Nonlinear Analysis of Fiber-Reinforced Composite Laminates Subjected to Uniaxial Compressive Load

Transcription

1 213 SIMULIA Reginal User Meeting Nnlinear Analysis f Fiber-Reinfrced Cmpsite Laminates Subjected t Uniaxial Cmpressive Lad Hsuan-Teh Hu Department f Civil Engineering, Natinal Cheng Kung University, Tainan, Taiwan 71, R.O.C. Wen-Pin Lin Department f Civil Engineering, Chinese Military Academy, Fengshan, Taiwan 83, R.O.C. Lung-Shen Ke Department f Civil Engineering, Natinal Cheng Kung University, Tainan, Taiwan 71, R.O.C. ABSTRACT A nnlinear cnstitutive mdel tgether with a mixed failure criterin fr a single lamina is develped t simulate the behavir f cmpsite laminates under uniaxial cmpressin. In the mdel, fiber and matrix are assumed t behave elastic-plastic and the in-plane shear t behave nnlinear with a variable shear parameter. The damage nset fr individual lamina is detected by a mixed failure criterin, which is cmpsed f the Tsai-Wu criterin and the maximum stress criterin. After damage is taken place within the lamina, fiber and in-plane shear are assumed t exhibit brittle behavir and matrix t exhibit degrading behavir. This material mdel has been tested against experimental data and gd agreement has been btained. 1. Intrductin Due t lightweight and high strength, the use f fiber-reinfrced cmpsite laminate materials (Fig. 1) in aerspace industry, mechanical engineering and applied engineering has increased rapidly in recent years. In numerus cases invlving the design f cmpsite structures, there is a need fr mre refined analysis that takes int accunt phenmena such as prgressive failure and inelastic r nnlinear defrmatin f the cmpsite materials 1,2. Such analysis is required nt nly t predict the defrmatinal respnse, but als t prvide a methd t evaluate the accurate stresses t be used in failure predictins. It is well knwn that unidirectinal fibrus cmpsites exhibit severe nnlinearity in their in-plane shear stress-strain relatins 3. In additin, deviatin frm linearity is als bserved with in-plane transverse lading but the degree f nnlinearity is nt cmparable t that bserved with the in-plane shear,5. Therefre, apprpriate mdeling f the nnlinear behavir f FRP becmes crucial. A significant number f macr-mechanical mdels have been prpsed t represent the cnstitutive relatin f fiber-reinfrced cmpsite materials such as nnlinear elasticity mdels 3,5,6, plasticity mdels 7-11, r damage thery cupled with elasticity 12. In additin, varius failure criteria have als been prpsed t predict the nset f damage in single layer within the fiber-reinfrced cmpsites, i.e. limit theries 13, plynmial theries 1,15, and direct mde determining theries As fr the pst-damage prcess f individual lamina, tw idealized types f failure mdes have been defined in the previus study 8 ; namely, brittle and ductile. Fr the brittle mde, the material is assumed t give up its entire stiffness and strength in the dminant stress directin as the damage is reached, whereas fr the ductile mde the material remains its strength but lses its verall stiffness in the damage directin. Figure 1 Material, element and structure crdinates f fiber reinfrced plastics Obviusly, a ratinally analysis f individual layer within the laminate under lading must include three parts; namely, pre-damage analysis, damage nset determining, and pst-damage 1/1

2 213 SIMULIA Reginal User Meeting analysis. In the pre-damage analysis, the prper cnstitutive mdel f lamina is a key tl t describe the real behavir f each layer within the laminate under lading. In the previus study, it is assumed that the fiber and matrix perfrm as elastic-plastic behavir 8 and the in-plane shear behaves nnlinear with a cnstant shear parameter 17. In this study, it is prpsed that the in-plane shear behaves nnlinearly with a variable shear parameter. The difference between these tw distinct types f shear parameter is investigated in this literature. In the past, the Tsai-Wu failure criterin 1 is the mst cmmn criterin used t determine the damage nset f individual layer. Hwever, Zhu and Sankar 2 and Lin and Hu 21 suggested that the cmbinatin f bth the Tsai-Wu criterin and the maximum stress criterin, which is called the mixed criterin, was a much better criterin fr damage determining f lamina. Thus, in this paper the mixed criterin is emplyed t determine the damage nset f individual layer within the laminate under lading. Fr the pst-damage analysis, a degrading mde fr matrix and brittle mdes fr fiber and in-plane shear are prpsed t simulate the pst damage behavirs f individual lamina. In this paper, a prpsed nnlinear analysis mdel included varius pst damage mdes is intrduced first. Secnd, a material cnstitutive mdel cnsidering the nnlinear in-plane shear behavir with variable shear parameter and the elastic-plastic behavir f fiber and matrix is develped. Third, varius failure criteria and pst damage mdes are reviewed, and a mixed failure criterin and the pst-damage mdes are prpsed. Furth, the laminate gverning equatins are built up t describe the incremental frce-strain relatins f the cmpsite laminates. Then, the ABAQUS finite element prgram 22 is used t carry ut numerical analyses fr laminates with varius cnfiguratins and varius ff-axis lads. Finally, numerical results predicted by the prpsed nnlinear analysis mdel are tested against the experimental data f Petit and Waddups 5 and cmpared with thse predicted by ther failure criteria and pst damage mdes. Fr a single lamina subjected t tensile lading, the stress-strain curves f the prpsed nnlinear analysis mdel are shwn in Figs. 2a and 2c. It is assumed that the material respnse can be represented by bilinear stress-strain curves in the principal material directins, i.e. 1 directin (fiber directin) and 2 directin (transverse directin), f the lamina. Let X yt and X ut be the yield strength and the ultimate strength f the lamina fr tensin in 1 directin, Y yt and Y ut be the yield strength and the ultimate strength f the lamina fr tensin in 2 directin. Fr the elastic regins, i.e. 1 X yt and 2 Y yt, the elastic mduli are dented by E iie (i 1,2). Fr the plastic regins, i.e. Xyt 1 Xut and Yyt 2 Yut, the elastic mduli are dented by E iip (i 1,2). Fr a lamina subjected t cmpressive lading, the stress-strain curves are shwn in Fig. 2b and 2d. It is bvius that X yc and X uc are the yield strength and the ultimate strength f the lamina fr cmpressin in 1 directin and that Y yc and Y uc are the yield strength and the ultimate strength f the lamina fr cmpressin in 2 directin. Let S be the ultimate in-plane shear strength. It is assumed that the in-plane shear in 1-2 directin can be mdeled by a nnlinear stress-strain curve as shwn in Fig. 2e. 2. Nnlinear Analysis Mdel 2.1 Prpsed stress-strain curves and pst damage mdel Figure 2 Stress-strain curves f the prpsed nnlinear failure mdel 2/1

