Finite Element Modeling

A FE model was developed for use in conjunction with this work, and is presented in Johnston et al. (2014), however the development itself was not part of this dissertation. The FE model is, however, integral to analyzing, understanding, and interpreting the experimental delamination toughness test data. For this reason, necessary results from Johnston et al. (2014) are presented herein.

4.3.1. Overview

All FE modeling in Johnston et al. (2014) was performed using Abaqus Version 6.8. All models were similar to the original three dimensional model developed and validated for analysis of the STB test (Davidson and Sediles, 2011). This earlier work described an extensive series of mesh refinement and validation studies that were performed before the model was applied to the original three dimensional STB configuration. Therefore, as a starting point, a model of the STB specimen was initially developed and constrained in accordance with the descriptions in their work. Energy release rate components were computed by the virtual crack closure technique (VCCT) (Rybicki and Kanninen, 1977; Krueger, 2004). For all cases considered, the average ERRs obtained by Johnston et al. (2014) were within 1% of those given by Davidson and Sediles (2011), and the graphical distributions of the ERR and its individual components were essentially

Following the above, the appropriate changes to the boundary conditions were first made. That is, the original STB configuration included pivot points exterior to the load tabs in order to allow rotations about the y-axis (cf. Figure 4.1). This was necessary to facilitate the mode I loading, but is not present in the mode III STB studied herein and was therefore removed. Next, a new and improved mesh was developed in the vicinity of the loading tabs, and additional mesh refinement studies were performed in order to select the final model used for the evaluations discussed herein.

4.3.2. Mesh and Boundary Conditions

Figure 4.4 presents a FE mesh that is typical of all models used in Johnston et al. (2014). As in Davidson and Sediles (2011), all load tabs utilize the same isotropic material properties (Young’s modulus, E = 209 GPa, Poisson’s ratio, ν = 0.3) and nominal dimensions as the steel load tabs used in the experiments. Based on measurements from physical test specimens, the epoxy bond-line thickness between the load tabs and the specimen was modeled as 0.15 mm, with isotropic material properties (E = 2.2 GPa, ν = 0.4) as taken from the manufacturer’s data sheet (Henkel, 2015). In comparison to the models of Davidson and Sediles (2011), the primary difference is the mesh in the tabbed region, which agrees with the physical geometry of the test specimens, as well as the way that the load pins are modeled. As indicated in Figure 4.4, each load pin extends through the thickness of the load tab, adhesive, and one of the cracked regions and connects to the corresponding nodes of the specimen and load tab along the inner curved surface. Each load pin is modeled as a rigid rod with a reference point, i.e., the point about which it can potentially rotate, at its own geometric center.

Figure 4.4. FE model of STB test with inset view of near-tip mesh.

For simplicity, the loading of Figure 4.1a is modeled. That is, one load block assembly is forced to translate in the y-direction and all other constraint locations are fixed with respect to y- direction translations. Thus, load is applied by imposing equal positive values of uy onto the two

lower load pins and by constraining uy = 0 for the upper load pins. The x-direction displacements

of all pins are fully constrained, and uz and all rotations of the pins are unconstrained. In the

above, uy and uz represent translational displacements in the y and z directions, respectively.

In addition to the above, and in order to simulate the constraints that the loading blocks impose on the load tabs, rigid surfaces are attached to all nodes that define the outer surface of each load tab. The reference point for each rigid surface is at its geometric center. The rigid surfaces impose the constraints that the outer surfaces of the load tabs cannot translate in the z- direction nor rotate about the x- or y-axes (θx = θy = 0). Translations in the x- and y-directions

are coupled to those of the load pins, and rotations about the z-axis are not constrained.

When the STB configuration is modeled, as shown in Figure 4.4, the end support and the center double roller and edge support (cf. Figure 4.1) are included via rigid rods attached to the corresponding nodes. Here, the horizontal rollers impose uz = θx = 0, and the vertical edge

x y

z

Lower load pin Upper load pin Delamination front

End roller

Center double roller and edge support

Crack h

h

supports impose uy = 0. For the SST, these constraints are not used. In all cases, the near-tip

mesh appears as shown in the insert of Figure 4.4 and is based on the original mesh refinement studies conducted by Davidson and Sediles (2011).

4.3.3. Mesh Refinement Studies

In order to develop and validate the mesh described above, Johnston et al. (2014) performed mesh refinement studies using the material properties of either IM7/977-3 or

T800S/3900-2B unidirectional carbon/epoxy, which are the two materials that are considered in the baseline experimental investigations. Material properties are presented in Table 4.1. Here, the values of the longitudinal modulus (E11) and in-plane shear modulus (G12) come from

experiments performed at the Syracuse University Composite Materials Laboratory (SU-CML), as they are the two properties to which the ERR is most sensitive (Davidson and Sediles, 2011). For T800S/3900-2B, the remaining properties were obtained from Davidson et al. (2007). For IM7/977-3, the remaining properties were obtained from Gregory and Spearing (2006), the supplier’s data sheet (Cytec, 2012), and the assumption of transverse isotropy.

Two separate mesh refinement studies were conducted to establish meshing across the specimen’s width and along its length. To investigate the former issue, meshes were considered that contained from 48 to 92 constant-width elements across the specimen’s width. A difference of approximately 1% in average ERRs was observed between these two extremes. The results of the 48-element model were also indistinguishable from those obtained from the variable-width element model that was developed to agree with the modeling approach of Davidson and Sediles (2011), described above. Thus, due to the simplicity and accuracy of the approach, all models were meshed in the y-direction with 48 constant width elements.

Table 4.1. Material properties (moduli in GPa).

Material E11 E22 E33 G12 G13 G23 ν12 ν13 ν23

IM7/977-3 163.8 8.34 8.34 4.95 4.95 2.98 0.27 0.27 0.40

T800S/3900-2B 147.6 7.58 7.58 4.31 4.31 2.87 0.32 0.32 0.32

The lengthwise, or x-direction meshing of all models was as shown in Figure 4.4. Considering the delamination plane in the cracked regions, the element length gradually

transitions from h/16 at the delamination tip to approximately 1.4h at the load tab, where h is the thickness of one of the cracked regions. A similar variation in element length is used in the uncracked region. To study refinement in this direction, Johnston et al. (2014) created a new model where this x-direction mesh density was doubled. Differences in ERRs of less than 0.5% were observed, so the mesh of Figure 4.4 was retained. For SST models with delamination lengths greater than 32 mm, more of the longest elements (e.g., next to the load tab) were included, with an element length never exceeding 1.5h.