The four-node quasi-conforming shell element was extended in the present article to the case of geometrically non-linear behavior of the FGM plates and shells. The high stress occurring in the FGM structures will affect its integrity and the structures is susceptible to failure. Therefore, we focus on the effect of volume fraction of the constituent materials in the mechanical behavior of FGM plates and shells. The material properties are assumed to be varied in the thickness direction according to a sigmoid function in terms of the volume fraction of the constituents. The series solutions of sigmoid FGM (S-FGM) plates, based on the first-order shear deformation theory and Fourier series expansion are provided as the reference solution for the numerical results. In quasi-conforming formulation, the tangent stiffness matrix is explicitly integrated. This makes the element computationally efficient in the non-linear analysis. Several selected examples of non-linear analysis of FGM shells are included in the article for the illustration of possibilities of the presented formulation. It is seen that the present results for the non-linear behavior of FGM plates and shells can provide a useful benchmark to check the accuracy of related numerical solutions.
Hirano, T. and Yamada, T. (1988). Multi-Paradigm Expert System Architecture Based Upon The Inverse Design Concept, In: International Workshop on Artificial Intelligence for Industrial Applications, Hitachi, Japan.
2.
Woo, J. and Meguid, S.A. (2001). Nonlinear Analysis of Functionally Graded Plates and Shallow Shells, International Journal of Solids and Structures , 38: 7409—7421.
3.
Praveen, G.N. and Reddy, J.N. (1998). Nonlinear Transient Thermoelastic Analysis of Functionally Graded Ceramic—Metal Plates, International Journal of Solids and Structures, 35: 4457—4476.
4.
Bao, G. and Wang, L. (1995). Multiple Cracking in Functionally Graded Ceramic/Metal Coatings, International Journal of Solids and Structures, 32: 2853—2871.
5.
Jin, Z.H. and Paulino, G.H. (2001). Transient Thermal Stress Analysis of an Edge Crack in a Functionally Graded Material , International Journal of Fracture, 107: 73—98.
6.
Delale, F. and Erdogan, F. (1983). The Crack Problem for a Nonhomogeneous Plane, ASME Journal of Applied Mechanics, 50: 609—614.
7.
Erdogan, F. and Chen, Y.F. (1998). Interfacial Cracking of FGM/Metal Bonds, In: Kokini K, (ed.), Ceramic Coating .
8.
Chung, Y.L. and Chi, S.H. (2001). The Residual Stress of Functionally Graded Materials , Journal of Chinese Institute of Civil and Hydraulic Engineering , 13: 1—9.
9.
Chi, S.H. and Chung, Y.L. (2002). Cracking in Sigmoid Functionally Graded Coating, Journal of Mechanics , 18: 41—53.
10.
Rousseau, C.E. (2006). Critical Examination of the Use of Coherent Gradient Sensing in Measuring Fracture Parameters in Functionally Graded Materials, Journal of Composite Materials, 40(19): 1763—1782.
11.
Tang, L.M., Chen, W.J. and Liu, Y.X. (1983). String net Approximation and Quasi-Conforming Technique, In: Alturi, S.N., Gallagher, R.H. and Zienkiewicz, O.C. (eds), Hybrid and Mixed Finite Element Methods, John Wiley, Chichester.
12.
Tang, L.M., Chen, W.J. and Liu, Y.X. (1984). Formulation of Quasi-Conforming Element and Hu—Washizu Principle, Computers and Structures, 19(1—2): 247—250.
13.
Guan, Y. and Tang, L.M. (1992). A Quasi-Conforming Nine-Node Degenerated Shell Finite Element, Finite Element Analysis and Design, 11: 165—176.
14.
Guan, Y. and Tang, L.M. (1995). A geometrically Non-Linear Quasi-Conforming Nine-Node Quadrilateral Degenerated Solid Shell Element, International Journal for Numerical Methods in Engineering, 38: 927—942.
15.
Shi, G. and Voyiadjis, G.Z. (1991). A simple C0 quadrilateral Thick/Thin Shell Element Based on the Refined Shell Theory and the Assumed Strain Fields, International Journal of Solids and Structures , 27: 283—298.
16.
Shi, G. and Voyiadjis, G.Z. (1991). Efficient and Accurate Four-Node Quadrilateral C 0 Plate Bending Element Based on Assumed Strain Fields, International Journal for Numerical Methods in Engineering , 32: 1041—1055.
17.
Shi, G. and Voyiadjis, G.Z. (1991). Geometrically Nonlinear Analysis of Plates by Assumed Strain Element with Explicit Tangent Stiffness Matrix, Computers and Structures, 41: 757—763.
18.
Shi, G. and Voyiadjis, G.Z. (1992). A Simple Non-Layered Finite Element for the Elasto-Plastic Analysis of Shear Flexible Plates, International Journal for Numerical Methods in Engineering , 33: 85—99.
19.
Shi, G. and Voyiadjis, G.Z. (1993). A Computational Model for FE Ductile Plastic Damage Analysis of Plate Bending, ASME J Applied Mechanics, 60: 749—758.
