Elishakoff, Isaac

In this thesis, we will explore different kinds of metamaterial or architectural structural problems, including structures composed of heterogeneous media with bi periodic sub-structures, discrete structures with sub-elements or continuous structures with discrete attached sub-elements. The thesis is composed of seven parts. After having introduced the specificities of metamaterial mechanics, the second chapter is devoted to the vibration of discrete beam problems called Hencky bar-chain model in a stochastic framework. It is shown that the lattice beam behaves as a nonlocal continuous beam problem, both in the deterministic and the non-deterministic analyses. The third chapter considers the vibration of continuous beams with the introduction of shear effects and attached periodically oscillators. A discussion on beam modelling, for example Timoshenko beam models or truncated Timoshenko beam models is included. It is shown that the bandgap phenomenon observed for metamaterial beams can be accurately captured by a truncated Timoshenko beam model which means the last term in the Timoshenko equation is not that important.

This thesis deals with the analytical study of vibration of carbon nanotubes and graphene plates. First, a brief overview of the traditional Bresse-Timoshenko models for thick beams and Uflyand-Mindlin models for thick plates will be conducted. It has been shown in the literature that the conventionally utilized mechanical models overcorrect the shear effect and that of rotary inertia. To improve the situation, two alternative versions of theories of beams and plates are proposed. The first one is derived through the use of equilibrium equations and leads to a truncated governing differential equation in displacement. It is shown, by considering a power series expansion of the displacement, that this is asymptotically consistent at the second order. The second theory is based on slope inertia and results in the truncated equation with an additional sixth order derivative term. Then, these theories will be extended in order to take into account some scale effects such as interatomic interactions that cannot be neglected for nanomaterials. Thus, different approaches will be considered: phenomenological, asymptotic and continualized. The basic principle of continualized models is to build continuous equations starting from discrete equations and by using Taylor series expansions or Padé approximants. For each of the different models derived in this study, the natural frequencies will be determined, analytically when the closed-form solution is available, numerically when the solution is given through a characteristic equation. The objective of this work is to compare the models and to establish the eventual superiority of a model on others.

In this study, a variational derivation of the simpler and more consistent version of Bresse-Timoshenko
beams equations, taking into account both shear deformation and rotary inertia in vibrating beams, is
presented. Whereas Timoshenko gets his beam equations in terms of the equilibrium, the governing
equations and the boundary conditions are here derived using the Hamilton’s principle. First, a list of
the different energy contributions is established, including the shear effect and the rotary inertia.
Second, the Hamilton’s principle is applied demanding the stationary of an appropriate functional,
leading to two different equations of motion. The resolution of these equations provides the governing
differential equation. It turns out that an additional term appears. The derived equations are intended for
dynamic stability applications. Specifically, the parametric vibrations will be studied when the axial force
varies periodically. This problem has important aerospace applications.

It is demonstrated in this thesis that the interval mathematics is a powerful tool to deal with uncertain phenomena especially when the uncertainty in bounded. In this thesis, we apply interval mathematics to several engineering problems, apparently for the first time in the world literature. The following topics are included: (1) The application of interval mathematics in several applied mechanics problems. A brief review of basis concepts is given, and some problems are presented to illustrate the application of interval mathematics. (2) The stability and dynamic response of viscoelastic plate are studied. The effect of viscoelastic parameters on critical velocity is elucidated. (3) The application of Qiu-Chen-Elishakoff theorem in uncertain string and beam problems is investigated.

The vibrational behavior of inhomogeneous beams and circular plates is studied, utilizing the semi-inverse method developed by I. Elishakoff and extensively discussed in his recent monograph (2005). The main thread of his methodology is that the knowledge of the mode shape is postulated. The candidate mode shapes can be adopted from relevant static, dynamic or buckling problems. In this study, the exact mode shapes are sought as polynomial functions, in the context of vibration tailoring, i.e. designing the structure that possesses the pre-specified value. Apparently for the first time in the literature, several closed-form solutions for vibration tailoring have been derived for vibrating inhomogeneous beams and circular plates. Twelve new closed-form solutions for vibration tailoring have been derived for an inhomogeneous polar orthotropic plate that is either clamped or simply supported around its circumference. Also, the vibration tailoring of a polar orthotropic circular plate with translational spring is analyzed. There is considerable potential of utilizing the developed method for design of functionally graded materials.

