Compression failure and fiber-kinking modeling of laminated composites

Transcription

1 Steel and Composite Strutures, Vol. 1, No. 1 (011) Compression failure and fier-kinking modeling of laminated omposites A. Kairi Ataaadi 1, S. Ziaei-Rad 1, and H. Hosseini-Toudeshky 1 Department of Mehanial Eng., Isfahan University of Tehnology, Isfahan , Iran Department of Aeronatial Eng., Amirkair University of Tehnology, Hafez Ave., Tehran, Iran (Reeived August 18, 010, Revised Otoer 13, 011, Aepted Otoer 1, 011) Astrat. In this study, the physially-ased failure models for matrix and fiers in ompression and tension loading are introdued. For the 3D stress ased fier kinking model a modifiation is proposed for alulation of the fier misalignment angle. All of these models are implemented into the finite element ode y using the advantage of damage variale and the numerial results are disussed. To investigate the matrix failure model, purely in-plane transverse ompression experiments are arried out on the speimens made y Glass/Epoxy to otain the frature surfae angle and then a omparison is made with the alulated numerial results. Furthermore, shear failure of (±45) s model is investigated and the otained numerial results are disussed and ompared with availale experimental results. Some experiments are also arried out on the woven laminated omposites to investigate the frature pattern in the matrix failure mode and shown that the presented matrix failure model an e used for the woven omposites. Finally, the otained numerial results for stress ased fier kinking model and improved ones (strain ased model) are disussed and ompared with eah other and with the availale results. The results show that these models an predit the kink and angle approximately. Keywords: laminated omposite; ompression; fier-kinking; matrix failure. 1. Introdution During the reent years, many researhes fous on the damage and failure analyses of omposite laminates due to the inrease of demand for omposite strutures in the industries. Delamination, matrix ompressive failure, fier ompressive failure, matrix tensile failure and fier tensile failure are typial damages in the omposite strutures. Investigations on the ompressive failure modes of omposite materials are not as extensive as the other failure modes. Therefore, a theoretial and experimental understanding of this omplex failure mode is required to improve the performane of these materials whih is also the aim of the present study. Initially, Puk and Shurmann (1998 and 00) proposed a matrix ompressive failure model ased on the Mohr-oulom riterion and further developments were later arried out y Davila et al. for LaRC0/ 03 failure riteria (Davila et al. 003 and Davila and Camanho 003). Pinho (005) implemented these failure riteria into a finite element ode and proposed a modifiation for the onsideration of frition stresses. This modifiation leads to more onservative failure load preditions. Depending on the material and geometry, different fier ompressive failure modes, suh as miro Corresponding author, Ph, D. Student, a.kairiataaadi@me.iut.a.ir

