Stiffness Reduction and Fracture Evolution of Oblique Matrix Cracks in Composite Laminates

Transcription

1 Stiffness Redution and Frature Evolution of Oblique Matri Craks in Composite Laminates Daniel H. Cortes and Ever J. Barbero Mehanial and Aerospae Engineering, West Virginia Universit, Morgantown, WV Abstrat A onstitutive model to predit the onset and evolution of matri raking and the subsequent stiffness redution is derived analtiall. The formulation is valid for smmetri laminates with otherwise arbitrar staking sequene and matri raks in one or two diretions. The proposed model alulates the redution of the mehanial properties of the damaged laminate as funtion of rak densities. The onset and evolution of matri raks are predited b the model in terms of undamaged lamina properties and the ritial strain energ release rates for modes I and II (G IC and G IIC ). Therefore, there is not need for postulating damage evolution funtions and no need for empiriall adjusting the assoiated material parameters. The model formulation was speialized for the partiular ase of unidiretional loading. Comparison with eperimental data showed an eellent predition of rak initiation and evolution for a variet of laminate staking sequenes. The ombination of onstitutive and damage evolution equations formed an integrated, mehanisti damage model with no adjustable parameters. ke words: Composites, Matri Craks, Stiffness Redution, Damage Evolution. Introdution Laminate omposite materials are prone to variet of failure modes. However, matri raking parallel to the fiber diretion on off-ais plies is usuall the first damage mode observed for in-plane loading. Although matri raking is not a atastrophi failure mode, its presene triggers the initiation of other damage modes suh as delamination, fiber-matri debonding, and fiber breakage that usuall lead to frature. Furthermore, matri raks drastiall inrease the permeabilit of the material, thus allowing aess to liquid and gas ontaminants that ma degrade the fibers and the fiber-matri interfae. Also, it failitates leakage of liquid and gas that would otherwise be ontained in tanks, pressure vessels, and other similar strutures. In addition, matri raks lead to stiffness redution and stress redistribution to adjaent laminae. Therefore, the understanding and predition of matri raking has held onsiderable attention in the literature. Graduate Researh Assistant Professor and Chairman, orresponding author. The original publiation is available at :

2 Matri ranking is usuall observed when a tensile stress perpendiular to the fibers is present. Usuall, unidiretional loading of ross-pl laminates produes matri raking of the 90 pl when the load is applied in 0 diretion. However this not the onl ase. Varna et al. (999) found in eperiments for balaned [0/ 4 /0 / ]s laminates that matri raks appear for angles as low as = 40. Yokozeki et al. (005a, 005b) observed matri raking for pl angles as low as 30 for unbalaned [0//90]s laminates. In addition to this, the showed matri raking initiation due to matri raks in adjaent plies. Other eperimental observations an be found in several reviews (Nairn 000; Nairn and Hu 004). Continuum Damage Mehanis (CDM) is a ommon methodolog for the predition of the stiffness redution in the presene of one or several damage modes. In CDM, damage is represented b internal state variables. Evolution equations for these variables are postulated in terms of additional parameters, whih must be adjusted for eah material and eah onfiguration using eperiments. A review on this methodolog an found on Talreja (994) and Barbero (007), with more reent publiations inluded in (Miami, et al. 007a; Li, et al. 998; Barbero, et al. 005; Barbero and DeVivo 00; Barbero and Lonetti 00, 00; Lonetti, et al. 003; Miami, et al. 007b; Shueker and Pettermann 006). A disadvantage of CDM is the diffiult in obtaining the model parameters from eperimental data. This issue have been partiall solved b ombining CDM with other tehniques suh as miromehanis (Lundmark and Varna 005; Singh and Talreja 008; Varna et al. 00b; Varna et al. 00a). Other methods inlude the use of frature mehanis (Joffe et al. 00) and in-situ damage effetive funtions (Kashtalan and Soutis 00). Another approah is the analtial solution of the stress and strains in a representative volume ell (Nuismer and Tan 988; Yokozeki and Aoki 005), this method is partiularl interesting beause the material properties of the damaged laminate depend elusivel on the rak densit and no additional parameters or funtions are needed. However, the solutions presented in (Nuismer and Tan 988; Yokozeki and Aoki 005) are limited to ver partiular ases. In (Nuismer and Tan 988) the Laminate Staking Sequene (LSS) is restrited to [0/90 n ]s where onl the 90-lamina an damage. In (Yokozeki and Aoki 005) the LSS is restrited to [ n / m ]s with raks on both diretions. One the mehanial behavior of damaged laminates is modeled b a onstitutive equation, a damage evolution equation has to be defined. Several failure riteria have been proposed for the raking initiation (Camanho, et al. 006; Davila, et al. 005; Kashtalan and Soutis 007) based on strength models and energ balane methods. Other studies were foused on the matri raking evolution as a funtion of the applied load (Katerelos, et al. 007; Li, et al. 998; Tan and Nuismer 989; Zhang, et al. 99). Failure riteria are usuall based on stress, strain, or strain energ present in the laminate during loading. Therefore, the predition of damage evolution depends on how auratel the elasti properties of the raked plate an be predited. In this stud, an analtial model is proposed to predit both stiffness redution and matri raking evolution for smmetri laminates with arbitrar staking sequene and raks in one or two diretions. The proposed model alulates the redution of the

