Research Article New Two-Dimensional Polynomial Failure Criteria for Composite Materials

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1 Advanes in Materials Siene and Engineering, Artile ID , 7 pages Researh Artile New Two-Dimensional Polynomial Failure Criteria for Composite Materials ShiYangZhaoandPuXue Shool of Aeronautis, Northwestern Polytehnial University, Xi an 7007, China Correspondene should be addressed to Shi Yang Zhao; haiyangrongyang@sina.om Reeived 6 Marh 04; Revised 7 April 04; Aepted 8 April 04; Published 8 May 04 Aademi Editor: Gonzalo Martínez-Barrera Copyright 04 S. Y. Zhao and P. Xue. This is an open aess artile distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. The in-plane damage behavior and material properties of the omposite material are very omplex. At present, a large number of two-dimensional failure riteria, suh as Chang-Chang riteria, have been proposed to predit the damage proess of omposite strutures under loading. However, there is still no good riterion to realize it with both enough auray and omputational performane. All these riteria annot be adjusted by experimental data. Therefore, any speial properties of omposite material annot be onsidered by these riteria. Here, in order to solve the problem that the riteria annot be adjusted by experiment, new two-dimensional polynomial failure riteria with four internal parameters for omposite laminates are proposed in the paper, whih inlude four distint failure modes: fiber tensile failure, fiber ompressive failure, matrix tensile failure, and matrix ompressive failure. In general, the four internal parameters should be determined by experiments. One example that identifies parameters of the new failure riteria is given. Using the new riteria an redue the artifiialness of hoosing the riteria for the damage simulation of the failure modes in omposite laminates.. Introdution Fiber-reinfored polymer (FRP) omposites are widely applied in aerospae, marine, and many other industries due to their lightweight, high stiffness, strength, and damping properties []. In order to get strong and reliable strutures, it is very important to study the mehanial behavior of FRP. In the studies, the finite element method is proved to be effetive and suessful whih an predit the behavior of omposite strutures under various loading onditions. From the literatures, many researhes an be found on the behavior of omposites, suh as bukling loads, modal harateristis, damage, and failure. Zhang and Yang [] gave omprehensive reviews and some of the future researh on omposite laminated plates. Orifii et al. [3] also gave a ritial review to assess the state of the art in material onstitutive modeling and omposite failure theories. They summarized the various theories and approahes within the ontextofthedissipatedenergyframework.thedissipated energy funtion, with units of energy per unit volume, ould be determined from experimental testing and was postulated to be a property of the material. Based on the ontinuum damage mehanis (CDM), Liu and Zheng [4] reviewed thedamageonstitutivemodeling,thefailureriteria,and the finite element implementation in the progressive failure analysis whih predited the stiffness degradation and failure strengths of omposite laminates. Speially, the methodologies to solve the numerial onvergene problems due to the loss of element stiffness in the finite element analysis were disussed. The failure behavior and damage analysis of the omposite laminated strutures by the finite element method are paid the most attention by researhers. Mihopoulos s work [5] shows that there is a relationship among many riteria. Hinton et al. [6 ]listed9theoriesoffailureriteria proposed for omposite laminates and gave a omparison of their preditive apabilities. They showed similarities and differenes between the preditions of the 9 theories by omparing the initial and final failure envelopes and representative stress-strain urves. They also explained the

