Effect Of Bimodularity Research Articles

Large deformation problem of hyperbolic shallow shells with bimodular effect: A perturbation‐variation method

AbstractAs an important analytical method, the perturbation‐variation method combines the advantages of perturbation method and variational method, and is widely used for the solution of nonlinear plate and shell problems. In this study, the perturbation‐variation method is used to solve the large deformation problem of a flexible hyperbolic thin shallow shells with different moduli in tension and compression (bimodular effect). First, we establish the governing equations expressed in terms of three displacement components, in which the bimodular effect of materials is considered in physical equations and the large deformation of shell is considered in geometrical equations. Next, we use the perturbation‐variation method to solve the governing equations established, obtaining the perturbation‐variation solution up to third‐order. During the application of the method, the displacements are expressed into perturbation formulas of all levels first, and the unknown constants and functions in the perturbation formulas are determined by the extreme value conditions of the functional established (variational principle), hence the perturbation‐variation method gets its name from this practice. The numerical results from FEM basically agree with the perturbation‐variation solution obtained. The method proposed may be extended into the solution of similar partial differential equations established in applied fields.

Large deformation analysis of functionally graded revolutionary shallow thin shells with bi-modular effect: Snap-through buckling under different boundary constraints

Functionally graded materials (FGMs) enable structures to achieve a smooth transition between different material properties, providing an optimized combination of functions to meet the diversity of engineering needs. However, the presence of bi-modular effect in FGMs is often neglected due to the complexity in the analysis. In this paper, the curvature correction equivalent method based on the von Kármán theory will be applied to the theoretical study for the large deformation problem and the resulting snap-through buckling of functionally graded revolutionary shallow thin shells with bi-modular effect under different boundary constraints. Furthermore, a bi-modular FGMs user material subroutine (UMAT) is developed for the first time to simulate the real material model, thus validating the theoretical solution. The results show that except for the bi-modular effect of materials, boundary constraints of shells also have important influences on the relationship of load vs. central deflection and snap-through buckling. The research of boundary constraints brings new conclusions to snap-through buckling: the rotation constraint and radial constraint at the shell edge have opposite effects on snap-through buckling. This work will contribute to the analysis of the snap-through buckling of functionally graded revolutionary shallow thin shells with an obvious bi-modular effect and under various boundary constraints.

Variational Solution and Numerical Simulation of Bimodular Functionally Graded Thin Circular Plates under Large Deformation

In this study, the variational method and numerical simulation technique were used to solve the problem of bimodular functionally graded thin plates under large deformation. During the application of the variational method, the functional was established on the elastic strain energy of the plate while the variation in the functional was realized by changing undetermined coefficients in the functional. As a result, the classical Ritz method was adopted to obtain the important relationship between load and maximum deflection that is of great concern in engineering design. At the same time, the numerical simulation technique was also utilized by applying the software ABAQUS6.14.4, in which the bimodular effect and functionally graded properties of the materials were simulated by subareas in tension and compression, as well as the layering along the direction of plate thickness, respectively. This study indicates that the numerical simulation results agree with those from the variational solution, by comparing the maximum deflection of the plate, which verifies the validity of the variational solution obtained. The results presented in this study are helpful for the refined analysis and optimization design of flexible structures, which are composed of bimodular functionally graded materials, while the structure is under large deformation.

Application of the Variational Method to the Large Deformation Problem of Thin Cylindrical Shells with Different Moduli in Tension and Compression

In this study, the variational method concerning displacement components is applied to solve the large deformation problem of a thin cylindrical shell with its four sides fully fixed and under uniformly distributed loads, in which the material that constitutes the shell has a bimodular effect, in comparison to traditional materials, that is, the material will present different moduli of elasticity when it is in tension and compression. For the purpose of the use of the displacement variational method, the physical equations on the bimodular material model and the geometrical equation under large deformation are derived first. Thereafter, the total strain potential energy is expressed in terms of the displacement component, thus bringing the possibilities for the classical Ritz method. Finally, the relationship between load and central deflection is obtained, which is validated with the numerical simulation, and the jumping phenomenon of thin cylindrical shell with a bimodular effect is analyzed. The results indicate that the bimodular effect will change the stiffness of the shell, thus resulting in the corresponding change in the deformation magnitude. When the shell is relatively thin, the bimodular effect will influence the occurrence of the jumping phenomenon of the cylindrical shell.