2 54 A. Kairi Ataaadi, S. Ziaei-Rad and H. Hosseini-Toudeshky ukling, kinking and fier failure may our in the omposite laminates. Some researhers onsidered the kinking mode as a onsequene of miro ukling, while others onsidered that as a separate failure mode (Shultheisz and Waas 1996, Singh and Kumar 010). The main fous of this artile is on the fierkinking damage mode in unidiretional omposite laminates. Kinking an e defined as the loalized shear deformation of the matrix. In this failure mode whih an e loalized in a and aross a speimen, the fiers are rotated y a large amount and the matrix is usually sujeted to a large shear deformation. Rosen (1965) proposed a D mehanial model to desrie fier miro ukling. He otained the fier s ukling stresses y minimizing the internal energy. The otained results y Rosen s model were typially 1.5 times higher for oron/epoxy, to 3 times higher for aron/epoxy and 4 times higher for glass/epoxy omposites when ompared with the experimental results. These differenes in failure stress predition may e due to the disregarding of 3D effets as disussed y Lager and June (1969), de Ferran and Harris (1970) and assumption of perfetly straight fiers as mentioned y Shultheisz and Waas (1996). Furthermore, the laminate onstrution may lead to different fier arrangements within the plane of the lamina and through the thikness diretion. Fier misalignment angles have first een reported to e smaller in the through the thikness diretion omparing with the plane of lamina (Davis 1975), ut more reent results reported that they are similar in magnitude (Shuart 1989). Argon (197) onsidered the fier-kinking as a separate failure mode. He assumed that failure ours as a result of matrix shear failure y an initial fier misalignment. Budiansky (1983) later extended Argon s analysis and in an analytial work with Flek (Budiansky and Flek 1993), he inluded the effet of strain hardening, shear loads, kink and inlination and finite fier stiffness, performing a non losed form solution for failure load predition. Kyriakides et al. (1995) arried out miro mehanial D FE models for the analysis of kinking phenomena, inluding matrix nonlinear ehavior and initial imperfetions. Liu et al. (004) onsidered the effet of random waviness of the fiers using a Cosserat smeared out finite element method. Budiansky et al. (1998) used unidiretional analysis to define two kind of steady state kink propagation of: and roadening, in whih an estalished kink and grows in the diretion of loading, and transverse propagation, in whih a kink and travels aross the width of the omposite speimen. Their theoretial study was on the ases of various physial models and inluded the effets of fier ending resistane in the analyses. Christensen and Deteresa (1997) inorporated a simple strain ased yield/failure riterion into a kink and analysis of ompressive failure. Considering of the real onditions of fier misalignment they predited ompressive failure loads of aout one-fifth of the ideal value and with the kink and inlinations of aout 0 deg. Davila et al. (003), used a omination of Argon s approah and the LaRC0/03 matrix failure riterion. Essentially, Davila et al. (003) supposed that the fiers might e misaligned and that further rotation will our y applying of the ompressive loading. Then they omputed the stresses in the updated misalignment frame to hek the matrix failure using LaRC0/03 matrix failure riterion. A D and 3D stress ased kinking model has also proposed y Pinho (005). In this model, whih is ased on Argon s approah and the later developments y Davila et al. (003), the stresses are alulated from the strains in the gloal oordinate and then they transformed into the misalignment loal oordinate to determine the applied shear stress on the fiers. Taking the advantage of the shear onstitutive law, inreasing of the misalignment angle an e determined. In the misalignment oordinate the matrix failure riteria is heked and if it is satisfied, then the stresses in the kink and are degraded. In this artile, the physially-ased failure models for matrix and fiers in ompression and tension are implemented into a finite element ode. Experiments are also arried out on the woven laminated omposites to investigate the frature patterns in the matrix failure mode. The 3D stress ased kinking model is also

3 Compression Failure and Fier-Kinking Modeling of Laminated Composites 55 implemented into a finite element ode. This kinking model is ritiized and a modifiation is proposed for alulation of the fier misalignment angle. The otained results are explained for several laminates.. Matrix failure riteria.1. Matrix ompressive failure riterion Experimental results (Pinho 005) under purely in-plane transverse ompression load show that the frature surfae ours in an angle with respet to the thikness diretion as shown in Fig. 1. The experimental value for this angle is generally φ o 53 o ± o for most of the omposite materials (Puk and Shurmann1998 and 00). For a general loading situation, the angle of frature plane, φ, is different from the one for pure ompression, φ o, as shown in Fig.. In 3D formulation, tration omponents are Cos ( φ) + Sin( φ) T Sin( φ) + Cos( φ) a Cos( φ) + a Sin( φ) L (1) In the aove equations the susript a refers to the fier diretion, the susript refers to the in-plane transverse diretion and the susript refers to the through-the-thikness diretion. The Mohr-Coulom riterion an e expressed in several forms for matrix ompressive failure. Puk and Shurmann (1998 and 00) initially proposed the following riterion f m T L S T µ T S L µ L () Whereas Davila et al (003) proposed the following riterion L + µ L f m T µ T S T S L (3) Fig. 1 Frature surfae angle for pure transverse ompression loading (Pinho 005)

4 56 A. Kairi Ataaadi, S. Ziaei-Rad and H. Hosseini-Toudeshky Fig. Frature surfae angle for a general loading situation Where, the operator < > is the M-Cauley raket defined as <x> max{o, x}, x R. Puk and Shurmann (1998 and 00) finally deided to use the following expression, elieving that it fits the experimental data in etter agreement f m T L S T µ T S T S L µ L S L (4) Pinho (005) improved Eq. (3) and otained the following riterion f < T + µ T Cos( θ )> m < L + µ L Sin( θ )> S T S L (5) Where T, L and are the tration omponents on potential frature planes as shown in Fig. and Y t and S L are the transverse tensile and longitudinal shear strengths, respetively. µ T and µ L are frition oeffiients in the transverse and longitudinal diretions and S T is frature plane resistane auses y transverse shear. The following relations etween the aove parameters and frature plane angle are also defined tan( φ o ) µ T (6) S T Y tan( ) φ o (7) µ L ---- S L µ T S T (8) Where Y is the transverse ompressive strength. Due to the existed simpliity in Eq. (), it has een preferred to use y researhes suh as Pinho... Matrix tensile failure riterion For a general stress state, the quadrati interation form etween the tration omponents, expressed in

