There is much confusion about what interlaminar stresses are and how to calculate them using composites analysis. Interlaminar stresses are the source of failure mechanisms uniquely characteristic of composite materials; their existence is a major reason that laminated composites tend to delaminate near free edges, such as an edge of a plate or around a hole. However, there seems to be a lot of misunderstanding of how to define interlaminar stresses and how to predict them using finite element analysis.

**Interlaminar Stresses - What Are They?**

Referring to the picture below, interlaminar stresses are the out-of-plane stresses σz, τxz, and τyz, defined at the interfaces between layers in a laminated composite material. From the figure, it is clear that a tensile σz along an interface would tend to separate the layers along the interface, and the interlaminar shear stresses would tend to shear apart the interface in the corresponding directions.

##

**That Doesn't Sound So Bad, So What's the Big Deal?**

There are actually two main complications associated with interlaminar stresses. The first is that delaminations generally initiate at a free edge, so these are the critical regions where we would like to predict the interlaminar stresses. Unfortunately, it has been shown that interlaminar stresses are singular at free edges, i.e. as you approach the free edge, the interlaminar stresses tend to infinity based on the theory of elasticity. The second issue is that accurate interlaminar stresses, even in regions away from free edges, can be a challenge to predict using finite element analysis. Both of these issues will be addressed in this article.

**The Free Edge Singularity**

The free edge singularity is graphically shown in the figure below. Simply put, the value of the interlaminar stress is typically well-behaved away from a free edge, but tends to infinity as it nears the free edge. Thus, any prediction of interlaminar stresses at a free edge using finite elements will be mesh-dependent, i.e. as the mesh is refined, the value of the interlaminar stress at the free edge will continue to increase and is indeed unbounded. Because it is this behavior that leads to delamination, how does one address this issue? There is no globally accepted solution, but the most common approaches are to characterize the near edge stress behavior, such as using the value some representative distance from the edge (often equal to a layer thickness) and compare to test or other configurations to correlate the behavior using composites analysis.

##

There are other approaches used to predict delamination, such as the cohesive zone model and virtual crack closure technique, but these do not directly use the interlaminar stresses, so we won't discuss them in this article.

**Predicting Interlaminar Stresses Using Finite Element Analysis**

The most direct way to accurately predict interlaminar stresses using composites analysis is to create a mesh of 3D elements, with at least one and ideally more than one element through the thickness of each layer of the laminate. The interlaminar stresses can then be extracted directly from the full stress tensor, noting that σ_{z}, τ_{xz}, and τ_{yz} are all continuous across the layer boundaries (the corresponding strains are not necessarily continuous across layer boundaries). Of course, for most realistic layered structures, this method can result in a finite element analysis model with so many elements that it is not a reasonable approach.

Another more efficient approach, assuming that the composite structure is relatively thin, is to use layered shell elements. The efficiency comes from the fact that only one shell element is required through the thickness, because shell theory is built into the element formulation. The disadvantages of this approach include:

(1) Shell theory assumes σ_{z} is zero, so this interlaminar stress component will not be calculated.

(2) Shell theory also assumes that the out-of-plane shear strains are zero, i.e. γ_{xz} and γ_{yz} are zero, which means the interlaminar shear stresses cannot be calculated directly from the stress-strain relationship.

There is no getting around the first item, it is a consequence of shell theory. But some finite element analysis codes get around the second issue by using the equilibrium equations to distinguish the interlaminar shear stresses from the in-plane stresses. For example, one can solve for T_{xz} using the equilibrium equation below:

##

By assuming that the interlaminar stresses are zero at the bottom of the shell (free surface), this equation can be integrated through the thickness to calculate the interlaminar shear stress on a layer by layer basis through the entire thickness of the shell. This approach actually does a pretty good job of calculating the interlaminar shear stresses for layered shells.

Which is great, except what if the composite structure is not thin and cannot be assumed to behave as dictated by shell theory? In that case, there is another element formulation available: layered solids.

Layered solids, unfortunately, have their own problems. The two main ones are:

- Unlike layered shells that require only one element to describe the through-thickness behavior, layered solids will often require stacking to ensure good global deflection and stiffness characterization. This complicates the interlaminar stress calculation because the zero stress assumption at the top and bottom may not be clear for a complex, curved geometry.

- The finite element analysis code is internally calculating the strain tensor in each element from the nodal displacements, and all the strains, including the interlaminar strains, are assumed continuous. Once the material properties are applied to obtain layer stresses, the interlaminar stress distribution through the layers is discontinuous, which is exactly opposite of what it should be.

Unfortunately, these issues lead to interlaminar stress predictions that are generally poor. Consider a two-layer thin shell loaded in shear. The shell consists of two equal-thickness isotropic layers, one stiffer than the other. The plot below shows the prediction of τ_{xz} through the two layers, with the value of τ_{xz} at the interface containing the interlaminar shear value. A model using eight solid elements through the thickness (four elements through each layer), and a model using layered shell elements correlate reasonably well. The result from a layered solid is much less accurate, and the interlaminar stress prediction is particularly troubling since it is discontinuous at the interface.

**Composites Analysis Recommendations**

If you want to calculate interlaminar stresses using finite element analysis, your options are:

- The best approach is to use multiple solid elements through each layer. Unfortunately, this is rarely practical.

- If the structure is thin and behaves like a shell, layered shell elements typically provide a good prediction of interlaminar shear stresses. Note that the interlaminar normal stress is assumed zero.

- If the structure is not thin, and you cannot use multiple or even one solid element through each layer, your best bet is to use layered solids to predict the global response, and then submodeling techniques using several solid elements through the layer thickness to obtain accurate interlaminar stresses in the critical regions. Alternatively, special postprocessing composites analysis tools can be used to extract the correct stresses from the layered solids.

And perhaps most importantly, understand that at a free edge, the interlaminar stresses are singular, and thus a comparative approach should be considered.