation of the th sub-layer then becoes: d d { F } = [ G ] { F } { B }. 18 After repeating the above process for all the individual sub-layers and with appropriate continuity requireents iposed at all the real and fictitious interfaces, a solution for the entire lainate can be forulated. In order to find the solution of the proble, the two unknown displaceent coponents, U and V in Eq. 15 ust be deterined first. If the sub-layers of the lainate are all sufficiently thin, it is reasonable to assue that U and V within the thin layer are linearly distributed in the direction, i.e. U V = U = V 1 U h h 1 V h h, [, h ], =1,,, N, 19 where U, U, V and V are the values of U and at the top and botto surfaces of the th thin layer, respectively. Inserting Eq. 19 into Eqs. 15d and 15e, vector { B in Eq. 18 can be epressed as { B } L 1 U } = [,, ε α α α 1 h 5 4 U π U U 1 h 5 L 4 V π 6 y V V 6 1 h α ΔT V h V, [, h ] T,,,], a { B } =,,,,, =, 4, 6. b The solution of Eq.18 at =h is { F h } [ D h ] { F } { H h } h h T =. 1 By introducing the following continuity conditions at all interfaces, i.e., { F } 1 = { F h }, =1,,, N-1, and then using Eqs.1 and recursively, a relationship between the state vectors on the top and botto surfaces of the lainate is established as follows: { F } [ D ] { F } { H } hn where, 3a N = N 1 9

11 D N =, 3b =N 1 [ ] [ D h ] { H } = [ D ] { H } [ D] 1 { H } L { H } N = N 3 = N. 3c { F h N } N and { F } 1 are, respectively, the state vectors at the top and botto surfaces of the lainated coposite. The traction free conditions at the top and botto surfaces yields T T [ X Y Z ] = [ ] 1 [ ] T = [ ]. 4 T X h Y h Z h N N N N Substituting Eq. 4 into Eq. 3 results in the following linear algebra equation syste D D D where D D D D D D D i and U V W i H = H H , 5 H are the atri eleents in [ ] D and H } of Eqs.3b- 3c that are related to the three displaceent coponents at the botto surface, respectively. Eq.5 is a set of linear algebra equations in ters of the three displaceent coponents, { U, V and W, at the top surface. The ters on the right-hand side of Eq. 5, H 4, H 5 and H 6, contain 4 N unknown constants, U, U, V, and V =1,, N, introduced in Eq.19. Because of the continuity of U and V at the interface between the th and the 1th sub-layers, = U U 1 and V = V 1 =1,, N-1. Hence, the nuber of unknown constants is then reduced to N1. These constants are deterined by introducing appropriate boundary conditions along the transverse edges. Ply-crack boundary conditions When a general angle-ply lainate is subected to an in-plane etension perpendicular to the 9 fibers, transverse ply cracks appear parallel to the fibers and across the entire width fro edge to edge. For eaple, subect to a unifor biaial etension, ε and F, and a shear loading, S, the [θ /9 n /φ s ] lainate shown in Fig. displays an array of periodic cracks in the 9 n layers, where the subscripts 1

12 denote the nuber of the real plies within a ply group. In reality, atri cracks can occur in any plies, but there are a large group of lainates, in which transverse cracking in 9º plies is the doinated daage ode, and therefore the inor atri cracking in non-9º plies is ignored in the present odel. Other daage odes, e.g. delaination and fibre breakage usually occur at high crack densities, so the current work focuses on low and interediate crack densities. Assuing that the cracks are equally spaced, a representative volue eleent Fig. 3 can be taken fro any two neighboring cracks to predict the stress and displaceent fields. For the cracked layers at =, L the boundary conditions are traction free, i.e., =. For the un-cracked layers at =, L, due to the fact that the = lainate is subected to unifor etension and shear loading, the displaceents of an uncracked layer in the and y directions, u and v, reain constant, that is u,y, = u v,y, = v. 6a Substituting Eq. 6a and Eq. 14 into Eq yields at = and y= u,, = v,, = = = U sin ξ U V sin ξ V 1 = u L 1 ε = v L 6b Fro also the equilibriu of the internal and eternal forces, the following equation eists d = F h = d S h 7a or, fro Eq.11 11