5 Compression Failure and Fier-Kinking Modeling of Laminated Composites 57 the potential frature plane is onsidered (Pinho (005)) f mt L n Y t T S T S L (9) 3. Fier-kinking failure riterion Fier-kinking failure usually has een oserved in the high fier-volume fration omposite materials when they are under ompressive load. A shemati representation of a kink and is shown in Fig. 3. In this setion, the stress ased fier-kinking model whih has een proposed y Pinho (005), with an improvement in alulation of the fier misalignment angle is explained. In this stress ased fier-kinking model, kinking results from matrix failure mehanism in the plane parallel to the misaligned fiers or from instaility mehanism due to the loss of shear stiffness for large shear strain values. A unidiretional omposite with misaligned region under ompression is onsidered as depited in Fig. 4. The stresses in the misalignment frame for a general plane stress loading are a m m a m m a a Cos ( θ) + a Sin( θ ) a a Cos( θ) a Sin( θ ) a Sin( θ ) + a Cos( θ) (10) For a D ondition under pure ompression in whih the matrix failure mehanism is ourred, y using the riterion in Eq. () the value of misalignment angle at the failure point, θ, is alulated from the following equation X ( Sin( θ )Cos( θ ) µ L Sin ( θ )) S L (11) Where, X is the axial ompressive strength. Davila et al. (003) solved this equation for θ and found the following expression Fig. 3 Mirograph from experiments in CFRP and kink and geometri parameters (Pinho et al. 006 and Pimenta et al. 009)

6 58 A. Kairi Ataaadi, S. Ziaei-Rad and H. Hosseini-Toudeshky Fig. 4 (a) kink and, () fier misalignment frame θ S L S L µ X L ---- X artan S L µ L X (1) Using the shear onstitutive law, the shear strain γ m an e otained from the shear stress m and the initial misalignment angle, θ i, an e alulated. From the onstitutive law, the shear stress at failure point (and in the material axes) is a funtion of the shear strain m f l ( γ m ) (13) For a D ondition under pure ompression, the shear stress at an angle θ is --Sin( θ )X m 1 (14) Comining of the two aove equations for the shear strain at failure point in pure ompression, γ m, leads to γ m f l 1 --Sin( θ )X 1 (15) Now, the initial misalignment angle an e alulated y the following equation θ i θ γ m (16) In the ondition that the failure ours due to the instaility mehanism, efore ourrene of matrix raking the following onditions are satisfied 1 f CL ( γ m ) --Sin( ( θ i + γ m ))X ( ) X Cos( ( θ i + γ m )) f CL γ m γ m (17)

7 Compression Failure and Fier-Kinking Modeling of Laminated Composites 59 Fig. 5 3D kinking model:(a) solid under general loading ondition, () fier-kinking plane, () stresses in (a,,) oordinate system, (d) stresses in (a,, ) oordinate system, (e) stresses in misalignment frame and (f) matrix failure plane (Pinho (005)) The aove equations represented that the in-plane shear stress urve and shear stress urve otained from the stress transformation are tangent and any inreasing in the load auses their divergene. By solving the Eq. (17), the initial misalignment angle of fiers and the shear strain at the failure point an e alulated. A 3D kinking model was also proposed y Pinho (005). A unidiretional lamina under a general ompressive state of stress is onsidered as shown in Fig. 5(a). The fier-kinking plane is assumed to e at an angle with the axis (), as shown in Fig. 5(). It has een assumed that and are the prinipal diretions in the - plane. Therefore, an e alulated y the following equation (see also Fig. 5()) tan( ) (18) Eq. (19) an e used to transform the stresses to the potential fier-kinking plane as follows Ψ a a Cos ( ) + Sin( ) a Cos( ) + a Sin( ) Sin( ) + Cos( ) a Cos( ) a Sin( ) (19) The fier misalignment angle is alulale y θ a ( θ i + γ m ) a (0) Where, the strain γ m is otained y solving the following equation iteratively