Axisymmetric large deformation problems of thin shallow shells with different moduli in tension and compression

When materials are in tension or in compression, their mechanical properties are essentially different, however, in existing studies it is generally ignored because of the complexity of the analysis. In this paper, a theoretical study of the problem of axisymmetric large deformation of thin shallow shells with different moduli in tension and compression (bimodular thin shallow shells) under uniformly vertical loading is presented. Based on the Föppl–von Kármán equations of flexible thin plates, the governing equation of the axisymmetric large deformation of bimodular thin shallow shells are established by introducing an appropriate curvature modification term from thin shallow shells and a bending stiffness term from bimodular plates. By selecting the central deflection as a perturbation parameter, the perturbation solution of the axisymmetric large deformation problem is derived using the perturbation method. The applications of variation method as well as the implementation of numerical simulation verify, from the theoretical and numerical perspectives, the correctness of the perturbation solution obtained. This study shows that the introduction of different moduli in tension and compression not only affects the relationship of loads vs. central deflection, but also affects the critical values of the rise-thickness ratio of bimodular thin shallow shells when the snapping phenomenon occurs. In particular, the bimodular effect may accelerate or slow down the occurrence of the snapping phenomenon of thin shallow spherical shells, according to the relative magnitude of tensile and compressive moduli. The work presented here will be helpful to analyze the mechanical response of heavily loaded thin shallow shells with obvious bimodular effect.

Solution of the Thermoelastic Problem for a Two-Dimensional Curved Beam with Bimodular Effects

In classical thermoelasticity, the bimodular effect of materials is rarely considered. However, all materials will present, in essence, different properties in tension and compression, more or less. The bimodular effect is generally ignored only for simple analysis. In this study, we theoretically analyze a two-dimensional curved beam with a bimodular effect and under mechanical and thermal loads. We first establish a simplified model on a subarea in tension and compression. On the basis of this model, we adopt the Duhamel similarity theorem to change the initial thermoelastic problem as an elasticity problem without the thermal effect. The superposition of the special solution and supplement solution of the Lamé displacement equation enables us to satisfy the boundary conditions and stress continuity conditions of the bimodular curved beam, thus obtaining a two-dimensional thermoelastic solution. The results indicate that the solution obtained can reduce to bimodular curved beam problems without thermal loads and to the classical Golovin solution. In addition, the bimodular effect on thermal stresses is discussed under linear and non-linear temperature rise modes. Specially, when the compressive modulus is far greater than the tensile modulus, a large compressive stress will occur at the inner edge of the curved beam, which should be paid with more attention in the design of the curved beams in a thermal environment.

One- and Two-Dimensional Analytical Solutions of Thermal Stress for Bimodular Functionally Graded Beams under Arbitrary Temperature Rise Modes

In this study, we analytically solved the thermal stress problem of a bimodular functionally graded bending beam under arbitrary temperature rise modes. First, based on the strain suppression method in a one-dimensional case, we obtained the thermal stress of a bimodular functionally graded beam subjected to bending moment under arbitrary temperature rise modes. Using the stress function method based on compatibility conditions, we also derived two-dimensional thermoelasticity solutions for the same problem under pure bending and lateral-force bending, respectively. During the solving, the number of unknown integration constants is doubled due to the introduction of bimodular effect; thus, the determination for these constants depends not only on the boundary conditions, but also on the continuity conditions at the neutral layer. The comparisons indicate that the one- and two-dimensional thermal stress solutions are consistent in essence, with a slight difference in the axial stress, which exactly reflects the distinctions of one- and two-dimensional problems. In addition, the temperature rise modes in this study are not explicitly indicated, which further expands the applicability of the solutions obtained. The originality of this work is that the one- and two-dimensional thermal stress solutions for bimodular functionally graded beams are derived for the first time. The results obtained in this study may serve as a theoretical reference for the analysis and design of beam-like structures with obvious bimodular functionally graded properties in a thermal environment.

