Large Deformation Problem of Bimodular Functionally-Graded Thin Circular Plates Subjected to Transversely Uniformly-Distributed Load: Perturbation Solution without Small-Rotation-Angle Assumption

General perturbation solution of large-deflection circular plate with different moduli in tension and compression under various edge conditions

Materials which exhibit different elastic moduli in tension and compression are known as bimodular materials. The bimodular materials model, which is founded on the criterion of positive-negative signs of principal stress, is important for the structural analysis and design. However, due to the inherent complexity of the constitutive relation, it is difficult to obtain an analytical solution of a bimodular bending components except in particular simple problems. Based on the existent simplified model, this paper solves analytically bending thin plates with different moduli in tension and compression. By using the continuity conditions of stress components in unknown neutral layer, we determine the location of the neutral layer, and derive the governing differential equation for deflection, the flexural rigidity, and the internal forces in the thin plate. We also use a circular thin plate with bimodulus to illustrate the application of this solution derived in this paper. The results show that the introduction of different moduli has influences on the flexural stiffness of the bending thin plate.

In this study, a set of Föppl–von Kármán equations for a bimodular functionally graded thin plate subjected to a uniformly distributed load is established, and its general perturbation solution in axisymmetric case is also obtained under different boundary conditions. First, the equation of equilibrium of the plate is established on the existence of the neutral layer when considering different properties in tension and compression. During the derivation of the consistency equation, the tensile effect in the thin plate with bimodular effect is fully taken into account. The perturbation method is used to solve the set of governing equations under different edge constraints, in which the central deflection and the load of the plate are taken as a perturbation parameter, respectively. The results indicate that the two selections for perturbation parameters are valid and consistent, and the two solutions are convenient for engineering application. This study also shows that the bimodular effect will modify the relation of load versus central deflection of the plate to some extent, and under the same edge constraint, the capacities resisting deformation in different cases of moduli depend on the relative magnitudes among the tensile modulus, the neutral layer modulus, and the compressive modulus.

Composite materials often exhibit different stiffnesses or moduli under tension loading than under compression loading. This behavior is modeled with a bilinear stress-strain curve having a modulus E t in tension and E c in compression as an approximation to the real nonlinear behavior. Under bending loads, laminated composites have both tensile and compressive stresses and hence are not subject to the same behavioral rules as ordinary single modulus materials. The resulting transcendental equilibrium equation is dependent upon the unknown neutral surface. This neutral surface is found and, hence, the equilibrium problem is solved with an iteration technique. The approach is applied to laminates ordinarily thought to be symmetric, antisymmetric, and unsymmetric about the middle surface. All laminates are found to exhibit coupling between bending and extension under bending in contrast to the usual concepts of symmetry and antisymmetry for single modulus laminates. Several approximate approaches are investigated for treating the multimodulus laminate problem. The effect of coupling due to different moduli in tension and compression on stresses and deflections is found to be generally significant for common composite materials such as boron/epoxy and graphitc/epoxy as well as carbon-carbon.

Based on the numerically iterative technique, we introduce the shear modulus pattern used for accelerating convergence and rebuild the iterative program. Using this program, we analyze the bending problem of a bimodular deep beam under the action of uniformly distributed loads, in which the different ratios of span length to height of the beam, and of elastic modulus in compression to that in tension, are considered. The results show that the bimodularity have great influences on the bending stress and the deflection of deep beam.

Large static deformation analysis of thin cylindrical shells with different moduli in tension and compression: An application in pressure hulls

A new approach is given to the theory of non-linear elastic materials which have different behaviour in tension and compression. Two applications are made to incompressible non-linear materials using general forms for the strain energy functions. The linear form of the theory is shown to be equivalent to that used by previous writers.

The buckling of columns made of materials which have different moduli in tension and compression is examined. A new criterion for buckling is proposed whereby the strain energy of the unbuckled column is compared to that of the column in a buckled configuration, and these are related to the end displacement of the column. These considerations are extended to the formulation of a principle of similarity for the buckling of a geometrically similar series of structures. In a specific case, buckling of the column is shown to occur in two stages. Limited buckling first takes place at the Euler critical load calculated from the compressive modulus, following which further loading is possible until a second less sharply defined critical point is reached.

A new computational framework for materials with different mechanical responses in tension and compression and its applications

The buckling of columns made of materials which have different moduli in tension and in compression is investigated. The energy of the buckled column is calculated for an assumed load and mode shape by means of an iterative procedure, and buckling is assumed to take place when this energy is smaller than that of a straight column under the same load. Results are presented indicating the existence of an elastic type of hysteresis and snap-through for high ratios of EcEr

When materials that exhibit different mechanical behaviors in tension and compression must be analyzed, Ambartsumyan's bimodular model for isotropic materials can be adopted. It deals with the principal stress state in a point, which is particularly important in the analysis and design of reinforced concrete structures. However, due to the inherent complexity of the constitutive relation, it is difficult to solve analytically for bending components with bimoduli except in particular simple problems. Here we propose a simplified mechanical model, based on the classical Kirchhoff hypothesis, used for the solution of the bimodular thin plates in bending. We first use the Kirchhoff hypothesis to judge the existence of the elastic neutral layers of bimodular thin plates in small-deflection bending. Based on the existent neutral layers, we extend the solution from the case of pure bending into the case of lateral force bending. We use the displacement variation method to illustrate the application of the proposed model, and compare it with FEM results strictly based on Ambartsumyan's materials model. The comparisons show that the proposed mechanical model is valid and helpful for analyzing bending structures with bimodularity.