13 [ ξu 9ξV 1Z ] [ 3ε α 3α y 9α ΔT ] h U L h 9 U L [ 9ξU 14ξV 6Z ] [ 1ε 9α 1α y 14α ΔT ] 9 V L L 14 V d = F d = S 7b where F and S are the forces per unit length Fig. 3. Fro the introduction of the boundary conditions, it can be seen that the solution can be found to the required accuracy by increasing the total nuber of the thin layers. Propagation of transverse cracking in lainates The total copleentary potential energy of a representative eleent Fig. 3 shows a representative volue eleent which is taken fro between two neighboring cracks in a [θ /9 n /φ s ] coposite lainate. Assuing that stress analysis has been carried out on this idealied eleent fro the previous section, where the lainate was assued to consist of N fictitious sub-layers. By using the present stress analysis, the total copleentary potential energy is easier to obtain than the total potential energy because the stresses, used to calculate the copleentary strain energy, are deterined iediately after solving the state space equations. The copleentary strain energy U c of this representative eleent is U c U c = N = 1 8 where the superscript denotes the th sub-layer and U c is the copleentary strain energy of the th sub-layer of the representative volue eleent. Using the stresses fro the previous section, U c per unit length in the y direction can be obtained as U c = 1 L T 1 T [ { } [ ] { } ΔT{ α} { } ] h dd 9a where { } = [ 9b T yy y ] 1

14 onsidering that the lainate is under a uniforly prescribed strain ε in the y direction, the potential of the prescribed strain ε can be calculated as N N L V c = V c = [ yy ] = 1 h = 1 ε dd 3 The total copleentary potential energy of this representative volue eleent is given as the difference of the copleentary strain energy U c and the potential of prescribed displaceents V c. h = U c V c = N L 1 T 1 T { } [ ] { } ΔT{ α} { } = 1 ε yy dd 31 The energy release rate due to transverse cracking onsider a coposite lainate subected to eternal loading and there are a sufficiently large nuber of transverse cracks in the 9 layers. The entire length of the lainate is L e and the thickness of cracked layers is H c. Fig. 4 shows the propagation process of the transverse cracks fro state a to state c. In state a, it is assued that there eist k uniforly spaced transverse cracks in the lainate. Therefore the crack density in this state is ρ k = k L e 3 With the changes of eternal loading, a new transverse crack fors and the crack pattern changes fro state a to state b. The nuber of the transverse cracks increases fro k to k1. Although in reality a new crack foration is randoly distributed, the overall crack distribution tends to be unifor when the nuber of cracks is large. In order to siplify the analysis, state b is idealied to state c, in which the k1 cracks are also equally spaced. The siplification of the crack spacing here is also due to that the focus of the present study is the effect of crack density on degradation of aterial properties. The crack density of state c is then ρ k 1 k 1 = L e 33 During the transverse cracking process, the crack surface area increent is H c. The energy release rate fro state a to state b is 13

15 d G = da Γ ρ = Le ρ = k 1 k 1 Γ ρ k k 1 Γ r ρ = H H r c Γ ρ k 1 L ρ Γ ρ H c e k r k k 1 c kγ ρ r k 34 where Γ ρ k 1 and Γ ρ k are the total copleentary potential energies of the entire lainate at crack densities ρ k 1 and ρ k, respectively; Γr ρk 1 and Γ r ρ k are the respective total copleentary potential energies of the representative volue eleent at crack densities ρ k 1 and ρ k. The transverse crack propagation criterion A new crack will for if the energy released due to crack foration reaches the critical energy release rate G c, i.e. G G c 35 G c is a aterial property and has units of energy per unit area. It can be easured by an eperiental ethod. Fig. 5 is a flowchart showing how to deterine the critical cracking load for a given crack density. For a given load, stress analysis is carried out by using the state space odel. The energy release rate G of a representative volue eleent is then coputed by Eq. 34 and copared with G c. If G>G c, the current load is reduced to achieve a saller energy release rate until G=G c. If G<G c, the current load is increased to obtain a larger energy release rate until G=G c. If G=G c, the current load is taken as the critical cracking load for the given crack density. Nuerical results The forulations and criterion proposed above are applied to predict transverse cracking in coposite lainates with different configurations, including syetric cross-ply lainates, syetric angle-ply lainates and general non-syetric lainates. The aterial properties and diension of these lainates Liu and Nairn, 199; Joffe et al., 1 are given in Table 1. Effects of residual theral stresses are included in the analysis. ΔT is the difference between the roo teperature and the cure teperature. Table 1 also lists the critical energy release rate G c for each aterial. 14