3 213 SIMULIA Reginal User Meeting Fr the pst-damage regin, the strengths are drpped t zer (brittle mdes) in 1 directin (Figs. 2a and 2b) and 1-2 directin (Fig. 2e). Hwever, the elastic stiffness is assumed t have a negative mdulus E22 f (degrading mde) in 2 directin (Figs. 2c and 2d). This means that the damaged lamina unlads in the transverse directin thrugh a negative tangent mdulus until n lad remains in the lamina. 2.2 Nnlinear cnstitutive mdel f the lamina Fr fiber-cmpsite laminate materials, each lamina can be cnsidered as an rthtrpic layer in a plane stress cnditin. Taking int accunt the elastic-plastic behavirs in the 1 directin and 2 directin and the nnlinear behavir n the 1-2 plane within the lamina, the strain-stress relatins fr an rthtrpic lamina in the material crdinates (1,2) can be written as 3 : 1 21 E11 E E11 E G12 2 S (1) 12 where 1, 2, and 12 represent the strains in 1 directin, 2 directin and 1-2 plane, respectively. 1, 2 and 12 dente the stresses in 1-directin, 2-directin and 1-2 plane, respectively. The 12 and 21 are Pissn s ratis and E 11 and E 22 are the elastic mduli in 1 directin and 2 directin. If the lamina is in the elastic stage in 1 directin r 2 directin, then E11 E11 e r E22 E22 e. If the lamina is in the plastic stage in 1 directin r 2 directin, then E 11 E 11p r E22 E22 p. The G 12 is the shear mdulus and S 6666 is a shear parameter t accunt fr the in-plane shear nnlinearity. The value f S 6666 can be determined by a curve fit t pure shear test data. The incremental stress-strain relatins fr a nnlinear rthtrpic lamina can be given as fllws: { '} [Q ' 1 ] {'} (2) { ' } [Q ' ' t 2] { t} (3) T where { '} { 1, 2, 12}, ' T T { t} { 13, 23}, {'} { 1, 2, 12}, T { ' t} { 13, 23} and E11 12E ' 21 E11 E Q 22 1 () /G12 3S G13 Q ' 2 2G (5) 23 The terms 1 and 2 are the shear crrectin factrs and are taken t be.83 in this study 23. It is assumed that the transverse shear stresses always behave linearly and d nt affect the nnlinear in-plane behavir f individual lamina. 3. Failure Criterin and Degradatin f Stiffness 3.1 Review f failure criteria As previusly mentined, failure criteria fall int three basic categries: (1) limit theries 13, (2) plynmial theries 1,15, and (3) direct mde determining theries Amng them, the maximum stress criterin 13, the Tsai-Wu failure criterin 1, the Chang failure criterin 17, the Rtem failure criterin 18 and the Edge failure criterin 19 are selected t be reviewed and numerical results based n these failure criteria are cmpared with each ther Maximum stress criterin The maximum stress criterin is the dminant member f the limit failure thery categry 13. Fr the plane stress cnditin, the maximum stress criterin fr an rthtrpic material can be expressed as fllws: 1 1 r 1 1 (6) X ut X uc 2 1 r 2 1 (7) Y ut Y uc 12 1 (8) S Tsai-Wu failure criterin The Tsai-Wu failure criterin has a general 3/1

