5 Designing Brittle Fracture Specimens to Investigate Environmentally Assisted Crack Growth 27 ˛ D . 2 C1/ . 1 C1/ . 2 C1/ C. 1 C1/ (5.3) ˇ D . 2 1/ . 1 1/ . 2 C1/ C. 1 C1/ (5.4) where D 1/ 2, D3 4v for plane strain and D(3 v)/(1Cv) for plane stress, is the shear modulus and v is the Poisson’s ratio. The First Dundurs’ parameter ˛ is the measure of the difference in the stiffness of the upper material to the bottom material of the bimateral beam. ˛> 0 when the top material is stiffer than bottom material of bimaterial beam and ˛< 0 when the top material is more compliant than the bottom material. The sign of ’is reversed when material 1 and material 2 are interchanged. The ’ parameter can be seen to be between 1 and 1 by letting one of the stiffness values go to zero [8]. Suo and Hutchinson [7] show that sub-interface cracking is only weakly dependent on the second Dundur’s parameter so the current parameter study focuses on ’only. It has been shown that ’parameter can easily be simplified by expressing it in terms of Elastic moduli of the two materials that makes up the bonded material [9]. The simplified form of ˛ is expressed in Eq. (5.5). ˛ D EC2 EC1 EC2 CEC1 (5.5) Where the expression for ECi is given in Eq. (5.6). ECi D ( Ei .1 vi 2/ .plane strain/ Ei .plane stress/ (5.6) Ei , vi (for i D1, 2) represent the elastic moduli and Poisson’s ratios of the material components of the bonded material. 2 dimensional (2D) plane stress finite element simulations were performed using FE modeling capabilities of the commercial FEA code ABAQUS [10]. The materials chosen for this research were only selected based on the availability of material data and required values of ’ parameter. The material properties required for the FE modeling are Elastic moduli, Poisson’s ratios, and CTEs. These materials’ data were obtained from Sandia National Laboratories and from relevant literature. Each of the material was assumed to be isotropic and linearly elastic and initial crack length of 15 mm was assumed to be present in the substrate. Boundary conditions were applied to restrict rigid body modes and displacements were otherwise free. All FE models were discretized using quadratic Quadrilateral (Q8) elements with a mesh focused mesh structure and quadratic quarter point elements at the crack tip. The method of J contour integral was applied the to evaluate stress intensity factors. For earlier validation of the Finite element model, the SSC in brittle substrates beneath adherent films were modeled for ’ values of 0.8, 0.6, 0.4, 0.2, 0, 0.2, 0.4, 0.6 and 0.8 and were compared to the analytical results obtained from [7]. Substrate dimensions used in the analysis were 150 mm length by 25.4 mm thick and the initial crack length of 15 mm was used for all models. The film and the substrate have equal length while the thickness of the film was gradually increased by increasing the film to substrate thickness ratio£ from 0 up to 0.1. The SSC depth and KI whenKII D0 were determined and the results were compared to the Analytical results. The FE model was extended to study SSC in the bottom material of bimaterial beam shown in Fig. 5.3. The materials used for this study were Schott B270 (a variant of soda-lime glass) as the top material and standard soda-lime glass as the bottom material. The material properties are shown in Table 5.1. It is important to note ’ D0 for the bimaterial beam because the Elastic moduli of the component are very close. The material components were picked because of the ready availability of the materials. The FE modeling of the bimaterial beam was accomplished by increasing the thickness ratio £ from 0.1 to 1 while retaining the initial crack length, thickness and length of the bottom material as15, 25.4 and 150 mm. The SSC depth andKI whenKII D0 were determined. Table 5.1 Material property for the biomaterial beam Material Schott B270 Soda-lime Elastic modulus (GPa) 71.5 74 Poisson’s ratio 0.22 0.24 CTE (20–600 ıC) 1.11185 10 5 1.0156 10 5
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