3 mehanial properties due to matri raking as a funtion of rak densities onl. Although previous studies have also proposed similar onstitutive relations for the stiffness redution, in this stud the proedure has been generalized to onsider arbitrar staking sequenes. Additionall, a damage evolution formulation, whih allows predition of the rak densities as a funtion of the applied load is derived. An important harateristi of the proposed model is that the rak evolution is determined based on the ritial values of the strain energ release rate for modes I and II (G IC and G IIC ). Therefore, there is not need for postulating damage evolution funtions and no need for empiriall adjusting the assoiated material parameters. Consequentl, a omplete damage model is obtained b ombining the onstitutive relations and the damage evolution formulation. Stiffness Redution In this setion the overall stiffness redution of a laminated omposited is studied analtiall. The analsis presented here is valid for a smmetri omposite with a general LSS and matri raks parallel to the fibers, oriented at one or two off-ais angles. A Repetitive Unit Cell (RUC) is hosen as the volume enlosed b the laminate mid surfae, laminate top surfae and b the parallelogram formed b the rossing of two suessive raks oriented at angles and (Fig. ). The lengths l and l of the sides of parallelograms depend on the rak densities and for raks at angles and, respetivel, as follows l sin( ) (a) l sin( ) (b) Orthogonal and oblique global oordinate sstems, respetivel denoted b -Y-Z and - -z are used for the present analsis. The and aes for the oblique oordinate sstem are aligned with the two set of matri raks (Fig. ). When onl one rak orientation is onsidered, one in-plane ais is aligned with the raks and the other in-plane ais an be arbitraril oriented. Two oordinate transformations (ovariant and ontravariant) an be defined between the orthogonal and oblique oordinate sstems for seond-order tensors (Fung 965) T A = TAT (ovariant) (a) T A = T AT (ontravariant) (b) os sin 0 T os sin 0. () 0 0 In order to obtain phsiall valid epressions, displaements and strains transformations were hosen to be ovariant, and fores and stresses transformations to be ontravariant. Transformation of the mehanial properties to the oblique oordinate sstem an be done using Voigt notation for strains and stresses (see Appendi A). Hereinafter, the proedure to obtain the stiffness redution due to matri raks is presented in the oblique oordinate sstem. The following assumptions were adopted. Due to the stress-free onditions on the top and bottom surfaes of a thin laminate we have 3

