Creep bending of thin-walled shells and plates by consideration of finite deflections

Abstract

A phenomenological constitutive model for the characterization of creep-damage processes of metals is applied to the numerical analysis of thin-walled shells and plates. The governing equations of the theory of shallow shells are used taking into account geometrical nonlinearities connected with ®nite time-dependent de¯ections by moderate bending. The solutions of the initial-boundary value problem are obtained for thin rectangular plates in order to show the in¯uence of geometrical nonlinearity on results of time-dependent deformation and stress redistribution as well as on estimations of the failure time. IntroductionAnalysis and design of thin-walled structures operating at high temperature conditions require the consideration of time dependent creep behaviour. Creep strains cause signi®cant stress redistributions in the structure and can give rise to creep buckling even by moderate loading. Another type of creep failure is associated with the progressive material softening during tertiary creep stage. For an adequate prediction of long-time strength of thin shells and plates it is necessary to formulate constitutive models giving an accurate approximation of the creep strain rate and the irreversible changes of material properties (such as nucleation and growth of microcavities) as they depend on the stress state. As shown in Altenbach et al. (1986), Betten and Borrmann (1987) stress redistributions in shells of revolution are strongly in¯uenced if the dependence of the creep strain rate on the kind of loading (e.g. tension, compression) is taken into account. Similar effects caused by sensitivity of the damage evolution to tensile and compressive stress states have been established by Bondar and Chrzanowski (1991), Naumenko (1996) by rupture life-time prediction of rectangular plates.On the other hand, suitable shell theories including geometrically nonlinear equations of kinematics and equilibrium have to be used in creep analysis. Geometrically nonlinear equations have been conventionally used in numerical models of creep-buckling analysis of shells, e.g. Miuazaki (1987) and in shell problems associated with large-strain plasticity, e.g. Kra Ètzig and On Äate (1990). In the present paper a geometrically nonlinear model will be formulated in order to predict creep-damage evolution of thin shells and plates.By means of the continuum damage mechanics approach, see Hayhurst (1994), Lemaitre and Chaboche (1990) we introduce the initial-boundary value problem which contains creep-damage material equations with a single scalar damage parameter and governing equations of the shell theory undergoing moderate rotations, e.g. Grigolyuk and Kabanov (1978). The loads are assumed to be suf®ciently small that the instantaneous deformation is linear-elastic and can be determined by use of geometrically linear theory. In order to obtain the time-dependent response of shells ®nite de¯ections caused by small strain creep-damage material behaviour are considered. Numerical solutions of creep problems will be demonstrated ...

Analysis of creep deformation and creep damage in thin-walled branched shells from materials with different behavior in tension and compression

“…For example, the cost of the calculations in the creep damage applications by the implicit backward Euler method is twice as high as the cost of calculations by the explicit Runge-Kutta-Merson's method with automatic time step control (Ling et al, 2000). Thus, the time integration algorithm with an implicit backward Euler method used in the creep analysis of shells by Altenbach and Naumenko (1997), Providakis (2002), and Shariyat and Eslami (1996) is computationally expensive. Furthermore, the varying time step algorithm based on the Runge-Kutta-Merson's method is much faster than its constant time step version, because it does not need small time steps and it concentrates its computational effort only on those time intervals that are needed, taking large strides over intervals.…”

“…However, they assumed the same creep deformation and same creep damage development in tension and compression for the materials of shells (Altenbach and Naumenko, 1997;Hayhurst, 1981;Hyde et al, 2003;Kojic and Bathe, 1987;Miuazaki, 1987;Morachkovskii and Zolochevskii, 1980;Penny and Marriott, 1995;Rabotnov, 1969;Shariyat and Eslami, 1996;Sichov, 1998Sichov, , 2003Takezono and Fujioka, 1981;Zolochevskii and Morachkovskii, 1982;Morachkovsky, 1978, 1979). For the first time, the creep deformation of shells with non-branched meridian taking into account different behavior of materials in tension and compression has been analyzed by Zolochevsky (1980Zolochevsky ( , 1982, and subsequently by Betten and Borrmann (1987) and Altenbach and Zolochevsky (1991).…”

Modeling of Creep for Structural Analysis

“…Some examples are discussed in [22,230]. The first iteration in the above introduced scheme is the forward difference predictor.…”

“…The creep behavior was assumed to have only primary and secondary creep stages and the Norton-Bailey-Odqvist creep constitutive equation sometimes extended by strain or time hardening functions was applied. the finite difference method [262] or direct variational methods [22,80]. In [196] the biharmonic equation describing a deflection surface of the Kirchhoff plate taking into account the given distribution of creep strains has been derived.…”

“…For elastic plates the governing equations (finite deflection model) were originally proposed by von Kármán [327]. In [20,21,22,33] we applied a geometrically-nonlinear theory to the creep-damage analysis of rectangular plates and cylindrical shells. Problems of long term stability and long term strength have required the use of refined geometrically-nonlinear structural mechanics models.…”