20.
Kim, K.D. , Lomboy, G.R. and Voyiadjis , G.Z. (2003). A 4-node Assumed Strain Quasi-Conforming Shell Element with 6 D.O . F., International Journal for Numerical Methods in Engineering , 58: 2177—2200.
21.
Park, T.H., Kim, K.D. and Han, S.C. (2006). Linear Static and Dynamic Analysis of Laminated Composite Plates and Shells using a 4-node Quasi-Conforming Shell Element , Composites Part B-Engineering, 37(2—3): 237—248.
22.
Kim, K.D. and Lomboy, G.R. (2006). A co-rotational Quasi-conforming 4-node Resultant Shell Element for Large Displacement of Elasto-Plastic Analysis , Computer Methods in Applied Mechanics and Engineering, 195(44—47): 6502—6522.
23.
Kebari, H. and Cassel, A.C. (1992). A Stabilized Nine-node Non-Linear Shell Element , International Journal for Numerical Methods in Engineering , 35: 37—61.
24.
Bathe, K.J. (1996). Finite Element Procedures, Prentice Hall, New Jersey.
25.
Reddy, J.N. (2000). Analysis of Functionally Graded Plates, International Journal for Numerical Methods in Engineering, 58: 2177—2200.
26.
Lee, Y.D. and Erdogan, F. (1995). Residual/thermal Stress in FGM and Laminated Thermal Barrier Coatings, International Journal of Fracture, 69: 145—165.
27.
Kanok-Nukulchai, W. (1979). A simple and Efficient Finite Element for General Shell Analysis, International Journal for Numerical Methods in Engineering, 14: 179—200.
28.
Allman, D.J. (1988). A quadrilateral Finite Element Including Vertex Rotation for Plane Elasticity Analysis, International Journal for Numerical Methods in Engineering, 26: 717—730.
29.
Hughes, T.J.R. and Brezzi, F. (1989). On Drilling Degrees of Freedom, Computer Methods in Applied Mechanics and Engineering , 72: 105—121.
30.
Simo, J.C., Fox, D.D. and Hughes, T.J.R. (1992). Formulations of Finite Elasticity with Independent Rotations, Computer Methods in Applied Mechanics and Engineering , 95: 277—288.
31.
Ibrahimbegovic, A. (1994). Stress Resultant Geometrically Nonlinear Shell Theory with Drilling Rotations-PartI. A Consistent Formulation, Computer Methods in Applied Mechanics and Engineering, 118: 265—284.
32.
Sansour, C. and Bednarczyk, H. (1995). The Cosserat surface as a Shell Model, Theory and Finite-Element Formulation, Computer Methods in Applied Mechanics and Engineering, 120: 1—32.
33.
XFINAS 2.0. (2005). Asian Institute of Technology & Konkuk University, Korea, http://www.xfinas.com/.
34.
Chi, S.H. and Chung, Y.L. (2006). Mechanical Behavior of Functionally Graded Material Plates Under Transverse Load — Part II: Numerical Results, International Journal of Solids and Structures, 43(13): 3675—3691.
35.
Crisfield, M.A. (1981). A Fast Incremental/Iterative Solution Procedure That Handles Snap-Through, Computers and Structures, 13: 55—62.
36.
Rhiu, J.J. and Lee, S.W. (1988). A Nine-node Finite Element for Analysis of Geometrically Non-Linear Shells, International Journal for Numerical Methods in Engineering, 24: 581—604.
37.
Kim, K.D. , Liu, G.Z. and Han , S.C. (2005). A Resultant 8-Node Solid-Shell Element for Geometrically Nonlinear Analysis , Computational Mechanics, 35(5): 315—331.
38.
Saleeb, A.F. , Chang, T.Y., Graf , W. and Yingyeunyong , S. (1990). A Hybrid/Mixed Model for Non-Linear Shell Analysis and its Applications to Large-Rotation Problem, International Journal for Numerical Methods in Engineering, 29: 407—446.
39.
Simo, J.C., Fox, D.D. and Rifai, M.S. (1990). On a Stress Resultant Geometrically Exact Shell Model. Part III: Computational Aspects of the Nonlinear Theory, Computer Methods in Applied Mechanics and Engineering, 79: 21—70.
40.
Lee, S.J. and Kanok-Nukulchai , W. (1998). A Nine-Node Assumed Strain Finite Element for Large Deformation Analysis of Laminated Shells, International Journal for Numerical Methods in Engineering, 42: 777—798.
41.
Sansour, C. and Kollmann, F.G. (2000). Families of 4-Node and 9-Node Finite Elements for a Finite Deformation Shell Theory. An Assesment of Hybrid Stress, Hybrid Strain and Enhanced Strain Elements, Computational Mechanics, 24: 435—447.
42.
Fontes Valente, R.A., Natal Jorge, R.M., Cardoso , R.P.R., César de Sá , J.M.A., and Grácio, J.J.A. (2003). On the Use of an Enhanced Transverse Shear Strain Shell Element for Problems Involving Large Rotations, Computational Mechanics, 30: 286—296.