Theoretical Study on Thermal Stresses of Metal Bars with Different Moduli in Tension and Compression

Extensive studies have shown that engineering materials, including metals and their oxides, will present different mechanical properties in tension or compression; however, this difference is generally neglected due to the complexity of the analysis. In this study, we theoretically analyze the thermal stress of a metal bar with a bimodular effect. First, the common strain suppression method is used to obtain a one-dimensional thermal stress expression. As a contrast with the one-dimensional solution, a two-dimensional thermoelasticity solution is also derived, based on the classical Duhamel theorem concerning body force analogy. Results indicate an important phenomenon that the linear temperature rise mode will produce thermal stress in a bimodular metal bar, whereas there is no thermal stress in the case of singular modulus. If the equilibrium relation is needed to be satisfied, the variation trend between different moduli and different thermal expansion coefficients in tension and compression should be opposite. In addition, the amplitude of stress variation, from the maximum tensile stress to the maximum compressive stress, increases dramatically. There exists an inevitable link between one- and two-dimensional solutions. These results are helpful to the refined analysis and measurements of the thermophysical properties of metals and their oxides.

Effect of bimodularity and thermomechanical stresses from composite curing on mixed-mode fracture behavior of functionally graded skin-stiffener runout

ABSTRACT The present work illustrates the influence of bimodulus polarity and curing stress on interface delamination in skin stiffener runout. In contrast to unimodular assumptions, the bimodulus skin stiffener’s fracture behavior is much more unsymmetrical depending on the sequential thermo-mechanical loading. The numerical study has been done to demonstrate debonding and delamination with the help of the Virtual Crack Closure Technique on compliant runout configuration under varying bimodular ratios. The modified design of the runout has been displayed, and different design configurations have been compared to determine the merits of the design of skin-stiffener runout: a baseline configuration, a tip-tapered configuration, and notch-tapered configuration with the functionally graded bimodular property. A three-dimensional finite element-based analysis has been performed on these configurations to evaluate the strength and damage tolerance. Also, the Strain Energy Release Rate has been assessed for non-identical loading and boundary conditions. It has been found that bimodularity significantly reduces failure growth. This analysis aims to decelerate the interfacial fracture extension rate by introducing functionally graded bimodularity in the structure.

Application of Variational Method to Stability Analysis of Cantilever Vertical Plates with Bimodular Effect.

In the design of cantilevered balconies of buildings, many stability problems exist concerning vertical plates, in which reaching a critical load plays an important role during the stability analysis of the plate. At the same time, the concrete forming vertical plate, as a typical brittle material, has larger compressive strength but lower tensile strength, which means the tensile and compression properties of concrete are different. However, due to the complexities of such analyses, this difference has not been considered. In this study, the variational method is used to analyze stability problems of cantilever vertical plates with bimodular effect, in which different loading conditions and plate shapes are also taken into account. For the effective implementation of a variational method, the bending strain energy based on bimodular theory is established first, and critical loads of four stability problems are obtained. The results indicate that the bimodular effect, as well as different loading types and plate shapes, have influences on the final critical loads, resulting in varying degrees of buckling. In particular, if the average value of the tensile modulus and compressive modulus remain unchanged, the introduction of the bimodular effect will weaken, to some extent, the bending stiffness of the plate. Among the four stability problems, a rectangular plate with its top and bottom loaded is most likely to buckle; next is a rectangular plate with its top loaded, followed by a triangular plate with its bottom loaded. A rectangular plate with its bottom loaded is least likely to buckle. This work may serve as a theoretical reference for the refined analysis of vertical plates. Plates are made of concrete or similar material whose bimodular effect is relatively obvious and cannot be ignored arbitrarily; otherwise the greater inaccuracies will be encountered in building designs.