4 213 SIMULIA Reginal User Meeting nature, because it cntains almst all ther plynmial theries as special cases. Under the plane stress cnditin, the Tsai-Wu failure criterin has the fllwing frm 1 : F1 1 F 22F F12 1 2F 222F (9a) where F1, F11, Xut Xuc XutXuc F2, F22 Yut Y uc Yut Y, uc 1 F66. (9b) 2 S The stress interactin term F 12 in Eq. (9a) is difficult t be determined and Narayanaswami and Adelman 2 suggested that F 12 culd be set equal t zer fr practical engineering applicatins. Therefre, F 12 = is used in this study Chang failure criterin Chang failure criterin 17 is a direct mde determining failure criterin, which prvides separate failure equatin fr each mde f failure. In a plane stress space, Chang uses five distinct plynmials t describe five mdes f failure, which are listed belw: (1) Fiber breakage mde: 1 1 (1) X ut (2) Fiber buckling failure mde: 1 1 (11) X uc (3) Matrix tensile cracking mde: S G12 1 (12) Y 2 ut S 3 S6666S 2G12 () Matrix cmpressin failure mde: S G12 1 (13) Y 2 uc S 3 S6666S 2G12 (5) Fiber-matrix shearing failure mde: S G12 1 (1) X 2 ut S 3 S6666S 2G12 r S G12 1 (15) X 2 uc S 3 S6666S 2G12 It shuld be nted that S 6666 is a cnstant in the Chang failure criterin Rtem failure criterin Fr the Rtem failure criterin 18, the failure criterin cmbines tw separate criteria, namely a fiber failure criterin and a matrix failure criterin. The fibers, being stiffer and strnger than the matrix, can nly fail by lads acting in the axial directin and its failure criterin can be expressed in the fllwing frms: 1 X ut r 1 X uc (16) The matrix failure is cntrlled by the plane stress cmpnents and takes the frm m Sm Y j S (17) where m indicates the lngitudinal stress f matrix and S m dentes the ultimate strength f matrix. The Yj Yut if 2 and Yj Yuc if 2. On the left hand side f Eq. (17), since the cntributin f the first term is much less than the ther tw terms, the first term f Eq. (17) usually may be neglected Edge failure criterin The Edge failure criterin 19 divides the failure f lamina int initial failure and final failure. In general, transverse tensin and cmpressin failures are regarded as initial failure and fiber failure and in-plane shear failure are regarded as final failure. Failure is predicted if the fllwing cnditins are btained. (1) Initial failure (a) Transverse tensin 2 Y ut (18) (b) Transverse cmpressin 2 Y uc (19) (c) Cmbined shear and transverse tensin /1

5 213 SIMULIA Reginal User Meeting (2) Yut S (d) Cmbined shear and transverse cmpressin (21) Yuc S (2) Finalfailure (a) Lngitudinal tensin 1 X ut (22) (b) Lngitudinal cmpressin 1 X uc (23) (c) In-plane shear 12 S (2) (d) Cmbined lngitudinal tensin and shear (25) Xut S (e) Cmbined lngitudinal cmpressin and shear (26) Xuc S 3.2 Prpsed mixed failure criterin Althugh the Tsai-Wu failure criterin is widely used in determining the damage nset f a lamina, there are sme drawbacks with it. Amng them is the fact that the failure stress f fiber in a lamina may exceed the strength f material fr the case f symmetric angle-ply laminates with small fiber angle (say 2 ) subjected t ff-axis tensin 21. In rder t eliminate this unreasnable phenmenn, the limitatin f the maximum stress f the lamina in fiber directin is added int the Tsai-Wu failure criterin t btain a mixed failure criterin, which has the fllwing frmulatins: F1 1 F 22 F 11 1 F 222 F (27) and 1 1 r 1 1 (28) X ut X uc 3.3 Nrmalized failure stresses and failure cntributin The Tsai-Wu failure criterin and the prpsed mixed failure criterin cnsider the cupling effect f in-plane stresses, 1, 2 and 12, in the lamina when the cllapse ccurs. In rder t figure ut the individual stress rati in the lamina, the nrmalized failure stresses are defined, which represent the stress ratis (failure stresses/crrespnding strengths) in the lamina fr varius stresses at the nset f cllapse. The expressins fr the nrmalized failure stresses are described as fllw: ( 11f 11f ) n r ( 11f 11f ) n (29a) X ut X uc ( 22f 22f ) n r ( 22f 22f ) n (29b) Y ut Y uc ( 12f 12f ) n (29c) S where ( 11f) n, ( 22f) n and ( 12f) n dente the nrmalized failure stresses in 1 directin, 2 directin, and 1-2 plane f the lamina, respectively. The 11f, 22f and 12f are the stresses f the lamina in 1 directin, 2 directin and 1-2 plane at the nset f failure. 3. Degradatin mdels Upn damage within the lamina ccurring, the material prperties begin t degrade. Material degradatin within the damaged area is evaluated based n the mde f failure predicted by the failure criterin. Therefre, the residual stiffnesses f cmpsites strngly depend n the mde f failure in each layer. Accrding t the literature, the degradatin mdels fr each layer can be separated int three idealized types f failure mdes named as brittle, ductile 8 and degrading mde 21. Fr the brittle mde, the material is assumed t lse its entire stiffness and strength in the dminant stress directin. Fr the ductile mde the material retains its strength but lses all f its stiffness in the failure directin. Fr the degrading mde the material is assumed t lse its stiffness and strength in the failure directin gradually until the stress in that directin is reduced t zer. Fr the Tsai-Wu failure criterin, it cannt distinguish the failure mdes and cannt cnsider the pst failure cnditin. Althugh the maximum stress criterin, the Rtem failure criterin and the Edge failure criterin can distinguish the failure mdes f material, their failure mdes are all assumed t be brittle types in this investigatin Chang's degradatin mdel Fr the Chang failure thery, it nt nly can distinguish the failure mdes but als can cnsider 5/1

