FAILURE CRITERIA FOR FRP LAMINATES IN PLANE STRESS

Transcription

1 FAIURE RIERIA FOR FRP AINAE IN PANE RE arlos G. Dávila Aerospace Engineer, Analytical and oputational ethods Branch, enior eber, AIAA. NAA angley Research enter, Hapton, VA 368 Navin Jaunky enior cientist, National Institute of Aerospace, eber, AIAA. Hapton, VA 3666 anjib Goswai National Research ouncil Research Associate. NAA angley Research enter, Hapton, VA 368 Abstract A new set of failure criteria for fiber reinforced polyer lainates that are being developed at NAA angley Research enter is described. Derived fro Hashin s criteria for unidirectional lainates and Puck s action plane concept, the physically based ar criteria predict atrix and fiber failure accurately without requiring curve-fitting paraeters. For atrix failure under transverse copression, the fracture plane is calculated by axiizing the ohr- oulob ective stresses. A criterion for fiber kinking is obtained by calculating the fiber isalignent under load, and applying the atrix failure criterion in the coordinate frae of the isalignent. he criteria are applied to a few exaples to predict failure load envelopes and to predict the failure ode for each region of the envelope. he analysis predictions are copared to the predictions of other available failure criteria and with experiental results. Predictions with ar correlate well with the experiental results. Introduction he ai of daage echanics, the atheatical science dealing with quantitative descriptions of the physical events that alter a aterial when it is subjected to loads, is to develop a fraework that describes the aterial response caused by the evolving daage state. he greatest difficulty in the developent of an accurate and coputationally icient nuerical procedure to predict daage growth has to do with how to analyze the aterial icro-structural changes and how to relate those changes to the aterial response. everal theories have been proposed for predicting failure of coposites. While significant progress has been ade in this area, there is currently no single theory that accurately predicts failure at all levels of analysis, for all loading conditions, and for all types of fiber reinforced polyer (FRP) lainates. While soe failure theories have a physical basis, ost theories represent attepts to provide atheatical expressions that give a best fit of the available experiental data in a for that is practical fro a designer s point of view. o the structural engineer, failure criteria ust be applicable at the level of the laina, the lainate, and the structural coponent. Failure at these levels is often the consequence of an accuulation of icro-level failure events. herefore, it is also necessary to have an understanding of icro-level failure echaniss in order to develop the proper failure theories. he World Wide Failure Exercise (WWFE) conceived and conducted by Hinton and oden -7 provides a good picture of the status of currently available theoretical ethods for predicting aterial failure in fiber reinforced polyer coposites. he recently published coparison of the predictions by the WWFE participants with experiental results indicates that even when analyzing siple lainates that have been studied extensively over the past 4 years, the predictions of ost theories differ significantly fro the experiental observations. 3 he uncertainty in the prediction of initiation and progression of daage in coposites has led to the his aterial is a work of the U.. Governent and is not subject to copyright protection in the United tates.

2 undertaking of an ort at angley Research enter to revisit existing failure theories and to develop new ones where necessary. he well-established failure criteria were revisited and their prediction capabilities were copared and their qualities were exained. his paper describes a newly developed set of nonepirical criteria for predicting failure of unidirectional FRP lainates that is derived fro Puck s action plane concept. All the calculations shown here are at the laina level with plane stress assuptions. First, a new criterion for atrix shear and copression is presented. hen, a fiber kinking failure criterion for fiber copression is developed by applying the atrix failure criterion to the configuration of the kink. Finally, soe exaples of failure envelopes are presented. he predicted results are copared to the results of other available failure criteria as well as with experiental results. trength-based Failure riteria trength-based criteria are coonly used with the finite eleent ethod to predict failure events in coposite structures. A large nuber of continuubased