16 Syetric lainates subected to tension The crack density as a function of applied average stresses for syetric cross-ply lainates is plotted in Fig.6. Herein, the applied average stress is the aial tension per unit length in the direction divided by the height H of the lainate. The test and variational results of Liu and Narin 199 are also shown in these figures for coparisons. The nuerical results for Material 1 with [ /9 ] s and [ /9 4 ] s laination profiles are plotted in Figs. 6a and 6b, respectively. Using a single value of G c, the predictions of the two lainates agree well with the eperiental results, which indicates that the critical energy release rate can be used as a aterial property that characteries transverse crack propagation in coposite aterials. The dependence of crack density on the applied average stress for syetric angleply lainates with layups [±θ /9 4 ] s are presented in Figs. 7a and 7b. The lainates are coposed of Material. The present results are copared with those obtained by Monte-arlo siulations and eperiental data in Joffe et al. 1. Once again very good agreeents are observed in these figures. It can be seen that cracks occur earlier in the [±3 /9 4 ] s than in the [±15 /9 4 ] s lainates. Non-syetric lainates subected to tension After successful validation for cross-ply syetric lainates, the ethod is applied to predict transverse crack propagation in two non-syetric lainates. The first one is constructed by replacing one set of 9 4 layers in the above [±3 /9 4 ] s lainate with 4 layers. Thus the new profile, [±3 /9 4 / 4 / 3 ], is now non-syetric. The stress-crack density relation of the non-syetric lainate is plotted in Fig. 8a. In coparison with Fig. 7b, the onset crack stress of the non-syetric lainate is significantly increased. The second lainate has a lay-up of [3 /9 /3 /9 ] and is coposed of Material 3. Both the 9 layers are assued to have transverse cracks and the crack distributions in both layers are identical. Fig. 8b shows the crack density as a function of the applied average stress in the lainate. No coparisons have been ade in Figs 8a and 8b, because the present solutions are believed to be the first ones in the literature on predicting transverse crack propagation in non-syetric lainates. These results, therefore, can be used as bencharks for testing new odels. 15

17 Lainates subected to tension and shearing In this section, the present ethod is further used to study the effects of shearing on transverse cracking. A syetric and a non-syetric cracked lainates under a cobination of tension and shearing are analyed, respectively. It is assued that the lainates are coposed of Material 3 fro Table 1. A series of curves are shown in Fig.9 to deonstrate the effects of shear stresses on the transverse cracking process of a syetric [3 /9 /9 /3 ] lainate. The lainate is subected to a cobined action of unifor tension and shearing. The applied average stress, which is the average stress on the cross-section perpendicular to the ais Fig, increases, while the shear stress keeps constant. The applied shear stresses are -1, -5,, 5 and 1 MPa, respectively. It can be seen that both the agnitude and the direction of shear stresses have significant effect on the initiation and developent of transverse cracks. The negative shear stresses advance and the positive stresses delay the transverse cracking. This is because the [3 /9 /9 /3 ] lainate has a negative shear strain in the -y plane when the lainate is subected to a single tension in the direction Fig. If a negative shear stress is also applied, the agnitude of the negative shear strain increases. On the contrary, applying a positive shear stress decreases the agnitude of the shear strain. As a result, the energy release rate in the case of applying negative shear stress is higher than that in the cases of applying no shear stress or positive stresses. The sae set of shear stresses are also applied on a non-syetric [3 /9 /3 /9 ] lainate and the obtained stress-crack density relations are shown in Fig. 1. As can be seen, the effect of shearing on cracking is siilar to the syetric case. Nevertheless, under the sae loading condition the crack initiation stress of the non-syetric lainate is slightly higher than that of the syetric one. This is because the 9 layers in the syetric lainate are thicker and a thicker 9 layer is ore prone to crack foration. To the authors best knowledge, coparable solutions to the results presented in Figs. 9 and 1 are not available in the literature. The results can again be used as benchark solutions for future developent of new theories. oncluding rearks 16

