Airy Stress Function Method Research Articles

Static analysis for multi-layered piezoelectric cantilevers

Taking the bonding layers and electrodes into account, the multi-layered piezoelectric cantilevers are studied based on the theory of elasticity. Different from the traditional investigations based on the elementary theory of elasticity, the Airy stress function method is used in the present paper. The stress function and induction function are proposed and determined, and then the exact solutions of the static governing equations are found. The material properties and thickness of different layers may be different in the present investigations. As two special cases, the exact static solutions for both unimorph and bimorph are directly obtained by using the present general solutions. The exact solutions obtained in the present paper are compared with the numerical results and others’ investigations, and good agreements are found. In addition, the effects of the properties of both bonding layers and electrodes are discussed. Moreover, the present solution can be used for function graded piezoelectric cantilever beams when the thickness of each bonding layer is taken as zero.

Bending behavior of 2–2 multi-layered piezoelectric curved actuators

The objective of this paper is to study the bending behavior of 2–2 multi-layeredpiezoelectric curved actuators. Unlike the traditional investigations based on theelementary theory of elasticity, the governing equations of a curved actuator are solvedbased on the general theory of elasticity and piezoelectricity. The Airy stress functionmethod is utilized and both the piezoelectric coefficient and the thickness for differentlayers are taken as variables in the analysis, which makes it easy to consider the effect ofelastic layers or electrodes on the bending behavior of the actuators. By introducingseveral recurrence formulae, the exact solutions are obtained. Furthermore, goodagreement is shown by comparing the theoretical results with numerical ones.

Bending analysis of a functionally graded piezoelectric cantilever beam

A new analysis based on Airy stress function method is presented for a functionally graded piezoelectric material cantilever beam. Assuming that the mechanical and electric properties of the material have the same variations along the thickness direction, a two-dimensional plane elasticity solution is obtained for the coupling electroelastic fields of the beam under different loadings. This solution will be useful in analyzing FGPM beam with arbitrary variations of material properties. The influences of the functionally graded material properties on the structural response of the beam subjected to different loads are also studied through numerical examples.

Static analyses for 2–2 multi-layered piezoelectric curved composites

Based on the general theory of elasticity, the static behavior of 2–2 multi-layered piezoelectric curved composites is investigated. The exact solutions of the mechanical and electrical fields of the curved composites are obtained by utilizing Airy stress function method. Both the piezoelectric coefficient and the thickness for different layers are taken as variables. Different from other investigations based on the elementary theory of elasticity, the present investigation gives exact solutions for these variables by introducing several recurrence formulas, which make the present results very useful. The theoretical results obtained in the present paper are compared with the numerical results and good agreements are found.

Analytical solution for fixed-fixed anisotropic beam subjected to uniform load

The analytical solutions of the stresses and displacements were obtained for fixed-fixed anisotropic beams subjected to uniform load. A stress function involving unknown coefficients was constructed, and the general expressions of stress and displacement were obtained by means of Airy stress function method. Two types of the description for the fixed end boundary condition were considered. The introduced unknown coefficients in stress function were determined by using the boundary conditions. The analytical solutions for stresses and displacements were finally obtained. Numerical tests show that the analytical solutions agree with the FEM results. The analytical solution supplies a classical example for the elasticity theory.

Use of automated photoelasticity to determine stress intensity factors of bimaterial joints

A new systematic experimental procedure has been developed to obtain the stress intensity factors governing the singular stress field that occurs near the intersection between the interface and free edges of bimaterial joints. A preliminary theoretical study of the singular stress field is carried out by the well-known Airy stress function method. The obtained stress laws are properly combined with the basic law of photoelasticity in order to define a procedure that permits the zone dominated by the singularity to be located and the stress intensity factors (SIFs) to be computed on the basis of full field data provided from automated photoelasticity. In particular, a systematic error analysis is used to determine the model zone where the experimental data have to be collected in order to obtain accurate SIF evaluation. As an example, the proposed method is applied to determine the SIFs of various aluminium/ PSM-1 specimens under different external loading conditions using Fourier transform photoelasticity. The experimental results have been compared to those obtained by an independent procedure, based on a boundary element approach, in order to validate the accuracy of the proposed procedure.