criteria have been derived to relate internal stresses and experiental easures of aterial strength to the onset of failure. In the following sections, the Hashin criteria are briefly reviewed, and iproveents proposed by un and Puck over Hashin s theories are exained. Hashin riteria D (98) Hashin 8, 9 can be credited with establishing the need for failure criteria based on failure echaniss. In the work in 973, he used his experiental observations of tensile speciens to propose two different failure criteria, one related to fiber failure and the other related to atrix failure. he criteria assue a quadratic interaction between the tractions associated with the plane of failure. In 98, he introduced fiber and atrix failure criteria that distinguish between tension and copression failure. Given the difficulty in obtaining the plane of failure in the 3D case, Hashin developed the ore recent criteria using a quadratic interaction between stress invariants. uch derivation was based on logical reasoning rather than icroechanics. Although they were developed for unidirectional lainates, the Hashin criteria have also been applied successfully to progressive failure analyses of lainates by taking into account the constraining interactions between the plies with in-situ unidirectional strengths. he D version of the failure criteria proposed by Hashin in 973 and 98 are suarized in able. Over the past decade, nuerous studies have indicated that the stress interactions proposed by Hashin do not always fit the experiental results, especially in the case of atrix or fiber copression. It is well known, for instance, that oderate aounts of transverse copression ( <) increases the apparent shear strength of a ply, which is not well predicted by Hashin. In addition, Hashin s fiber copression criterion does not account for the ects of in-plane shear, which reduce significantly the ective copressive strength of a ply. everal authors have proposed odifications to Hashin s criteria to iprove their predictive capabilities, two of which are exained below. Iproved riteria for atrix opression un et al. proposed an epirical odification to Hashin s 973 criterion for atrix copression failure to take into account the beneficial role that copressive has on atrix shear strength. he criterion is: Y + η () where η is an experientally deterined constant and ay be regarded as an internal aterial friction paraeter. he denoinator η can be considered an ective in-plane shear strength that increases with transverse copression (see sign convention in Fig. ). As in Hashin s theories, no attept was ade to calculate the angle of the fracture plane. Fiber orientation n α Figure. Fracture of a unidirectional laina subjected to transverse copression and shear.

3 able. Hashin riteria 8, 9 for plane stress atrix Failure atrix tension, Y + Fiber Failure Fiber tension, 973: 98: atrix copression, < Y + Y + Fiber copression, < F Y F + where, Failure Index, with the subscripts and F indicating atrix and fiber failure, respectively ij coponents of noral and shear stresses, with i,j,, strength in fiber direction under tension and copression, respectively (ongitudinal strength) Y, Y strength noral to the fiber direction under tension and copression, respectively (ransverse strength) and are the in-plane and transverse shear strengths, respectively. + Puck s action plane proposal 5 represents the beneficial influence of transverse copression on atrix shear strength by increasing the shear strength by a ter proportional to the noral stress n acting at the fracture plane shown in Fig.. In this forulation, the atrix failure criterion under transverse copression is: η n + η n () In Eq., the direct contribution of has been eliinated by assuing that the initiation of fracture at the failure plane is independent of the transverse copressive strength. Internal aterial friction is characterized by the coicients η and η, which are deterined experientally. One key to Puck s proposal is the calculation of the angle of the fracture plane, α, shown in Fig.. Puck deterined that atrix failures doinated by in-plane shear occur in a plane that is noral to the ply and parallel to the fibers (α). For increasing aounts of transverse copression, the angle of the action plane α changes to about 4, and increases with copression to 53 ± for pure transverse copression. In the WWFE, Puck s predicted failure envelopes correlated very well with the test results 6. However, the Puck s phenoenological approach uses several aterial paraeters that are not physical and ay be difficult to quantify without considerable experience with a particular aterial syste. A