18 By using the energy ethod, an approach based on the state space stress analysis to predict the propagation of transverse cracking in general coposite lainates has been proposed. The proposed ethod inherits the advantages of the state space ethod, by which an accurate stress distribution and, hence, an accurate estiate of strain energy can be coputed. The ethod can also deal with both syetric and non-syetric lainates. In conunction with the stress analysis, the energy release rate due to transverse cracking was derived in a lainate with an idealied unifor crack distribution. A new crack fors when the energy release rate approaches the critical energy release rate. Nuerical results for syetric lainates were copared with alternative nuerical solutions and eperiental results. The solution was etended to the analysis of non-syetric lainates under tension, and then to the analysis of general lainates subected to both tension and shearing. This provided new nuerical solutions that are hardly found in the literature. Fro the new results, it was found that shearing had significant effect on the cracking process. It is noted that in this work the transverse cracking process was siplified as a crack density increent in a uniforly spaced state, while the nature of the crack ultiplication in reality is stochastic. Though good coparisons with test results have been observed for the globe relationship between the applied stresses and crack density, a statistical approach should be resorted to odeling transverse cracking in future work in order to gain deeper understanding of the cracking process. References Bailey J E and Parvii A 1981 On fiber debonding effects and the echanis of tranverse-ply failure in cross-ply lainates of glass fiber/theroset coposites Journal of Materials Science Dvorak G J and Laws N 1987 Analysis of progressive atri cracking in coposite lainates. II. First ply failure Journal of oposite Materials Fan J and Zhang J 1993 In-situ daage evolution and icro/acro transition for lainated coposites oposites Science and Technology Flaggs D L 1985 Prediction of tensile atri failure in coposite lainates Journal of oposite Materials

19 Flaggs D L and Kural M H 198 Eperiental deterination of the in situ transverse laina strength in graphite/epo lainates Journal of oposite Materials Garrett K W and Bailey J E 1977 Multiple transverse fracture in 9 o ; cross-ply lainates of a glass fibre-reinforced polyester Journal of Materials Science Herakovich,.T., Mechanics of fibrous coposites. Wiley, New York. Joffe R, Krasnikovs A and Varna J 1 OD-based siulation of transverse cracking and stiffness reduction in [θ/9 n ] s lainates oposites Science and Technology Laws N and Dvorak G J 1988 Progressive transverse cracking in coposite lainates Journal of oposite Materials 9-16 Leblond P, El Mahi A and Berthelot J M 1996 D and 3D nuerical odels of transverse cracking in cross-ply lainates oposites Science and Technology Li S-H and Li S 5 Energy release rates for transverse cracking and delainations induced by transverse cracks in lainated coposites oposites Part A Applied Science and Manufacturing Liu S and Nairn J A 199 The foration and propagation of atri icrocracks in cross-ply lainates during static loading Journal of Reinforced Plastics and oposites Mcartney L N 1998 Predicting transverse crack foration in cross-ply lainates oposites Science and Technology Mcartney L N Model to predict effects of triaial loading on ply cracking in general syetric lainates oposites Science and Technology 6 55 Mcartney L N 4 Effect of ied ode loading on ply crack developent in lainated coposites: theory and application. Nat. Phys. Lab., Teddington, UK p iv Mcartney L N 5 Energy-based prediction of progressive ply cracking and strength of general syetric lainates using an hoogenisation ethod oposites Part A Applied Science and Manufacturing Nairn J A 1989 The strain energy release rate of coposite icrocracking: a variational approach Journal of oposite Materials Nairn J A oprehensive oposite Materials, ed Z Anthony Kelly and arl Oford: Pergaon p 43 Nairn J A and Hu S 199 The foration and effect of outer-ply icrocracks in crossply lainates: a variational approach Engineering Fracture Mechanics

20 Nairn J A, Hu S and Bark J S 1993 A critical evaluation of theories for predicting icrocracking in coposite lainates Journal of Materials Science Parvii A, Garrett K W and Bailey J E 1978 onstrained cracking in glass fibrereinforced epo cross-ply lainates Journal of Materials Science Sirivedin S, Fenner D N, Nath R B and Galiotis 6 Viscoplastic finite eleent analysis of atri crack propagation in odel continuous-carbon fibre/epo coposites oposites Part A: Applied Science and Manufacturing Sith P A and Ogin S L 1999 On transverse atri cracking in cross-ply lainates loaded in siple bending oposites - Part A: Applied Science and Manufacturing Soldatos, K.P. and Ye, J.Q Wave propagation in anisotropic lainated hollow cylinders of infinite etent. Journal of Acoustics Society of Aerica. 965, Part I, Wang A S D, Kishore N N and Li A 1985 rack developent in graphite-epo cross-ply lainates under uniaial tension oposites Science and Technology 4 1 Ye J Q Lainated oposite Plates and shells: 3D Modelling London: Springer Ye J Q and Sheng H Y 3 Free-edge effect in cross-ply lainated hollow cylinders subected to aisyetric transverse loads International Journal of Mechanical Sciences Ye J Q, Sheng H Y and Qin Q H 4 A state space finite eleent for lainated coposites with free edges and subected to transverse and in-plane loads. oputers and Structures 815/ Ye J Q and Soldatos K P 1994a Three-diensional stress analysis of orthotropic and cross-ply lainated hollow cylinders and cylindrical panels oputer Methods in Applied Mechanics and Engineering Ye J Q and Soldatos K P 1994b Three-diensional vibration of lainated cylinders and cylindrical panels with syetric or antisyetric cross-ply lay-up oposites Engineering Ye J Q and Soldatos K P 1995 Three-diensional buckling analysis of lainated coposite hollow cylinders and cylindrical panels International Journal of Solids and Structures Ye J Q and Soldatos K P 1996 Three-diensional vibration of lainated coposite plates and cylindrical panels with arbitrarily located lateral surfaces point supports International Journal of Mechanical Sciences

