Stitch Interface Models

Several models for analyzing stitched interfaces with a propagating delamination have been proposed in the literature. Section 2.2.1 provides a brief review of these analysis techniques. All these models replace the stitches with an equivalent traction vs. displacement law that is assumed to govern the tow tractions introduced to the sub-laminates at the delamination interface as the delamination propagates and displaces the tow. Both experimental characterization techniques and micromechanical models have been proposed to determine the traction law and are reviewed in Section 2.2.2.

Failure at a stitched interface occurs through a propagating delamination that is bridged by one or more stitches in the delamination wake. The propagation

0.02 in

Deformed tow (Stitch) Undeformed tow

(Stitch)

Delamination crack tip

Tow provides closure tractions Resin

interface

of the delamination can be analyzed using the Griffith energy balance, Equation (2.2), where is the energy dissipated by the stitches such that

(2.26)

where is the energy dissipated by the stitches relative to a change in crack area, and the Griffith fracture criterion, Equation (2.5), becomes

(2.27)

where it is noted that LEFM is not applicable because the assumption that all energy dissipation is confined to the crack tip is violated [108, 208, 209].

Massabò et al. showed the importance of considering LSB in analysis of bridged delaminations under Mode II and subsequently mixed-mode load conditions [208, 210, 211].

The combination of the two terms on the right hand side of Equation (2.27) has been considered by several authors as an apparent fracture toughness where the first term is the contribution of energy released due to delamination extension and the second term is due to irreversible failure of the stitches [198, 212]. Such an approach effectively smears the response of the discrete stitches as an equivalent uniformly distributed toughness, which is only valid when one of the two mechanisms (delamination of the resin interface or stitch failure) dominates the response [213]. In addition, FEA of a DCB showed that discrete representation of stitches is necessary when the stitch spacing is on the order of the laminate thickness [214, 215]. While in the case of Z-pining a smeared

representation may be suitable because of the use of relatively densely placed TTR, typical spacing used for stitching is much larger and thus necessitates discrete representation of the stitches.

Several FE representations have been used to model stitches discretely.

Analyses of stitched interface delamination propagation have evolved from nonlinear springs with VCCT to 3D cohesive elements for both resin interface delamination and stitch failure, as shown in Figure 2.12. Typically, the sub-laminates are represented by shell elements with their reference surfaces offset to the interface between the two sub-laminates as shown in Figure 2.12b-d. Figure 2.12b shows the relatively simple representation where the stitch is modeled by a spring element and the delamination is advanced using VCCT, which is a model that has been considered in several studies [108, 216–219]. In all cases, the traction-displacement law for the stitch is assumed to govern the stitch behavior.

Some studies treated the opening and shearing displacements independently by using connector elements, e.g., [108, 216]. As has already been discussed in Section 2.1, the delamination crack tip is not adequately described by LEFM because of the crack closure tractions in the delamination wake. Therefore, recalling that VCCT is an LEFM technique, Dantuluri et al. proposed using the CZM to model the delamination propagation instead of VCCT (Figure 2.12c) and demonstrated the effectiveness of such a model in a 2D analysis with Mode I loading conditions [220].

One difficulty with a spring representation is that the stitch tractions are introduced to the sub-laminate at a single node, which leads to a mesh-size

dependency [221]. In these models, the nodal force exerted on the sub-laminate shell elements due to the stitch is averaged over an area equivalent to the sub-laminate element size. Grassi et al. used an element size such that the area of the element was similar to the stitch cross sectional area, which provides a reasonable engineering approximation. However, a mesh-size independent model is desirable, which motivated representation of the stitches with cohesive elements so that the stitch traction is introduced at several nodes, as shown in Figure 2.12d [222, 223]. This representation used two separate cohesive traction-separation laws: one for the delamination and one for the stitches. The model was demonstrated for Mode I and Mode II loadings and found to be in good agreement with test results [222, 223]. A direct comparison between spring and cohesive elements representations of the stitches revealed that the analysis using springs converged slower and thus it took four times more CPU time, though both techniques yielded similar results [223].

An additional advantage of using cohesive elements instead of nonlinear spring elements to represent the stitches is that cohesive elements incorporate irreversible damage mechanics in a thermodynamically consistent implementation. A model that uses cohesive elements to represent stitches ensures proper energy dissipation under all loading conditions and histories. In contrast, the energy dissipation of a model with a nonlinear spring representation of stitch damage is only valid under monotonic opening at the stitch locations.

Figure 2.12. Various FE representation for modeling delamination of stitched interfaces.

Actual engineering structures rarely undergo delamination under a single mode of loading; typically the loading conditions are mixed-model. Cui et al.

extended Bianchi and Zhang's model for mixed-mode analysis [154]. The cohesive element model is readily extended to mixed-mode conditions assuming a stitch traction-displacement law is available for mixed-mode loadings. Cui et al.

used a micromechanical model to determine a suitable traction-displacement law for a variety of mode-mixities. They proposed that damage evolution for tension and shear was uncoupled, despite acknowledging that the physical process is coupled [154]. Nevertheless, this modeling approach was used to analyze stitched and Z-pinned T-joints subjected to a bending load. The plane strain analysis

(a) Idealization (b) FE representation: 1D spring

1D spring element

(c) FE representation: 1D spring w/ CZM

Cohesive elements

showed good agreement with the test results [224]. This analysis represents the state-of-the-art in delamination propagation of stitched interfaces.