detailed discussion on Hashin, Puck, un and any other criteria has been presented by París, who also discusses the ad hoc nature of the forulation of ost strength-based criteria. ar Plane tress riteria for atrix Failure In this section, a new set of criteria is proposed for atrix fracture based on the concepts proposed by Hashin, cobined with the fracture plane concept proposed by Puck. In the case of atrix tension, the fracture planes are noral to the plane of the plies and parallel to the fiber direction. he quadratic interaction between in-plane shear and transverse noral tension in Hashin s criterion correlates well with test results. For atrix copression, the plane of fracture ay not be noral to the ply, and Hashin was not able to calculate the angle of the plane of fracture. In the present proposal, ohr-oulob ective stresses 3 are used to calculate the angle of fracture. riterion for atrix Failure in opression ( <) he ohr-oulob (-) criterion 3 is coonly used in applications where the fracture under tension loading is different fro fracture under copression loading, such as in soil echanics or in the fracture of cast iron. he application of the - criterion to 3

4 ultiaxial failure of epoxy resins was studied by Kawabata 4 based on correlation with his own test results. While studying the failure of chopped glassfiber/epoxy at lainates under confining pressures, Boehler 5 found the sai-wu criterion to be inadequate, and forulated a shearing criterion based on the - criterion that fit his experiental easureents well. aliercio 6 used the - criterion within a nonlinear icroechanical odel to predict the acroscopic strength properties of fiber coposites. he - criterion is represented geoetrically by the diagra illustrated in Fig.. he ohr s circle represents a state of uniaxial copression he angle of the plane of fracture α is chosen here to be 53, which is a typical fracture angle for coposites [Puck 5 ]. he line AB is the tangent to ohr s circle at A and is called the oulob fracture line. he - criterion postulates that in a state of biaxial noral stress, fracture occurs for any ohr s circle that is tangent to the oulob fracture line. he ective stress is related to the stresses and n acting on the fracture plane by the expression n. In the literature, tan ( η) is called the angle of internal friction and it is assued to be a aterial constant. When η, the - criterion is equivalent to the resca condition. 3 Diandro 7 explains the role of internal friction on the strength of carbon-fiber coposites by noting the absence of cheical bonds between fiber and atrix, and that adhesion is attributed to the Van der Waal s interactions. When subjected to an external load, the shear slipping of the two phases is prevented until the shear stress at the fiber-atrix interface reaches a liiting value. Diandro also notes that η is an epirical factor that encopasses all cheical-physical interactions, including the theral residual shrinkage of the atrix around the fiber. arson 8 exained the relative ects of interfacial friction and roughness on the length of interfacial sliding which proceeds fro the tip of an ipinging fracture oriented perpendicular to the interface. According to arson, sliding is key to the cracking behavior of fibrous brittle atrix coposites in that it affects the stress concentration on the fibers, the atrix crack spacing, and, therefore, the global toughness of a coposite aterial. In general, the fracture plane can be subjected to transverse as well as in-plane stresses, in which case ective stresses ust be defined in both orthogonal directions as shown in Eq. 3. n n (3) -Y c α53 α 53 n -Y A n B η.378 Y c - tan( ) α where the ters η and η are referred to as coicients of transverse and longitudinal influence, respectively, and the operand x x if x ; otherwise x. atrix failure under copression loading is assued to result fro a quadratic interaction between the ective shear stresses acting on the faces of a fracture plane. he failure index () is written as + (4) Figure. ohr s circle for uniaxial copression and the ective transverse shear. where and are the transverse and in-plane shear strengths, respectively. he stress coponents acting on the fracture plane can be expressed in ters of the in-plane stresses and the angle of the fracture plane, α (see Fig. ). 4