6 213 SIMULIA Reginal User Meeting the pst failure cnditins. The Chang s degradatin mdels 17 fr each lamina are described as fllw: (1) Fr the matrix tensile r cmpressive failure mde, the in-plane prperties in the failed layer are reduced as E11 Q ' 1 (3) 1 1/G S (2) Fr the fiber breakage r buckling failure mde, the material in that regin cannt sustain furthermre lad. Thus the material prperties fr the failed layer and all ther layers are reduced t zer. Q ' 1 (31) (3) Fr the fiber-matrix shearing failure mde, the material can still carry lad in the fiber directin and in the matrix directin, but shear lads can n lnger be carried. This is mdeled by reducing the shear mdulus and the Pissn's ratis 12 and 21 tzer. E11 Q ' 1 E22 (32) 3..2 Prpsed degradatin mdel In this investigate, it is prpsed that the pst damage mde are idealized as the brittle behavir fr 1 and 12 and the degrading behavir fr 2 (Fig. 2). The fllwing three rules are used t determine whether the ply failure is caused by matrix fracture, shear failure, r fiber breakage r buckling 26 : (1) If a ply fails in the cnditin f Xuc 1 Xut and S12 S, the damage is assumed t be matrix induced. Cnsequently, the degradatin f transverse stiffness ccurs. Due t the interlck actin with the neighbring plies, the damaged ply gradually lses its capability t supprt transverse stress, until the fracture in shear r the breakage r buckling in fiber n the same ply. But, the lamina remains t carry the lngitudinal and shear stresses. In this case, the cnstitutive matrix f the lamina becmes E11 Q ' 1 E22f (33) 1 1/G S where E22 f is a negative tangent mdulus in transverse directin f lamina after matrix damage. In the prpsed mdel, the shear parameter S 6666 has variable value. (2) If the ply fails in the cnditin f Xuc 1 Xut, and 12 S r 12 S, the damage is assumed t be shear induced. Cnsequently, the damaged lamina lses its capability t supprt transverse and shear stresses, but remains t carry lngitudinal stress. In this case, the cnstitutive matrix f the lamina becmes E11 Q ' 1 (3) (3) If the ply fails with 1 X ut r 1 X uc, the ply failure is caused by the fiber breakage r buckling and a ttal ply rupture is assumed. Thus, the cnstitutive matrix f the lamina becmes Q ' 1 (35). Laminate Gverning Equatins The frging nnlinear failure analysis mdel fr fiber-reinfrced cmpsite lamina can be cmbined with classical laminatin thery t frm the fllwing incremental laminate frce-strain relatins: n Δ N Qi t i (36) i1 where N N x,n y,nxy T and T { x, y, xy} are the vectrs f the incremental in-plane frces and the incremental strains in the verall laminate crdinate system (x, y), respectively. The term t i is the thickness f the i-th layer, n is the ttal number f layers. The matrix [Q] i stands fr cnstitutive matrix fr the i-th layer and can be btained by prper rtatin f the [Q ' 1 ] matrix f that layer 26. 6/1

7 213 SIMULIA Reginal User Meeting 5. Numerical Analysis 5.1 Numerical simulatins and material prperties The afrementined nnlinear cnstitutive mdel cmbined with varius failure criteria and varius pst damage mdes fr cmpsite materials are implemented int a FORTRAN subrutine and linked t the Abaqus finite element prgram 22. The analyzed laminate plate is simply supprted arund all edges and subjected t uniaxial cmpressive lad in lngitudinal directin nly (Fig. 3). The length f the plate L is equal t 12 cm and the width f the plate W is equal t 2 cm. The laminate plate cntains plies with the thickness t f each ply equal t.116 mm. The laminae are assumed t be perfectly bunded and n slipping ccurs within the laminate. Since the stress field is unifrm thrughut the laminate plate, nly ne eight-nde isparametric shell elements with six degrees f freedm per nde (three displacements and three rtatins) is used t mdel the plate. The reduced integratin rule tgether with hurglass stiffness cntrl is emplyed t frmulate the element stiffness matrix 22. Figure 3 Gemetry and bundary cnditins f cmpsite laminates In Abaqus prgram, stresses and strains in material crdinates (1,2,3) are calculated at each incremental step, and evaluated by the failure criteria t determine the ccurrence f failure and the mde f failure. Mechanical prperties f each lamina in the damaged area are reduced, accrding t prper degradatin mdels. Stresses and strains are then recalculated t determine any additinal damage as a result f stress redistributin at the same lad. This prcedure cntinues until n additinal damage is fund. Then, the next increment f lad is pursued. The final cllapse lad is determined when the cmpsite plates cannt sustain any additinal lad. In Abaqus prgram, stresses and strains in material crdinates (1,2,3) are calculated at each incremental step, and evaluated by the failure criteria t determine the ccurrence f failure and the mde f failure. Mechanical prperties f each lamina in the damaged area are reduced, accrding t prper degradatin mdels. Stresses and strains are then recalculated t determine any additinal damage as a result f stress redistributin at the same lad. This prcedure cntinues until n additinal damage is fund. Then, the next increment f lad is pursued. The final cllapse lad is determined when the cmpsite plates cannt sustain any additinal lad. In rder t verify the prpsed nnlinear failure analysis mdel, numerical results generated frm the mdel are cmpared with the test data f Brn/Epxy cmpsites 5. The material prperties f Brn/Epxy cmpsites used in the analysis are E11e 27 GPa, E11p 18 GPa, E22e 21.2 GPa, E22p 15.9 GPa, E22f GPa, G GPa, 12.3, 3 S GPa (cnstant) r -3 S exp(- 12 /.337) GPa (variable), Xyt 828 MPa, Xyc 136 MPa, Yyt 57.9 MPa, Yyc 97.3 MPa, Xut 137 MPa, Xuc 2787 MPa, Yut 86.3 MPa, Yuc 262 MPa, S = MPa. It shuld be nted that the shear parameter S 6666 has tw types, a cnstant type and a variable type. The variable shear parameter is btained by curve fitting frm the pure shear test data, Verificatin f the prpsed nnlinear cnstitutive mdel It is necessary t assure that the prpsed cnstitutive mdel can crrectly simulate the stress-strain relatins in the principal directins and in pure shear f a lamina befre predicting the mechanical behavir and failure stresses f cmpsite laminates under varius lading. Figure shws the numerical results fr a single lamina subjected t pure shear lading against the experimental data 5. It is bvius that the shear stress-strain curve simulated by the prpsed cnstitutive mdel with variable S 6666 mdel agrees with the test data well and is much better than thse with linear shear mdel ( S 6666 =) and nnlinear shear mdel with S 6666 beingcnstant. 7/1

