Laminate free-edge effect

In the present paper the Boundary Finite Element Method is presented as a boundary discretization method for the numerical investigation of interfacial stress concentrations in composite laminates. In contrast to the classical boundary element method, the element formulation is finite element based, which avoids the necessity of a fundamental solution. Comparative results from finite element calculations show good agreement both for the laminate free-edge effect and for the example of the stress concentrations near cracks in composite laminates. 

Keywords Boundary Finite Element Method, composite laminates, stress localization, laminate free-edge effect, free-edge stresses, crack problems, transverse matrix crack, numerical methods, boundary discretization... 

Because of this importance of stress localizations many investigations have been dedicated both to the laminate free-edge effect, starting with the finite difference analyses of Pipes and Pagano 1970 [2] and the matrix crack problem. For both cases, closed-form analytical solutions are of a more or less approximate character. On the other hand, numerical methods as the finite elements (FEM) require a high discretizational effort because of... 

In the following results are presented for the application of the Boundary Finite Element Method both for the case of the laminate free-edge effect and for the case of a single transverse matrix crack in the framework of linear elasticity theory. 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

Failure of matrix-dominated ( [ 0]2s [ 0/9O]g ) laminates and fiber- dominated tc/4 laminates has been studied. A new multiple-mode failure criterion was proposed. This new criterion, which took the free-edge effect into account, was found capable of predicting laminate strengths for the matrix-dominated laminates and delamination on-set stress for tc/4 laminates. 

Wang, S.S. and Choi, 1. (1982). Boundary layer effects in composite laminates. Part I free-edge stress singularities J. Appl. Mech. 49, 541-548. 

Since the in-plane stresses (Cn, CT22 12) ibe surface ply near the free edge are not constant, the average stresses over a distance of 2t from the free edge were used in the Tsai-Hill criterion for failure prediction. Further, since the surface plies are partially free from constraints (the lamination effect), the in-plane shear strength should be lower than that measured with [ 45]2s specimen. Thus, we took the value S = 14.4 ksi (for AS4/3501-6) reported in most literature. For T300/5208 graphite/epoxy composite we found S = 8.2 ksi was suitable. 

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