Homogenized and non-classical beam theories in ship structural design – challenges and opportunities

The paper gives an overview of the recent developments on the application of homogenized, non-classical beam theories used to predict the micro- and macrostructural stresses in the design of marine structures. These theories become important when ultra-lightweight marine structures are developed and one needs to explore the regions where the length scales of beam openings are in the range of the characteristic lengths of the beams or when lattice/frame-type beams are used to reduce the weight of ship structures. The homogenized beam models are based on non-classical continuum mechanics that allow local bending inside the beams. This added feature allows the treatment of size effects with great accuracy. The resulting analytical and finite element models have special features in terms of shape functions and iterative solutions in non-linear problems. Non-classical beam models enable localization processes that recover the microstructural effects from homogenized solutions accurately and the models are able to handle limit states of serviceability and ultimate strength. The non-classical models are validated by experiments and 3D FE simulations of periodic beams and plates. The non-classical beam theories converge to the physically correct solutions for wider range of beam parameters than the classical beam theories do

Theories and analyses of functionally graded circular plates

This paper presents the governing equations and analytical solutions of the classical and shear deformation theories of functionally graded axisymmetric circular plates. The classical, first-order, and third-order shear deformation theories are presented, accounting for through-thickness variation of two-constituent functionally graded material, modified couple stress effect, and the von Kármán nonlinearity. Analytical solutions for bending of the linear theories, some of which are not readily available in the literature, are included to show the influence of the material variation, boundary conditions, and loads

Bending of Euler-Bernoulli beams using Eringen's integral formulation: A paradox resolved

The Eringen nonlocal theory of elasticity formulated in differential form has been widely used to address problems in which size effect cannot be disregarded in micro- and nano-structured solids and nano-structures. However, this formulation shows some inconsistencies that are not completely understood. In this paper we formulate the problem of the static bending of Euler-Bernoulli beams using the Eringen integral constitutive equation. It is shown that, in general, the Eringen model in differential form is not equivalent to the Eringen model in integral form, and a general method to solve the problem rigorously in integral form is proposed. Beams with different boundary and load conditions are analyzed and the results are compared with those derived from the differential approach showing that they are different in general. With this integral formulation, the paradox that appears when solving the cantilever beam with the differential form of the Eringen model (increase in stiffness with the nonlocal parameter) is solved, which is one of the main contributions of the present work.This work was supported by the Ministerio de Economía y Competitividad de España (grants numbers DPI2011-23191 and DPI2014-57989-P). Prof. J.N. Reddy acknowledges the support of Universidad Carlos III de Madrid with a Cátedra de Excelencia funded by Banco Santander during academic year 2014–2015

Inert gas clearance from tissue by co-currently and counter-currently arranged microvessels

To elucidate the clearance of dissolved inert gas from tissues, we have developed numerical models of gas transport in a cylindrical block of tissue supplied by one or two capillaries. With two capillaries, attention is given to the effects of co-current and counter-current flow on tissue gas clearance. Clearance by counter-current flow is compared with clearance by a single capillary or by two co-currently arranged capillaries. Effects of the blood velocity, solubility, and diffusivity of the gas in the tissue are investigated using parameters with physiological values. It is found that under the conditions investigated, almost identical clearances are achieved by a single capillary as by a co-current pair when the total flow per tissue volume in each unit is the same (i.e., flow velocity in the single capillary is twice that in each co-current vessel). For both co-current and counter-current arrangements, approximate linear relations exist between the tissue gas clearance rate and tissue blood perfusion rate. However, the counter-current arrangement of capillaries results in less-efficient clearance of the inert gas from tissues. Furthermore, this difference in efficiency increases at higher blood flow rates. At a given blood flow, the simple conduction-capacitance model, which has been used to estimate tissue blood perfusion rate from inert gas clearance, underestimates gas clearance rates predicted by the numerical models for single vessel or for two vessels with co-current flow. This difference is accounted for in discussion, which also considers the choice of parameters and possible effects of microvascular architecture on the interpretation of tissue inert gas clearance

Static and vibration analysis of functionally graded beams using refined shear deformation theory

Static and vibration analysis of functionally graded beams using refined shear deformation theory is presented. The developed theory, which does not require shear correction factor, accounts for shear deformation effect and coupling coming from the material anisotropy. Governing equations of motion are derived from the Hamilton's principle. The resulting coupling is referred to as triply coupled axial-flexural response. A two-noded Hermite-cubic element with five degree-of-freedom per node is developed to solve the problem. Numerical results are obtained for functionally graded beams with simply-supported, cantilever-free and clamped-clamped boundary conditions to investigate effects of the power-law exponent and modulus ratio on the displacements, natural frequencies and corresponding mode shapes

Finite element computation of magnetohydrodynamic nanofluid convection from an oscillating inclined plate with radiative flux, heat source and variable temperature effects

The present work describes finite element computations for radiative magnetohydrodynamic convective Newtonian nanofluid flow from an oscillating inclined porous plate with variable temperature. Heat source/sink and buoyancy effects are included in the mathematical model. The problem is formulated by employing Tiwari-Das nanofluid model and two water - based nanofluids with spherical shaped metal nano particles as copper and alumina are considered. The Brinkman and Maxwell-Garnetts models are used for the dynamic viscosity and effective thermal conductivity of the nanofluids respectively. An algebraic flux model, the Rosseland diffusion approximation is adopted to simulate thermal radiative flux effects. The dimensionless, coupled governing partial differential equations are numerically solved via the finite element method with weak variational formulation by imposing initial and boundary conditions with a weighted residual scheme. A grid independence study is also conducted. The finite element solutions are reduced to known previous solutions in some limiting cases of the present investigation and are found to be in good agreement with published work. This investigation is relevant to electromagnetic nanomaterial manufacturing processes operating at high temperatures where radiation heat transfer is significant

Oscillatory dissipative conjugate heat and mass transfer in chemically-reacting micropolar flow with wall couple stress : a finite element numerical study

High temperature non-Newtonian materials processing provides a stimulating area for process engineering simulation. Motivated by emerging applications in this area, the present article investigates the time-dependent free convective flow of a chemically-reacting micropolar fluid from a vertical plate oscillating in its own plane adjacent to a porous medium. Thermal radiative, viscous dissipation and wall couple stress effects are included. The Rosseland diffusion approximation is used to model uni-directional radiative heat flux in the energy equation. Darcy’s model is adopted to mimic porous medium drag force effects. The governing two-dimensional conservation equations are normalized with appropriate variables and transformed into a dimensionless, coupled, nonlinear system of partial differential equations under the assumption of low Reynolds number. The governing boundary value problem is then solved under physically viable boundary conditions numerically with a finite element method based on the weighted residual approach. Graphical illustrations for velocity, micro-rotation (angular velocity), temperature and concentration are obtained as functions of the emerging physical parameters i.e. thermal radiation, viscous dissipation, first order chemical reaction parameter etc. Furthermore, friction factor (skin friction), surface heat transfer and mass transfer rates have been tabulated quantitatively for selected thermo-physical parameters. A comparison with previously published paper is made to check the validity and accuracy of the present finite element solutions under some limiting cases and excellent agreement is attained. Additionally, a mesh independence study is conducted. The model is relevant to reactive polymeric materials processing simulation

Approximation of the critical buckling factor for composite panels

This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented