### Markku Palantera

#### Latest posts by Markku Palantera (see all)

- HyperWorks Composites Go Multiscale with Expanded Classical Capability - August 28, 2018
- At the origins of delamination stresses – Free edge analysis gives insight - January 12, 2013

When you make a tensile test for [±45/0/90]s and [90/±45/0]s laminates you may observe different failure strengths. This may seem first surprising since the laminates are basically the same – only the stacking sequence is different, which should not influence on the in-plane values.

Test samples are not infinite, however. At the free edges of a laminate, out-of-plane stresses arise even when the laminate is subjected to in-plane loading. Out-of-plane stresses are caused by the mismatch of material properties in the bonded adjacent layers. The image below illustrates the Poisson’s ratio (green line) for a CFRP ply as a function of the layer orientation.

The mismatch of Poisson’s ratios for the adjacent ±θ layers is identically zero for all angles. If a laminate includes an interface of ±θ and 90 plies, where θ is approximately 25, the mismatch of Poisson’s ratios is maximized.

When the laminate is loaded in the x-direction, the stress resultant is zero in the transverse direction, but the layer level stresses are non-zero. Layer level stresses can be calculated using the Classical Lamination Theory (CLT) and the results are valid in the middle of the laminate. Layer level stresses are influenced by the mismatch of Poisson’s ratios and the stiffness of the plies.

At the free edge, the out-of-plane normal stress σ_{z} originates from σ_{y} through the moment balance. High internal transversal stresses result in high out-of-plane normal stresses at the free edge. This is illustrated for the three [±θ/90]s laminates, where θ is 25, 45 and 65, respectively, in the image below. When comparing the transversal layer level stresses of the three laminates using the CLT, it is obvious that the stresses are the largest for [±45/90]s laminate and smallest for the [±65/90]s laminate, respectively. This is explicitly seen from the σ_{z} values. Edge singularities exist both at the ±θ and -θ/90 interfaces, which can be seen from the image.

Out-of-plane shear stresses τ_{yz} and τ_{xz} originate from σ_{y} and τ_{xy}, respectively, through the force balance. Off-axis layers of a laminate are known to shear in the plane of the laminate when subjected to axial strain. This effect can be described with the coefficient of mutual influence η_{x,xy} defined by Lekhnitskii. The previous image illustrates the Lekhnitskii’s coefficient (red line) for a CFRP ply as a function of the layer orientation. Contrary to the Poisson’s ratio, the Lekhnitskii’s coefficient is an odd function of θ. Consequently, the mismatch of the Lekhnitskii’s coefficient for the adjacent ±θ layers is non-zero. Under the same axial strain, two adjacent ±θ layers are forced to distort into opposite directions due to the difference in the Lekhnitskii’s coefficient. As the layers are perfectly bonded, the displacement continuity at the layer interfaces require the existence of non-zero shear stresses τ_{xy} and τ_{xz}. The relation of the Lekhnitskii’s coefficient to τ_{xz} can be seen from the out-of-plane shear stresses depicted below at the free edge. For the [±25/90]s laminate, the mismatch in Lekhnitskii’s coefficients for the adjacent ±θ layers is the largest and for the [±65/90]s laminate it is basically zero.

Back to the original [±45/0/90]s and [90/±45/0]s laminates. They indeed have a very different stress distribution shape and, for example, the magnitude of the σ_{z} in the middle of the laminate is approximately five times larger for the [±45/0/90]s laminate when compared to the [90/±45/0]s laminate. This can be the reason why the first laminate has a lower tensile strength. The failure propagation has initiated from the onset of delamination.

Out-of-plane stresses near the free edge can be controlled through the choice of materials, ply orientations, stacking sequence, and ply thickness. Out-of-plane stresses can conveniently be studied with the ESAComp free edge analysis tool. The analysis is made for a selected laminate and the load object can be a combination of uniaxial in-plane and flexural loads and hygrothermal loads varying in the thickness direction.