8 60 A. Kairi Ataaadi, S. Ziaei-Rad and H. Hosseini-Toudeshky f CL ( ) γ m a Sin( ( θ i + γ m )) + a Cos( ( θ i + γ m )) (1) If Eq. (1) does not have a solution, then failure has taken plae y instaility. Then, the failure envelope for this mode is defined y a f CL ( γ m ) Sin( ( θ i + γ m )) + a Cos( ( θ i + γ m )) f CL ( γ m ) ( γ a )Cos( ( θ i + γ m )) a Sin( ( θ i + γ m )) m () Now, using the misalignment angle, θ, the stresses an e transformed in to the misalignment frame as follows a m a a Cos( θ) + a Sin( θ ) + m a a Ψ a m m a Sin( θ ) + a Cos( θ ) m m Cos( θ ) a Sin( θ ) a m a Cos( θ ) + m Sin( θ ) (3) At this stage, the matrix failure an also e heked f kink , m T T S T µ T S L µ L (4) f kink ---- n Y t T S T L + + 1, m > S L (5) In the two aove equations, the tration omponents on the frature plane are given y + m T m L m Cos ( φ) + m Sin( φ) Sin( φ) + m Cos( φ) m m a Cos( φ ) + a msin( φ ) (6)

9 Compression Failure and Fier-Kinking Modeling of Laminated Composites 61 The aove proposed 3D model for kinking is a stress ased model and the most important parameter that must e alulated, is the misalignment angle, θ. For alulating the misalignment angle the equations must e solved iteratively. Thus Pinho onsidered a modifiation and used the following equation to alulate the misalignment angle 1 γ mi f CL a Sin ( θ i ) + a Cos( θ i ) (7) In this equation, the initial misalignment angle, θ i, is dedued from experimental data y solving the following equation iteratively θ i θ C f L 1 --Sin( θ i )X 1 (8) And the misalignment angle θ eomes as θ a ( θ i + γ mi ) a (9) 4. Finite element implementation 4.1. Material ehavior Before the ourrene of any damage, the in-plane shear deformation is only onsidered to ehave nonlinearly. For definition of the in-plane shear response, the maximum shear strain is defined as (Pinho (006), Part II) γ a max () t max{ γ a( t ) }, t t (30) And the inelasti shear strain as in γ a max γ a f L max ( γ a )/G a (31) The funtion f L (γ a ) represents the value of shear stress for γ a 0. The material shear deformations shown in Fig. 6 an e presented as a γ a f CL ( γ a ) γ a γ a γ a γ a in G a γ a γ a γ a γ a γ a max max (3) Where G a is the shear modulus in the (a, ) plane.

10 6 A. Kairi Ataaadi, S. Ziaei-Rad and H. Hosseini-Toudeshky Fig. 6 (a) Typial experimental nonlinear shear ehavior, () irreversiility: loading, unloading and reloading paths 4.. Damage formulation A typial unidiretional laminate loaded in transverse tension is shown in Fig. 7(a). After damage initiation, damage variale is used to explain the material ehavior. In this study in order to otain the damage variale, a ilinear softening onstitutive law is assumed as shown in Fig. 7(). Aording to Fig. 7(), the maximum strain, ε f, an e otained as a funtion of the energy per unit area of damage or frature surfaes, Γ, material strength, o, and the element dimension, L (Pinho (006), Part II) ε f Γ l (33) The damage variale, d, is defined to onsider a linear degradation for the relevant stress omponents. The instantaneous value of the damage variale, d inst, is also defined as d inst max 0 min 1 ε f ε ε,, ε( ε f ε ) (34) Considering the irreversiility ondition, the damage variale is defined as d() t max{ d inst ( t )}, t t (35) Matrix failure modeling In this damage mode when the matrix failure riterion is satisfied, the tration omponents on the Fig. 7 (a) Typial unidiretional omposites loaded in transverse tension up to omplete failure, () Material onstitutive law with linear softening

11 Compression Failure and Fier-Kinking Modeling of Laminated Composites 63 frature plane are degraded < T ( 1 d mat ) T L ( 1 d mat ) L 1 d n >,, mat n (36) To derive the damage variale, the strain variale, ε mat, the magnitude of tration on the frature plane, mat, and the harateristi length, L mat, have to e alulated. The magnitude of tration on the frature plane is (see Fig. ) + mat < > mat (37) In whih the shear omponent, mat, defined as mat T + L (38) The elasti strain omponents on the frature plane are -- [( ε + ε ) + ( ε ε )Cos( φ ) + γ Sin( φ) ] ε n 1 ( )Sin( φ ) + γ Cos( φ ) γ T ε ε γ L el el γ a Cos( φ ) + γ a Sin( φ ) (39) Where the elasti omponent of the in-plane shear strain is el γ a a G a (40) It is notale that the elasti internal energy in the element at the onset of failure is only ontriuted to the frature proess. Therefore, the elasti part of γ a is only onsidered. The strain ating on the diretion of mat is defined as Where ε n Sin( ω) + γ mat Cos( ω ) ε mat < > (41) λ γ mat γ T Gos( λ ) + γ L elsin( λ ) artan L, w ar tan T < > mat (4) The onset stress and strain are determined from the value of mat and ε mat at the onset of failure mat F, ε mat mat1 mat ε mat F mat 1 (43)