5 cos α n sinα cosα cosα (5) Using Eqs. 3 and 5, the ective stresses for an angle of the fracture plane α between and 9 are cosα (sinα η cosα ) cosα ( cosα ) alculation of oicients η and η and trength (6) he coicients of influence η and η are obtained fro the case of uniaxial transverse copression ( <, ). At failure, the in-plane copressive stress is equal to the atrix copressive strength, -Y. he ective transverse shear stress at failure is Y cosα ( sin α η cosα) (7) Under uniaxial transverse copression, fracture occurs at a fracture angle α that axiizes the ective transverse shear. aking the derivative of the transverse shear stress, Eq. 7, with respect to α gives α sin α cos α + cos α sin ( sin α ( cos α α olving Eq. 8 for η gives η η sin α cos α ) sin α ) cos α (8) (9) tan α Puck 5 deterined that when they are loaded in transverse copression, ost unidirectional graphite/epoxy coposites fail by transverse shear along a fracture plane oriented at α 53 ±. herefore, the coicient of transverse influence is in the range. η.36. Note that if the fracture plane were oriented at α 45, the coicient of transverse influence would be equal to zero. he transverse shear strength is a quantity that is difficult to easure experientally. However, substituting Eq. 8 into Eq. 7 gives an expression relating the transverse shear strength to the transverse copressive strength: cosα Y cos α ( sin α + ) () tan α For a typical fracture angle of 53,.378 Y, as was shown in Fig.. Note that in soe textbooks, the transverse shear strength is often approxiated as.5 Y, which iplies fro Eqs. 7 and 9 that the fracture plane is at α 45 and that η. Also note that with this approxiation, Hashin s 98 D criterion for atrix failure in copression becoes identical to his 973 criterion. he coicient of longitudinal influence, η, can be deterined fro shear tests with varying degrees of transverse copression. In the absence of biaxial test data, η can be estiated fro the in-plane and transverse shear strengths, as proposed by Puck: η η η η Y Y cosα cos α cos ( tan α sin α + cosα ) α Deterination of the Angle of the FracturePlane () he angle of the fracture plane for a unidirectional lainate loaded in transverse copression is a aterial property that is easily obtained fro experiental data. However, under cobined loads, the angle of the fracture plane is unknown. he correct angle of the fracture plane is the one that axiizes the failure index,, in Eq. 4. In the present work, the fracture angle is obtained by searching for the axiu of the failure index (Eq. 4) within a loop over the range of possible fracture angles: <α <α o. A graphical representation of atrix failure envelopes at various fracture angles is shown in Fig. 3 for a unidirectional E-Glass/Y556 coposite subjected to transverse copression and inplane shear. As seen in the figure, the fracture angle that axiizes the for sall transverse stresses is α. When the applied transverse stress has a agnitude equal to approxiately /3 of the transverse copressive strength, Y, the angle of the critical frac- 5

6 ture plane switches fro α to α4, and then rapidly increases to αα, the angle of fracture for uniaxial transverse copression α3 α α5 α45 α , Pa, Pa Figure 3. atrix failure envelopes for a typical unidirectional glass/e. laina subjected to in-plane copression and shear loading. riterion for atrix Failure in ension > Hashin s criterion for atrix tension failure provides adequate predictions of the interaction of in-plane shear and transverse tension for atrix failure. herefore, the ar criterion for atrix failure in tension is identical to the criterion proposed by Hashin and is written as Y + ar riteria for Fiber Failure riterion for Fiber ension () he ar criterion for fiber tension failure is a non-interacting axiu allowable strain criterion. ε riterion for Fiber opression ε (3) opressive failure of aligned fiber coposites occurs fro the collapse of the fibers as a result of shear kinking and daage of the supporting atrix (ee for instance the representative works of Fleck 9 and outis, and hultheisz review of icroechanical copressive failure theories). Fiber kinking occurs as shear deforation leading to the foration of a kink band. Argon was the first to analyze the kinking phenoenon. His analysis was based on the assuption of a local initial fiber isalignent. he fiber isalignent leads to shearing stresses between fibers that rotate the fibers, increasing the shearing stress and leading to instability. ince Argon s work, the calculation of the critical kinking stress has been significantly iproved with a ore coplete understanding of the geoetry of the kink band as well as the incorporation of friction and aterial nonlinearity in the analysis odels. 9-, 3 In the present approach, the copressive strength is assued to be a known aterial property that can be used in the ar atrix daage criterion (Eq. 4) to calculate the fiber isalignent angle that would cause atrix failure under uniaxial copression. - ϕ - Figure 4. Iperfection in fiber alignent idealized as local region of waviness. alculation of Fiber isalignent Angle he iperfection in fiber alignent is idealized as a local region of waviness, as shown in Fig. 4. he ply stresses in the isalignent coordinate frae shown in Fig. 4 are sin cos sinϕ cosϕ ϕ + sin ϕ ϕ + cos ϕ + sinϕ cosϕ + sinϕ cosϕ sinϕ cosϕ + (cos ϕ sin ϕ) (4) At failure under axial copression,, and. ubstituting those values into Eq. 4 gives sin ϕ and sinϕ cosϕ, 6

