Keywords: zig-zag models, transverse normal deformability, Hermitian degrees of freedom, sublaminates approach, damaged interfaces, slip-locking.

The use of advanced composite materials, typical of modern Aerospace projects, leads to the need for theories capable to capture a number of non-classical effects: the transverse shear flexibility, the need for fulfilling the continuity requirement of transverse shear stresses at the layer interfaces, the possibility to model interfacial bonding damage between the constituent laminae. The linear and cubic zig-zag models proposed by Di Sciuva [1] can fulfil the above described requirements without increasing the number of degrees of freedom with respect to the FSDT. An improvement of the model (the Hermitian zig-zag model) is presented in this paper; the term Hermitian is due to the fact that the through-the-thickness shape functions are a generalization of the classical Hermite polynomials. The presented model offers some interesting capabilities: (i) through-the-thickness linear transverse displacement, (ii) evaluation of the transverse normal deformability, (iii) traction equilibrium condition on the external surfaces and (iv) use of the displacements and transverse shear stresses of the external surfaces as degrees of freedom. But, perhaps, the most important characteristic of the Hermitian model is that it can be used to obtain a through-the-thickness description of displacements, strains and stresses (in particular of the normal ones) that is more detailed than what is allowed by the displacement field. Applying the sublaminates approach proposed by Averill [2], a beam finite element based on the Hermitian model has been formulated; then a discretizing and assembling procedure has been used that enables to divide the laminate thickness in a number of elements-sublaminates. In this way, for example, we can study a laminated beam using a piecewise linear transverse displacement instead of a simply linear one (that is what we can do with the Hermitian model by itself).

The problem of interlayer slip is a big challenge for 2-D laminate models. The classical cubic zig-zag model can be used to study damaged beams and plates [1]: it is in fact possible to introduce jumps in the in-plane displacements at the damaged interfaces and, moreover, these jumps are proportional to the interlaminar transverse shear stress (linear slip law). This capability is still present in the Hermitian model. It is to be said that this way to describe damage is not valid when high levels of debonding are experienced. As we will show in this work, in fact, when the interlaminar damage is severe, the classical zig-zag models experience what we have called slip-locking: we use the term locking because the laminated structure behaves in a stiffer and stiffer way when the slip damage level is increasing. In this work the origin of the phenomenon is explained. When we use the classical zig-zag model or the Hermitian one but with only a sublaminate, the through-the-thickness distribution of the transverse shear stress exhibits a shape that does not change with the damage level and this is an incorrect behaviour; as a consequence, when the damage is very high, the whole transverse shear stress distribution vanishes. The correct behaviour should be characterized by the vanishing of the shear stress only in correspondence to the damaged interface and this can be obtained only if the shape of the through-the-thickness distribution changes with the intensity of the slip defect. The use of the sublaminates approach allow to divide the thickness in such a way that the damaged interface is a boundary between adjacent elements-sublaminates. The damage effects are introduced imposing a particular relation (not only linear) between the in-plane displacement jump and the transverse shear stress for the nodes where the debonding defect is present. This procedure can be easily introduced because the degrees of freedom used for the elements are the displacements and the shear stresses of the external surfaces of each sublaminate.

In this work it is shown that the Hermitian model is valid in describing local through-the-thickness responses (in particular in terms of transverse quantities) when used together with the FEM/sublaminates approach. A particular assembling scheme is used to describe localized or diffused interlayer slips; a comparison between the results obtained with the sublaminates approach and those obtained with the classical and Hermitian (one sublaminate) zig-zag models show that the latter are unable to describe severe damages because of the slip-locking phenomenon. Some numerical results are presented to show the potentialities of the method.

1

M. Di Sciuva, "A Geometrically Nonlinear Theory of Multilayered Plates with Interlayer Slips", AIAA Journal, 35(11), 1753-1759, 1997. doi:10.2514/2.23

2

R. C. Averill, Y. C. Yip, "Thick Beam Theory and Finite Element Model with Zig-Zag Sublaminate Approximations", AIAA Journal, 34(8), 1627-1632, 1996. doi:10.2514/3.13281

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