2 Advanes in Materials Siene and Engineering differenes between the preditions. They did not intend to prove whih theory is the best but just larified their differenes between their preditions. Despite a large number of published papers on the damage and failure behavior of the omposite laminates under various loadings, a very effetive riterion to predit the failure behavior of omposites has not ahieved yet. Several widely used phenomenologial failure riteria, suh as the maximum stress, Hoffman, Tsai-Wu, and Tsai- Hill failure riteria [8, 4], are proposed to desribe the whole ply failure of omposite material strutures. Other popular phenomenologial failure riteria, suh as Hashin riteria, Hou riteria, Chang-Chang riteria, Linde riteria, LaRC04 riteria, and Maimí riteria [5 ], an desribe the damage proess of omposite strutures in detail, suh as fiber breakage, fiber bukling, matrix raking, therefore, matrix ompression. All these riteria annot be adjusted by experiments. Therefore, any speial properties of omposite material annot be onsidered by these riteria. By studying the problems of dynami bending failure of omposite laminated beams, Santiuste et al. [] suggested that Hashin riteriamaybesuitablefordutileompositematerial,and Hou riteria may be suitable for brittle material in unidiretional laminated beams. Therefore, for different material and different strutures, just one failure riterion whih annot be adjusted by experiments ould not satisfy various requirements. When we use some known riteria in our study diretly, it has used a supposition that material properties used in your own engineering are the same as that in the study whih proposed the known riteria. In fat, the failure riteria should be deided by experiments. If one riterion is deided by experiments, the riterion an onsider speial properties of any material used in engineering. And also, you an use the new failure riteria in numerial simulation to get better numerial results. Here we try to propose new two-dimensionalpolynomialfailureriteria,whihanbe adjusted by experiments. New two-dimensional polynomial failure riteria with four internal parameters for omposite laminates are proposed in the paper. Four distint failure modes inluding fiber tensile failure, fiber ompressive failure, matrix tensile failure, and matrix ompressive failure are onsidered in the extended riteria. Meanwhile, when some speial values are given to the four internal parameters, the extended riteria will beome some lassial failure riteria suh as Chang- Chang riteria, Hashin riteria, and Hou riteria in two dimensional ase. Finally, the features of the failure riteria are disussed in detail. And one example that hooses proper values for the parameters in the riteria is given.. New Two-Dimensional Failure Criteria The damage of omposite laminates under loading inludes intraply failure and interply failure [3]. Intraply failure modes are fiber kinking, fiber frature, matrix raking, and matrix-fiber debonding. Interply failure modes are interfae raks between plies. New two-dimensional polynomial failure riteria for omposite laminates are with four internal parameters. And eah failure mode has a parameter. Therefore, four failure modes all an be adjusted by experiments. And almost speial properties of eah failure mode an be onsidered. New two-dimensional polynomial failure riteria are obtained by omparing ommon and harateristi parts of the present failure riteria, suh as Chang-Chang riteria, Hou Criteria, Hashin riteria, and Linde riteria... Definition of Variables. The related variables inlude stress omponents, σ,σ,σ ; the tensile damage variable of fiber, d ft ;theompressivedamagevariableoffiber,d f ; the tensile damage variable of matrix, d mt ;theompressive damage variable of matrix, d m ;somematerialproperties: longitudinal failure stress strength, X T,X C ; transverse failure stress strength, Y T,Y ; in-plane shear failure stress strength, S ;thesubsripts T and C refer to tension and