4 () i z 0, (3) where (i) denotes the i th lamina. Due to smmetr of the LSS we have ( i) ( i) w w 0, (4) where w (i) is displaement in the z diretion of the i th lamina. Finall, a linear variation of shear stress in the z diretion is assumed i, i z z ( i) i, i i, i i, i z z z z () i h (5a) i, i z z ( i) i, i i, i i, i () i h (5b), where ii is the shear stress at the interfae between the i th and i+ th plies, and i, i is the value of the z oordinate at the interfae between the i- th and i th laminae. Cruial to the analsis is finding averaged displaements in the laminate for a given in-plane loading ondition. An average of the mehanial parameters aross the lamina is defined as ( i) ( i) dz () i () i h, (6) h where an over-line denotes an averaged quantit. Notie that the average shown in Eq. (6) is for the pl (i) and should not be onfused with the average for the entire laminate whih will be denoted b. The overall redued stiffness properties an be obtained appling unit normal and shear loads and alulating the indued deformations. The analsis begins b stating the fundamental equations (Hooke s law and equilibrium equations) in terms of averaged quantities. The Hooke s law for in-plane stresses an be written in terms of averaged in-plane displaements, after appling ondition (4), as ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) Q u Q v Q u v,, 6,, Q u Q v Q u v ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i),, 6,, Q u Q v Q u v ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) 6, 6, 66,, where ( ), and ( ), represent partial derivatives with respet to and diretions, () respetivel, and Q i are the stiffness matri of a lamina in the oblique oordinate ij sstem. The transformation of the properties from orthogonal to oblique oordinate sstem is desribed in Appendi A. Hooke s law for out-of-plane shear strains and stresses an be epressed in terms of interfae shear stresses and averaged displaements b taking a weighted average of these equations (see Appendi B). These equations are usuall alled shear lag equations, and the are written as follows. ( i) ( i) ( i) ( i) i, i ( i) ( i) ( i) ( i) i, i u u h S45 S55 h S45 S55 h S45 S55 ( i) ( i) i, i i, i v v 6 S44 S 45 z 3 S44 S 45 3 S44 S 45 z () i () i h S45 S55 6 S44 S 45 ii, ii, z z (7) 4

5 (8) where S 44, S 45 and S 55 are the omponents of the out-of-plane shear ompliane matri in oblique oordinates (Appendi A). Finall, the equilibrium equations in the and diretion an be written as ( i ) ( i ) i, i i, i ( i ) (9a),, z z h 0 (9b) ( i ) ( i ) i, i i, i ( i ),, h 0 Replaing equations (7) and (8) into equations (9) leads to a seond-order sstem of partial differential equations in terms of the pl-averaged displaements. The analtial solution of that sstem an be epressed as () u a b () a b u ( n) u an bn sinh sinh () v a n b n (0) () v an bn ( n) v an bn where n is half the number of plies in the laminate;,, are onstant deformations that appear during the integration proedure; and a i, b i, and are unknown onstants with lak of phsial meaning. Replaing (0) into (9) leads to two eigenvalue sstems where a i - and b i - are the pairs of eigenvetors and eigenvalues for eah sstem. The final solution is the linear ombination of all partiular solutions, as follows () u a b () a b u ( n) n n u an bn A sinh () j j B j sinh j v j a n j b n () () v an bn ( n) v an b j n j where A j, B j, and are unknown quantities. However, two i and two i eigenvalues are equal to zero, so the number of unknowns is redued to 4n-. To find the values of these onstants, the following boundar onditions are used. First, stress-free onditions are assumed at the surfaes of the raks l ( ) d 0 l, at = l (a) l l ( ) d 0 l, at = l (b) l 5

6 l ( ) d 0 l, at = l l () l ( ) d 0 l, at = l l (d) where,, are the labels for laminae with raks along and diretions, respetivel (Figure ). Equation () represents a set of N N independent onditions, where N and N are the number of - and -laminae, respetivel. Seond, the stresses in the unraked surfaes are related to the eternal loads b nn ( u) h l ( u) d h l l u, at = l (3a) h nn ( u) l ( u) d h l u l, at = l (3b) nn u h l ( u) d h l u l, at = l (3) where, and are the omponents of the eternal load applied to the laminate; and u and u are the labels for unraked pl surfaes normal to and diretions, respetivel. Three independent onditions are obtained from Eq. (3). Finall, the displaements in the and diretions are assumed to be the same for all unraked pl surfaes l ( ) l ur ( u) u d u d l 0 0 l at = l u ur (4a) l ( ) l ur ( u) v d v d l 0 0 l at = l u ur (4b) l ( ) l ur ( u) u d u d l 0 0 l at = l u ur (4) l ( ) l ur ( u) v d v d l 0 0 l at = l u ur (4d) where (ur) and (ur) are two unraked laminae taken as referene. From Eqs. (4a) n N ; and (4b), the number of independent onditions that an be obtained is on the other hand, n N onditions are obtained from Eqs. (4) and (4d). Equations ()-(4) represent the set of 4n- boundar onditions, whih allows us to alulate all the unknowns. Then, the average strains an be alulated as l l ( ) u, dd 4ll l l l l ( ) v, dd 4ll l l l l ( ) ( ) v, u, dd 4ll l l (5a) (5b) (5) 6

