Singularities near three-dimensional corners in composite laminates
Tóm tắt
The high interlaminar stresses, which appear in laminated composites due to the boundary layer effect near the free edge, play an important role in the analysis and design of advanced structures. Moreover, they are also the dominant effect causing delamination. Even if the singular behavior of such structures is investigated in many works, most of them deal either with 2D, or with pseudo-3D problems, i.e. problems of two variables in a three-dimensional space. However, some numerical and experimental findings indicate that laminated plates exhibit a tendency to delaminate at corners, an effect impossible to be determined by a two-dimensional analysis. The aim of the present paper is to investigate stress singularities in a laminated composite wedge under consideration of real three-dimensional corner effects. A weak formulation, as well as a finite element approximation technique introduced in the past for isotropic problems is extended here to cover anisotropic material properties. This formulation leads to a quadratic eigenvalue problem, which is solved iteratively using the Arnoldi method. The first singular terms in the asymptotical expansion of the linear-elastic solution near the vertex of the wedge are obtained as eigenpairs of this eigenvalue problem. The order and mode of singularity are reported for all wedge angles and different fiber orientations for angle and cross-ply laminates. All calculations are based on a typical for some high modulus graphite-epoxy systems orthotropic material model.
Tài liệu tham khảo
Altus, E., Rotem, A. and Shmueli, M. (1980). Free edge effect in angle ply laminates – a new three dimensional finite difference solution. Journal of Composite Materials 14, 21–30.
Apel, T., Sändig, A.-M. and Solov'ev, S. (2001). Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes. Preprint SFB393/01-33, Technical University Chemnitz, Germany.
Bogy, D. (1968). Edge-bounded dissimilar orthogonal elastic wedges under normal and shear loading. Journal of Applied Mechanics 35, 460–466.
Czarnek, R., Post, D. and Herakovich, C. (1983). Edge effect in composites by moire interferometry. Experimental Techniques 7, 18–21.
Dauge, M. (1988). Elliptic boundary value problems on corner domains. smoothness and asymptotics of solutions, Lecture Notes in Mathematics, Vol. 1341, Springer-Verlag, Berlin.
Dimitrov, A. and Schnack, E. (2002). Asymptotical expansion in non-Lipschitzian domains – a numerical approach. Numerical Linear Algebra with Applications (to appear).
Dimitrov, A., Andrä, H. and Schnack, E. (2001). Efficient computation of order and mode of corner singularities in 3D-elasticity. International Journal for Numerical Methods in Engineering 52, 805–827.
Ghahremani, F. (1991). A numerical variational method for extracting 3d singularities. International Journal of Solids and Structures 27, 1371–1386.
Griffin, O. (1988). Three-dimensional thermal stresses in angle-ply composite laminates. Journal of Composite Materials 22, 53–69.
Griffin, O. and Roberts, J. (1983). Numerical/experimental correlation of three dimensional thermal stress distributions in graphite/epoxy laminates. Journal of Composite Materials 17, 539–548.
Grisvard, P. (1992). Singularities in Boundary Value Problems, Recherches en Mathematiques Appliquées. Masson, Springer-Verlag, Berlin.
Groth, H. (1985). A method to predict fracture in an adhesively bonded joint. International Journal of Adhesion Adhesives 5, 19–22.
Groth, H. (1988). Stress singularities and fracture at interface corners in bonded joint. Int. J. Adhesion Adhesives 8, 107–113.
Gu, L. and Belytschko, T. (1994). A numerical study of stress singularities in a two-material wedge. International Journal of Solids and Structures 31, 865–889.
Hein, V. and Erdogan, F. (1971). Stress singularities in a two-material wedge. International Journal of Fracture Mechanics 7, 317–330.
Herakovich, C. et al. (1985). Free edge strain concentrations in real composite laminates: experimental-theoretical correlation. Journal of Applied Mechanics 52, 787–793.
Kassapoglou, C. (1986). An efficient method for the calculation of interlaminar stresses in composite materials. Journal of Applied Mechanics 53, 744–750.
Kondratiev, V. (1967). Boundary problems for elliptic equations in domains with conical or angular points. Transactions Moscow Mathematics Society 16, 227–313.
Leguillon, D. and Sanchez-Palencia, E. (1987). Computation of Singular Solutions in Elliptic Problems and Elasticity, Masson, Wiley, New York.
Leguillon, D. and Sanchez-Palencia, E. (1999). On 3D cracks intersecting a free surface in laminated composites. International Journal of Fracture 99, 25–40.
Oplinger, D., Parker, B. and Chiang, F. (1974). Edge effect studies in fiber-reinforced laminates. Experimental Mechanics 14, 347–354.
Pageau, S. and Biggers, S. (1996). A finite element approach to three-dimensional singular stress states in anisotropic multi-material wedges and junctions. International Journal of Solids and Structures 33, 33–47.
Pipes, R. and Daniel, I. (1971). Moire analysis of the interlaminar shear edge effect in laminated composites. Journal of Composite Materials 5, 255.
Pipes, R. and Pagano, N. (1970). Interlaminar stresses in composite laminates under uniform axial extension. Journal of Composite Materials 4, 538–548.
Reddy, J. (1996). Mechanics of Laminated Composite Plates: Theory and Analysis, CRC Press, Boca Raton.
Rybicki, E. (1971). Approximate three-dimensional solutions for symmetric laminates under inplane loading. Journal of Composite Materials 5, 354–360.
Somaratna, N. and Ting, T. (1986). Three-dimensional stress singularities in anisotropic materials and composites. International Journal of Engineering Sciences 24, 1115–1134.
Spilker, R. and Chou, S. (1980). Edge effects in symmetric composite laminates: importance of satisfying the traction-free-edge condition. Journal of Composite Materials 14, 2–20.
Wang, A. and Crossman, F. (1977). Some new results on edge effect in symmetric composite laminates. Journal of Composite Materials 11, 92–106.
Wang, J. and Dickson, J. (1978). Interlaminar Stresses in symmetric composite laminates. Journal of Composite Materials 12, 390–402.
Wang, S. and Choi, I. (1982). Boundary-layer effects in composite laminates: Part 1 and 2. Journal of Applied Mechanics 49, 541–560.
Whitcomb, J., Raju, I. and Goree, J. (1982). Reliability of the finite element method for calculating free edge stresses in composite laminates. Computers and Structures 15, 23–37.
Yosibash, Z. (1997). Computing edge singularities in elastic anisotropic three-dimensional domains. International Journal of Fracture 86, 221–245.