12 64 A. Kairi Ataaadi, S. Ziaei-Rad and H. Hosseini-Toudeshky Finally, the expression for f ε mat is f ε mat Γ mat L mat (44) The harateristi length, L mat, is defined as L 1 L L 3 L mat L a L φ (45) The required variales to define the L mat are introdued in Fig. 8. In partiular, if no mixed mode data is known, a simple weighted average of Γ, Γ T and Γ L an e used to determine the frature toughness, Γ. Γ Γ < n >+Γ T T Γ T + L mat mat mat (46) Where Γ is the mode-i interalaminar frature toughness and Γ T Γ L G IIC (Puk and Shurmann 1998) Modeling of fier-kinking failure mode After satisfation of Eq. (4) or (5) and kinking failure initiation, the shear stresses on the kink and ( m, m m ) as well as the stress normal to the kink and (depending on the sign of m C a a a ) are degraded using the damage variale, d kink < m m a ( 1 d kink ) m m m a a ( 1 d kink ) m m a a 1 d a m>,, kink a m a m (47) This failure proess is assoiated with the rotation of the fiers in the kink and, whih is due to the Fig. 8 Determination of the harateristi length within an element: (a) Rotation of an angle β, () rotation of an angle φ, () Rotation of an angle, (d) rotation of an angle θ

13 Compression Failure and Fier-Kinking Modeling of Laminated Composites 65 shear stress, m m a. Therefore at the sale of the kink and, failure propagation is ontrolled y the driving el stress, kink m m. Also, in this mode, the elasti omponent of γ a, ( γ a a a ), is only onsidered el el G a for the driving strain, whih is defined as ε kink γ m m, where γ m m a a is otained y rotation of the elasti strains. The onset stress and strain are defined as kink,ε mat kink f kink 1 ε kink fkink 1 (48) And the expression for the final strain f ε kink is f ε kink Γ kink kink L kink (49) The harateristi length L kink in the aove equation shown in Fig. 8 is defined as L 1 L L 3 L kink L a ml (50) Finally, the frature toughness or energy density for kinking, Γ kink, an e otained from the experiments (Pinho et al. 006). 5. Modifiation and improvement In the previously proposed stress ased kinking model the shear stress onstitutive law whih is used in the gloal frame (Eq. (3)) will e used in the loal frame (misalignment frame) to alulate the misalignment angle (Eq. (1)). In this paper, alulation of misalignment angle of fiers, θ, is improved on the ase of strains and implemented into an expliit finite element ode (Aaqus). It is worth to note that the strains in eah inrement are usually availale, therefore implementation of models ased on the strains are easier than those ased on the stresses. In order to alulate the misalignment angle at failure, θ, initial misalignment angle, θ i, and the fier-kinking plane, Eqs. (1), (16) and (18) are used. Now, the important parameter of misalignment angle, θ, an e alulated using the following equation θ θ i + γ m (51) In whih, θ i is a material parameter and an e positive or negative and γ m is the shear strain in the misalignment frame whih an e determined as ε 1 -- ( ε + ε + ( ε ε ) os( ) + γ sin( ) ) (5) γ a γ a os( ) + γ a sin( ) (53) ( ) sin( θ i ) + γ θ a os( i ) γ m ε a ε (54)

