In this paper, second order statistics of thermally induced post buckling response of elastically supported piezoelectric laminated composite plate using micromechanical approach is examined. A Co finite element has been used for deriving eigenvalue problem using higher order shear deformation theory (HSDT) with von-Karman nonlinearity. The uncertain system properties such as material properties of fiber and matrix of composite and piezoelectric, fiber volume fraction, plate thickness, lamination angle and foundation are modeled as random variables. The temperature field considered to be uniform temperature distributions through the plate thickness. A direct iterative based nonlinear finite element method combined with mean-centered second order perturbation technique (SOPT) is used to find the mean and coefficient of variance of the post buckling temperature. The effects of volume fraction, fiber orientation, and length to thickness ratio, aspect ratios, foundation parameters, position and number of piezoelectric layers, amplitude and boundary conditions with random system properties on the critical temperature are analysed. It is found that small amount of variations of uncertain system parameters of the composite plate significantly affect the initial and post buckling temperature of laminated composite plate. The results have been validated with independent Monte Carlo simulation (MCS) and those available in literature.
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Classical models of clusters’ fission have failed to fully explain strange phenomenons like the phenomenon of shattering (Ziff et al., 1987) and the sudden appearance of infinitely many particles in some systems with initial finite particles number. Furthermore, the bounded perturbation theorem presented in (Pazy, 1983) is not in general true in solution operators theory for models of fractional order γ (with 0 < γ ≤ 1). In this article, we introduce and study a model that can be understood as the fractional generalization of the clusters’ fission process.We make use of the theory of strongly continuous solution operators for fractional models (analogues of C0-semigroups for classical models) and the subordination principle for fractional evolution equations (Bazhlekova, 2000, Prüss, 1993) to analyze and show existence results for clusters’ splitting model with derivative of fractional order. In the process, we exploit some properties of Mittag-Leffler relaxation function (Berberan-Santos, 2005), the He’s homotopy perturbation (He, 1999) and Kato’s type perturbation (Banasiak, 2006) methods. The Cauchy problem for multiplication operator in the fractional dynamics is first considered, before we perturb it. Some additional concepts like Laplace transform, Hille-Yosida theorem and the dominated convergence theorem are use to finally show that there is a solution operator to the full fractional model that is positive and contractive.
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