A Two-Dimensional Thermoelasticity Solution for Bimodular Material Beams under the Combination Action of Thermal and Mechanical Loads

A typical characteristic of bimodular material beams is that when bending, the neutral layer of the beam does not coincide with its geometric middle surface since the mechanical properties of materials in tension and compression are different. In the classical theory of elasticity, however, this characteristic has not been considered. In this study, a bimodular simply-supported beam under the combination action of thermal and mechanical loads is theoretically analyzed. First, a simplified mechanical model concerning the neutral layer is established. Based on this mechanical model, Duhamel’s theorem is used to transform the thermoelastical problem into a pure elasticity problem with imaginary body force and surface force. In solving the governing equation expressed in terms of displacement, a special solution of the displacement equation is found first, and then by utilizing the stress function method based on subarea in tension and compression, a supplement solution for the displacement governing equation without the thermal effect is derived. Lastly, the special solution and supplement solution are superimposed to satisfy boundary conditions, thus obtaining a two-dimensional thermoelasticity solution. In addition, the bimodular effect and temperature effect on the thermoelasticity solution are illustrated by computational examples.

Effect of Bimodularity on the Stress State of a Variable Cross-Section Reinforced Beam

The article is dedicated to the effect of different modulus of the material on the stress state of a beam of the variable rectangular cross section. The height of the beam varies linearly along its length. Formulas for calculating the maximum compressive and tensile stresses and determining the neutral line are obtained. The maximum tensile and compressive stresses are determined for the clamped and simply supported beams. The dependence of the maximum normal stress on the number of reinforcing bars located in the stretched zones is numerically investigated. The stress state of the beam is compared with and without consideration of the bimodularity of the material for simply supported and cantilever beams. It is shown that taking into account the bimodularity of the material significantly affects the maximum tensile and compressive stresses. The magnitude of the tensile stresses is increased by 30%; the magnitude of the compressive stresses is reduced by 21%. As a bimodular material, fibro foam concrete is considered in work.

One-Dimensional Theoretical Solution and Two-Dimensional Numerical Simulation for Functionally-Graded Piezoelectric Cantilever Beams with Different Properties in Tension and Compression.

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.

One-Dimensional and Two-Dimensional Analytical Solutions for Functionally Graded Beams with Different Moduli in Tension and Compression

The material considered in this study not only has a functionally graded characteristic but also exhibits different tensile and compressive moduli of elasticity. One-dimensional and two-dimensional mechanical models for a functionally graded beam with a bimodular effect were established first. By taking the grade function as an exponential expression, the analytical solutions of a bimodular functionally graded beam under pure bending and lateral-force bending were obtained. The regression from a two-dimensional solution to a one-dimensional solution is verified. The physical quantities in a bimodular functionally graded beam are compared with their counterparts in a classical problem and a functionally graded beam without a bimodular effect. The validity of the plane section assumption under pure bending and lateral-force bending is analyzed. Three typical cases that the tensile modulus is greater than, equal to, or less than the compressive modulus are discussed. The result indicates that due to the introduction of the bimodular functionally graded effect of the materials, the maximum tensile and compressive bending stresses may not take place at the bottom and top of the beam. The real location at which the maximum bending stress takes place is determined via the extreme condition for the analytical solution.

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

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.

An electroelastic solution for functionally graded piezoelectric material beams with different moduli in tension and compression

In this study, we derived an electroelastic solution of functionally graded piezoelectric beams with bimodular effect under the combined action of uniformly distributed loads, concentrated loads, and bending moments. We first presented the simplified mechanical model established in previous study, and this model is based on a complete partition of tension and compression. Based on the theory of piezoelectric elasticity and functional graded property, we assumed that material parameters vary exponentially along the thickness, and the exponential functions adopt two different forms in the tensile zone and the compressive zone. Using stress function method, we obtained an analytical solution of the problem in which unknown constants are determined by boundary conditions and continuity conditions on the neutral layer. The results indicate that the introduction of exponential functions makes us be able to effectively distinguish the relative magnitude of the tensile modulus, the compressive modulus and the modulus on the neutral layer. Thus, the variation in the stress and the displacement including the electric displacement with the thickness are discussed in detail. The results obtained are helpful to refined analysis and design of functionally graded piezoelectric materials and structures with obvious bimodular effect.

