Institute of Fundamental Technological Research

We present a nonlinear finite element analysis to investigate the buckling and post-buckling behaviour of functionally graded material (FGM) plates resting on the elastic foundation. The material properties are assumed to vary gradually across the thickness according to a power law distribution. The starting point of the investigation is the generalized third-order plate theory and the Vlasov model of elastic foundation having properties varying throughout the depth. The plates are subjected to bending to verify the formulation and compression loads including buckling and post-buckling analysis to investigate the influence of various parameters on the structural response. Key words: FGM plate, elastic foundation, post-buckling, nonlinear finite element analysis.

FGM plate, elastic foundation, post-buckling, nonlinear finite element analysis.

We present a nonlinear finite element analysis to investigate the nonlinear behaviour of functionally graded materials (FGM) plates. The material properties are assumed to vary gradually across the thickness according to a power law distribution. The starting point of the investigation is the generalized third-order plate theory which is modified in the present analysis to include the position of the neutral surface and enhanced with additional terms to represent the distribution stresses better. The plates are subjected to bending and compression loads including buckling and post-buckling analysis.

FGM plate, Nonlinear finite element analysis, Bending, Post-buckling

In this paper, some of the displacement-based plate theories are used to investigate the elastic-plastic analysis of plates in the framework of the finite element method including the buckling and postbuckling effects with the focus on the general third-order plate theory (GTPT). The plate calculation results were compared with the results obtained using 64-nodes solid elements involving Lobatto integration scheme. The problem is solved using the Newton–Raphson method applying modified Crisfield constant arc-length procedure. The results show good agreement of results and the GTPT can be acknowledged to fulfill essential criteria for application to the elastic-plastic analysis of thin and thick plates.

buckling and postbuckling, elasto-plastic plates, third-order plate theory

The aim of the work is to show both the similarities and differences in the formation of deformation-induced roughness in contact compression in the presence of oil and the problem of free surface roughing during uniaxial stretching in a plastic area. The relationships between changes in the roughness are caused by the deformation of the sample and the viscosity of oil at the contact area. It has been shown that normal contact loading with the presence of oil initially leads to an increase in surface roughness, then to its smoothening. The results of the experimental research have been compared with numerical simulation made using FSI (Fluid Structure Interaction) and ABAQUS systems. Using finite element calculations, it was possible to explain the phenomenon of roughness formation on the surface of a smooth steel sample. The changes in the structure of the smooth surface resulting from compression in the presence of oil are caused by the rotation and deformation of surface grains. The roughness of this structure is dependent on the viscosity of oil: the more viscous the liquid is, the rougher texture is formed.

deformation-induced roughening, viscosity, finite element method

We present a nonlinear finite element method to investigate the nonlinear stability of stiffened functionally graded materials (FGM) plates considered as a whole unit. The plates are subjected to mechanical and thermal loads. The material properties are assumed to be temperature dependent and varied gradually across the thickness according to a power law distribution. The nonlinear equations of FGM plates are based on the first-order shear order plate theory. The influence of material, geometrical properties of stiffeners and initial deflections on the buckling and post-buckling response of the stiffened plates are studied in detail. Including the latest information no work has been oriented towards post-buckling analysis of stiffened FGM plates considered as a whole unit.

FGM stiffened plate, nonlinear finite element analysis, post-buckling

The geometrically nonlinear free vibration of functionally graded thick plates resting on the elastic Pasternak foundation is investigated. The motion equations are derived applying the Hamilton principle. We consider the first order shear deformation plate theory (FSDT), in which the modified shear correction factor is required. A 16-noded Mindlin plate element of the Lagrange family which is free from shear locking due to small thickness of the plate used. The material properties are assumed to be temperature-dependent and expressed as a nonlinear function of temperature. Because the FGM plates are not homogeneous, the basic equations are calculated in the equivalent physical neutral surface which differs from the geometric mid-plane. In the pre-buckling range natural frequencies decrease ultimately reaching zero for critical stress in the bifurcation point.

FGM plates, Two-parameter elastic foundation, Nonlinear free vibration, Finite element method

The aim of the present study is to provide a methodology for improving the design of a shrink-fitted assembly by focusing on two objectives: (i) taking into account the surface roughness of components, (ii) introducing a non-classical hardening friction law in which tangential micro-slip displacements are considered. The calculations have been made for elasto-plastic properties of component in axisymmetric conditions. Both the loading and unloading processes have been considered. After finite element simulations, it is observed that overall static behaviour of the assembly is influenced considerably by the surface finish of the mating components. The current numerical results of the shrink fitted assembly received from the coarse mesh could be treated as contact benchmark for further finite element analyses using as closely as possible identical input data.

Shrink-fit, Roughness, Finite elements, Elasto-plasticity

First, we discuss characteristics of functionally graded materials and describe methods of their manufacturing. Then, we provide an overview of analytical and numerical methods for calculating plates, with characteristics of functionally graded materials, resting on elastic foundation. The presented numerical results have been obtained by the finite elements method, referring to post-bifurcation problems of thermally loaded plates. The first-order shear deformation theory (FSDT) has been employed. In numerical calculations we have used a new 16-node plate element, free of problems related to shear locking.