7 where the angle ϕ is the isalignent angle for the case of axial copression loading only. o calculate the failure index for fiber kinking, the stresses and are substituted into the criterion for atrix copression failure (Eq. 4). he ode of failure in fiber kinking is doinated by the shear stress, rather than by the transverse stress. he angle of the fracture plane is then equal to, and. he atrix failure criterion becoes ϕ G + ( G ) ϕ + (9) Fiber copression failure by foration of a kink band is predicted using the stresses fro Eq. 4 and the failure criterion for atrix tension or atrix copression. For atrix copression ( < ), the criterion is the ohr-oulob criterion given in Eq. 4, with α and. he criterion for fiber kinking becoes: ( ϕ cosϕ η sin ) sin ϕ (5) olving for ϕ leads to the quadratic equation: F +η () tan tan + ϕ ϕ (6) he saller of the two roots of Eq. 6 is 4 ϕ tan (7) Note that if η were neglected and ϕ were assued to be a sall constant angle, Eq. 5 would give ϕ, which is the expression derived by Argon to estiate the fiber isalignent angle. he total isalignent angle ϕ can be decoposed into an initial (constant) isalignent angle ϕ that represents a anufacturing iperfection, and an additional rotational coponent ϕ R that results fro shear loading. he angles ϕ and ϕ R can be calculated using sall angle approxiations and Eqs. 4 ϕ ϕ R ϕ G ϕ ϕ R ϕ ϕ G + ϕ G ϕ G + ( ϕ ) (8) ince ϕ ϕ + ϕ R, it is now possible to solve Eqs. 8 for ϕ in ters of ϕ. For fiber copression with atrix tension, the transfored stresses of Eq. 4 are substituted into the atrix tensile failure criterion given in Eq. to get the following criterion for fiber kinking: F Y + () It is interesting to note that for, the fiber failure criterion in Eq. becoes F + k with k > ϕ η ϕ () he linear interaction between and in Eq. is identical to the for used by Edge for the WWFE 7. For 3/94, Edge suggests using an epirical value of k.5. However, Edge also indicates that other researchers have shown excellent correlation with experiental results with k. Using the WWFE strength values of 9 Pa, 8 Pa, Y Pa; an assued fracture angle in transverse copression of 53 ; and Eqs. and 7; we get: η.34, ϕ 5.3. With the approxiation ϕ ϕ, Eq. gives k.7. atrix Daage in Biaxial opression In the presence of high transverse copression cobined with oderate fiber copression, atrix daage can occur without the foration of kink bands 7

8 or daage to the fibers. his atrix daage ode is calculated using the stresses in the isaligned frae in the failure criterion in Eq. 4, which gives: + (3) As for all atrix copressive failures herein, the stresses and are functions of the fracture angle α, which ust be deterined iteratively. All the failure odes represented by ar in Eqs. 3-3 (see also the Appendix) can be represented in a failure envelope in the - plane. An exaple is shown in Fig. 5 for a E-glass/Y75 epoxy laina. Puck s analysis results, which showed the best correlation with experiental results in the WWFE 3, is also shown for coparison. In the figure, there is good agreeent between ar and Puck in all quadrants except biaxial copression, where ar predicts an increase of the axial copressive strength with increasing transverse copression. esting for biaxial loads presents a nuber of coplexities, and experiental results are rare. However, Waas et al. 4 present a nuber of references in which ultiaxial copression was studied by superposing a hydrostatic pressure in addition to the copressive loading. For all aterials considered, there is a significant increase in copressive strength with increasing pressure. In particular, the results of Wronsky and Parry 5 on glass/epoxy show a strength increase of 3.3 Pa per Pa of hydrostatic pressure, which gives an actual strength increase of 4.3 Pa per Pa of applied transverse biaxial stress. ore recently, igley et al. 6 found 3% to 7% increases in copressive strength per Pa superposed pressure. he results of Wronsky and igley can be copared qualitatively to the 4.3 Pa/Pa increase in copressive strength for the plane stress failure envelope in Fig. 5. Verification Probles Exaple. Unidirectional coposite E-Glass/Y556 