Bending and local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube

This paper studies the pure bending and bending-induced local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube (SWNT). The Airy stress-function method was employed to analyse the deformation of the matrix, and the cross-sectional change of the SWNT in bending was taken into account. A particular consideration was given to the effect of the SWNT’s radial flexibility on the strain/stress states and buckling. It was found that in thicker matrix layers the SWNT buckles locally at smaller bending angles and greater flattening ratios. This causes higher strains/stresses in the surrounding matrix and in turn degrades the strength of the nanocomposite structure.

Bending Behavior of Functionally Gradient Piezoelectric Cantilever

Based on the theory of elasticity, the bending behavior of piezoelectric cantilever with gradient property for piezoelectric parameter g31is studied. By use of Airy stress function method, the solution of cantilever subjected to a couple and an electric field is obtained. The results obtained in present paper are compared with that from both experiment and other analytical models based on BaTiO3-ceramics. It is found that the tip deflection of cantilever obtained in present papers agrees well with the result of N-morph cantilever when the number N gets to∞. Present paper also gives a theoretical demonstration that all the mechanical stresses will be vanishing in FGM actuators. Finally addressed in present paper is a method of parameter identification.

General solution of a density functionally gradient piezoelectric cantilever and its applications

We have used the plane strain theory of transversely isotropic bodies to study a piezoelectric cantilever. In order to find the general solution of a density functionally gradient piezoelectric cantilever, we have used the inverse method (i.e. the Airy stress function method). We have obtained the stress and induction functions in the form of polynomials as well as the general solution of the beam. Based on this general solution, we have deduced the solutions of the cantilever under different loading conditions. Furthermore, as applications of this general solution in engineering, we have studied the tip deflection and blocking force of a piezoelectric cantilever actuator. Finally, we have addressed a method to determine the density distribution profile for a given piezoelectric material.

914 不均質円筒の平面非軸対称等温および熱弾性問題に関する解析的研究

In this paper, isothermal elastic and thermoelastic behaviors for a nonhomogeneous hollow cylinder are considered. Assuming that the cylinder has a nonhomogeneous material property such as shear modulus of elasticity G varying with the radial coordinate with an arbitrary power product form, we have analyzed asymmetrical plane strain problem subject to non-uniform external forces or non-uniform heat supplies from the inner and outer boundaries. Making use of Airy's stress function method, theoretical results for deformations and stresses are obtained. Thereafter, numerieal calculations are carried out for the various nonhomogeneous parameters, and the results obtained are shown graphically.

Two displacement methods for in-plane deformations of orthotropic linear elastic materials

Two displacement formulation methods are presented for the plane strain and plane stress problems of orthotropic linear elastic materials having the three planes of symmetry at x1 = 0, x2 = 0 and x3 = 0. The first method starts with solving the two governing partial differential equations simultaneously, while the second method begins with solving one equation and ends with enforcing the other. The former follows the approach of Eshelby, Read and Shockley, whereas the latter is based on an extended version of Green's theorem and thus has similarities with Airy's stress function method. The two displacement methods lead to the same characteristic equation that is identical to the one obtained by Lekhnitskii using a stress formulation method. The general solutions resulting from the two displacement methods can be used to solve plane elasticity problems of orthotropic materials with displacement or mixed boundary conditions.

