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Keywords: metis element, interlaminar stresses, composite laminate, free edge.

Summary

The "metis" finite element method was developed by Nguyen Dang [1]. It is a special class of hybrid finite element. The unisolvent condition of the external displacement field permits the transformation of the integral of contour into an integral of volume. This is a special class of the classic hybrid model. The present paper deals with the use the "metis" displacement method to calculate the interlaminar stresses in composite laminate structures. The results for the symmetric composite laminates under a uniform axial strain are presented and compared to many other studies to demonstrate the accuracy and efficiency of the present method.

Consider a composite material structure subjected to a constant axial extension. Several attempts have been made to obtain accurate stress distributions in this structure: finite element method, hybrid stress finite element, pure equilibrium, complex series expansion in conjunction with boundary collocation or singular hybrid finite element. This paper develops the metis displacement finite element method to solve this problem.

Based on Lekhnitskii's elasticity theory [2], the formulae of generalized plane strain state for composite laminate structure are established. The problem in three dimensions is reduced to a problem in two and a half dimensions: The geometry is a two dimensional domain and the displacement and stress fields are in three dimensions.

Then, we use the metis finite element method to establish the variational formulation for free edge problem of composite laminate. After that, the discretized formulation for metis displacement finite element is presented. In the metis element method, the Lagranges multiplier field, which is interpreted physically as the displacement field is prolonged to the interior of each element in a continuous manner (unisolvent principle). This principle permits the transformation of a contour integral into a volume integral or into a surface integral.

In the application part of the paper, a composite laminate combined from four symmetrical layers of a high-modulus graphite-epoxy system is considered. The axial strain is assumed to be uniform. As a result of geometric symmetry, only a quarter of the cross sectional area is considered. The triangular element in second degree and a mesh of 236 elements with a refinement growing towards the free edge are used.

The results for composite laminates [90^o/0^o]_s, [0^o/90^o]_s, [45^o/-45^o]_s, [30^o/-60^o]_s, [60^o/-30^o]_s, [15^o/-75^o]_s, [75^o/-15^o]_s are presented and compared with references [3,4,5]. The results obtained show that the interlaminar stresses are generally confined to a boundary region or layer near or at the free edge, and the interlaminar stress components can be singular at the intersection of the free edge with the interface between two distinct layers. Because of the stress state in this boundary region of laminate it is not a plane stress state; the classical laminate theory is inadmissible in this region.

The conclusion is that metis displacement method provided very good results in the treatment of a difficult problem comprising a singularity. The accuracy of the current solution is supported by some independent studies. It appears that the metis method provides a new method to calculate the interlaminar stresses in the free edge problem of composite laminates.

References

1: Nguyen Dang Hung, "On the monotony and the convergence of a special class of hybrid finite element: the mongrel element. Variational methods in mechanics of solids", Proceeding of the IUTAM Symposium organised in Northwestern, University of Evanston, Illinois, U.S.A., Ed. by S. Nemat-Nasser, Pergamon, 208-213, 1980.
2: S.G. Lekhnitskii, "Theory of elasticity of an anisotropic body", Holden-day, San Fransisco, California, USA, 1963.
3: W.L. Yin, "Free-Edge effect in anisotropic laminates under extension bending and twisting, part II: Eigenfunction analysis and the results for symmetric laminates", Journal of Applied Mechanics, 61, 416-421, 1994. doi:10.1115/1.2901460
4: S.S. Wang and I. Choi, "Boundary-layer effects in composite laminates", Journal of Applied Mechanics, 49, 549-560, 1982.
5: S.S. Wang and F.G. Yuan, "A singular hybrid finite element analysis of boundary layer stresses in composite laminates", International Journal Solids and Structures, 19 (9), 825-837, 1983. doi:10.1016/0020-7683(83)90075-6

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 83 PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro Paper 266 Metis Element Model for Interlaminar Stresses in Composite Laminates N.T. Duong¹ and H. Nguyen Dang² ¹Welding and Metals' Technology Department, Hanoi University of Technology, Vietnam ²Department of LTAS - Fracture Mechanics, University of Liège, Belgium doi:10.4203/ccp.83.266 purchase the full-text of this paper Full Bibliographic Reference for this paper N.T. Duong, H. Nguyen Dang, "Metis Element Model for Interlaminar Stresses in Composite Laminates", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 266, 2006. doi:10.4203/ccp.83.266 Keywords: metis element, interlaminar stresses, composite laminate, free edge. Summary The "metis" finite element method was developed by Nguyen Dang [1]. It is a special class of hybrid finite element. The unisolvent condition of the external displacement field permits the transformation of the integral of contour into an integral of volume. This is a special class of the classic hybrid model. The present paper deals with the use the "metis" displacement method to calculate the interlaminar stresses in composite laminate structures. The results for the symmetric composite laminates under a uniform axial strain are presented and compared to many other studies to demonstrate the accuracy and efficiency of the present method. Consider a composite material structure subjected to a constant axial extension. Several attempts have been made to obtain accurate stress distributions in this structure: finite element method, hybrid stress finite element, pure equilibrium, complex series expansion in conjunction with boundary collocation or singular hybrid finite element. This paper develops the metis displacement finite element method to solve this problem. Based on Lekhnitskii's elasticity theory [2], the formulae of generalized plane strain state for composite laminate structure are established. The problem in three dimensions is reduced to a problem in two and a half dimensions: The geometry is a two dimensional domain and the displacement and stress fields are in three dimensions. Then, we use the metis finite element method to establish the variational formulation for free edge problem of composite laminate. After that, the discretized formulation for metis displacement finite element is presented. In the metis element method, the Lagranges multiplier field, which is interpreted physically as the displacement field is prolonged to the interior of each element in a continuous manner (unisolvent principle). This principle permits the transformation of a contour integral into a volume integral or into a surface integral. In the application part of the paper, a composite laminate combined from four symmetrical layers of a high-modulus graphite-epoxy system is considered. The axial strain is assumed to be uniform. As a result of geometric symmetry, only a quarter of the cross sectional area is considered. The triangular element in second degree and a mesh of 236 elements with a refinement growing towards the free edge are used. The results for composite laminates [90^o/0^o]_s, [0^o/90^o]_s, [45^o/-45^o]_s, [30^o/-60^o]_s, [60^o/-30^o]_s, [15^o/-75^o]_s, [75^o/-15^o]_s are presented and compared with references [3,4,5]. The results obtained show that the interlaminar stresses are generally confined to a boundary region or layer near or at the free edge, and the interlaminar stress components can be singular at the intersection of the free edge with the interface between two distinct layers. Because of the stress state in this boundary region of laminate it is not a plane stress state; the classical laminate theory is inadmissible in this region. The conclusion is that metis displacement method provided very good results in the treatment of a difficult problem comprising a singularity. The accuracy of the current solution is supported by some independent studies. It appears that the metis method provides a new method to calculate the interlaminar stresses in the free edge problem of composite laminates. References 1 Nguyen Dang Hung, "On the monotony and the convergence of a special class of hybrid finite element: the mongrel element. Variational methods in mechanics of solids", Proceeding of the IUTAM Symposium organised in Northwestern, University of Evanston, Illinois, U.S.A., Ed. by S. Nemat-Nasser, Pergamon, 208-213, 1980. 2 S.G. Lekhnitskii, "Theory of elasticity of an anisotropic body", Holden-day, San Fransisco, California, USA, 1963. 3 W.L. Yin, "Free-Edge effect in anisotropic laminates under extension bending and twisting, part II: Eigenfunction analysis and the results for symmetric laminates", Journal of Applied Mechanics, 61, 416-421, 1994. doi:10.1115/1.2901460 4 S.S. Wang and I. Choi, "Boundary-layer effects in composite laminates", Journal of Applied Mechanics, 49, 549-560, 1982. 5 S.S. Wang and F.G. Yuan, "A singular hybrid finite element analysis of boundary layer stresses in composite laminates", International Journal Solids and Structures, 19 (9), 825-837, 1983. doi:10.1016/0020-7683(83)90075-6 purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £140 +P&P)
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