Transient thermal analysis of functionally graded shallow shells by the MLPG

• Autorzy: Sladek J. , Sladek V. , Solek P.

Warianty tytułu

PL

Analiza termiczna powłok z funkcjonalnych materiałów gradientowych z wykorzystaniem metody MLPG

• Języki publikacji: EN

Abstrakty

EN

In recent years the demand for construction of huge and lightweight shell and spatial structures is increasing. To minimize the weight of shell structures a layered profile of the shell is utilized frequently. In such a case a delaminating of individual layers may occur due to a jump change of the material properties. To remove this phenomenon the functionally graded materials (FGMs) has been introduced recently. FGMs are multi-phase materials with a pre-determined property profile, where the phase volume fractions are varying gradually in space. This results in continuously nonhomogenous material properties at the macroscopic structural scale. Often, these spatial gradients in the material behaviour render FGMs as superior to conventional composites because of their continuously graded structures and properties. FGMs may exhibit isotropic or anisotropic material properties, depending on the processing technique and the practical engineering requirements. Many linear bending studies are focused only to a lateral pressure load with assumption of uniformly distributed temperature in the whole shell. However, in shells with FGM properties the role of thermal loading is more imperative. Therefore, it is interesting to analyze shells under a general thermal load. Literature sources on this subject are poor and they are mostly restricted to analyses of plates. Due to the high mathematical complexity of the boundary or initial-boundary value problems, analytical approaches for FGMs are restricted to simple geometry and boundary conditions. The choice of an appropriate mathematical model together with a consistent computational method is important for such kind of structures. Most significant advances in shell analyses have been made using the finite element method (FEM). It is well known that numerical results by standard displacement-based type shell element are over stiff with yielding the shear locking phenomena in thin shells. Locking problems arise due to inconsistencies in discrete representations of the transverse shear energy and the membrane energy. The boundary element method (BEM) has emerged as an alternative numerical method to solve plate and shell problems. It is a very powerful computational method if a fundamental solution is available for considered problem. However, the fundamental solution for a thick orthotropic shell wit continuously varying material properties is not available according to the best of the author?s knowledge. Meshless approaches for solution of problems of continuum mechanics have attracted much attention during the past decade. One of the most rapidly developed meshfree methods is the Meshless Local Petrov-Galerkin method (MLPG). The solution of the uncoupled problem in the present paper is split into two tasks. In the first task the temperature distribution in the shell has to be obtained by solving the diffusion equation. The temperature distribution in shell has to be analyzed as 3-D problem. The MLPG is applied to transient heat conduction equations in a continuously nonhomogeneous solid. The Laplace transform technique is used to eliminate the time variable. Several quasi-static boundary value problems must be solved for various values of the Laplace-transform parameter. The Stehfest?s inversion method is applied to obtain the time-dependent solution. In the second task, the set of governing differential equations for Reissner-Mindlin shell bending theory with Duhamel-Neumann constitutive equations is solved. Since thermal changes in solids are relatively slow with respect to elastic wave velocity, the inertial terms in Reissner-Mindlin governing equations are not considered. The problem is considered as quasi-static with time dependent thermal forces. The MLPG method is applied again to solution of that problem with the meshless Moving Least-Squares (MLS) approximation of primary field variables. The nodal points are spread freely in the analyzed domain and on its boundary. The essential boundary conditions are satisfied by collocation of approximated fields at nodes with prescribed values. In other nodes, the governing PDEs are considered on subdomains around these nodes in the local weak-form with using unit test functions. The resulting local integral equations are discretized within the assumed approximation of field variables. Numerical results for simply supported and clamped square shells are presented to illustrate the efficiency of the present computational method.