A coparison of results fro various failure criteria with the experiental results in the - stress plane obtained fro the WWFE 7 is shown in Fig. 6. It can be observed that within the positive range of, all the quadratic failure criteria and ar give satisfactory results. ince the axiu tress criterion does not prescribe interactions between stress coponents, its failure envelope is rectangular. he ost interesting behavior develops when becoes copressive. Hashin s 973 criterion gives an elliptical envelope with diinishing as copressive increases, while the experiental data shows a definite trend of shear strength increase as goes into copression ar F, Eq., Eq. 3 Puck F, Eq. 5 Y c 5, Eq., Eq. 4 F, Eq. 3 Figure 5. Biaxial - failure envelope of E- glass/y75 epoxy laina. he envelope for Hashin s 98 criteria was calculated using a transverse strength obtained fro Eq. 9, and it provides a odest iproveent in accuracy copared to the 973 criterion. Of the criteria shown in Fig. 6, un (Eq. ), ar (Eq. 4), and Puck (Eq. ) capture the shear strength increase at the initial stage of copressive. he failure envelope for un s criterion (Eq. ) was calculated using η.336 fro Eq.. he results indicate a significant iproveent over Hashin s criteria. An even better fit would have been achieved using higher value for η. Puck s envelope, which was extracted fro the WWFE 6, is the ost accurate, but it relies on fitting paraeters based on the sae test data. he ar curve uses the stiffnesses and strengths shown in able, an assued α 53, and no other epirical or fitting paraeter. 8

9 able. Properties of E-Glass/Y556 7 ( Pa) E E G ν Y Y 53,48 7,7 5, able 3. Properties of A ( Pa) E E G ν Y 7,6,3 6, ar WWFE test un 96 Puck Hashin 73 Hashin , Pa, Pa 5 ax. stress ar Hashin opressive stress, Pa ar est, huart 89 Hashin 73 α α44 α 53 Figure 6. Failure envelopes and WWFE test data for unidirectional coposite E-Glass/Y556. Exaple. ross-ply lainates huart 8 studied the copression failure of [±θ] s lainates and found that for θ <5, the doinant failure ode in these lainates is interlainar shearing; for 5 <θ <5, it is in-plane atrix shearing; and for θ >5, it is atrix copression. Fiber scissoring due to atrix aterial nonlinearity caused the switch in failure ode fro in-plane atrix shearing to atrix copression failure at larger laination angles. he aterial properties shown in able 3 were used for the analysis. he angle α was assued to be a typical 53. For other laination angles, the fracture angle was obtained by searching nuerically for the angle that axiizes the failure criterion in Eq. 4. he results in Fig. 7 indicate that the predicted strengths using ar criteria correlate well with the experiental results. he copressive strength predicted using Hashin 973 criteria is also shown in Fig. 7. For θ <, the Hashin criteria result in an overprediction of the failure load because the criterion does not account for the ect of inplane shear on fiber failure. For laination angles near 7, the use of the Hashin criteria result in an underprediction of the failure load because the criteria do not account for the increase in shear strength caused by transverse copression. aination angle, degrees Figure 7. opressive strength as a function of ply orientation for [±θ] s A4/35 lainates. oncluding Rearks he results of the recently concluded World Wide Failure Exercise indicate that the existing knowledge on failure echaniss needs further developent. any of the existing failure odels could not predict the experiental response within a tolerable liit. In fact, differences of up to an order of agnitude between the predicted and experiental values were not uncoon. In this paper, a new set of physicsbased failure criteria devoid of epirical variables is proposed. he criteria for atrix and fiber copression failure are based on a ohr-oulob interaction of the stresses associated with the plane of fracture. Failure envelopes for unidirectional lainates in the - and - stress planes were calculated, as well as the copressive strength of cross-ply lainates. he predicted failure envelopes indicate good correlation with experiental results. he proposed criteria represent a significant iproveent over the coonly used Hashin criteria, especially in the cases of atrix or fiber failure in copression. 9

