Non-Linear Bending of Functionally Graded Thin Plates with Different Moduli in Tension and Compression and Its General Perturbation Solution

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, the large deformation problem of a functionally-graded thin circular plate subjected to transversely uniformly-distributed load and with different moduli in tension and compression (bimodular property) is theoretically analyzed, in which the small-rotation-angle assumption, commonly used in the classical Föppl–von Kármán equations of large deflection problems, is abandoned. First, based on the mechanical model on the neutral layer, the bimodular functionally-graded property of materials is modeled as two different exponential functions in the tensile and compressive zones. Thus, the governing equations of the large deformation problem are established and improved, in which the equation of equilibrium is derived without the common small-rotation-angle assumption. Taking the central deflection as a perturbation parameter, the perturbation method is used to solve the governing equations, thus the perturbation solutions of deflection and stress are obtained under different boundary constraints and the regression of the solution is satisfied. Results indicate that the perturbation solutions presented in this study have higher computational accuracy in comparison with the existing perturbation solutions with small-rotation-angle assumption. Specially, the computational accuracies of external load and yield stress are improved by 17.22% and 28.79% at most, respectively, by the numerical examples. In addition, the small-rotation-angle assumption has a great influence on the yield stress at the center of the bimodular functionally-graded circular plate.

The existing studies indicate polymers will present obviously different properties in tension and compression (bimodular effect) which is generally ignored because of the complexity of the analysis. In this study, a functionally graded piezoelectric cantilever beam with bimodular effect was investigated via analytical and numerical methods, respectively, in which a one-dimensional theoretical solution was derived by neglecting some unimportant factors and a two-dimensional numerical simulation was performed based on the model of tension-compression subarea. A full comparison was made to show the rationality of one-dimensional theoretical solution and two-dimensional numerical simulation. The result indicates that the layered model of tension-compression subarea also makes it possible to use numerical technique to simulate the problem of functionally graded piezoelectric cantilever beam with bimodular effect. Besides, the modulus of elasticity E* and the bending stiffness D* proposed in the one-dimensional problem may succinctly describe the piezoelectric effect on the classical mechanical problem without electromechanical coupling, which shows the advantages of one-dimensional solution in engineering applications, especially in the analysis and design of energy harvesting/sensing/actuating devices made of piezoelectric polymers whose bimodular effect is relatively obvious.

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.

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.

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.

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.

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

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.

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

This paper develops micromechanics models to estimate the tensile and compressive elastic moduli of elastic solids containing randomly distributed two-dimensional microcracks. The crack faces are open under tension and closed under compression. When the crack faces are closed, they may slide against one another following the Coulomb's law of dry friction. The micromechanics models provide analytical expressions of the tensile and compressive moduli for both static and dynamic cases. It is found that the tensile and compressive moduli are different. Further, under dynamic loading, the compressive and tensile moduli are both frequency dependent. As a by-product, the micromechanics models also predict wave attenuation in the dynamic case. Numerical simulations using the finite element method (FEM) are conducted to validate the micromechanics models.

Identification of temperature-dependent parameters for an elastic–plastic-damage material model of 3D-printed PETG

The purpose of this study is to analyze the large deflection problem of bimodular functionally graded truncated thin conical shells under the transverse mechanical load and non-uniform thermal load, in which two different boundary constraints of the truncated shell with two ends simply supported and fully fixed are considered. It is assumed that the temperature distribution along the thickness direction satisfies the Fourier law of heat transfer, and the material properties change exponentially along the thickness direction while different properties in tension and compression are considered. The geometric equation of the conical shell is established based on the equivalent method of curvature correction of von-Kármán deformation theory, and the analytical solution of the problem is obtained by Ritz method. Numerical simulation of bimodular functionally graded conical shells under the thermal and mechanical loads is carried out by Abaqus, and the numerical solution agrees with the theoretical solution. The results show that the introduction of bimodular functionally graded material will affect the maximum displacement and this effect has different rules under the mechanical load and thermal load. In addition, factors such as the cone apex angle and the truncated distance have a great influence on the maximum displacement and its location of the conical shell.