ompression... Fiber Tensile Failure. Under tension loading, fiber is the main portion to arry load. In omposites with high fiber volumefrationandthosewhosefailurestrainofthematrix is higher than that of the reinforing fiber, suh as arbonepoxy omposites, longitudinal failures start by isolated fiber fratures in weak zones [0]. The loalized fratures inrease the normal and interfaial shear stresses in adjoining fiber, and the loal stress onentration promotes matrix raking and fiber and matrix debonding [4]. Fiber failure riteria in tension are the most simple damage form for omposite laminates. After the aumulation of frature energy, individual fiber will fail until that the whole ply failure happens. And the whole laminates will arry any loads. In general, the maximum stress strength or strain strength is used to predit fiber failure. A new fiber failure riterion in tensile is d ft =k ( σ ) +( k X ) σ T X T +β σ /G +3/4ασ 4 S /G +3/4αS 4, σ 0, where k is a parameter of the riteria of fiber failure; β is used to onsider the interation form involving in-plane shear,whihantaketwovalues:0or;α is a parameter for the shear nonlinear effet, whih an take two values: 0 or..3. Fiber Compressive Failure. Under fiber ompression loads, laminate failure modes are omplex and depend on the material properties. The modes may be mirobuking, kinking, and fiber failure [5, 6]. The strutures may be in the state of ollapse. The riteria for fiber ompressive failure are almost the same as the fiber tensile failure, although the failure phenomenon of fiber ompressive failure is more omplex than the fiber tensile failure. ()

3 Advanes in Materials Siene and Engineering 3 A new fiber failure riterion in ompression is d f =k ( σ ) +γ (k X ) σ X +β ( σ S ), σ 0, where k is a parameter; β is used to onsider the interation form involving in-plane shear, whih an take two values: 0 or ; γ is a parameter whih an take two values: and Failure Criteria for Matrix. Under transverse loading, thebehaviorofthematerialismoreomplexthanthefiber failure. A nonlinear stress-strain response will be observed ifthelaminateisloadedinthetransversediretionorin the shear diretion. The famous Chang-Chang failure riteria onsidered the nonlinear shear effets and introdued a oeffiient to haraterize the nonlinearity. The nonlinear onstitutive relation proposed by Rihard and Blaklok [7] introdued an exponential degradation form. In general, the frature plane is perpendiular to the mid-plane of the ply under transverse tensile load. But the angle of the frature plane will hange if transverse stress inreases under the ompressive load. Generally, for arbon-epoxy and glassepoxyompositesloadedinpuretransverseompression,the frature plane is at an angle of 53 ±3 with respet to the thikness diretion [0, 8]. The LaRC04 riteria onsider the alterative influene of the angle. However, the omputational effiieny dereases if the frature angle hanges frequently, so Maimíetal.assumedthatthefratureangleanonlytake oneofthetwodisretevalues:0 or 53 [0]. Here, two-dimensional matrix failure riteria for omposite materials are expressed by a quadrati polynomial. They will also not inlude the variable of the mutative fratural angle..4.. Matrix Tensile Failure. The riteria for matrix tensile failure involve an interation between the tensile normal and in-plane shear stresses. A new matrix failure riterion in tension is d mt =k 3 ( σ ) +( k Y 3 ) σ + σ /G +3/4ασ 4 T Y T S /G +3/4αS 4, σ 0, where k 3 is a parameter; α is used to onsider the nonlinear interation form involving in-plane shear..4.. Matrix Compressive Failure. The riteria for matrix tensile failure are omplex in Hashin riteria, Hou riteria, and Chang-Chang riteria. The oeffiients of quadrati polynomials are different. Therefore, it is advisable to define thematrixfailureriterioninompressionwithparameters. Parameters should be hosen by experiments. () (3) A new matrix failure riterion in tension is as follows: d m =k 4 ( σ ) +(k Y 4 ) σ + σ /G +3/4ασ 4 Y S /G +3/4αS 4, σ 0, where k 4 is a parameter; α is used to onsider the nonlinear interation form involving in-plane shear. So far, the two-dimensional failure riteria for omposite materials are given. 