14 66 A. Kairi Ataaadi, S. Ziaei-Rad and H. Hosseini-Toudeshky After alulation of the misalignment angle, θ, the stress omponents in the misalignment frame an e determined using Eqs. (19) and (3). At this stage, the matrix failure is heked using the riteria defined y Eqs. (4) and (5) whih in these equations the tration omponents on the frature plane are determined y Eq. (6). The angle φ is otained y trying a small numer of tentative angles in the interval of 0 φ <π. In the proposed modifiation, only the proess of fier misalignment angle alulation is onsidered. However, similar to the stress ased fier kinking model, the improvement an e applied to other steps of the proedure suh as frature surfae seletion and stress alulations. 6. Numerial results The aron/epoxy material properties used for numerial analysis are shown in Tale 1. The availale experimental in-plane shear stress versus strain ehavior (Pinho 005) was used in the numerial analysis. This ehavior has een fitted y the logarithmi funtion of, k 1 ln(k γ + 1) with k 1 46 and k 100. The frature angle for pure transverse ompression is onsidered as φ 0 53 o. Regarding the through-thethikness diretion, it is assumed that the omposite material ehave transversely isotropi and the interlaminar toughness is Γ 0.kj / m, the parameter Γ kink is 79.9 kj/m and Γ L Γ T 1.1 kj/m. (Pinho (005)). In the aove tale, E a, E and, G a are the elasti modulus in the fier diretion, the elasti modulus in the transverse diretion and the shear modulus in the (a, ) plane respetively and Y t, Y, X, X t, S a are the tension and ompressive strength in the transverse diretion, ompressive and tension strength in the fier diretion and shear strength in the plane (a, ) respetively and ν and ν a are the poison ratios Matrix failure under purely in-plane transverse ompression load For verifying the matrix failure model implemented into the finite element ode, a numerial analysis is arried out. For this purpose a model with 60 mm length, 15 mm width and 10 mm thikness and material properties mentioned in Tale 1, is generated and transversely loaded until damage proess is ompleted. In the plane of lamina, element size should e small enough to ensure that the harateristi length, L mat in f Eq. (45), is adequate for alulating of ε mat in Eq. (44). In this model the in-plane element size is onsidered to e aout 0.6 mm. A typial frature surfae otained from pure transverse ompression test has een shown in Fig. 1. The experimental frature surfae has an angle of aout 53º from the thikness diretion. The results of numerial analysis also have een shown in Figs. 9(a) and (). The angle of the failure and whih an e oserved in the model is etween 51º-54º whih is in good agreement with the previously otained experimental results. For a further verifiation, the authors arried out purely in-plane transverse ompression experiments on speimens made y Glass/Epoxy (Type C) ASNA 4197 Prepreg unidiretional omposite. The speimens had 13.6 mm length, 9.8 mm width (in the fier diretion) and 6 mm thikness approximately. In order to ensure that the speimens are in full ontat with the loading devies and to eliminate the potential ending Tale 1 Material properties for T300/913 (Pinho 005) Material properties E a (GPa) E (GPa) G a (GPa) Y t (MPa) Y (Mpa) X (MPa) X t (MPa) S a (MPa) ν ν a T300/

15 Compression Failure and Fier-Kinking Modeling of Laminated Composites 67 Fig. 9 (a) Numerial results for matrix failure under purely in-plane transverse ompression load, () present the results after eliminating of the damaged elements moments, a spherial self adjusting devie (similar to Bing and Sun work (008)) was manufatured and used. The otained experimental results are shown in Fig. 10. The frature surfae angle is measured etween 49 o to 58 o with an average of 53.8 o through the thikness whih is losed to that found y others for Caron/Epoxy omposite (see Fig. 10(a)). The otained experimental transverse ompressive strength of Glass/Epoxy (Type C) ASNA 4197 speimens was aout 15 MPa. 6.. Shear failure modeling of (±45)s test speimen Pinho (005) arried out shear tests with (±45)8s lay-ups (zero fier angle is in loading diretion) aording to the ASTM standard D3518/D3518M-01 and found ±45 o failure and as shown in Fig. 11(a). Numerial analyses of different models shown that the frature surfae predition may e independent of the dimensions and numer of lay-ups (n in (±45)ns ) of finite element model. The presented material properties in Tale 1 were also used for this numerial analysis. In Figs. 11(a), (), () and (d) the ±45º failure and an e oserved whih are in good agreement with Fig. 10 Frature surfae angles for transversely ompression tests on Glass/Epoxy (Type C) ASNA 4197 omposite

16 68 A. Kairi Ataaadi, S. Ziaei-Rad and H. Hosseini-Toudeshky the experimental results. In the numerial model shown in Fig. 11 no effort is arried out to speify the failure initiation point. The results show that the failure is initiated from the edges. However, if the weekend element or pre-rak is used to speify the failure initiation point, the ±45º failure and will e oserved as shown in Fig. 1. In Fig. 1, the model has a pre-rak in the middle and a quarter of the model is used for symmetry. In order to investigate the failure of woven omposite laminates, the authors performed similar experiments for this type of laminates. The speimens were made y hand lay-up tehnique with 8 layer Eglass-polyester (marine grade) material. The size of the speimens seleted aording to the ASTM standard D3039/D3039M-00 and depited in Tale. The speimens were manufatured with fiers diretions of ±45º, similar to the speimen shown in Fig. 11(a). Figs. 13(a), () and () show the speimens efore and after testing. As shown in Figs. 13() and () the angle of failure and is aout 45º and similar ehavior for these two types of omposites ((±45)s unidiretional laminated omposite and woven laminated omposite) has een oserved Fier ompression failure modeling In this setion, the formation of kink-ands is predited using the two explained stress ased and strain ased failure models. For T300/913 omposite laminate, Pinho oserved that the kink and is formed within 5±5º for standard axial ompression speimens (see Fig. 14(a)). A numerial model with 10 mm length, 10 mm width and mm thikness is used for this analysis. The otained results from the implementation of stress ased failure model for the failure analyses of T300/913 omposite laminates are Fig. 11 (a) shear test speimen (Pinho 005), () model of shear speimen, () predition of 45 o failure and in with (±45) 8s model, (d) the result of matrix failure after eliminating the damaged elements Fig. 1 ±45 o failure and oservation in numerial analysis with the model whih has a prerak (a quarter of the model is shown for symmetry)