PL

Tematem niniejszej pracy jest wykorzystanie lokalnej beziastkowej metody Petrova-Galerkina (MLPG) do problemu odkształceń termicznych powłok Reissnera-Mindlina. W studium wykorzystano model funkcjonalnych materiałów gradientowych z założeniem ciągłej zmiany własności na grubości elementu powłokowego. Forma słaba równań występujących w teorii Reissnera-Mindlina została przeniesiona na zbiór równań całkowych rozwiązywanych w obszarze zdefiniowanych poddomen. Cylindryczne poddomeny losowo otaczają wygenerowane punkty węzłowe. W rozwiązaniu wykorzystano beziatkową aproksymację metody Moving Least-Squares (MLS).

Słowa kluczowe

EN

Meshless local Petrov-Galerkin method (MLPG); Mindlin theory; orthotropic properties; transient thermal load; Laplace transform

• Wydawca: Wydawnictwa AGH
• Czasopismo: Computer Methods in Materials Science
• Rocznik: 2009
• Tom: Vol. 9, No. 2
• Strony: 171--177
• Opis fizyczny: Bibliogr. 16 poz., rys.

Twórcy

autor

Sladek J.

autor

Sladek V.

autor

Solek P.

• Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia,

Bibliografia

• 1.         Atluri, S.N., The Meshless Method, (MLPG) For Domain & BIE Discretizations, Tech Science Press, 2004.
• 2.         Bapu Rao, M.N., Thermal bending of thick rectangular plates, Nuclear Engineering and Design, 54, 1979, 115-118.
• 3.         Belytschko,  T.,  Krogauz,  Y.,  Organ,   D.,   Fleming,   M.,Krysl, P., Meshless methods; an overview and recent developments, Comp. Meth. Appl. Mech. Engn., 139, 1996, 3-47.
• 4.         Das, M.C., Rath, B.K., Thermal bending of moderately thick rectangular plates, AIAA Journal,  10,  1972,  1349-1351.
• 5.         Jarak, T., Soric, J., Hoster, J., Analysis of shell deformation responses by the Meshless Local Petrov-Galerkin (MLPG) approach, CMES: Computer Modeling in Engineering & Sciences, 18,2007,235-246.
• 6.         Krysl, P., Belytschko, T., Analysis of min shells by the element-free Galerkin method, Int. J. Solids and Structures, 33, 1996,3057-3080.
• 7.         Long, S.Y., Atluri, S.N., A meshless local Petrov Galerkin method for solving the bending problem of a thin plate, CMES: Computer Modeling in Engineering & Sciences, 3, 2002,11-51.
• 8.         Mindlin, R.D., Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates, Journal of Ap­plied Mechanics ASME, 18, 1951,31-38.
• 9.         Praveen, G.N., Reddy, J.N., Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates,Int. J. Solids and Structures, 35, 1998, 4457-4476.
• 10.        Qian, L.F., Batra, R.C., Chen, L.M., Analysis of cylindrical bending thermoelastic deformations of functionally graded plates by a meshless local Petrov-Galerkin method, Computational Mechanics, 33, 2004, 263-273.
• 11.        Qian, L.F., Batra, R.C., Three-dimensional transient heat conduction in a functionally graded thick plate with a higher-order piąte theory and a meshless local Petrov Galerkin method, Computational Mechanics, 35, 2005, 214-226.
• 12.        Reissner, E., Stresses and small displacements analysis of shallow shells-II, Journal Math. Physics, 25,  1946, 279-300.
• 13.        Sladek, J., Sladek, V., Krivacek, J., Aliabadi, M.H., Local boundary integral eąuations for orthotropic shallow shells, Int. J. Solids and Structures, 44, 2007, 2285-2303.
• 14.        Sladek, J., Sladek, V., Solek, P., Wen, P.H., Thermal bending of Reissner-Mindlin plates by the MLPG, CMES:Computer Modeling in Engineering & Sciences, 28, 2008, 57-76.
• 15.        Stehfest, H., Algorithm 368: numerical inversion of Laplace transform, Comm. Assoc. Comput. Mach., 13, 1970, 47-49.
• 16.        Suresh, S., Mortensen A., Fundamentals of Functionally Graded Materials, Institute of Materials, London, 1988.