w. Lee et aL. / Composites Science and Technology 66(2006)435-443 Outer Layer(Material 2) Edge crack 2h Innner Layer(Material #1 Outer Layer(Material 2 Edge Crack Fig. 2. Geometry of the modelled laminate structure used for the analysis (a c: remotely applied mechanical load) It would be practically more convenient to measure these changes in strains of the layers and estimate the y≡tan (10) misfit strain to characterise the residual stress Thus stress.Thus, where K, and K, are opening and shear stress intensity the relative misfit strain E with respect to the mechanical factors respectively, were calculated by estimating K1 ad-induced strain, eapp, as and K2 using the interaction integral method [26]. Finite element mesh used for the calculations is shown in Fig 3. Typical 8-node isoparametric elements were used and quarter-point node elements [27]were Positive I corresponds to compressive residual stress in employed around the crack tip region to model crack the inner intact layer (layer #1) and tensile residual tip singularity. To model symmetry boundary cond stress in the outer cracked layers(layer#2), and vice ver- tions, normal displacement degrees of freedom(DOF) sa. This term will be used to describe the residual stress of the node points on the symmetry lines were fixed state of the layered system considered in the current whilst their tangential displacement DOFs were re- study(Fig. 2) leased. Dundurs parameter varied from -0.85 to 0.85 whilst keeping B equal to zero for simplicity since it has been demonstrated and suggested that effect of B is 3. Numerical analysis not significant [25, 28]. Applied load was simulated by displacing the node points in the right end face of the Double edged notch specimen geometry shown in model along x-direction by a predetermined distance Fig. 2, loaded under a uniform tensile loading on its Residual stress due to the misfit strain was simulated end faces by a constant displacement 8, was chosen for by applying stresses estimated from Eq.(6) within the the numerical analysis for the convenience of modelling. range of r between -l0 and 1.0. It is believed that r The laminate shown in Fig. 2 consists of two outer lay in this range covers er estimated for some practical lam- ers with the thickness h and an inner layer with the inate systems such as SiC/C, SiC/B,C, SiC/TiB2 and thickness 2h. It was assumed that the primary cracks ZrOz/Al]o3 laminates [l]. A general-purpose finite ele- in the outer layers, terminated at the interface, are dou- ment analysis package ABAQUS was used for the bly deflected as usually observed in many real systems calculations [14, 18-20]. Only the quarter section of the geometry was modelled for the analysis due to the symmetry The residual stress terms(Eq (6) were separately cal- culated for various Is and superposed to the mechanical Before discussing the effect of residual stress on the loading term using principle of superposition to obtain crack deflection criteria, it is first necessary to confirm their combined contributions to the fracture parameters whether the numerical solution converges to the analyt governing crack deflection such as strain energy release ical solutions obtained for infinite geometry since the rates, a and p(and equivalent stress intensity factor numerical model used for the analysis herein is based and phase angle of loading. a and s, were calculated on the geometry having finite dimensions. This was done through the estimation of J-integral [21, 22] based on by estimating the a and p for various lengths of infin the virtual crack extension/domain integral methods tesimal crack penetration ap and crack deflection ad rel 23, 24] assuming linear plane strain elasticity. Phase an- ative to a reference global length scale, the thickness of gle of loading, which is an indicator of the relative con the outer layer h which is equal to the length of the pri tribution of shear and opening terms to the total crack driving force and defined as [25] Product of hibbit. Karlsson soreIt would be practically more convenient to measure these changes in strains of the layers and estimate the misfit strain to characterise the residual stress. Thus, magnitude of the residual stress is defined herein as the relative misfit strain er with respect to the mechanical load–induced strain, eapp, as C ¼
er eapp : ð9Þ Positive C corresponds to compressive residual stress in the inner intact layer (layer #1) and tensile residual stress in the outer cracked layers (layer #2), and vice ver￾sa. This term will be used to describe the residual stress state of the layered system considered in the current study (Fig. 2). 3. Numerical analysis Double edged notch specimen geometry shown in Fig. 2, loaded under a uniform tensile loading on its end faces by a constant displacement d, was chosen for the numerical analysis for the convenience of modelling. The laminate shown in Fig. 2 consists of two outer lay￾ers with the thickness h and an inner layer with the thickness 2h. It was assumed that the primary cracks in the outer layers, terminated at the interface, are dou￾bly deflected as usually observed in many real systems [14,18–20]. Only the quarter section of the geometry was modelled for the analysis due to the symmetry (Fig. 2). The residual stress terms (Eq. (6)) were separately cal￾culated for various Cs and superposed to the mechanical loading term using principle of superposition to obtain their combined contributions to the fracture parameters governing crack deflection such as strain energy release rates, Gd and Gp (and equivalent stress intensity factors) and phase angle of loading. Gd and Gp were calculated through the estimation of J-integral [21,22] based on the virtual crack extension/domain integral methods [23,24] assuming linear plane strain elasticity. Phase an￾gle of loading, which is an indicator of the relative con￾tribution of shear and opening terms to the total crack driving force and defined as [25] W tan
1 K2 K1
; ð10Þ where K1 and K2 are opening and shear stress intensity factors respectively, were calculated by estimating K1 and K2 using the interaction integral method [26]. Finite element mesh used for the calculations is shown in Fig. 3. Typical 8-node isoparametric elements were used and quarter-point node elements [27] were employed around the crack tip region to model crack tip singularity. To model symmetry boundary condi￾tions, normal displacement degrees of freedom (DOF) of the node points on the symmetry lines were fixed whilst their tangential displacement DOFs were re￾leased. Dundurs
parameter a was varied from
0.85 to 0.85 whilst keeping b equal to zero for simplicity since it has been demonstrated and suggested that effect of b is not significant [25,28]. Applied load was simulated by displacing the node points in the right end face of the model along x-direction by a predetermined distance. Residual stress due to the misfit strain was simulated by applying stresses estimated from Eq. (6) within the range of C between
1.0 and 1.0. It is believed that C in this range covers er estimated for some practical lam￾inate systems such as SiC/C, SiC/B4C, SiC/TiB2 and ZrO2/Al2O3 laminates [1]. A general-purpose finite ele￾ment analysis package ABAQUS1 was used for the calculations. 4. Results and discussion Before discussing the effect of residual stress on the crack deflection criteria, it is first necessary to confirm whether the numerical solution converges to the analyt￾ical solutions obtained for infinite geometry since the numerical model used for the analysis herein is based on the geometry having finite dimensions. This was done by estimating the Gd and Gp for various lengths of infin￾itesimal crack penetration ap and crack deflection ad rel￾ative to a reference global length scale, the thickness of the outer layer h which is equal to the length of the pri￾Outer Layer (Material # 2) Edge Crack Symmetry Line Edge Crack Innner Layer (Material #1) h 2h h Outer Layer (Material # 2) δ8 δ8 Fig. 2. Geometry of the modelled laminate structure used for the analysis (r1: remotely applied mechanical load). 1 Product of Hibbit, Karlsson & Sorensen, Inc. W. Lee et al. / Composites Science and Technology 66 (2006) 435–443 437