17 Compression Failure and Fier-Kinking Modeling of Laminated Composites 69 Fig. 13 (a) test speimens, () the otained failure and during the test, () test speimen after failure Tale Nominal dimensions of speimens Overall length (mm) Average thikness (mm) Average width (mm) depited in Fig. 14. It is worth to note that updating of the misalignment frame after damage initiation in this type of modeling leaded to the less aurate results. Therefore, the analyses were performed without updating the misalignment frame. The otained kink-and angle for this stress ased failure model is etween 18º to 0º, however Pinho et al. (006) reported that aout 15º in numerial analyses. The implemented strain ased model into the finite element ode is also used for the analyses of the same laminate. The misalignment frame updating is easer here eause the formulation of the strain ased model is simpler than the stress ased one. Therefore, the numerial analyses for this model were performed with the misalignment frame updating after damage initiation. Fig. 15 shows that the otained kink-and angle is etween 0º to 3 o whih is in etter agreement with the experimental result when ompared with the stress ased fier kinking model (Pinho 005). Similar to the stress ased model, some failure sattering is oserved in the results whih an e improved y updating the frature surfae and stress in every step of the proedure (Setion 5). In another attempt, the mesh dependeny of kink-and formation is also examined for stress ased model. For this purpose, the model with mm in dimensions is analyzed using three different meshes with the elements size of mm (oarse), mm (medium) and mm (fine). The results for kink-and width and kink-and angle are shown in Fig. 16. The strength and kink-and angle preditions are not signifiantly hanged etween these three models. However more kink-and sattering is oserved as the mesh eomes finer. Fig. 14 (a) kink-and in the failed longitudinal ompression test speimen (Pinho 005), () initial kink-and angle aout 0 o for stress ased kinking model, () propagation of kink-and with an angle of aout 0 o, (d) kink-and roadening with an angle aout 18 o (stress-ased model)

18 70 A. Kairi Ataaadi, S. Ziaei-Rad and H. Hosseini-Toudeshky Fig. 15 Strain ased model results (a) initial kink-and angle aout 3 o, () propagation of kink-and with an angle of aout 0 o. Fig. 16 (a) fier-kinking analysis with oarse mesh, () medium mesh and () refined mesh (stress-ased model) The results of these two models are different in displaement whih auses the initiation of kinking failure proess. Furthermore, their damage variale ontours are not similar and their predited kink-and angles are also different (see Figs. 14 and 15). Therefore, more experimental and numerial investigations on the effets of various parameters on kink-and formation are required to evaluate these models. 7. Conlusions From the results of this researh, the following onlusions are otained: - The investigated matrix failure models predit the failure and angle in ompression effiiently. The result of numerial analysis showed that the failure and angle predited y these models is in good agreement with the experimental results. - The experiments arried out on the woven laminated omposite showed that the ehavior of these kind of omposites are similar to the unidiretional laminated omposites in matrix failure mode, therefore the matrix failure models whih are used for unidiretional laminated omposites an e improved for the woven laminated omposites failure. - The experiments arried out on glass and aron fier reinfored omposites showed that these two omposite materials fail through the same mehanisms, and the same failure models an e used for oth materials. - The stress ased kinking model proposed y Pinho (005) an predit kink-and angle approximately. This model, without updating the misalignment frame, gives etter ontour for damage variale in the numerial analysis. - Investigation of element size affetion for the stress ased model shown that the ompressive strength