7 To obtain the ompliane matri S of the raked omposite laminate, three unit-load ases are onsidered ( a) ( b) ; and 0 ; (6) 0 0 and the deformations obtained for eah ase are the omponents of the inreased ompliane matri S in the oblique oordinate sstem ( a) ( b) ( ) S (7) The ompliane matri in the orthogonal oordinate sstem, S, is obtained using the oordinate transformation T S T ST, T os sin os sin (8) os sin os sin ossin os sin os sin ossin The overall elasti properties for the raked laminate an be written in terms of the omponents of the matri S as S E, E Y, vy and G Y (9) S S S S33 A ver important harateristi of the proedure presented here is that properties of the damaged lamina are funtion onl of the original elasti properties and the rak densities and. Therefore, there is not need for defining damage evolution funtions in terms of additional, thus no need to adjust suh parameters using additional eperimental data. Frature Evolution Analsis In the previous setion, the elasti properties of the damaged laminate were alulated as a funtion of the rak densities. In this setion the strain Energ Release Rate (ERR) and a damage evolution analsis are formulated based on the results presented above. The ERR assoiated with Mode I and Mode II an be epressed as U I GI (0a) A U II GII (0b) A where U I and U II are the strain energ for Mode I and Mode II, respetivel; and A is the rak area. The rak area of the - and -laminae depend on the rak densities λ and λ, respetivel. For -raks, G I and G II an be epressed in terms of the rak densit as U I GI, (a) A ( ) 7

8 G II U II. (b) A where A is the area of raks in the -laminae and U I and U II are the Mode I and Mode II strain energies assoiated with the -laminae. For -raks, an epression similar to Eq. () an be easil obtained in terms of. For eample, onsidering - raks, the strain energ omponents U I and U II an be epressed in terms of the average (applied) stresses and strains in material oordinates as V u GI (a) A V u GII (b) A where V u is the volume of the unit ell. The stresses and strains in the material oordinates an be obtained b transformation from orthogonal oordinates os sin os sin sin os os sin Y, (3a) os sin os sin os sin Y os sin os sin sin os os sin Y, (3b) os sin os sin os sin Y where, Y, Y,, Y and Y are the omponents of the applied stresses and strains in the global (orthogonal) oordinates. In order to obtain a relation between the eternal loading and the rak densit, a failure riterion must be applied. Several failure riteria in terms of ERR have been proposed for omposite materials. In this stud, the riterion proposed b Hahn (Hahn 983) was hosen. It an be epressed as GI GI GII ( r) r, (4) G G G IC IC IIC where r GIC GIIC and G IC and G IIC are the ritial values of the ERR for Mode I and II. The proedure shown in Eqs. (0)-(4) an be used to predit the rak evolution on smmetri laminates with otherwise arbitrar LSS under general loading. The proedure is illustrated first for the partiular ase of balaned laminates and later for unbalaned laminates. For balaned laminates the raked plies will have opposite angles ( ). Additionall, if onl aial loading is applied, the applied stress and the average strains redue to E Y 0, (5a) Y 0 8

9 Y Y. (5b) Y 0 Notie that the stresses and strains in material oordinates (Eq. (3)) for the and - plies will have the same magnitude; therefore, it is epeted that the and - plies develop the same rak densit ( ). The ERR for mode I and II an be obtained b substituting Eqs. (3) and (5) into Eq. () Vu E 4 4 vy GI os vy sin os sin E sin os A (6a) Vu E vy GII vy sin os E sin os. (6b) A The normalized ERR for Modes I and II an be defined as G I GI (, ) (7a) GII GII (, ). (7b) A frature evolution equation an be obtained replaing Eq. (7) into Eq. (4) as follows G I GI G II r r (8) GIC GIC GIIC Equation (8) represents the relation between the aial strain ( ) and the rak densit on the -laminae. This equation is valid for smmetri balaned laminates under aial loading. Similar relations an be obtained for other failure riteria. Another ase of interest is the aial loading of unbalaned laminates ( ). In this ase, the average (applied) stresses and strain an be epressed as E Y 0, (9a) Y 0 Y Y. (9b) Y Y _ The ERR for modes I and II for the -laminae is obtained using Eqs. (), (3) and (9) as follows Vu E 4 4 GI os os sin Y sin os sin os sin Y _ E os sin os sin os sin Y Y _ A 9

10 G V E u II os sin os sin Y os sin (os sin ) Y _ Y Y _ E os sin os sin (os sin ). A (30a) (30b) The rak evolution relation for the -lamina is obtained substituting Eq. (30) into Eq. (4) as follows G G G I I II ( r) r GIC GIC GIIC The same proedure an be followed in order to obtain evolution of the -laminae in terms of as follows G G G I I II ( r) r GIC GIC GIIC G I and (3) G II and the rak Equations (3) and (3) desribe the rak evolution for unbalaned laminates under unidiretional loading. Notie that Eqs. (3) and (3) must be satisfied simultaneousl. Multiple N laminae with thikness t and angle as well as multiple N laminae with thikness t and angle an be handled easil b appling N + N boundar onditions Eq () and taking into aount that -laminae and -laminae have rak densities and, respetivel. Results and Disussion In this setion, omparisons between results obtained using the proposed formulation and eperimental data reported in the literature are presented. The mehanial properties of the laminate as a funtion of rak densit for several LSS are presented and ompared with eperiments. Finall, omparisons of damage evolution between analtial preditions and eperimental results for several LSS are presented. To validate the predition of redued mehanial properties due to the presene of obliquel-rossed matri raks, a omparison between analtial results from this formulation and that presented b Yokozeki and Aoki (005) is presented in Fig.. For this omparison a [45] S IM600/#33 omposite was seleted. The material properties an be found in Table. The hange of the Young s modulus and the Poisson s ratio was obtained as funtion of the rak densit of the = 45 o lamina; the rak densit of the = -45 o lamina was kept onstant and equal to rak/mm just to reprodue eatl the partiular results presented b Yokozeki and Aoki (005). A perfet math is observed between the two formulations. This was epeted sine the formulation presented here redues to theirs for [ m / n ] S laminates. Varna, et al. (999) reported a series of eperiments for unidiretional loading of [0/ 4 /0 / ] S E-Glass/Epo laminates. The material properties for the plies an be found (3) 0

11 on Table. A omparison of normalized elasti modulus E and Poisson s ratio v Y for = 90, 70 and 55 is presented in Figs The eperimental data for = 55 reported b Varna et al. (999) displas an offset from epeted behavior. The authors of that stud believed that shear damage was the ause of this offset. In this stud, the data for the 55 ase (Fig. 5) was shifted b a onstant value, so that the normalized values of E and v Y approah one when the rak densit approahes zero. In general, good agreement was observed between the eperiments and analtial results. The predition of the properties for = 70 was aeptable, but not as good as those for 90 and 55. It is observed that the predited values of v Y for = 55 inrease with rak densit as epeted. Yokozeki and Aoki (005) also reported an inrease of Poisson s ratio for [] S laminates for values of lower that 60. An important harateristi of the formulation presented here is that the predition is funtion onl of the rak densit. Nuismer and Tan (988) and Yokozeki and Aoki (005) also presented formulations whih are funtion onl of the rak densit. However, those formulations are limited [ m,90 n ] S and [ m / n ] S LSS, respetivel. Thus, those are speial ases of the formulation presented here for arbitrar LSS. Varna et al. (999) and Singh and Talreja (008) presented formulations for the predition of the elasti properties of damaged laminates. Their methodolog inluded finding several parameters using numerial and/or eperimental results. In addition, their methods produed linear preditions for [0/ 4 /0 / ] S omposites, whih ma be inappropriate for large values of rak densities. It an be observed from the preditions of the formulation presented here that the values of E and v Y derease asmptotiall to a given value. This behavior is epeted sine the undamaging laminae eventuall take the entire load. Linear preditions do not predit this epeted behavior. Comparison for the damage evolution (rak densit vs. applied strain) between this formulation and eperimental data reported b Varna et al. (999) is presented in Figs The values of G IC and G IIC (75 J/m and 500 J/m, respetivel) were taken from the literature for eperimental results for Glass Fiber/Epo omposites (Rikards et al. 998). A ver good predition of the damage initiation and evolution (rak densit vs. strain) is obtained for all the ases. However, the predition of the onset strain for the ase of 70 was not as good as those for the other angles. It is believed that this ma be aused b numerial inauraies whih were tpiall observed at small rak densities in laminates with large angles between the oblique oordinate aes. The damage evolution for balaned laminates an be desribed b onl one value of the rak densit; whih orresponds to the 90 or laminae. However, when the laminate is not balaned, eah lamina ma have its own raking onset strain and rak densit depending on the lamina angle and the applied load. An eample of this behavior an be observed in Fig. 9, where an aial load was applied to a [0/-45 4 /55 4 /0 / ] S laminate with the materials properties of Varna et al. (999) (Table ). It an be observed that matri raks appear first in the 55 lamina at a strain of.8%. This was epeted sine the 55 o lamina has the largest tensile stress perpendiular to the fibers for aial loading. The