10 References ) oden, P.D., Hinton,.J., and Kaddour, A.., "A oparison of the Predictive apabilities of urrent Failure heories for oposite ainates," oposites cience and echnology, Vol. 58, No. 7, 998, pp ) oden, P.D., Hinton,.J., and Kaddour, A.., "Biaxial est Results for trength and Deforation of a Range of E-Glass and arbon Fibre Reinforced oposite ainates: Failure Exercise Benchark Data," oposites cience and echnology, Vol. 6, No. -3,, pp ) Hinton,.J., Kaddour, A.., and oden, P.D., "A oparison of the Predictive apabilities of urrent Failure heories for oposite ainates, Judged against Experiental Evidence," oposites cience and echnology, Vol. 6, No. -3,, pp ) Hinton,.J., and oden, P.D., "Predicting Failure in oposite ainates: he Background to the Exercise," oposites cience and echnology, Vol. 58, No. 7, 998, pp. -. 5) Puck, A., and churann, H., "Failure Analysis of FRP ainates by eans of Physically Based Phenoenological odels," oposites cience and echnology, Vol. 58, No. 7, 998, pp ) Puck, A., and churann, H., "Failure Analysis of FRP ainates by eans of Physically Based Phenoenological odels," oposites cience and echnology, Vol. 6, No. -3,, pp ) Edge, E.., "tress-based Grant-anders ethod for Predicting Failure of oposite ainates," oposites cience and echnology, Vol. 58, No. 7, 998, pp ) Hashin, Z., and Rote, A., "A Fatigue riterion for Fiber-Reinforced aterials," Journal of oposite aterials, Vol. 7, 973, pp ) Hashin, Z., "Failure riteria for Unidirectional Fiber oposites," Journal of Applied echanics, Vol. 47, 98, pp ) hahid, I., and hang, F.-K., "An Accuulative Daage odel for ensile and hear Failures of ainated oposite Plates," Journal of oposite aterials, Vol. 9, 995, pp ) un,.., Quinn, B.J., and Oplinger, D.W., "oparative Evaluation of Failure Analysis ethods for oposite ainates," DO/FAA/AR-95/9, 996. ) París, F., "A tudy of Failure riteria of Fibrous oposite aterials," NAA/R--66, Hapton, VA, arch. 3) alençon, J., Handbook of ontinuu echanics: General oncepts, heroelasticity, pringer, Berlin; New York,. 4) Kawabata,., "trength of Epoxy Resin under ultiaxial tress Field," Proceedings of the I-IV, okyo, 98, pp ) Boehler, J.P., and Raclin, J., "Failure riteria for Glass-Fiber Reinforced oposites under onfining Pressure," J. truct. ech., Vol. 3, No. 3 & 4, 985, pp ) aliercio, A., and agraoso, P., "Uniaxial trength of Polyeric-atrix Fibrous oposites Predicted through a Hoogenization Approach," International Journal of olids and tructures, Vol. 3, No. 4, 995, pp ) Diandro,., and Pegoraro,., "arbon Fibre- heroplastic atrix Adhesion," Journal of aterial cience, Vol., 987, pp ) arson,.., and iles, H.F., "On the Effects of Friction, Roughness and oughness on Interfacial liding in Brittle oposites," echanics of aterials, Vol. 7, No., 998, pp ) Fleck, N.A., and iu, D., "icrobuckle Initiation fro a Patch of arge Aplitude Fibre Waviness in a oposite under opression and Bending," European Journal of echanics - A/olids, Vol., No.,, pp ) outis,., ith, F.., and atthews, F.., "Predicting the opressive Engineering Perforance of arbon Fibre-Reinforced Plastics," oposites Part A: Applied cience and anufacturing, Vol. 3, No. 6,, pp ) chultheisz,.r., and Waas, A.., "opressive Failure of oposites, Part : esting and icroechanical heories," Progress in Aerospace ciences, Vol. 3, 996, pp. -4. ) Argon, A.., Fracture of oposites, reatise of aterials cience and echnology, Vol., Acadeic Press, New York, 97. 3) Budiansky, B., Fleck, N.A., and Aazigo, J.., "On Kink-Band Propagation in Fiber oposites," Journal of the echanics and Physics of olids, Vol. 46, No. 9, 998, pp ) Waas, A.., and chultheisz,.r., "opressive Failure of oposites, Part II: Experiental tudies," Progress in Aerospace ciences, Vol. 3, No., 996, pp

11 5) Wronski, A.., and Parry,.V., "opressive Failure and Kinking in Uniaxially Aligned Glass- Resin oposite under uperposed Hydrostatic Pressure," Journal of aterial cience, Vol. 7, 98, pp ) igley, R.H., Wronski, A.., and Parry,.V., "Axial opressive Failure of Glass-Fibre Polyester oposites under uperposed Hydrostatic Pressure: Influence of Fibre Bundle ize," oposite cience and echnology, Vol. 43, 99, pp ) oden, P.D., Hinton,.J., and Kaddour, A.., "aina Properties, ay-up onfigurations and oading onditions for a Range of Fibre- Reinforced oposite ainates," oposites cience and echnology, Vol. 58, No. 7, 998, pp. -. 8) huart,.j., "Failure of opression-oaded ultidirectional oposite ainates," AIAA Journal, Vol. 7, No. 9, 989, pp Appendix able A. uary of ar criteria for plane stress. atrix racking atrix tension, atrix copression, Y + < Y + < Y + Fiber Failure Fiber tension, Fiber copression, ε F ε F < +η < F Y + cosα (sinα η cosα) he ective stresses are cosα ( + cos ) η α cosα he transverse shear strength is Y cos α ( sin α + ) tan α η cos α and η or fro test data. tan α Y cos α he stresses in the fiber isalignent frae are cos ϕ sin ϕ where sinϕ cosϕ ϕ G + sin ϕ + cos ϕ + sinϕ cosϕ + ( G ) ϕ + + sinϕ cosϕ sinϕ cosϕ + (cos ϕ sin ϕ) and ϕ tan 4