The free edge effect is an inter-galactic phenomenon found on the outer reaches of the cosmos that leads to a parallel…well…no, not exactly. What I’m referring to is a region within a laminate that produces a fully three-dimensional stress field at the free edge, then decays quite rapidly to a two-dimensional stress field as the distance from the free edge increases. The three-dimensional stress field is responsible for delamination at a free edge, and is commonly referred to as inter-laminar stresses. A composite analysis is typically confined to classical lamination theory. Unfortunately, this assumes that all out-of-plane stresses are zero, making the determination of transverse stresses impossible. Consequently, this assumption is not valid when one is interested in calculating out-of-plane stresses in regions near or at a free edge.
What is an Interlaminar stress?
"Near free edges, both shear and normal stresses may arise between the layers. These interlaminar stresses may significantly alter the stress field existing away from the free edge and, importantly, may cause separation (delamination) of adjacent layers."
What can cause an interlaminar stress?
“The principal reason for the existence of interlaminar stresses is the mismatch of Poisson’s ratios nxy and coefficients of mutual influence mx and my between adjacent laminas.”
The interlaminar stress is produced by a maximum shear stress occurring at a free edge of a laminate between two adjacent layers. This shear stress effect (which can be high) extends over a region equivalent to a distance approximately equal to the thickness of the laminate. Figure 1 below shows the free edge effect using a flat plate and is illustarted by a hatched region.
Analysis of interlaminar stresses begins by addressing the causes in a bit more detail. In this article, I will elaborate on Poisson’s mismatch…huh? No, I’m not talking about two different types of fish! I will instead, start with a sub-laminate having all the lamina oriented in the 90-degree direction. It is here you will note that the major Possion’s Ratio vxy is equal to about 0.02 (See Figure 2). Now, I will add a sub-laminate made of only angle plies: its Possion’s Ratio ranges between 0.25 to 0.7 (Vxy-0 ≠ Vxy-90). Interestingly, this is an order of magnitude greater than the 90-degree ply group.
Poisson’s mismatch for vxy is greatest when the angle between the 90-degree and angle-ply group is between 25 and 35 degrees (See Figure 2). For angles greater than 35-degrees, the Poisson’s mismatch decreases.
Now, when the sub-laminates are bonded together, and a tensile loading is applied (let’s say Nx), the Poisson’s mismatch will induce a coupling between axial extension and bending. Since the angle plies (which are aligned closer to the 0-degree direction) are more compliant in the transverse direction, they will contract transversely more than the 90-degree plies. Meaning, the pulling action (or tension) results in an interlaminar stress near the laminate’s mid-surface that is quantified with Sigma Z (See Figure 1).
Imagine if these two sub-laminates were allowed to move freely…they both would deform by different amounts in their respective transverse directions.
Now, let us put this all together by leveraging Figure 3. First take note of the locations of both the inter-laminar normal and shear stresses. When a plate’s stress is located “away” from the free edge, the inter-laminar shear stress Tau-yz will disappear. However, at the free edge both stresses appear and could yield ply delamination. Why? Because sigma-y at the free edge becomes unbalanced. To equilibrate this unbalanced stress, a Tau-yz is produced. Consequently, these two stresses produce a moment that must be reacted with an additional normal stress (defined as sigma-z) to achieve balance…and uh-oh that’s where the potential problem exists: delamination at the free edge.
Engineer’s should use caution when assuming linear elasticity in a interlaminar stress calculation. The result will typically yield an infinite stress at the laminate's free edge.
It’s important that the engineer check for inter-laminar stresses when analyzing composite designs that include cutouts, re-entrant corners, notches, ply-drops, thick composites or material discontinuities at the interfaces. There are many scenarios that contribute to the generation of inter-laminar stresses, and I have only briefly highlighted some of those issues. As always, the intent in this article is to create an awareness while steering clear of most of the technical minutia when considering the possible pitfalls, that on occasion, are over looked during the design or analysis of a composite system.
- Effects of Interlaminar Stress Gradients on Free Edge Delamination in Composite Laminates, Simon Chung, Drexel University
- Mechanics of Composite Structures, George S. Springer, Stanford University
- Fiber-Reinforced Composites, P.K. Mallick