The dynamic behavior of straight cantilever pipes conveying fluid is studied, establishing the conditions of stability for systems, which are only limited to move in a 2D-plane. Internal friction of pipe and the effect of the surrounding fluid are neglected. A universal stability curve showing boundary between the stable and unstable behaviors is constructed by finding solution to equation of motion by exact and high-dimensional approximate methods. Based on the Boobnov-Galerkin method, the critical velocities for the fluid are obtained by using both the eigenfunctions of a cantilever beam (beam functions), as well as the utilization of Duncan's functions. Stability of cantilever pipes with uniform and non-uniform elastic foundations of two types are considered and discussed. Special emphasis is placed on the investigation of the paradoxical behavior previously reported in the literature.

This dissertation deals with the non-perturbative finite element methods for stochastic structures and conditional simulation techniques for random fields. Three different non-perturbative finite element schemes have been proposed to compute the first and second moments of displacement responses of stochastic structures. These three methods are based, respectively, on (i) the exact inverse of the global stiffness matrix for simple stochastic structures; (ii) the variational principles for statically-determinate beams; and (iii) the element-level flexibility for general stochastic statically indeterminate structures. The non-perturbative finite element method for stochastic structures possesses several advantages over the conventional perturbation-based finite element method for stochastic structures, including (i) applicability to large values of the coefficient of variation of random parameters; (ii) convergence to exact solutions when the finite element mesh is refined; (iii) requirement of less statistical information than that demanded by the high-order perturbation methods. Conditional simulation of random fields has been an extremely important research field in most recent years due to its application in urban earthquake monitoring systems. This study generalizes the available simulation technique for one-variate Gaussian random fields, conditioned by realizations of the fields, to multi-variate vector random field, conditioned by the realizations of the fields themselves as well as the realizations of the fields derivatives. Furthermore, a conditional simulation for non-Gaussian random fields is also proposed in this study by combining the unconditional simulation technique of non-Gaussian fields and the conditional simulation technique of Gaussian fields. Finally, the dissertation incorporates the simulation technique of random field into the non-perturbation finite element method for stochastic structures, to handle the cases where only one-dimensional probability density function and the correlation function of the random parameters are available, the demanded two-dimensional probability density function is unavailable. Simulation technique is applied to generate the samples of random fields which are used to estimate the correlation between flexibilities over elements. The estimated correlation of flexibility is then used in finite element analysis for stochastic structures. For each proposed approach, numerous examples and numerical results have been implemented.

This dissertation deals with the identification of boundary conditions of elastic structures, and nonlinear random vibration analysis of elastic and viscoelastic structures through a new energy-based equivalent linearization technique. In the part of convex identification, convex models are utilized to represent the degree of uncertainty in the boundary condition modification. This means that the identification is actually the identification of the convex model to which the actual boundary stiffness profile belongs. Two examples are presented to illustrate the application of the method. For the beam example the finite element analysis is performed to evaluate the frequencies of a beam with any specific boundary conditions. For the plate example, the Bolotin's dynamic edge effect method, generalized by Elishakoff, is employed to determine the approximate natural frequencies and normal modes of elastically supported isotropic, uniform rectangular plates. In the part of nonlinear random analysis, first a systematic finite element analysis procedure, based on the element's energy formulation, through conventional stochastic linearization technique, is proposed. The procedure is applicable to a wide range of nonlinear random vibration problem as long as element's energy formulations are presented. Secondly, the new energy-based stochastic linearization method in finite element analysis setting is developed to improve the conventional stochastic linearization technique. The entire formulation is produced in detail for the first time. The theory is applied to beam problem subjected to space-wise and time-wise white noise excitations. Finally, the new energy-based stochastic linearization technique is applied to treat nonlinear vibration problems of viscoelastic beams.

This dissertation deals with the determination of buckling loads of composite cylindrical shell structures which involve uncertainty either in geometry, namely thickness variation, or in material properties. Systematic research has been carried out, which evolves from the simple isotropic cases to anisotropic cases. Since the initial geometric imperfection has a dominant role in the reduction of those imperfection-sensitive structures such as cylindrical shells, the combined effect of thickness variation and initial imperfection is also investigated in depth. Both analytic and numerical methods are used to derive the solutions to the problems and asymptotic formulas relating the buckling load to the geometric (thickness variation and/or initial imperfection) parameter are established. It is shown that the axisymmetric thickness variation has the most detrimental effect on the buckling load when the modal number of thickness variation is twice as much as that of the classical buckling mode. For the composite shells with uncertainty in material properties, the convex modelling is employed to evaluate the variability of buckling load. Based on the experimental data for the elastic moduli of the composite laminates, the upper and lower bounds of the buckling load are derived, which are numerically substantiated by the results from nonlinear programming. These bounds will be useful in practice and can provide engineers with a better view of the real load-carrying capacity of the composite structure. Finally, the elastic modulus is modeled as a function of coordinates to complete the study on variability of material property so that the result can be obtained to account for the situation where the elastic modulus is different from one place to another in the structure.