18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS SIZE EFFECTS IN PROGRESSIVE DAMAGE OF NOTCHED AND HOLED COMPOSITES B. Chen 1 , T.E. Tay 1 *, P.M. Baiz 2 , S.T. Pinho 2 1 Department of Mechanical Engineering, National University of Singapore, Singapore, 2 Department of Aeronautical Engineering, Imperial College London, London, UK * Corresponding author (mpetayte@nus.edu.sg) Keywords : Size effect; cohesive model; thickness dependence of fracture energy 1 Introduction applied to numerical modeling here and it will be The strength dependence of notched composites on shown later that this is critical to the prediction of size of specimens has been well documented in thickness size effect. literature. Wisnom et al. have performed extensive 2.2 Fiber failure modeling experimental studies on the size effect of V-notched σ > and open-hole composite specimens under tension Fiber tensile failure initiates when X where 1 t and compression [1-4]. It was shown that the X is the fiber direction lamina tensile strength. t strength of open-hole composite laminates depends Post-failure softening of fiber tensile failure is on in-plane scale, thickness scale and ply lay-ups. modeled by a linear cohesive softening law (Fig.1). Fiber direction mode I fracture energy has recently Fiber compressive failure is more complicated than been experimentally determined to be thickness tensile failure because fiber micro-buckling and dependent in [5]. However, the mechanisms of this kinking often happens before the material reaches its apparent dependence have not been fully explained. theoretical compressive strength, and the strength in This paper presents a computational study of the compression is found to be very difficult to obtain in prediction of in-plane strength of notched and holed experiments [6, 7]. Current studies focus on tension laminates, accounting for the thickness size problems and a simple maximum stress criterion dependence effect, using a cohesive failure model.. σ < − X is used for fiber compressive failure 1 c initiation. 2 Composite failure theory 2.3 Matrix failure modeling 2.1 Thickness dependence of composite fracture Matrix failure initiation is determined by Tsai-Wu toughness failure criterion. Since this work studies tension loading cases and matrix here does not undergo Literature reports large variations of the mode I fiber compressive failure, only tension and shear stresses tensile fracture energy G fc with respect to the are involved in the failure criterion. It is also thickness of the 0-plies in the tested specimen [5]. assumed that matrix cracks are all parallel to the The amount of fiber pull-out in [02/90]s laminate fiber direction and perpendicular to the lamina shell increases compared to that in [0/90]s and causes a plane (this assumption may not be valid for significant increase of energy dissipation. The compressive failure since the shear fracture plane in measured 0-ply fracture energy of [02/90]s is more compression generally will not be perpendicular to than twice of that of [0/90]s (Table 1). To the the lamina shell plane). Therefore the normal vector author’s knowledge, no numerical work has of potential crack surface is ê 2 and the stresses that employed this thickness dependence of fiber fracture σ τ energy. Although a quantitative description of mode apply to the potential crack surface are , 12 and 2 I fiber fracture energy in terms of thickness has not τ σ = σ σ is kept as where it where max( 0 , ) . 2 2 1 32 been well established, for the cases in this project, is in the original Tsai-Wu criterion form. The only the fracture energies of single ply and two complete form of the matrix failure initiation blocked plies are needed and they are available from criterion is: [5]. The thickness dependence of fracture energy is
σ + σ + σ + σ 2 2 G F F F F = 1 1 2 2 11 1 22 2 S B + (1) (5) + τ + τ + σ σ = G G 2 2 F F 2 F 1 N S 44 32 66 12 12 1 2 1 1 1 1 1 = + = − = + where F , F , F , 1 11 2 X X X t X Y Y 1 t c c t c = σ ε G l , N 2 2 e 1 1 2 = = = − (6) F , F F , 22 44 66 2 Y t Y S 1 c = τ γ + τ γ G ( ) l = − S 12 12 23 23 e 2 F F F .[8] 2 12 11 22 Y is the matrix tensile strength and c Y is the matrix t C C G is the mode I matrix fracture energy and G is n s compressive strength. S is the matrix shear strength. the mode II/III matrix fracture energy. η is set to 1. 44 τ 2 This form has an extra term compared to F 32 2.4 Delamination modelling classical plane-stress Tsai-Wu criterion. Numerical models here use continuum shell element which is The role of delamination on laminate strength has formulated based on thick shell theory where been extensively studied in experiments by Wisnom τ transverse shear is considered. Since acts on the et al.[11]. It is observed that for in-plane loading, 32 potential crack plane, it is therefore included in the significant delamination can occur, which releases criterion. The original derivation of Tsai-Wu the stress concentration of the 0-plies and affects the criterion in [8] was based on 3D stress state and overall strength of the laminate. It is therefore important to include delamination analysis even for 1 = F where Q and Q’ are positive and ′ in-plane loading simulation. Here delamination is 44 Q Q modeled by inserting ABAQUS cohesive elements negative shear strengths along the 2-3 plane. Here of very small thickness (0.001mm) between plies. the same shear strength is assumed for both positive The initiation of delamination is determined by a τ τ and negative shear loadings and for both 12 and , quadratic stress criterion. Propagation of 32 1 delamination is modeled by a linear cohesive = = thus F F . 44 66 2 softening law with mixed-mode fracture energy S defined by the B-K formula. Table 2 lists values of Matrix damage propagation is described by a the material parameters used for cohesive elements. cohesive softening law of effective stress Values of the penalty stiffness K nn (normal), K ss and (2) 2 K tt (two shear directions) are assumed to be very σ = σ + τ 2 + τ 2 eff 2 12 23 high as to simulate perfect bonding before delamination (Fig.2). Special care should be taken when assigning values to the above parameters so as with respect to effective displacement. to ensure that damage growth is always positive, i.e., (3) no healing of material is predicted[12]. A formula 2 = e ε + γ 2 + γ 2 u l eff 2 12 23 proposed in [12] states the relationship between K nn , K ss and K tt : l is the characteristic element length. Mixed-mode e 2 fracture energy G mc is defined by Benzeggagh- C G S = = n K K K Kenane (B-K) formula [9]: ss tt nn C N (7) G s η = + − (4) C C C ( ) G G G G B mc n s n In this project, K nn = 5×10 7 N/mm 3 and K ss = K tt = where 2.25×10 7 N/mm 3 . Delamination modeling using cohesive elements is very sensitive to element size [13,14]. The element size must be small enough to
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