8 213 SIMULIA Reginal User Meeting 12 (MPa) Linear shear 2 Nnlinear shear, cnstant S 6666 Nnlinear shear, variable S Figure Pure shear stress-strain curve fr Brn/Epxy lamina x (MPa) Cnstant S 6666 Variable S x Figure 5 Uniaxial tensile stress-strain curve fr [ 5] s Brn/Epxylaminate Prpsed mdel Figure 6 Lngitudinal cmpressive stress-strain curve fr Brn/Epxy lamina The results simulated by the variable S 6666 mdel and the cnstant S 6666 mdel fr a [ 5/ 5] s laminate subjected t uniaxial tensin lading against the experimental data 5 are shwn in Fig. 5. It can be seen that the result simulated by the prpsed nnlinear shear mdel with variable S 6666 exhibits better fit with the test data than that simulated by the nnlinear shear mdel with cnstant S Prpsed mdel Figure 7 Transverse cmpressive stress-strain curve fr Brn/Epxy lamina Figures 6 and 7 illustrate the numerical results fr a single lamina subjected t uniaxial lngitudinal cmpressive lading and uniaxial transverse cmpressive lading against the experiment data 5. It can be seen that the prpsed elastic-plastic behavir in the lngitudinal directin and transverse directin f the lamina exhibit quite gd crrelatin with the experimental dada. As the result, the prpsed material mdel with variable S 6666 is prved t mdel the nnlinear behavir f cmpsite laminates adequately. 5.3 Cmparisns amng varius pst failure mdes The lad-defrmatin behavir f a cmpsite laminate is greatly affected by the stress-strain behavir f individual layer within the laminate and the ultimate strength f a cmpsite laminate is greatly cntrlled by the pst-damage mde f damaged lamina within the laminate. In rder t verify the prpsed pst-damage mde in transverse directin f lamina is a suitable ne, three idealized pst failure mdes, brittle, ductile and degrading mdes, are taken int accunt. Figure 8 shws the uniaxial cmpressive stress-strain curves predicted by varius matrix pst failure mdes fr [ 2] s angle ply laminate against experimental data 5. The ultimate lads predicted by the brittle mde and the degrading mde are exactly the same and equal t.82 GPa, which is clse t the experimental data.75 GPa. Fr the ductile mde, the predicted ultimate lad is significantly verestimated and is misleading. 8/1

9 213 SIMULIA Reginal User Meeting x Brittle Ductile Degrading x Figure 8 Uniaxial cmp. stress-strain curves predicted by varius matrix pst failure mdes fr [ 2] s laminate x Brittle Ductile Degrading x Figure 9 Uniaxial cmp. stress-strain curves predicted by varius matrix pst failure mdes fr [ 3] s laminate x Brittle Ductile Degrading x Figure 1 Uniaxial cmp. stress-strain curves predicted by varius matrix pst failure mdes fr [ / 9] s laminate Figure 9 shws the uniaxial cmpressive stress-strain curves predicted by varius matrix pst failure mdes fr [ 3] s angle ply laminate against experimental data 5. The ultimate lad predicted by the degrading mde is.36 GPa, which is clse t the experimental data.3 GPa. Fr brittle mde, instability exists in the fiber directin immediately after the failure f the lamina ccurs. As the result, the cmpressive strain in the laminate is suddenly and significantly increased which is cntradicted t the experimental data. When the stress redistributin in the laminate is cmpleted, the stress in the laminate starts t increase again and the predicted ultimate lad is. GPa, which is verestimated. Fr the ductile mde, the predicted ultimate lad is again significantly verestimated. Hence, it is apprpriated and justified t use the degrading mde t mdel the pst failure f a lamina. Figure 1 shws the uniaxial cmpressive stress-strain curves predicted by varius matrix pst failure mdes fr [ / 9] s crss ply laminate against experimental data 5. While the ultimate lad predicted by the brittle mde is 1.5 GPa, the ultimate lads predicted by the ductile mde and by the degrading mde are the same and equal t 1.5 GPa. It can be seen that all these predicted ultimate lads are clse t the experimental data 1.7 GPa. x Brittle Ductile Degrading x Figure 11 Uniaxial cmp. stress-strain curves predicted by varius matrix pst failure mdes fr [15 / 75] s laminate Figures 11, 12, 13 shw the uniaxial cmpressive stress-strain curves predicted by varius matrix pst failure mdes fr [15 / 75] s, [3 / 6] s and [ 5] s crss ply laminates. Since there are n experimental data t cmpare, we can nly see the trends. It can be bserved that the predicted stress-strain curves as well as the ultimate lads with different pst failure mdes are exactly the same fr all these three crss ply laminates. Fr 9/1

