Issue 29

M. Marino et alii, Frattura ed Integrità Strutturale, 29 (2014) 241-250; DOI: 10.3221/IGF-ESIS.29.21 242 associated with stress localization: for many practical applications, structural joining (both adhesive and bolted) could represent a gap for the structural behavior because of their failure mode. Failure modes of bolted joints in laminated composite plates under tensile loads usually occur in four basic modes: cleavage, net-tension, shear-out and bearing modes. In detail, local bearing failure modes are characterized by a local laminate compressive failure caused by the bolt diameter which tends to crush the composite material. Pin-bearing failure mode of bolted FRP joints, locally associated with matrix cracks, is an important design problem that has attracted the interest of the international scientific community, as confirmed by the great number of researches carried out in the last years [1–10]. Results of these studies have highlighted that both geometric (e.g., bolt diameter, plate width and thickness, end distance) and material properties (e.g., fiber inclination angle, matrix type and fiber nature, stacking sequence) highly affect the strength and the failure mode of FRP-based jointed elements. Accordingly, a computational model able to give parametric indications on the mechanical performance of bolted FRP joints, as well as able to predict their failure mechanisms, would be a powerful and useful design tool for both civil and mechanical advanced applications. Aim of this paper is to develop a numerical model based on a non-linear finite-element formulation for the analysis of the progressive damaging and the failure modes in bolted joints between fiber-reinforced composite laminates. The numerical formulation, applied to a pin-plate system, is based on a plane-stress bidimensional model and on an incremental displacement-based approach driven by the pin position. Neglecting friction, the unilateral contact at the pin-plate interface has been treated through a surface-to-surface penalty method. In order to describe the damage evolution, the model implements two failure criteria available in the literature (by Rotem [13] and Huang [15]), involving different stress- strain measures at different material scales. The obtained results have been successfully compared with the experimental data in [10], allowing to show soundness and accuracy of the proposed formulation, as well as to highlight the effectiveness and/or possible limitations of the considered failure criteria. T HEORETICAL BACKGROUND RP composite laminates are made of layers (plies) bonded together to form a plate-like structural element. Each ply consists in unidirectional continuous fibers embedded in a polymeric matrix, with a preferred fiber direction. Accordingly, each composite ply exhibits a global constitutive response characterized by a transversely isotropic symmetry, with the isotropic plane orthogonal to the fiber direction. In the following, as a notation rule, for each layer the subscript A denotes the direction parallel to the fibers, T the transverse-to-the-fibers direction, and symbols +/- discriminate strength material properties in traction and compression, respectively. Furthermore the generic constituent is indicated by the subscript c ( c f  for fibers and c m  for matrix). The fiber’s undamaged material is assumed to be linearly elastic, with symmetry plane orthogonal to the fiber’s axis (with engineering constants A f E , T f E , AT f G , AT f  , T f  ), and the matrix’s undamaged material is isotropic linearly elastic (with engineering constants m E and m  ). FRP layers and laminated plates are assumed to be planar and characterized by small thicknesses. Accordingly, a plane-stress condition is assumed in the following. The ability to predict initiation and growth of damage in bolted FRP joints can only be offered by progressive damage modeling techniques. Failure analysis of laminate composites are made up of three main ingredients: stress analysis through homogenization theories, failure analysis by means of strength criteria for composite layers, and a material degradation law for describing the failure occurrence in composite constituents. Stress analysis In order to determine the mechanical properties of the laminate a refined homogenization procedure, that takes into account localization mechanisms, has been used. Accordingly, in agreement with the Huang’s indications, the Bridging Model [14] is herein employed. Addressing a single composite layer with mono-directional fiber direction, the 6 x 6 equivalent homogeneous compliance matrix   [ ] ij S s  , expressed in a local coordinate system ( A,T,T ), results in:         1 [ ] [ ] f m f m f m S v S v S A v I v A             (1) F