3. Features of the New Criteria 3.. Unified Expression of Fiber and Matrix Failure Criteria. The new failure riteria for primary damage failure behavior are given above. In fat, the new riteria an be rewritten as a tensor form as follows: d il =k i ( σ i ) +γ X i ( k) il σ i X il +β i ( σ /G +3/4α i σ 4 S /G +3/4α i S 4 ), where k i is a parameter; β i is used to onsider the interation form involving in-plane shear; γ i is a parameter whih an take two values: and 0; is the sign of the absolute value; i is used to denote the damage ategory: fiber or matrix; l is for the stress diretion: tensile or ompression. 3.. Some Speial Forms of the New Criteria. In fat, Chang- Chang riteria, Hou riteria, and so forth are proposed from some speial materials, suh as arbon/epoxy and graphite epoxy. Therefore, the validity of parameters in these riteria is very limited. All these known riteria are just good at simulating some speial material. If more aurate numerial results are needed, the failure riteria of every material should onsider their own material properties and be given one set of speial values of parameters. And all these parameters are from experiments. When some speial values got by experiments are given to four parameters of the new riteria, it will get good numerial results. Meanwhile, when some speial values are given to the four internal parameters, the new riteria will beome these lassial failure riteria suh as Chang-Chang riteria, Linde riteria, Hashin riteria, and Hou riteria in two dimensional ase. When k =, β =, α=0; k =, β =0, γ=0; k 3 =; k 4 =(Y //S 3 ), the new riteria will beome Hashin riteria in two-dimensional ase as follows: d ft =( σ ) +( σ X T S ), σ 0, d f =( σ ), σ X 0, (4) (5)

4 4 Advanes in Materials Siene and Engineering d mt =( σ ) +( σ ), σ Y T S 0, d m =( Y ) ( σ ) +[( Y ) ] σ S 3 Y S 3 Y +( σ ), σ S <0. (6) Table : Failure strengths of T300/94C. X T /Mpa X C /Mpa Y T /Mpa Y C /Mpa S /Mpa Table : Failure strengths of E-glass/Ly556/HT907/DY063 epoxy. X T /Mpa X C /Mpa Y T /Mpa Y C /Mpa S /Mpa G I /J/m When k =, β =, α=0; k =, β =, γ=0; k 3 =; k 4 =(Y //S ), the new riteria will beome Hou riteria in two dimensional ase as follows: d ft = ( σ ) + ( ) σ X +( σ T S ), σ 0, d f =( σ ) +( σ X T S ), σ 0, d mt =( σ ) +( σ ) +( σ 3 ), σ Y T S S 0, 3 d m = [( Y ) ]σ Y S + σ 4S +( σ ), σ S <0. When k =X T /X, β =0, α=0; k =X /X T, β =0, γ=; k 3 =Y T /Y ; k 4 =Y /Y T, the new riteria will beome Linde riteria in two dimensional ase as follows: d f =( )σ X T X + σ X T X, d m =( )σ Y T Y + σ Y T Y +(σ ). S When k =, β =, α=; k =, β =0, γ=0; k 3 =; k 4 =, the new riteria will beome Chang-Chang riteria as follows: d ft =( σ ) + σ /G +3/4ασ 4 X T S /G +3/4αS 4, σ 0, d mt =( σ ) Y T d m =( σ ) Y d f =( σ ), σ X 0, + σ /G +3/4ασ 4 S /G +3/4αS 4, σ 0, + σ /G +3/4ασ 4 S /G +3/4αS 4, σ 0. (7) (8) (9) When k 3 =G I /G II, α=0,(3) willbeonepartofthe failure riteria LaR04, shown in (0). G I,G II are the frature toughness of Mode I and Mode II, respetively, d mt = G I ( σ ) +( G I ) σ +( σ ), σ G II Y T G II Y t S 0. (0) From (5) (9), we find that the new riteria with four parameters have enough apability to math the failure riteria whih are used widely in the damage analysis of omposite materials. The parameters of the new failure riterion an be defined by experimental data to get better numerial results. 