10 213 SIMULIA Reginal User Meeting laminates with the [ / ( 9)] s crss ply layup, as the fiber angles are mre deviated frm and 9 degrees, the stress-strain curves f the laminates exhibit mre nnlinear behavir. This is due t the nnlinear in-plane shear effect. In additin, as the fiber angles are mre deviated frm and 9 degrees, the ultimate lads f the laminates becme lwer. x Brittle Ductile Degrading x Figure 12 Uniaxial cmp. stress-strain curves predicted by varius matrix pst failure mdes fr [3 / 6] s laminate x Brittle Ductile Degrading x Figure 13 Uniaxial cmp. stress-strain curves predicted by varius matrix pst failure mdes fr [ 5] s laminate 5. Cmparisns amng varius failure criteria The Tsai-Wu failure criterin [1] is the mst ppular failure criterin fr cmpsite laminate and has been extensively used in literature. As mentined previusly, with the Tsai-Wu failure criterin, the failure stress f fiber in a lamina may exceed the strength f material fr the case f symmetric angle-ply laminates with small fiber angle subjected t ff-axis tensin 21. Hence, it is replaced by the prpsed mixed failure criterin, i.e. the Tsai-Wu failure criterin cmbined with the maximum stress criterin. In this sectin, the prpsed mixed failure criterin is cmpared with ther ppular criteria such as the Rtem criterin, the Edge criterin and the Chang criterin against experimental data 5. x Rtem criterin Edge criterin Chang cruterin Mixed criterin x Figure 1 Uniaxial cmpressive stress-strain curves predicted by varius failure criteria fr [ 2] s laminate x Rtem criterin Edge criterin Chang criterin Mixed criterin x Figure 15 Uniaxial cmpressive stress-strain curves predicted by varius failure criteria fr [ 3] s laminate Figure 1 shws the uniaxial cmpressive stress-strain curves predicted by varius failure criteria fr [ 2] s angle ply laminate against experimental data 5. The predicted failure lads are -.82 GPa fr the Rtem criterin, -.78 GPa fr the Edge criterin, -.81 GPa fr the Chang criterin and -.82 GPa fr the prpsed mixed criterin and the experimental failure lad is.75 GPa. All the failure lads and the stiffnesses f the laminate predicted by the fur criteria are reasnably clse t the experimental data. Figure 15 shws the uniaxial cmpressive stress-strain curves predicted by varius failure 1/1

11 213 SIMULIA Reginal User Meeting criteria fr [ 3] s angle ply laminate against experimental data 5. The predicted failure lads are -.1 GPa fr the Rtem criterin, -.36 GPa fr the Edge criterin, -.37 GPa fr the Chang criterin and -.36 GPa fr the mixed criterin and the experimental failure lad is -.3 GPa. It can be seen that the Rtem criterin verestimates the stiffness and the failure lad f the laminate than the ther three criteria. Althugh, the failure lads f the laminate predicted by the ther three criteria are als higher than the experimental data, the stiffnesses f the laminate predicted by the ther three criteria are pretty well. x x Rtem criterin Edge criterin Chang criterin Mixed criterin Figure 16 Uniaxial cmpressive stress-strain curves predicted by varius failure criteria fr [ 6] s laminate x Rtem criterin Edge criterin Chang criterin Mixed criterin x Figure 17 Uniaxial cmpressive stress-strain curves predicted by varius failure criteria fr [ / 9] s laminate Figure 16 shws the uniaxial cmpressive stress-strain curves predicted by varius failure criteria fr [ 6] s angle ply laminate against experimental data 5. The predicted failure lads are -.23 GPa fr the Rtem criterin, -.23 GPa fr the Edge criterin, -.25 GPa fr the Chang criterin and -.25 GPa fr the mixed criterin and the experimental failure lad is -.39 GPa. Althugh, the failure lads f the laminate predicted by all criteria are significantly underestimated than the experimental data, the stress-strain curve predicted by the Rtem criterin still deviates frm thse predicted by the ther three criteria. Figure 17 shws the uniaxial cmpressive stress-strain curves predicted by varius failure criteria fr [ / 9] s crss ply laminate against experimental data 5. The predicted failure lads are -1.5 GPa fr the Rtem criterin, -1.5 GPa fr the Edge criterin, -1.5 GPa fr the Chang criterin and -1.5 GPa fr the mixed criterin and the experimental failure lad is -1.7 GPa. Again all the failure lads and the stiffnesses f the laminate predicted by the fur criteria are reasnably clse t the experimental data. Failure stress xf (MPa) Chang criterin Edge criterin Rtem criterin Tsai-Wu criterin Mixed criterin (degrees) Figure 18 Uniaxial cmpressive failure stresses predicted by varius failure criteria fr [ ] s angle ply laminates Figure 18 shws the uniaxial cmpressive failure stresses xf predicted by varius failure criteria fr [ ] s angle ply laminates against experimental data 5. Figure 19 shws the nrmalized material failure stresses predicted by the varius failure criteria fr [ ] s angle ply laminates subjected t uniaxial cmpressive lading. It can be bserved that the failure stress xf and the nrmalized failure stresses predicted by the Tsai-Wu failure criterin and the prpsed mixed failure criterin are almst the same nly with a little discrepancy in the neighbrhd f the angle 5. It shuld be nted that arund the regin 5, the predicted nrmalized failure stress ( 12) f f the Tsai-Wu failure criterin exceeds the 11/1