12 rak densit of 55 o lamina inreases until the -45 lamina starts raking, at a strain of.64%. After this point, the rak densities in the two laminae grow at different rates. Conlusion In this stud, a model to predit material properties and damage evolution for smmetri laminates is presented. The formulation is valid for laminates with an arbitrar staking sequene and matri raks in one or two diretions. An important harateristi of this formulation is that hanges on the elasti properties are predited as a funtion of the rak densities onl. Another important feature is that the evolution of damage is predited using a strain energ release rate riterion without needing additional eperimentall-adjusted parameters. Good omparison between the preditions and eperiments is observed. Appendi A The ovariant transformation of the strain tensor an be epressed in matri form as z os sin 0 Y Z os sin 0 os sin 0 Y YZ os sin 0 z z 0 0 Z YZ 0 0 Equation (A) an be rewritten using Voigt notation as os sin os sin os sin os sin Y z Z sin os 0 YZ sin os 0 z Z sinos os os sin sin Y ossin The ontravariant transformation for the stresses an be epressed as T (A) (A) os sin os sin os sin os sin Y z Z sin os 0.(A3) YZ sin os 0 z Z sinos os os sin sin Y ossin Equations (A)-(A3) are appliable to lamina-averaged, loal or global stress and strain tensors. B means of Eqs. (A) and (A3), the in-plane stiffness and ompliane matries for a lamina in the oblique oordinate sstem an be alulated, respetivel, as follows T ˆ Q T QT, (A4) ˆ T S TST, (A5) T

13 where ˆQ and Ŝ are the in-plane stiffness and ompliane in the material (orthogonal) oordinate sstem of the undamaged lamina and T os sin os sin os sin os sin ossin os sin os sin ossin In a similar wa, the out-of-plane shear stiffness and ompliane matries in the oblique oordinate sstem an be obtained T Q44 Q ˆ ˆ 45 os sin Q44 Q 45 os sin Q ˆ ˆ 45 Q 55 os sin Q os sin 45 Q (A6) 55 T S44 S45 os sin Sˆ ˆ 44 S 45 os sin S ˆ ˆ 45 S 55 os sin S os sin 45 S. (A7) 55 Appendi B The relationship between the out-of-plane shear stresses and the averaged displaements an be obtained b alulating a weighted average of in-plane deformation. Using Hooke s law, the shear strains an be epressed in terms of the shear stresses as ( i) ( i) ( i) ( i) ( i) S44 S45 z. (B) ( i) ( i) ( i) ( i) ( i) S S z z Equation (B) an be written in terms of displaements, and making use of Eqs. (3-4) the following equation is obtained 0 ( i) ( i) ( i) i, i i, i i, i hi z ( i) i, i i, i i, i hi z u, z w, S45 S ( i) 55 z z z ( i) h h (B) 0 ( i) ( i) ( i) i, i i, i i, i hi z ( i) i, i i, i i, i hi z v, z w, S44 S ( i) 45 z z z ( i) h h The following weighted average is defined hi () i hi zdz (B3) hi h Appling the operator shown in Eq. (B3) on both sides of Eq. (B), the following relationship is obtained ( i) ( i) ( i) ( i) ( i) h ( i) i, i h ( i) i, i h ( i) i, i h ( i) i, i u hi u S45 S45 S55 z S55 z (B4) ( i) ( i) ( i) ( i) ( i) h ( i) i, i h ( i) i, i h ( i) i, i h ( i) i, i vhi v S44 S44 S45 z S45 z where u(h i- ) and v(h i- ) are the in-plane displaements evaluated at z = h i-. Following a similar proedure for pl (i-), but using (z h i- ) as a weight, the following relation is obtained 3