Nonlinear problems of circular plates with variable thickness under central concentrated load

On the basis of the moving least-squares (MLS) approximation, the complex variable element-free Galerkin (CVEFG) method for two-dimensional elastic large deformation problems, such as large displacement and large rotation, is presented in this paper. The advantage of the CVMLS approximation is that the trial function of a two-dimensional problem is formed with one-dimensional basis function. Then the CVMLS approximation is applied to solve two-dimensional large deformation problems. The Galerkin weak form is employed to obtain the discretized system equation, the penalty method is used to apply the essential boundary conditions. And then the complex variable element-free Galerkin (CVEFG) method for two-dimensional large deformation problems based on the total lagrange formulation is formed, the corresponding formulae are obtained, and the Newton-Raphson iteration is used in the numerical implementation. Three numerical examples are given to show that this method in this paper is effective for elastic large deformation problems.

The aim of this work is to construct and analyze the resulting solutions for a circular orthotropic plate using analytical methods that allow integrating differential equations containing discontinuous functions. Discontinuous functions are understood here as a unit function, a delta function and its derivatives. This approach makes it possible to directly integrate the differential equations of plates with discontinuous load and stiffness characteristics. This eliminates the need to cut the plate into separate elements, each of which has continuous load and stiffness characteristics. The article shows how the resolving differential equation of a circular orthotropic plate is obtained from the equilibrium equations. The case of an annular load acting on an orthotropic plate is considered and a comparison with the results obtained for an isotropic plate is made. It is concluded that the bending moments at α2 < 1, that is, for the plates with radial stiffness D1 greater than circumferential stiffness D2, at the center, the plates turn to infinity, but at the same time, at α2 > 1, that is, in the case when D2 > D1, bending moments vanish. Both that and the other results do not correspond to the conditions of orthotropic plates’ real work. This means that for orthotropic circular plates, Kirchhoff hypotheses are not applicable at the point r = 0. The article gives recommendations on the calculation of circular plates under the action of a load distributed over a circle, as well as reinforced by several circumferential ribs, which are introduced into the original differential equations by delta functions.KeywordsCircular plateOrthotropic materialDeflectionBending momentsDiscontinuous functionsUnit functionDelta functionDifferential equationsIntegrationAxially-symmetric deformations

The deflections are analytically studied in a flat, deformable layer, firmly anchored to a rigid substrate, and frictionlessly indented by a rigid paraboloidal punch, having in mind the design of hip replacements possessing elastomeric layers. An existing perturbation solution of differential character, developed for a layer subject to plane strain conditions and indented by a rigid cylinder, and valid for high contact widths, is here extended to the analogous axisymmetric problem of a layer compressed by a rigid paraboloidal punch approximating a spherical indenter. Perturbed solutions up to the second order are obtained, and the contact pressure profiles activated by a paraboloidal punch are derived for both compressible and incompressible layers. Numerical examples exploring the sensitivity of the stress field to perturbations of the Poisson's ratio are also included. INTRODUCTION This paper deals with the deflections of a flat, deformable layer, firmly bonded to a rigid substrate and frictionlessly compressed by a rigid paraboloidal indenter, Fig. 1. This axisymmetric study is relevant in the design of hip replacements whose cup is covered with an elastomeric layer, and compressed by a ball replacing the femoral head, [1] . An analytical perturbation solution of differential character, developed in [2] for a layer subject to plane strain conditions and indented by a rigid cylinder, valid for high contact widths, is here extended to the axisymmetric problem of a layer compressed by a rigid paraboloidal punch approximating a spherical indenter. This paper is organized as follows. First, the equations describing the layer deflections as functions of an imposed axisymmetric pressure profile are developed up to the second perturbation order. A subsequent section solves the more relevant situation where the Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533 398 Contact Mechanics indenting profile is known, and the contact pressure has to be determined. Numerical examples exploring the sensitivity of the solution to perturbations of the Poisson's ratio end the paper. THE PERTURBED PRESSURE-DEFLECTION SOLUTION Following Armstrong [2], the subsequent steps are developed: a) the axisymmetric equilibrium equations are expressed in terms of the displacement field; b) the equilibrium equations are normalized with respect to proper variables, and a small parameter e is identified, which renders this problem amenable to perturbation solution techniques; c) an approximate solution is achieved for parameter € up to the second order, which connects the pressure (as well as its derivatives) acting upon an elastomeric layer to its local deflection. The axisymmetric equilibrium equations in terms of stresses are [3] : By introducing into (1) the Hooke law linking stresses to strains, expressed in cylindrical coordinates in terms of radial, u , and axial , w , displacements, [3], the equilibrium equations are formulated in terms of the displacement field as :