12 213 SIMULIA Reginal User Meeting value 1 (Fig. 19a), which is unreasnable and unlike t ccur. That is the reasn why the authrs prpse the mixed failure criterin, as the cases f laminates subjected t uniaxial tensile lads 21. Nrmalized failure stresses Nrmalized failure stresses (a) ( 22f ) n ( 11f ) n ( 12f ) n -1. Mixed criterin Tsai-Wu criterin Chang criterin (degrees) (b) ( 22f ) n ( 11f ) n ( 12f ) n -1. Mixed criterin Edge criterin Rtem criterin (degrees) Figure 19 Nrmalized material failure stresses predicted by varius failure criteria fr [ ] s angle ply laminates subjected t uniaxial cmpressive lading Cmparing the Chang failure criterin with the prpsed mixed criterin, we can find that the failure stress xf and the nrmalized failure stresses predicted with these tw criteria are abut the same except the regin 15. This is because the Chang failure criterin des nt cnsider the jint effect due t axial stress, transverse stress and in-plane shear stress acting simultaneusly. When 1, we can see that the failure f the angle ply laminate is due t axial failure stress ( 11f) n fr the Chang Criterin (Fig. 19a). If the jint effect f axial stress, transverse stress and in-plane shear stress is cnsidered, such as the Tsai-Wu failure criterin, the predicted failure stress xf and the nrmalized failure stresses (in abslute values) will nt be s high within 15 regin. Fr the Edge criterin, it can be seen that its predicted failure stress xf is usually lwer than thse predicted by ther criteria (Fig. 18). The reasn caused that is its predicted stresses 1, 2 and 12 are smaller (in abslute value) than thse predicted by ther criteria, which can be clear seen in Fig. 19b. Fr the Rtem criterin, the predicted failure stress xf is usually higher than thse predicted by ther failure criteria when 5 and lwer than thse predicted by ther failure criteria when 5 (Fig. 18). When 15, the Rtem criterin, same as the Chang criterin, des nt cnsider the jint effect due t axial stress, transverse stress and in-plane shear stress acting simultaneusly. Hence, it has the similar trend as the Chang criterin. When 15 8, the predicted transverse stress 2 by the Rtem criterin is lwer than that by mixed criterin and the predicted in-plane shear stress 12 by the Rtem criterin is higher than that by mixed criterin (Fig. 19b). This is due t the Rtem mdel treating the in-plane shear having a linear behavir. Therefre, it verestimates the in-plane shear stress. After the failure criterin is emplyed, the transverse stress is then underestimated. 6. Cnclusins This paper presents a material cnstitutive mdel fr simulating the mechanical respnse and predicting the ultimate strength f varius symmetrical cmpsite laminates subjected t uniaxial cmpressive lad. The mdel is cmpsed f three parts: (1) nnlinear cnstitutive mdel, (2) mixed failure criterin, and (3) pst-damage mde. In the nnlinear cnstitutive mdel, the fiber and matrix are simulated by elastic-plastic behavir and the in-plane shear is simulated by nnlinear behavir with variable shear parameter, which is a functin f in-plane shear strain. The mixed failure criterin is cmpsed f the Tsai-Wu failure criterin and the maximum stress criterin t determine the damage nset f a lamina. The mixed failure criterin can avided the verestimatin f stresses predicted by the Tsai-Wu failure criterin in a lamina. In the pst-damage regins, the fiber and the in-plane shear are simulated by a brittle mde and the matrix by a degrading mde. The validity f the cnstitutive mdel has been verified against experimental data 5 and reasnable accuracy has been achieved. It has been shwn that fr laminates with the 12/1