Thick plates, Functionally graded materials, Finite elements method

The Pasternak elastic foundation model is employed to study the statics and natural frequencies of thick plates in the framework of the finite element method. A new 16-node Mindlin plate element of the Lagrange family and a 32-node zero-thickness interface element representing the response of the foundation are used in the analysis. The plate element avoids ill-conditioned behaviour due to its small thickness. In the case of the eigenvalue analysis, the equation of motion is derived by applying the Hamilton principle involving the variation of the kinetic and potential energy of the plate and foundation. Regarding the plate, the firstorder shear deformation theory is used. By employing the Lobatto numerical integration in which the integration points coincide with the element nodes, we obtain the diagonal form of the mass matrix of the plate. In practice, diagonal mass matrices are often employed due to their very attractive timeintegration schemes in explicit dynamic methods in which the inversion of the effective stiffness matrix as a linear combination of the damping and mass matrices is required. The numerical results of our analysis are verified using thin element based on the classical Kirchhoff theory and 16-node thick plate elements.

Mindlin plate, two-parameter elastic foundation, Lobatto integration, bending and eigenvalue analysis

We have used an ultrasonic method to determine the normal and shear stiffness for three different surfaces. The degree of hysteresis for the loading/unloading and stiffness ratio is a function of roughness. Nonlinear contact stiffness characteristics are obtained. The ratio of tangential to normal stiffness KT/KN slowly increases in proportion to normal loading. The novelty of our setup is that at the same time we can measure the reflection coefficient, obtain results for three transducers simultaneously, and measure the approach as a function of load. The presented experimental results of normal contact stiffness measurements have been used for the verification of our theoretical model based on a fractal description of rough surfaces (Buczkowski et al., “Fractal Normal Contact Stiffness of Rough Surfaces,” Arch. Mech. (submitted for publication).

Surface roughness, Stress, Reflectance, Stiffness, Ultrasonic measurement, Waves

We used the fractal theory based on a single variable Weierstrass–Mandelbrot function to obtain the normal contact stiffness if rough and smooth isotropic surfaces are pressed against each other. Because in the original fractal theory the distribution of contact area is assumed geometrically, we propose the method in which the actual deformation of asperities and a correction due to asperity coupling (interaction) will be taken into account. This correction is equivalent to an increase of the effective separation by a quantity proportional to the nominal pressure and it has a significant effect on contact stiffness at larger normal loads (low separations). The numerical results demonstrate a nonlinear evolution of the contact stiffness with the normal load in particular in the first stage of loading at low squeezing pressures. We have compared the results of the theoretical contact stiffness using the fractal method with the experimental ultrasonic measurements. Experimental results made on real surfaces agree remarkably well with the theoretical predictions.

fractal model, contact stiffness, ultrasonic measurements

The present study is divided in two parts. In the first one the complete elasto-plastic microcontact model of anisotropic rough surfaces is given. Rough surfaces are modelled as a random process in which the height of the surface is considered to be a two-dimensional random variable. It is assumed that the surface is statistically homogeneous. The description of anisotropic random surfaces is concentrated on strongly rough surfaces; for such surfaces the summits are represented by highly eccentric elliptic paraboloids. The model is based on the volume conservation of asperities with the plasticity index modified to suit more general geometric contact shapes during plastic deformation process. This model is utilized to determine the total contact area, contact load and contact stiffness which are a combination of the elastic, elasto-plastic and plastic components. The elastic and elastoplastic stiffness coefficients decrease with increasing variance of the surface height about the mean plane. The standard deviation of slopes and standard deviation of curvatures have no observable effects on the normal contact stiffness. The part two deals with the solution of the fully three-dimensional contact/friction problem taking into account contact stiffnesses in the normal and tangential directions. An incremental non-associated hardening friction law model analogous to the classical theory of plasticity is used. Two numerical examples are selected to show applicability of the method proposed.

The complete elasto-plastic microcontact model of anisotropic rough surfaces is proposed. The description of anisotropic random surfaces is restricted here to strongly rough surfaces; for such surfaces the summits are represented by highly eccentric elliptic paraboloids having their semi-major axes oriented in the direction of the grain. The present model is based on the volume conservation of asperities in which the plasticity index is modified to suit more general geometric contact shapes during plastic deformation process. This model is utilized to determine the total contact area, contact load and contact stiffness which are a mixture of both the elastic and plastic components. For low nominal pressures both the elastic and elasto-plastic contact stiffness is found to be almost linear in relation to the normal load. The elastic and elasto-plastic stiffness coefficients decrease with increasing variance of the surface height about the mean plane. The standard deviation of slopes and standard deviation of curvatures have no observable effects on the normal contact stiffness.

Rough surfaces, Statistical modelling, Contact stiffness, Finite element method