Applied Mechanics and Materials Vols. 592-594 (2014) pp 1151-1154 © (2014) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.592-594.1151 Progressive Failure Analysis of Laminated Composite Plate by Using Higher Order Shear Deformation Theory Appaso M Gadade1, a *, Achchhe Lal2,b and Bhairu N Singh3,c 1 Department of Mechanical Engineering, Army Institute of Technology, Pune-411015, India 2 Department of Mechanical Engineering, S.V. National Institute of Technology, Surat-395007, India 3 Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur-721302, India a b c appagadade@yahoo.com, achchhelal@med.svnit.ac.in, bnsingh@aero.iitkgp.ernet.in Keywords: Progressive failure analysis, Higher order shear deformation theory, First ply failure load, Last ply failure load, Mode of failure. Abstract. A finite element analysis procedure is developed for progressive failure analysis of laminated composite plates under in plane tensile loading. A finite element model is based on higher order shear deformation theory (HSDT) with seven degrees of freedom. The degradation technique used for degradation of material properties of a failed ply is a ply discounting approach. The mode dependant Hashin and Lee failure criterion is used for the progressive failure analysis. The results of first ply failure load and last ply failure load are obtained for different stacking sequence of composite plate. The results shows that a considerable amount of strength remains un utilized in composite laminates after first ply failure of laminates. Introduction In the design of composite structures it is required to find out last ply failure load (ultimate strength) for better utilization of available strength of composite structures. Although literatures are reach in prediction of first ply failure load of composite structures by using classical lamination theory (CLT) and first order shear formation theory (FSDT), but very few literatures deals with first ply failure analysis of composite structures using higher order shear deformation theory (HSDT). Analytical solutions are available only with composite laminates with ideal geometry, boundary conditions and loading conditions. In case of composite laminates only prediction of first ply failure load does not give complete picture of strength analysis of composite laminates. A considerable amount of strength remains unutilized even after first ply failure and it is necessary to develop approximate computational technique which help to predict failure of composite laminate up to last ply failure. Kinematic Assumptions We begin with the displacement field u ( x, y, z ) = u ( x, y ) + zψ x ( x, y) + z 2ξ x ( x, y ) + z 3ζ x ( x, y ), v( x, y, z ) = v( x, y ) + zψ y ( x, y ) + z 2ξ y ( x, y ) + z 3ζ y ( x, y ), (1) w( x, y, z ) = w( x, y ) where u, v and w denote the displacements of a point (x, y) on the mid plane and Ψx and Ψy are the rotations of normal to mid plane about the y and x axes, respectively. The functions ξx, ζx, ξy and ζy will be determined using the condition that the transverse shear stresses σxz = σ5 and σyz = σ4 vanish on the plate top and bottom surfaces. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 117.239.210.99-13/07/14,06:15:34) 1152 Dynamics of Machines and Mechanisms, Industrial Research The displacement field becomes u = u + zψ − z 3 4 / 3h 2 (ψ + ∂w / ∂x) = u + f1 ( z )ψ + f 2 ( z )∂w / ∂x, x x x 3 2 v = v + zψ − z 4 / 3h (ψ + ∂w / ∂y ) = v + f1 ( z )ψ + f 2 ( z )∂w / ∂y , y y y (2) w = w. where f1 ( z ) = C1 z − C2 z 3 and f 2 ( z ) = −C4 z 3 , here C1=1 and C2=C4=4/3h2, It can be seen that the number of degrees of freedom (DOF) per node, by treating θ x and θ y as separate DOFs, increases from 5 to 7 for HSDT model. However, the strain vector will be having only first order derivatives, and hence a C0 continuous element would be sufficient for the finite element analysis [2]. This model is solved by using a finite element technique proposed by Lal et al [2], and it is extended successfully for progressive failure analysis. A ply discounting approach is used for degradation of material properties. Constitutive equation (stress-strain relation) Stress is related to strain by following relation {σ } = Q  {ε } σ x   σ 1  σ  σ   y   2  τ xy  = σ 6  = τ  σ   yz   4  τ xz  σ 5   Q 11   Q 12   Q 16  0   0 Q 12 Q 16 0 Q 22 Q 26 0 Q 26 Q 66 0 0 0 Q 44 0 0 Q 45 0 ε   x 0  ε y     0  γ xy  ,   Q 45  γ yz   Q 55   γ xz  (3) (4) where Q11 = Q11 cos 4 θ k + Q22 sin 4 θ k + 2 ( Q12 + 2Q66 ) sin 2 θ k cos 2 θ k , Q12 = (Q11 + Q22 − 4Q66 ) sin 2 θ k cos 2 θ + Q12 ( cos 4 θ k + sin 4 θ k ) , Q 22 = Q22 cos 4 θ k + Q11 sin 4 θ k + 2 ( Q12 + 2Q66 ) sin 2 θ k cos 2 θ k , Q16 = (Q11 − Q12 − 2Q66 ) sin θ k cos3 θ k + ( Q12 − Q22 + 2Q66 ) sin 3 θ k cos θ k , Q 26 = (Q11 − Q12 − 2Q66 ) sin 3 θ k cos θ k + ( Q12 − Q22 + 2Q66 ) sin θ k cos3 θ k , Q 44 = Q44 cos 2 θ k + Q55 sin 2 θ k , Q 45 = (Q55 − Q44 ) sin θ k cos θ k = Q 54 , Q 55 = Q55 cos 2 θ k + Q44 sin 2 θ , Q 66 = (Q11 + Q22 − 2Q12 − 2Q66 ) si, n 2 θ k cos 2 θ k + Q66 ( cos 4 θ k + sin 4 θ k ) , (5) here Qij are elastic material constants and θk is the angle of fiber orientation, of lamina. Failure Criterion Failure indices for the Hashin criteria involve four failure modes [7]. Tensile fibre failure – for σ11 ≥ 0 2 2 >1 2   ( e1t ) =  σX11  + σ12S+2σ13 = ≤ 1 failure  no  T 12 2 (6) Compressive fibre failure – for σ11 < 0  σ  > 1 failure =  11  =   X C  ≤ 1 no 2 c 2 1 (e ) Tensile matrix failure – for σ 22 + σ 33 > 0 (7) Applied Mechanics and Materials Vols. 592-594 2 ( e2t ) = (σ 22 + σ 33 ) 2 2 T Y + (σ 23 − σ 22σ 33 ) σ 122 + σ132 S 2 23 + 2 12 S 1153 > 1 failure = ≤ 1 no (8) Compressive matrix failure – for σ 22 + σ 33 < 0   Y  2   (σ + σ )  c 2 = e ( 2 )  2SC  − 1  22Y 33  C   223    2 + σ σ σ ( ) − σ σ σ 122 + σ 132 > 1 ( ) 23 22 33 + 22 2 33 + + = 4 S 23 S 232 S122 ≤ 1 failure no Lee’s failure criterion in terms of the four failure modes is as follows [4]: Tensile fiber mode: Tensile matrix mode: σ1 Xt σ2 Yt σ1 =1; Compressive fiber mode: − = 1; Compressive matrix mode: − Xc σ2 Yc (9) =1 (10) =1 Results and Discussion In the present study, a computer program in MATLAB 9.0 (R2009b) has been developed to compute first ply (FPL) and last failure (LPL) load of a composite plate acted upon by in plane tensile mechanical loading. A nine nodded Lagrange isoparametric element with 63 degree of freedoms (DOFs) per element for the present HSDT model has been used for discretizing the laminate and (4×4) mesh based on convergence has been used throughout the study. Table 1 Material properties of T300/5208 graphite/epoxy pre-peg [4] Properties E1 E2 E3 G12=G13 G23 ν12 = ν13 ν23 Value 132.5 Gpa 10.8 Gpa 10.8 Gpa 5.7 Gpa 3.4 Gpa 0.24 0.49 Properties Xt Xc Yt =Zt Yc=Zc R S=T Ply thickness h1 Value 1515 MPa 1697 MPa 43.8 MPa 43.8 MPa 67.6 MPa 86.9MPa 0.127mm Boundary conditions All edges simply supported (SSSS): v = w = θ2 = φ2 = 0, at ξ1 = 0, a; u = w = θ1 = φ1 = 0 at ξ2 = 0, b ; Where a and b are length and breadth of plate. Convergence and validation study Based on the convergence study shown in Table 2, finite element mesh used in present analysis is 4×4. Table 2 Convergence study and comparison of central deflection of square laminated [0/90]s cross ply plate subjected to non dimensionalised sinusoidal load with simply supported boundary conditions with different aspect ratios(aspect ratio s=a/h=10, E1/E2=25.0, G12/G23=2.5, υ12= υ23=0.25) Aspect Ratio (a/h) 10 20 Mesh density 2×2 3×3 4×4 5×5 6×6 2×2 3×3 Normalised central deflection Present Tolson et Ocha et al[4] al[5] 1.6919 1.693 -------1.6816 --------------1.6733 1.671 -------1.6733 --------------1.6733 -------1.790 1.1911 1.188 -------1.1826 --------------- Reddy [1] 1.534 ----------------------------1.136 -------- Reddy [6] 1.643 1.163 3D Solution 1.709 1.189 Elasticity 1154 Dynamics of Machines and Mechanisms, Industrial Research 4×4 5×5 6×6 1.1798 1.1734 1.1730 1.177 --------------- --------------1.216 ---------------------- Table 3 Ultimate failure load for a symmetrical laminate under uniaxial tensile load with a/h=150. theta 0 10 20 30 40 50 60 80 90 FPF stress 1507.7 1500.3 1437.8 988.79 744.20 574.25 389.09 230.09 223.46 Degradation rule -----------E1=0.001,G12=0.001,G23=0.001,V12=0.001 E2=0.001,G12=0.001,G23=0.001,V21=0.001 E2=0.001,G12=0.001,G23=0.001,V21=0.001 E2=0.001,G12=0.001,G23=0.001,V21=0.001 E2=0.001,G12=0.001,G23=0.001,V21=0.001 E2=0.001,G12=0.001,G23=0.001,V21=0.001 E2=0.001,G12=0.001,G23=0.001,V21=0.001 E1=0.001,G12=0.001,G23=0.001,V12=0.001 Mode TF TF CM,TM CM,TM CM,TM CM,TM CM,TM CM,TM CM,TM LPF Stress ----0.02 -1.50 -1.58 -1.66 1.62 208.27 220.29 224.39 Final LPF 1507.7 1500.3 1437.8 988.79 744.20 575.87 597.36 445.96 447.85 Progressive failure analysis results Table 3 shows validation of first ply and last ply failure load for square laminated plate under uniaxial tensile loading, for [θ4/04/θ4]s stacking sequence. Conclusions The progressive failure analysis results shows that the considerable amount of strength of composite laminate remains unutilized after first ply failure analysis. The variation in first ply failure and last ply failure increases with stacking angle. References [1] Reddy Y.S.N. and Reddy J. N. Linear and non-linear failure analysis of composite laminates with transverse shear, Composites Science and Technology 44 (1992) 227-255. [2] Achchhe Lal, B.N. Singh and Dipan Patel, Stochastic nonlinear failure analysis of laminated composite plates under compressive transverse loading, Composite Structures 94 (2012) 1211– 1223. [3] T. Y. Kam, H. F. Sher and T N Chao, Predictions of Deflection and First-ply Failure Load of Thin Laminated Composite Plates via the Finite Element Approach, International Journal of Solids and Structures Vol. 33, No. 3, pp. 375-398, 1996. [4] S. Tolson and N Zabaras, Finite Element Analysis of Progressive Failure in Laminated Composite Plates, Computers and Structures, Vol. 38, No.3, pp. 361-376, 1991. [5] Ozden O. Ochoa and John J. Engblom Analysis of Progressive Failure in Composites, Composites Science and Technology 28 (1987) 87-102. [6] J Reddy J.N., Pandey A.K., A first-ply failure analysis of composite laminates. Computers and Structures, 25(3), (1987), 371-93. [7] A.K., Hashin, Z. (1980). Failure Criteria for Unidirectional Fibre Composites, Journal of Applied Mechanics, 47: 329–334.