13 213 SIMULIA Reginal User Meeting [ / ( 9)] s crss ply layup and subjected t uniaxial cmpressive lad, as the fiber angles are mre deviated frm and 9 degrees, the stress-strain curves f the laminates exhibit mre nnlinear behavir and the ultimate lads f the laminates becme lwer. The prpsed cnstitutive mdel has been cmpared with ther ppular failure criteria fr [ ] s angle ply laminate subjected t uniaxial cmpressive lad. Fr the Chang criterin, the predicted failure stress xf is usually higher than the mixed criterin and the Tsai-Wu criterin when 15. Fr the Edge criterin, the predicted failure stress xf is usually lwer than the mixed criterin and the Tsai-Wu criterin. Fr the Rtem criterin, the predicted failure stress xf is usually higher than thse predicted by the mixed criterin and the Tsai-Wu criterin when 5 and lwer than thse predicted by the mixed criterin and the Tsai-Wu criterin when References 1 Hintn M. J., and Sden, P. D., Predicting Failure in Cmpsite Laminates: Backgrund t the Exercise, Cmpsite Science and Technlgy, Vl. 58, N. 7, 1998, pp Anyfantis, K. N., and Tsuvalis, N. G., Pst Buckling Prgressive Failure Analysis f Cmpsite Laminated Stiffened Panels, Applied Cmpsite Materials, Vl. 19, N. 3-, 212, pp Hahn, H. T., and Tsai, S. W., Nnlinear Elastic Behavir f Unidirectinal Cmpsite Laminae, Jurnal f Cmpsite Materials, Vl. 7, N. 1, 1973, pp Jnes, R. M., and Mrgan, H. S., Analysis f Nnlinear Stress-Strain Behavir f Fiber-Reinfrced Cmpsite Materials, AIAA Jurnal, Vl. 15, N., 12, 1977, pp Petit, P. H., and Waddups, M. E., A Methd f Predicting the Nnlinear Behavir f Laminated Cmpsites, Jurnal f Cmpsite Materials, Vl. 3, N. 1, 1969, pp Bgetti, T. A., Hppel, C. P. R., Harik, V. M., Newill, J. F., and Burns, B. P., Predicting the Nnlinear Respnse and Prgressive Failure f Cmpsite Laminates, Cmpsites Science and Technlgy, Vl. 6, N. 3-, 2, pp Sun, C. T., and Chen J. L., A Simple Flw Rule fr Characterizing Nnlinear Behavir f Fiber Cmpsite, Jurnal f Cmpsite Materials, Vl. 23, N. 1, 1989, pp Vaziri, R., Olsn, M. D., and Andersn, D. L., A Plasticity-Based Cnstitutive Mdel fr Fibre-Reinfrced Cmpsite Laminates, Jurnal f Cmpsite Materials, Vl. 25, N. 5, 1991, pp Nanda, A., and Kuppusamy, T., Three-Dimensinal Elastic-Plastic Analysis f Laminated Cmpsite Plates, Cmpsite Structures, Vl. 17, N. 3, 1991, pp Griffin, O. H., Kamat, M. P., and Herakvich, C. T., Three-Dimensinal Inelastic Finite Element Analysis f Laminated Cmpsites, Jurnal f Cmpsite Materials, Vl. 15, N. 6, 1981, pp Kenaga, D., Dyle, J. F., and Sun, C. T., The Characterizatin f Brn/Aluminum Cmpsite in the Nnlinear Range as an Orthtrpic Elastic-Plastic Material, Jurnal f Cmpsite Materials, Vl. 21, N. 6, 1987, pp Allen, H., Harris, C. E., and Grves, S. E., A Thermmechanical Cnstitutive Thery fr Elastic Cmpsites with Distributed Damage-I, Theretical Develpment, Internatinal Jurnal f Slids Structures, Vl. 23, N. 9, 1987, pp Rwlands, R. E., Strength (Failure) Theries and Their al Crrelatin, Failure Mechanics f Cmpsites, Edited by Sih, G. C., and Skudra, A. M., Elsevier, Amsterdam, 1985, pp Tsai, S. W., and Wu, E. M., A General Thery f Strength fr Anistrpic Materials, Jurnal f Cmpsite Materials, Vl. 5, N. 1, 1971, pp Hffman, O., The Brittle Strength f Orthtrpic Materials, Jurnal f Cmpsite Materials, Vl. 1, N. 2, 1967, pp Lee, J. D., Three Dimensinal Finite Element Analysis f Damage Accumulatin in Cmpsite Laminate, Cmputers & Structures, Vl. 15, N. 3, 1982, pp Chang, F. K, and Lessard, L. B., Damage Tlerance f Laminated Cmpsite Cntaining an Open Hle and Subjected t Cmpressive Ladings: Part I-Analysis, Jurnal f Cmpsite Materials, Vl. 25, N. 1, 1991, pp Rtem, A., Predictin f Laminate Failure with the Rtem Failure Criterin, Cmpsites Science and Technlgy, Vl. 58, N. 7, 1998, pp Edge, E. C., Stress Based Grant-Sanders Methd fr Predicting Failure f Cmpsite Laminates, Cmpsites Science and Technlgy, Vl. 58, N. 7, 1998, pp Zhu, H., and Sankar, B. V., Evaluatin f Failure Criteria fr Fiber Cmpsites Using Finite Element Micrmechanics, Jurnal f Cmpsite Materials, Vl. 32, N. 8, 1998, pp Lin, W.-P., and Hu, H.-T., Nnlinear Analysis f Fiber-Reinfrced Cmpsite Laminates Subjected t Uniaxial Tensile Lad, Jurnal f Cmpsite Materials, Vl. 36, N. 12, 22, pp Abaqus, Inc., Abaqus Analysis User's Manuals and Example Prblems Manuals, Versin 6.11, Prvidence, Rhde Island, Mindlin, R. D., Influence f Rtatry Inertia and Shear n Flexural Mtins f Istrpic Elastic Plate, Jurnal f Applied Mechanics, Vl. 18, 1951, pp /1

14 213 SIMULIA Reginal User Meeting 2 Narayanaswami, R., Adelman, H. M., Evaluatin f the Tensr Plynmial and Hffman Strength Theries fr Cmpsite Materials, Jurnal f Cmpsite Materials, Vl. 11, N., 1977, pp Lin, W.-P., Nnlinear Failure Analysis Mdel fr Fiber-Reinfrced Cmpsite Laminate under Uniaxial and Biaxial Tensile Lads, Ph.D. Thesis, Department f Civil Engineering, Natinal Cheng Kung University, Tainan, Taiwan, R.O.C., Hu, H.-T., Influence f In-plane Shear Nnlinearity n Buckling and Pstbuckling Respnses f Cmpsite Laminate Plates and Shells, Jurnal f Cmpsite Materials, Vl. 27, N. 2, 1993, pp /1