On displacement methods in planar anisotropic elasticity

Two displacement formulation methods are presented for problems of planar anisotropic elasticity. The first displacement method is based on solving the two governing partial differential equations simultaneously/ This method is a recapitulation of the orignal work of Eshelby, Read and Shockley [7] on generalized plane deformations of anisotropic elastic materials in the context of planar anisotropic elasticity. The second displacement method is based on solving the two governing equations separately. This formulation introduces a displacement function, which satisfies a fourth-order partial differential equation that is identical in the form to the one given by Lekhnitskii [6] for monoclinic materials using a stress function. Moreover, this method parallels the traditional Airy stress function method and thus the Lekhnitskii method for pure plane problems. Both the new approach and the Airy stress function method start with the equilibrium equations and use the same extended version of Green's theorem (Chou and Pagano [13], p. 114; Gao [11]) to derive the expressions for stress or displacement components in terms of a potential (stress or displacement) function (see also Gao [10, 11]). It is therefore anticipated that the displacement function involved in this new method could also be evaluated from measured data, as was done by Lin and Rowlands [17] to determine the Airy stress function experimentally. The two different displacement methods lead to two general solutions for problems of planar anisotropic elasticity. Although the two solutions differ in expressions, both of the depend on the complex roots of the same characteristic equation. Furthermore, this characteristic equation is identical to that obtained by Lekhnitskii [6] using a stress formulation. It is therefore concluded that the two displacement methods and Lekhnitskii's stress method are all equivalent for problems of planar anisotropic elasticity (see Gao and Rowlands [8] for detailed discussions).

OPTIMIZATION OF MATERIAL COMPOSITION OF NONHOMOGENEOUS HOLLOW CIRCULAR CYLINDER FOR THERMAL STRESS RELAXATION MAKING USE OF NEURAL NETWORK

In this study, a neural network is applied to optimization problems of material compositions for a nonhomogeneous hollow circular cylinder with arbitrarily distributed and continuously varied material properties such as those of a functionally graded material. Using the analytical procedure of a laminated hollow circular cylinder model, the analytical temperature solution for the cylinder is derived approximately. Further more, the thermal stress components are formulated under the mechanical condition as being traction-free and making use of the Airy's stress function method. As a numerical example, the hollow circular cylinder composed of zirconium oxide and titanium alloy is considered. In addition, for the optimization problem of minimizing the thermal stress distribution, the numerical calculations are made using a neural network and the optimum material composition is determined taking into account the effect of the temperature dependency of material properties. Furthermore, the results obtained by the neural network and the ordinary nonlinear programming method are compared.

Optimization of Material Composition of Hollow Circular Cylinder of Functionally Graded Material for Thermal Stress Relaxation Making Use of Genetic Algorithm.

In this paper, a genetic algorithm is applied to an optimization problem of material composition for a hollow circular cylinder of step-formed functionally graded materials. In the first place, the one dimensional transient temperature distribution for a laminated composite hollow circular cylinder is analyzed theoretically. Furthermore, making use of Airy's stress function method, the thermal stress components are formulated under the mechanical condition of being traction free. As a numerical example, the hollow circular cylinder composed of zirconium oxide and titanium alloy is considered. And, as the optimization problem of minimizing the thermal stress distribution, the numerical calculations are carried out making use of genetic algorithm without supposing a distribution function of material composition, and the optimum material composition of each layer is determined taking into account the effect of temperature-dependency of material properties.

THERMAL STRESS ANALYSIS OF A RECTANGULAR PLATE AND ITS THERMAL STRESS INTENSITY FACTOR FOR COMPRESSIVE STRESS FIELD

In this study, a transient thermal stress problem of a rectangular plate due to a nonuniform heat supply is treated theoretically and, thereafter, fracture behaviors of the plate with a crack are examined for compressive stress states. Assuming that a crack located on an arbitrary position, with an arbitrary direction, is sufficiently small and is closed because of the compressive stress field, a temperature field, in a transient state, is analyzed by taking into account the effect of relative heat transfer on both surfaces of the plate. Thereafter, the corresponding thermal stress analysis is developed on the basis of the two-dimensional plane stress problem using Airy's stress function method, and the stress intensity factor is analyzed for the biaxial stress state. As an analytical model, we consider mechanical boundary conditions of prescribed displacement and estimate the stress intensity factor of a crack tip using parameters of the crack configuration such as the location, direction, length, and coefficient of friction. These numerical results are shown in graphical form.