4. Identifying Parameters of the New Failure Criteria The proess of identifying parameters of the new failure riteria by experiments is shown here. Biaxial tests of E-glass and arbon fiber reinfored omposite laminates were performed by Soden et al. [9]. In biaxial tests, just σ,σ,andσ anbeobtained.therefore, just parameters k,k,k 3,andk 4 an be identified by experimental data. The other parameters β,β,α,γin the new riteria are not the main items. In fat, γ is used to onsidertheinflueneoflinearitem,σ,tothefibertensile damage; β is used to onsider the influene of shear stress, σ, to the fiber tensile damage; β is used to onsider the influene of shear stress, σ, to the fiber ompressive damage; α is used to onsider the influene of nonlinear fator of material. The orresponding real values of the parameters of β,β,α,γin Chang-Chang riteria, Linde riteria, Hashin riteria, and Hou riteria in two dimensional ase are almost the same. Therefore, based on the four failure riteria above, the parameters, β =, β =, α=0, γ=, are assumed. In order to redue the omplexity of identifying parameters, k,k,k 3,k 4,wejusthoosetheorrespondingoeffiients of those parameters from Hashin, Hou, Linde, and Chang- Chang riteria, suh as k =, X T /X C ; k =, (X /X T ) and X /X T ; k 3 =, Y T /Y and G I /G II ; k 4 =(Y /S ),or Y /Y T. 4.. Experiments. E-glass and arbon fiber reinfored omposite laminates were used in biaxial tests [9]. Their material properties are listed in Tables and,respetively. Biaxial failure for unidiretional T300/94C arbon/epoxy lamina is expressed under ombined longitudinal and

5 Advanes in Materials Siene and Engineering 5 σ (Mpa) σ (Mpa) σ (Mpa) σ (Mpa) k = k =X T /X C Experiment k = k =X C /X T k =(X C /X T ) Experiment Figure : Fiber tensile failure envelope. Figure : Fiber ompressive failure envelope. shear loading (σ versus σ ). The speimens were in the form of axially wound tubes made from prepreg T300/BSL94C arbon/epoxy. The tubes were tested under ombined axial tension or ompression and torsion. All the tubes were end reinfored and grips were used to transmit the torque to the tubes [9]. Biaxial failure for unidiretional E-glass/Ly556/HT907/ DY063 epoxy lamina is expressed under ombined transverse and shear loading (σ versus σ ). The tubes with filament wound by irumferentially wound were 60 mm in internal diameter and mm in thikness and were onstruted from 6% by volume Vetrotes K43 E-glass fiber (Gevetex) rovings and a Ciba-Geigy epoxy resin system Ly556/HT907/DY063 mixed in weight proportions of 00 : 85 : 4. The tubes were ured at 00 C for h and posturved at 50 Cforh[9]. 4.. Determination of Parameters. The parameters, β =, β =, α = 0, γ =, are assumed, whih are based on Chang-Chang riteria, Linde riteria, Hashin riteria, and Hou riteria in two dimensional ase. And () (4)ofthenew riteria should be rewritten as follows For Fiber Tensile Failure. Consider d ft =k ( σ ) +( k X ) σ +( σ ), σ T X T S 0, () where k is a parameter whih an take two values: and X T /X C.Theurvesof()areshowninFigure.Theurveof k =andtheurveofk =X T /X C are both lying between the experimental data, whih are both effetive. However, the urve of k = X T /X C islosetoalargenumberof experimental data above in Figure. Therefore, the urve of k = X T /X C is used for failure analysis of T300/94C arbon/epoxy lamina For Fiber Compressive Failure. Consider d f =k ( σ ) + (k X ) σ + ( σ ), σ X S 0, () σ (Mpa) k 3 = k 3 =Y T /Y C σ (Mpa) k 3 =G I /G II Experiment Figure 3: Matrix tensile failure envelope. where k is a parameter whih an take three values:, (X /X T ),andx /X T.Theurvesof() areshownin Figure. Theurveofk =isoutmost among all urves in Figure, and the experimental data is almost out of the envelope of all urves. The differene of the urve of k = with the data is minimal. Therefore, the value k = should be used for failure analysis of T300/94C arbon/epoxy lamina For Matrix Tensile Failure. Consider d mt =k 3 ( σ ) +( k Y 3 ) σ +( σ ), σ T S 0, (3) where k 3 is a parameter whih an take three values:, Y T /Y, and G I /G II.Theurvesof(3) areshowninfigure 3. The number of the experimental data is just five. The urve of k 3 = Y T /Y C is nethermost. Three data are just in the urve of k 3 = Y T /Y C, and the differene of the urve k 3 =Y T /Y C is minimal. Therefore, the value k 3 =Y T /Y C should be used for failure analysis of E-glass/Ly556/HT907/DY063 epoxy lamina.