14 ( i) ( i) ( i) ( i) ( i) h ( i) i, i h ( i) i, i h ( i) i, i h ( i) i, i u hi u S45 S45 S55 z S55 z ( i) ( i) ( i) ( i) ( i) h ( i) i, i h ( i) i, i h ( i) i, i h ( i) i, i vhi v S44 S44 S45 z S45 z (B5) Adding Eqs (B4) and (B5) and presenting the result in a matri form we obtain ( i) ( i) ( i) ( i) i, i ( i) ( i) ( i) ( i) i, i u u h S45 S55 h S45 S55 h S45 S55 ( i) ( i) i, i i, i v v 6 S44 S 45 z 3 S44 S 45 3 S44 S 45 z h 6 () i S S S S () i ii, ii, z (B6) 4

15 Tables Table. Material properties for the laminates onsidered in stud. IM600/#33 [Yokozeki and Aoki 005] E-Glass/Epo [Varna, et al. 999] E L (GPa) E T (GPa) v LT v TT 0.45* 0.4* G LT (GPa) Pl thikness (mm) G IC (J/m ) [Rikards et al. 998] G IIC (J/m ) [Rikards et al. 998] * Assumed value. 5

16 Figures Figure. Orthogonal and oblique oordinate sstems and geometr of the Repetitive Unit Cell. 6

17 a) b) Figure. Comparison between analti results from present formulation and those presented b (Yokozeki and Aoki 005): a) Normalized Young s Modulus vs. λ, b) Laminate Poisson s Ratio vs. λ. 7

18 a) b) Figure 3. Redued properties for the [0/90 8 /0 / ]s omposite, a) Nomarlized Young s modulus ( E / E ), b) Laminate Poisson s Ratio ( v Y ). 8

19 a) b) Figure 4. Redued properties for the [0/70 4 /0 / ]s omposite, a) Nomarlized Young s modulus ( E / E ), b) Laminate Poisson s Ratio ( v Y ). 9

20 a) b) Figure 5. Redued properties for the [0/55 4 /0 / ]s omposite, a) Nomarlized Young s modulus ( E / E ), b) Laminate Poisson s Ratio ( v Y ). 0

21 Figure 6. Comparison of the matri rak evolution between analtial results and those reported b (Varna, et al. 999).

22 Figure 7. Comparison of the matri rak evolution between analtial results and those reported b (Varna, et al. 999). Figure 8. Comparison of the matri rak evolution between analtial results and those reported b (Varna, et al. 999).