21 Yokoeki T, Aoki T and Ishikawa T Transverse crack propagation in the specien width direction of FRP lainates under static tensile loadings Journal of oposite Materials Yokoeki T, Aoki T and Ishikawa T 5 onsecutive atri cracking in contiguous plies of coposite lainates International Journal of Solids and Structures Zhang J, Fan J and Soutis 199 Analysis of ultiple atri cracking in [±θ /9 n ] s coposite lainates. 1. In-plane stiffness properties oposites Zhang D, Ye J Q and La D 7a Free-edge and Ply racking Effect in Angle-ply Lainated oposites subected to In-plane Loads Journal of Engineering Mechanics, Transactions ASE Zhang D, Ye J Q and La D 7b Properties degradation induced by transverse cracks in general syetric lainates International Journal of Solids and Structure

22 Type Table 1 Material properties and diensions Material 1 Material Material 3 Fiberite 934/T3 Glass/epo Graphite/epo E L 18 GPa GPa GPa E T 7. GPa 1.76 GPa 9.58 GPa ν LT ν TT G LT 4. GPa 5.8 GPa 4.97 GPa G TT.4 GPa 4.49 GPa 3.37 GPa / o / o N/A / o / o N/A ΔT -15 o -15 o o G c 69 J/ 61 J/ 9 J/ L e H ply

23 ε L Y Z, 3 1 X ε h Fig. 1 Noenclature of an off-ais laina.

24 F ε Y S H θ S 9 n φ s F X Z ε L Fig. Scheatic view of a [θ /9 n /φ s ] lainate with an array of transverse ply cracks in 9 n layers. 3

25 X θ Z 9 n H φ s L Fig. 3 A representative volue eleent of a [θ /9 n /φ s ] lainate with ply cracks in 9 n layers. 4

26 L e a H c New crack b c Fig. 4 Noenclature of the propagation process of transverse cracks and the idealised unifor distribution state. 5

27 Initial load Stresses within a representative eleent Decrease the load opute G Eq. 16 Increase the load G>G c opare G with Gc G<G c G=G c Output the critical cracking load Fig. 5. Flowchart of calculating critical cracking load for a given crack density. 6

28 1.8 rack Density 1/ ` Present. Variational Method Eperiental Data Applied Average Stress MPa rack Density 1/.6.4 ` Present. Variational Method Eperiental Data Applied Average Stress MPa a Fig. 6 Dependence of crack density on the applied average stress in a Fiberite 934/T3 [ /9 ] s lainate with transverse cracks. b Fiberite 934/T3 [ /9 4 ] s lainate with transverse cracks b 7

29 .6.6 rack Density 1/ ` Present.1 Monte-arlo Siulation Eperiental Data Applied Average Stress MPa rack Density 1/ ` Present.1 Monte-arlo Siulation Eperiental Data Applied Average Stress MPa a Fig. 7 Dependence of crack density on the applied average stress in a [±15 /9 4 ] s glass/epo lainate with transverse cracks. b [±3 /9 4 ] s glass/epo lainate with transverse cracks. b 8

30 rack Density 1/ ` Present rack Density 1/ ` Present Applied Average Stress MPa a Applied Average Stress MPa b Fig. 8. Dependence of crack density on the applied average stress in a [±3 /9 4 / 4 / 3 ] glass/epo lainate with transverse cracks; b [3 /9 /3 /9 ] graphite/epo lainate with transverse cracks. 9

31 1.5 =-1MPa rack Density 1/ =-5MPa ` =MPa =5MPa =1MPa Applied Average Stress MPa Fig.9. Dependence of crack density on the applied average stress and shear stresses in a [3 /9 /9 /3 ] graphite/epo lainate. 3

32 1.5 rack Density 1/ =-1MPa =-5MPa ` =MPa =5MPa =1MPa Applied Average Stress MPa Fig.1. Dependence of crack density on the applied average stress and shear stresses in a [3 /9 /3 /9 ] graphite/epo lainate. 31