One-Dimensional Transient Thermal Stress Problem for Nonhomogeneous Hollow Circular Cylinder and Its Optimization of Material Composition for Thermal Stress Relaxation

In this study, the one-dimensional transient thermal stress problem of a nonhomogeneous hollow circular cylinder with arbitrarily distributed and continuously varied material properties, such as functionally graded materials, is evaluated theoretically. Using the analytical procedure of a laminated hollow circular cylinder model, the analytical temperature solution for the cylinder is derived approximately. Furthermore, making use of Airy's stress function method, the thermal stress components are formulated under the mechanical condition of being traction-free. As a numerical example, a hollow circular cylinder composed of zirconium oxide (ZrO 2 ) and titanium alloy (Ti-6Al-4V) is considered. To optimize(i.e., minimize) the thermal stress distribution, numerical calculations are carried out, and the optimum material composition is determined taking into account the effect of the temperaturedependence of the material properties. Furthermore, taking into account the variation in the thickness of the cylinder, the temperature rise of the surrounding medium, and the relative heat transfer coefficients on the inner and outer surfaces, the optimum values of the material composition are obtained. Numerical data for the calculations are shown graphically.

Some Faults in the Stress Analysis of Power-Hardening Materials With Pseudo-Stress Function Method

A new analytic method was suggested and described by Lee(1987), which is known as the pseudo-stress function method similar to the Airy stress function method in linear elasticity. Here the serious errors of Lee's theory are presented

Theoretical Analysis of Two-Dimensional Thermoelastoplastic Bending Deformation of Plate Subject to Partially Distributed Heat Supply.

In this paper, making use of the strain increment theorem, the transient thermoelastoplastic bending problem of an infinite plate is discussed. In order to develop the analysis for the temperature field, the methods of Fourier cosine and Laplace transforms are introduced, and the theoretical solution is obtained. On the other hand, for the development of the thermoelastoplastic field, both the analytical technique of Airy's stress function method for thermoelastic behavior and the numerical technique of the finite difference method for the plastic behavior are applied. However, difficulty in applying the finite difference method may occur. It is clearly impossible to cover an infinite region with a finite difference grid. Thus we have introduced the assumption that plastic deformation is restricted to a local region. Thereafter, based on Dixon's experimental results that the stress field outside the plastic region is the same as the elastic stress field, namely, the elastic solution prevails outside the plastic region, the finite difference method is applied to the finite region predicted to show the plastic deformation. Some numerical results, such as the plastic deformation in the out-of-plane direction, are shown in figures and discussed briefly.

One-Dimensional Transient Thermal Stress Problem for Nonhomogeneous Hollow Circular Cylinder and Its Optimization of Material Composition for Thermal Stress Relaxation.

In this study, the one-dimensional transient thermal stress problem of a nonhomogeneous hollow circular cylinder with arbitrarily distributed and continuously varied material properties, such as functionally graded materials, is evaluated theoretically. Using the analytical procedure of a laminated hollow circular cylinder model, the analytical temperature solution for the cylinder is derived approximately. Furthermore, making use of Airy's stress function method, the thermal stress components are formulated under the mechanical condition of being traction-free. As a numerical example, a hollow circular cylinder composed of zirconium oxide (ZrO2) and titanium alloy (Ti-6Al-4V) is considered. To optimize (i. e., minimize) the thermal stress distribution, numerical calculations are carried out, and the optimum material composition is determined taking into account the effect of the temperature dependence of the material properties. Furthermore, taking into account The variation in the thickness of the cylinder, the temperature rise of the surrounding medium, and the relative heat transfer coefficients on the inner and outer surfaces, the optimum values of the material composition are obtained. Numerical data for the calculations are shown graphically.

Transient Thermal Stress Problems of a Rectangular Plate under Nonuniform Heat Supply and Its Fracture Behavior for Compression Field.

This paper is concerned with a transient thermal stress problem of a rectangular plate under nonuniform heat supply and the fracture behavior (stress intensity factor) of the plate with a crack under the axial compressive stress state. Assuming that the crack is sufficiently small and is closed for a compressive stress field, then the temperature field is not disturbed by the crack. Theoretical analyses for the temperature and the stress field are developed on the basis of the two-dimensional plane-stress problem using a heat conduction equation and Airy's stress function method, and the stress intensity factor is analyzed for the biaxial compressive stress field. As an analytical model, we consider the mechanical boundary condition of prescribed displacement, and estimate the stress intensity factor of the crack tip using parameters of the crack configuration such as the location, direction, length and coefficient of friction. These numerical results are shown in graphical form.