ABSTRACT
Fibre-reinforced and sandwich composites with laminated faces are the best candidate materials in many engineering fields by the viewpoint of the impact resistance, containment of explosions, protection against projection of fragments, survivability and noise and vibration suppression. A considerable amount of the incoming energy is absorbed through local matrix and fibre failure, or as an interaction of both, and accumulation of delamination damage. The most important energy dissipation mechanisms are the hysteretic damping in the matrix and in the fibers, that is dominant in undamaged composites vibrating at small amplitudes, and the frictional damping at the fiber-matrix interface. The dissipation of the incoming energy also partly takes place as a not well understood dissipation at the cracks and delamination sites. As self-evident, these local effects have to be accurately accounted to predict the structural response of composite materials. This task requires sophisticated computational models, but with an affordable computational effort and a representation suitable for design purposes. The widespread equivalent single layer models are of limited validity, since they cannot appropriately account for the layerwise kinematics and the stress fields inherent to the multilayer construction. They can merely provide a description of the overall behaviour. As a compromise between accuracy and simplicity, in this paper the piecewise cubic zig-zag model of Ref [1] is used in the optimization process that makes stationary the energy contributions under in-plane variation of material properties. The choice of this partial layerwise model is due to the intricate equations in terms of lay-up, fiber orientation, fiber volume fraction fraction , thickness of plies and choice of constituent materials involved, which make unsuitable the most accurate, but computationally intensive discrete-layer
models. The zig-zag model chosen, which provides a rather accurate description of the stored energy although it requires to be post-processed for providing accurate interlaminar stress distributions, postulates following distribution of in-plane displacements across the thickness:
with S as the current interface and N-l as the total number of interfaces, while the transverse displacement is assumed constant across the thickness. Owing to this last assumption the effects of the transverse normal stress and strain are disregarded in the constitutive equations. To overcome this drawback and improve the accuracy of interlaminar stresses, which is already satisfactory since the terms in the summation implies the fulfillment a priori of the interfacial contact conditions on these stresses and the terms F'\ and G\ enable the fulfillment of the traction-free boundary conditions at the top and bottom boundary surfces, a strain energy updating procedure is used. This procedure, see Ref. [2], makes the strain energy equal to that of a refined 3D zig-zag model (see, Ref. [3]) by an iterative process:
U(x, y, z) = u(x, y,z) + YJ <* V„ (*, y^-mZ+ K (3)
V(x, y, z) = v(x, y, z) + g <% (x, y){z-mZ * >/k (4)
W(x,y,z) = w(x,y,z) + £ <>,(*, ^ - " ' ^ K + £ "V, (*,}-)(z-(i)Z+ ^ k <5> In this way the direct use of the 3D zig-zag model, which has a prohibitive effort within the optimization process, is avoided although the effects of the transverse normal stress on strain energy and of the interfacial contact conditions on these stresses and on the transverse normal stress gradient are taken into account, as required by the elasticity theory. The mixed solid element (this term is used to indicate that the master fields are the internal fields) developed in Ref. [4], which is more accurate than the zig-zag models but also more computationally intensive, is used to assess the local effects of the optimized solutions and to predict their structural performance. As nodal d.o.f, it has the three displacement components and the three interlaminar stresses. Characteristic feature, these d.o.f are interpolated within the element domain using C°, tri-linear, standard serendipity shape functions. According, the intra-element equilibria are met in an approximate integral form. Although seldom used, this approximation was chosen because it does not affect accuracy and reduces the effort required to develop mixed elements. Thanks to this representation, the computational effort required is not larger than for displacement-based counterpart solid elements, while accuracy and convergence are dramatically improved (see, e.g. Loubignac et al. [5]). The readers find a compendium on mixed elements in the recent book by Hoa and Feng [6] and a comprehensive discussion of the details here omitted on solvability, stability and convergence characteristics in Ref. [4]. This element appeared stable, accurate, quite fast convergent and capable to predict smooth, continuous interlaminar stresses despite abrupt changes in the material properties of layers and the presence of a stress singularity.