6 6 Advanes in Materials Siene and Engineering σ (Mpa) 50 k 4 = k 4 =(Y C /S /) k 4 =Y C /Y T Experiment Figure 4: Matrix ompressive failure envelope For Matrix Compressive Failure. Consider d m =k 4 ( σ ) +(k Y 4 ) σ +( σ ), σ Y S <0, (4) where k 4 is a parameter whih an take three values: (Y /S ),,ory /Y T.Theurvesof(4) areshownin Figure4. Beause of the differene between the strength obtained through experiments and that used in material property, three urves are all not perfet. From Figure 4, the urve of k 4 =Y /Y T is lose to experimental data. Therefore, the value of k 4 =Y /Y T should be used in failure analysis of E-glass/Ly556/HT907/DY063 epoxy lamina. From the disussion above, it is found that it is not enough for the damage riterion that has only one set of the parameters. The failure riteria should be adjusted by experimental data for the materials to meet higher aurate requirement. 5. Conlusions In this work, new two-dimensional failure riteria with four parameters for omposite materials are given. The new riteria inlude fiber failure riteria in tension and ompression and matrix failure riteria in tension and ompression. The new riteria an be adjusted with four parameters to onsider different material properties in engineering. In general, when we use some known riteria in our study, it has to assume that our material properties are the same as that in the study whih proposed the known riteria. Those problems above an be solved by the new riteria proposed in this paper. Meanwhile, when some speial values are given to the four internal parameters, the new riteria have the same expression as some lassial failure riteria, suh as Chang- Chang riteria, Hashin riteria, Linde riteria, and Hou riteria in two dimensional ase, whih are used frequently in the researh of damage proess for the omposite strutures. In general, the four internal parameters should be hosen by experiments. One example that hooses proper values for the parameters is given, whih also validates that there is no one and only set of value for the parameters in any riterion. σ (Mpa) Conflit of Interests The authors delare that there is no onflit of interests regarding the publiation of this paper. Aknowledgment ThisworkisfinaniallysupportedbytheNationalNatural Siene Foundation of China under Grant 070. Referenes [] S. Agrawal, Kalyan Kumar Singh, and P. K. Sarkar, Impat damage on fibre-reinfored polymer matrix omposite a review, Composite Materials,vol.48,no.3,pp.37 33,04. []Y.X.ZhangandC.H.Yang, Reentdevelopmentsinfinite element analysis for laminated omposite plates, Composite Strutures,vol.88,no.,pp.47 57,009. [3] A. C. Orifii, I. Herszberg, and R. S. Thomson, Review of methodologies for omposite material modelling inorporating failure, Composite Strutures,vol.86,no. 3,pp.94 0,008. [4]P.F.LiuandJ.Y.Zheng, Reentdevelopmentsondamage modeling and finite element analysis for omposite laminates: a review, Materials and Design, vol. 3, no. 8, pp , 00. [5] J. G. Mihopoulos, On the reduibility of failure theories for omposite materials, Composite Strutures,vol.86,no. 3,pp , 008. [6] M. J. Hinton and P. D. Soden, Failure riteria for omposite laminates, Composites Siene and Tehnology,vol.58,no.7,pp , 998. [7] P. D. Soden, M. J. Hinton, and A. S. Kaddour, Lamina properties and lay-up onfigurations and loanding onditions of a range fiber reinfored omposite laminates, Composites Siene and Tehnology, vol. 58, pp , 998. [8] P. D. Soden, M. J. Hinton, and A. S. Kaddour, A omparison of the preditive apabilities of urrent failure theories for omposite laminates, Composites Siene and Tehnology, vol. 58, no. 7, pp. 5 54, 998. [9] M. J. Hinton, P. D. Soden, and A. S. Kaddour, Evaluation of failure predition in omposite laminates: bakground to Part B of the exerise, Composites Siene and Tehnology, vol. 6, pp , 00. [0] M. J. Hinton, A. S. Kaddour, and P. D. Soden, A further assessment of the preditive apabilities of urrent failure theories for omposite laminates: omparison with experimental evidene, Composites Siene and Tehnology,vol.64,no.3-4,pp , 004. [] A. S. Kaddour, M. J. Hinton, and P. D. Soden, A omparison of the preditive apabilities of urrent failure theories for omposite laminates: additional ontributions, Composites Siene and Tehnology,vol.64,no.3-4,pp ,004. [] O. Hoffman, The brittle strength of orthotropi material, Composite Materials,vol.9,no.,pp.00 05,975. [3] R. Narayanaswami and H. M. Adelman, Evaluation of the tensor polynominal and Hoffman strength theories for omposite materials, Composite Materials, vol., no. 4, pp , 977. [4] S. W. Tsai and E. M. Wu, A general theory for anisotropi materials, Composite Materials, vol. 5, pp , 97.

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