An analytical solution of bending thin plates with different moduli in tension and compression

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

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.

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.

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.

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

A bi-modulus material, which tension modulus is different from the compression one along the same direction, is widely used in engineering, e.g., concrete or cast iron. For the sake of the mechanical properties of a bi-material depend on local stress state, many times of structural reanalysis are required to obtain the accurate stress field. To improve the efficiency of topology optimization of a structure with bi-modulus material, the reference-interval with material replacement method is used and the effects of the mechanical properties of bi-material on the final material distribution of a structure are discussed. In the method, a reference interval of strain energy density (SED) is adopted to control the update of design variables. The interval changes frequently in iterations to satisfy the active constraints in optimization. On the other hand, a material-replacement scheme in which the bi-modulus material is replaced with two types of isotropic materials is suggested to simplify structural analysis. And then, the effective SED field of the structure with original material is obtained according to the stress and strain fields of the structure with replaced materials and the update of design variables can be carried out by comparing the local effective SED and the current reference interval of SED. Numerical results show that the difference between the tension modulus and the compression one of the bi-modulus material in a structure has obvious effects on the final material distribution, which is important for an engineer to find an appropriate initial design in practical engineering.

The research provides an innovative contribution to the interpretation of three-point and four-point bending tests on mortars by employing a bi-modulus material model, which assumes an asymmetric constitutive law, i.e., different elastic moduli in tension and in compression. To this aim, Euler–Bernoulli and Timoshenko bi-modulus beam models are defined, and the related displacement fields are reported for three-point loading, and provided for the first time for the four-point bending layout. A wide experimental campaign, including uni-axial tensile and compressive tests, three-point and four-point bending tests, and on notched specimens three-point tests for mode-I fracture energy, has been carried out on lime mortar specimens exploiting traditional contact (CE-DT) and contactless (DIC) measurement systems. Experimental results provided the values of tensile and compressive mechanical characteristics, which are employed to validate estimations of the analytical model. The tension-to-compression moduli ratio experimentally observed is on average 0.52. Experimental outcomes of the DIC analysis proved the bi-modulus behaviour during the four-point bending tests showing visible shifting of the neutral axis. The bi-modulus analytical model provides closer results to the experimental ones for the slender specimens subjected to four-point bending.

The stress-strain relationship of a dry lithium-ion graphite anode coating has been characterized by a bending test method. The method is based on U-shaped bending of single-side coated electrodes, which enables separate measurements of tensile and compressive properties of the electrode coating. The experiments reveal that the elastic modulus of the anode coating in compression is higher than the elastic modulus in tension and that the compressive stiffness increases with strain level. Contrary, the tensile modulus is approximately independent of strain. The quantitative results for compressive modulus, and in particular the stiffening effect with increasing strain, are believed to be new to the battery research community. The measured stiffness of the anode coating is compared to previously reported results for a cathode coating. It is found that the anode coating is stiffer in compression compared to the cathode coating despite a much larger particle stiffness of the cathode material in comparison to the anode. It is concluded that differences in porosity are the main reason for the observed behavior. The method also successfully captures the hysteresis effects, both in tension and compression, that are present due to the polymeric binder and the evolution of microstructural contacts. Relaxation experiments are as well conducted to characterize the time-dependent properties of the anode coating, and the response is modeled by a Prony series.

The load-carrying capacity for geocell with different tension and compression modulus is researched in this paper. The equilibrium equation of geocell with different tension and compression modulus is erected from the force diagram of geocell. The tension stress is solved from the equilibrium equation. The factors affecting the solution of geocell's tension stress are analyzed. The results indicate that the geocell's tension stress is affected strongly by the ratio of geocell's tension to compression modulus and the designers should take the influence of the different tension and compression modulus into account in the latter design of geocell.