A biparametric perturbation method for the Föppl–von Kármán equations of bimodular thin plates

In this study, a biparametric perturbation method is proposed to solve the Föppl–von Kármán equations of bimodular thin plates subjected to a single load. First, by using two small parameters, one describes the bimodular effect and another stands for the central deflection, we expanded the unknown deflection and stress in double power series with respect to the two parameters and obtained the approximate analytical solutions under various edge conditions. Due to the diversity of selection of parameters and its combination, by using the bimodular parameter and the load as two perturbation parameters, we elucidated further the application of this method. The use of two sets of parameter schemes both can obtain satisfactory perturbation solutions; the numerical simulations also verify this idea. The results indicate that in a biparametric perturbation method, the selection and its combination of parameters may reflect the combined effects introduced by nonlinear factors. The method proposed in this study may be used for solving other mathematical equations established in some application problems.

An elasticity solution of functionally graded beams with different moduli in tension and compression

ABSTRACTIn this study, we analytically solved the problem of a functionally graded beam with different moduli in tension and compression under the action of uniformly distributed loads. By determining the location of the unknown neutral layer of the beam, we first established a simplified mechanical model based on complete partition of tension and compression. Using boundary conditions and continuity conditions of the neutral layer, we obtained an elasticity solution of the problem, in which grade functions of tensile and compressive moduli of elasticity are assumed to be two different exponential expressions while Poisson's ratio is unchanged. The numerical results and comparison also verified the validity of the analytical solution. By changing the grade parameters of the material, the stress and displacement of the beam in three cases, i.e., the tensile modulus is greater than, equal to, or less than the compressive modulus, are discussed, respectively. The result shows that due to the introduction of bimodular effect and functional grade of materials, the maximum tensile and compressive bending stresses may not take place at the bottom and top of the beam, which should be given more attention in the analysis and design of structures made of functionally graded materials with bimodular effect.

A perturbation solution of von-Kármán circular plates with different moduli in tension and compression under concentrated force

ABSTRACTBy modifying classical von-Kármán equations, we established bimodular von-Kármán equations of thin plates with different moduli in tension and compression. Adopting central deflection as a perturbation parameter, we used a perturbation method to solve the equations under various boundary conditions, including rigidly clamped, loosely clamped, simply hinged, and simply supported. The relation of load versus central deflection and stress formulas were derived via the perturbation solution obtained. The numerical simulation also shows that the perturbation solution based on central deflection is overall valid. The results indicate that when the compressive modulus of materials is greater than the tensile one, the bearing capacity of the plate will be further strengthened, which should be considered in the analysis and design of plate-like structures with obvious bimodular effect. Moreover, by comparing with the case under uniformly distributed load, the plate-membrane transition under centrally concentrated force presents discontinuity to some extent.

Periodic response of bimodular laminated composite cylindrical panels with and without geometric nonlinearity

The combined influence of bimodularity, geometric nonlinearity and curvature on the dynamic response characteristics of bimodular material laminated composite cylindrical panels is investigated under periodic excitation. The analysis is carried out using Bert’s constitutive model and first order shear deformation theory based finite element. The geometrically nonlinear forced periodic response is obtained using shooting technique coupled with Newmark time marching, arc length/pseudo-arc length continuation algorithms. The effect of bimodularity, curvature, aspect ratio (a/b), radius to thickness ratio (r/h), boundary conditions, lamination scheme and small initial imperfection on the forced vibration response is analysed. The frequency response curves reveal significant difference in the positive and negative half cycle amplitudes for bimodular material panels which increases with the increase in curvature and bimodularity ratio. The through the thickness variations of normal strain/stress are presented to show the extent of assignment of tensile/compressive properties and restoring force due to combined influence of bimodularity and geometric nonlinearity. Period doubling and sub/super harmonic participations are observed for some cases.