23 Figure 9. Damage evolution of a [0/-45 4 /55 4 /0 / ]s laminate under aial loading. 3

24 Referenes Barbero E. J. (007) Finite Element Analsis of Composite Materials. CRC Press, Boa Raton, FL Barbero E. J., Abdelal G. F. and Caeres A. (005) A miromehanis approah for damage modeling of polmer omposites. Composite Strutures 67: Barbero E. J. and DeVivo L. (00) A onstitutive model for elasti damage in fiberreinfored PMC laminae. Journal of Damage Mehanis 0:73-93 Barbero E. J. and Lonetti P. (00) Damage model for omposites defined in terms of available data. Mehanis of Composite Materials and Strutures 8:99-36 Barbero E. J. and Lonetti P. (00) An inelasti damage model for fiber reinfored laminates. Journal of Composite Materials 36:94-96 Camanho P. P., Davila C. G., Pinho S., et al. (006) Predition of in situ strengths and matri raking in omposites under tension and in-plane shear. Composites Part A: Applied Siene and Manufaturing 37:65-76 Davila C. G., Camanho P. P. and Rose C. A. (005) Failure riteria for FRP laminates. Journal of Composite Materials 39: Fung Y. C. (965) Foundations of Solid Mehanis. Prentie-Hall, Englewood Cliffs, NJ,USA Hahn H. T. (983) A mied-mode frature riterion for omposite materials. Journal of Composites Tehnolog and Researh 5:6-9 Joffe R., krasnikovs A. and Varna J. (00) COD-based simulation of transverse raking and stiffness redution in [S/90 n ] s laminates. Composite Siene and Tehnolog 6: Kashtalan M. and Soutis C. (00) Analsis of loal delaminations in omposite laminates with angle-pl raks. International Journal of Solids and Strutures 39: Kashtalan M. and Soutis C. (007) Stiffness and frature analsis of laminated omposites with off-ais pl matri raking. Composites Part A: Applied Siene and Manufaturing 38:3-69 Katerelos D. T. G., Varna J. and Galiotis C. (007) Energ riterion for modelling damage evolution in ross-pl omposite laminates. Composite Siene and Tehnolog In press Li S., Reid S. R. and Soden P. D. (998) A otinuum damage model for transverse matri raking in laminate fibre-reinfored omposites. Philosophial Transation of the Roal Soiet of London. A 356:379-4 Lonetti P., Barbero E. J., Zinno R., et al. (003) Interlaminar damage model for polmer matri omposites. Journal of Composite Materials 37: Lundmark P. and Varna J. (005) Constitutive Relationships for Laminates with Pl Craks in In-plane Loading. International Journal of Damage Mehanis 4:35-59 Miami P., Camanho P. P., Maugo J. A., et al. (007a) A ontinuum damage model for omposite laminates: Part I - Constitutive model. Mehanis of Materials 39:

25 Miami P., Camanho P. P., Maugo J. A., et al. (007b) A ontinuum damage model for omposite laminates: Part II - Computation implementation and validation. Mehanis of Materials 39: Nairn J. (000) Matri Miroraking in Composites. In: R. Talreja and J. A. E. Manson (ed) Polmer Matri Composites, Elservier Siene, Amsterdan, pp Nairn J. and Hu S. (004) Matri Miroraking. In: R. Talreja (ed) Damage Mehanis of Composites Materials, Elsevier, Amsterdan, pp Nuismer R. J. and Tan S. C. (988) Constitutive Relations of a Craked Composite Lamina. Journal of Composite Materials :306-3 Rikards R., Buhholz F. G., Wang H., et al. (998) Investigation of mied mode I/II interlaminar frature toughness of laminated omposites b using a CTS tpe speimen. Engineering Frature Mehanis 6:35-34 Shueker C. and Pettermann H. E. (006) A ontinuum damage model for fiber reinfored laminates based on pl failure mehanisms. Composites Strutures 76:6-73 Singh C. V. and Talreja R. (008) Analsis of Multiple Off-Ais Pl Craks in Composite Laminates. International Journal of Solids and Strutures 45: Talreja R. (994) Damage Charaterization b Internal Variables. In: R. Talreja (ed) Damage Mehanis of Composite Materials, Elsevier Siene, Amsterdam, pp Tan S. C. and Nuismer R. J. (989) A Theor for Progressive Matri Craking in Composite Laminates. Journal of Composite Materials 3: Varna J., Joffe R., Akshantala N. V., et al. (999) Damage in Composite Laminates with Off-Ais Plies. Composite Siene and Tehnolog 59:39-47 Varna J., Joffe R. and Talreja R. (00a) Mied miromehanis and ontinuum damage mehanis approah to transverse raking in [S,90 n ] s laminates. Mehanis of Composite Materials 37:5-6 Varna J., Joffe R. and Talreja R. (00b) A snergisti damage-mehanis analsis of transverse raking in [ ] s laminates. Composite Siene and Tehnolog 6: Yokozeki T. and Aoki T. (005) Overall Thermoelasti Properties of Smmetri Laminates Containing Obliquel Crossed Matri Craks. Composite Siene and Tehnolog 65: Yokozeki T., Aoki T. and Ishikawa T. (005a) Conseutive Matri Craking in Contiguous Plies of Composite Laminates. International Journal of Solids and Strutures 4: Yokozeki T., Aoki T., Ogasawara T., et al. (005b) Effets of laup angle and pl thikness on matri rak interation in ontiguous plies of omposite laminates. Composites Part A: Applied Siene and Manufaturing 36:9-35 Zhang J., Fan J. and Soutis C. (99) Analsis of multiple matri raking in [ m /90 n ] s omposite laminates. Part : Development of transversel pl raks. Composites 3:9-98 5