19 Compression Failure and Fier-Kinking Modeling of Laminated Composites 71 and kink-and angle preditions are not signifiantly hanged with different element sizes. However, more kink-and sattering is oserved as the mesh eomes finer. - The strain ased model proposed in this researh an also predit kink-and angle approximately, ut ompared to the stress ased one, it is simpler and with updating the misalignment frame, the results are in etter agreement with the experimental results when ompared with those otained from the stress ased model ut failure sattering in the numerial results shows that the model needs to e improved. Referenes Argon, A. S. (197), Frature of omposites, in: Treatise on Materials Siene and Tehnology, Aademi Press, New York, ASTM Standard D3518/D3518M,(001), Standard test method for in-plane shear reponse of polymer matrix omposite materials y tensile test of a +/ 45 laminate, (001). ASTM Standard D3039/D3039M-00, Standard test method for tensile properties of polymer matrix omposite materials, (000). Budiansky, B. (1983), Miromehanis, Computers and Strutures 16(1-4), 3-1. Budiansky, B. and Flek, N. A. (1993), Compressive failure of fire omposites, J. Me. Phys. Soli. 41(1), Budiansky, B., Flek, N. A. and Amazigo, J. C. (1998), On kink-and propagation in fier omposites, J. Meh. Phys. Solid, 46(9), Bing, Q. and Sun, C.T. (008), Speimen size effet in off-axis ompression tests of fier omposites, Comp. Eng., Part B 39(1), 0-6. Christensen, R. M. and Deteresa, S. J. (1997), The kink and mehanism for the ompressive failure of fier omposite materials, J. Appli. Meh., 64(1). Dávila, C. G., Jaunky, N. and Goswami, S. (003), Failure riteria for FRP laminates in plane stress, in: 44th AIAA/ASME/ASCE/AHS/ASC Strutures, Strutural Dynamis, and Materials Conferene, AIAA Paper Dávila, C. G. and Camanho, P. P. (003), Failure riteria for FRP laminates in plane stress, Teh. Rep. NASA/ TM , National Aeronautis and Spae Administration, U.S.A. Davis Jr, J. G. (1975), Compressive strength of fier-reinfored omposite materials, in: Composite Reliaility, ASTM STP 580, Amerial Soiety for Testing and Materials, Philadelphia, De Ferran, M. E. and Harris, B. (1970), Compression strength of polyester resin reinfored with steel wires, J. Comp. Mat, 4(1), 6-7. Kyriakides, S., Arseuleratne, R. and Perry, E. J. (1995), On the ompressive failure of fier reinfored omposites, Inter. J. Solids. Strut., 3(6/7), Lager, J. R. and June, R. R. (1969), Compressive strength of oron-epoxy omposites, J. Comp. Mater. 3(1), Liu, D., Flek, N. A. and Sutliffe, M. P. F. (004), Compressive strength of fier omposite with random fier waviness, J. Meh. Phys. Solid 5(7), Puk, A. and Shurmann, H. (1998), Failure analysis of FRP laminates y means of physially ased phenomenologial models, Comp. Si.Tehno. 58(7), Puk, A. and Shurmann, H. (00), Failure analysis of FRP laminates y means of physially ased phenomenologial models, Comp. Siene. Tehno. 6(1-13), Pinho, S.T. (005), Modelling failure of laminated omposites using physially-ased failure models, Imperial College London. Pinho, S. T., Roinson, P. and Iannui, L. (006), Frature toughness of the tensile and ompressive fier failure modes in laminated omposites, Comp. Siene. Tehno., 66(13), Pinho, S.T., Iannui, L. and Roinson, P. (006), Physially ased failure models and riteria for laminated fire-reinfored omposites with emphasis on fire kinking. Part II: FE implementation, Applied siene and manufaturing, 37(1), Pimenta, S., Gutkin, R., Pinho, S. T. and Roinson P. (009), A miromehanial model for kink-and formation:

20 7 A. Kairi Ataaadi, S. Ziaei-Rad and H. Hosseini-Toudeshky Part II- Analytial modeling, Compo. Si. Tehno, 69(7-8), Rosen, V. W. (1965), Mehanis of omposite strengthening, Fier Composite Materials Amerian Soiety of Metals, Metals Park, Ohio Shultheisz, C. R. and Waas, A. M. (1996), Compressive failure of omposites, part I: Testing and miromehanial theories, Progress in Aerospae Sienes 3(1), 1-4. Shuart, M. J. (1989), Failure of ompression-loaded multidiretional omposite laminates, AIAA Journal 7, Singh, S.B. and Kumar, Dinesh (010), Postukling response and failure of symmetri laminated plates with retangular utouts under in-plane shear, Strutural Engineering and Mehanis, An Int\'l Journal 34(). CC