Elasto-plastic postbuckling of damaged orthotropic plates

Based on the elasto-plastic mechanics and continuum damage theory, a yield criterion related to spherical tensor of stress is proposed to describe the mixed hardening of damaged orthotropic materials. Its dimensionless form is isomorphic with the Mises criterion for isotropic materials. Furthermore, the incremental elasto-plastic damage constitutive equations and damage evolution equations are established. Based on the classical nonlinear plate theory, the incremental nonlinear equilibrium equations of orthotropic thin plates considering damage effect are obtained, and solved with the finite difference and iteration methods. In the numerical examples, the effects of damage evolution and initial deflection on the elasto-plastic postbuckling of orthotropic plates are discussed in detail.

PERTURBATION ANALYSIS FOR THE POSTBUCKLING OF RECTANGULAR ORTHOTROPIC PLATES

In this paper, applying perturbation method to von Karman-type nonlinear large deflection equations of orthotropic plates by taking deflection as perturbation parameter, the postbuckling behavior of simply supported rectangular orthotropic plates under in-plane compression is investigated. Two types of in-plane boundary conditions are now considered and the effects of initial imperfections are also studied. Numerical results are presented for various cases of orthotropic composite plates having different elastic properties. It is found that the results obtained are in good agreement with those of experiments.

Symplectic system based analytical solution for bending of rectangular orthotropic plates on Winkler elastic foundation

This paper analyses the bending of rectangular orthotropic plates on a Winkler elastic foundation.Appropriate definition of symplectic inner product and symplectic space formed by generalized displacements establish dual variables and dual equations in the symplectic space.The operator matrix of the equation set is proven to be a Hamilton operator matrix.Separation of variables and eigenfunction expansion creates a basis for analyzing the bending of rectangular orthotropic plates on Winkler elastic foundation and obtaining solutions for plates having any boundary condition.There is discussion of symplectic eigenvalue problems of orthotropic plates under two typical boundary conditions,with opposite sides simply supported and opposite sides clamped.Transcendental equations of eigenvalues and symplectic eigenvectors in analytical form given.Analytical solutions using two examples are presented to show the use of the new methods described in this paper.To verify the accuracy and convergence,a fully simply supported plate that is fully and simply supported under uniformly distributed load is used to compare the classical Navier method,the Levy method and the new method.Results show that the new technique has good accuracy and better convergence speed than other methods,especially in relation to internal forces.A fully clamped rectangular plate on Winkler foundation is solved to validate application of the new methods,with solutions compared to those produced by the Galerkin method.

A THREE-DIMENSIONAL SOLUTION FOR LAMINATED ORTHOTROPIC RECTANGULAR PLATES WITH VISCOELASTIC INTERFACES

When a body consists completely or even partly of viscoelastic materials, its response under static loading will be time-dependent. The adhesives used to glue together single plies in laminates usually exhibit a certain viscoelastic characteristic in a high temperature environment. In this paper, a laminated orthotropic rectangular plate with viscoelastic interfaces, described by the Kelvin-Voigt model, is considered. A power series expansion technique is adopted to approximate the time-variation of various field quantities. Results indicate that the response of the laminated plate with viscoelastic interfaces changes remarkably with time, and is much different from that of a plate with spring-like or viscous interfaces.

Interaction formulae for buckling and failure of orthotropic plates under combined axial compression/tension and shear

An interaction function was constructed based on the axial compression/tension and shear loads of orthotropic plates.The coefficients of the polynomial function were determined by uniaxial test results.Buckling interaction and failure interaction formulae under combined axial tension/compression and shear loads were established.Based on the uniaxial load test results of orthotropic plates,the buckling load and bearing capacity under any proportion of the combined loads could be predicted by using the proposed interaction formulae.The buckling interaction curves and failure envelopes predicted by the proposed interaction formulae were in excellent agreement with the test results.

AN EXACT SOLUTION FOR THE BENDING OF POINT-SUPPORTED ORTHOTROPIC RECTANGULAR THIN PLATES

A closed series solution is proposed for the bending of point-supported orthotropic rectangular thin plates. The positions of support points and the distribution of transverse loadare arbitrary. If the number of simply supported points gradually increases the solution can infinitely approach to Navier's solution. For the square plate simply supported on the middle of each edge and free at each corner, the results are very close to the numerical solutions in the past.

BOUNDARY ELEMENT METHOD FOR ORTHOTROPIC THICK PLATES

The fundamental solutions of the orthotropic thick plates taking into account the transverse shear deformation are derived by means of Hrmander's operator method and a plane-wave decomposition of the Dirac δ-function in this papey.The boundary integral equations of the thick plates have been formulated which are adapted to arbitrary boundary conditions and plane forms.The numerical calculation of the fundamental solutions is discussed in detail.Some numerical examples are analyzed with BEM.

AN INVESTIGATION ON METERIAL PROPERTIESOF NON-HOMOGENEOUS CYLINDRICALLYORTHOTROPIC CIRCULAR PLATES

It has been been reported that the reduced stiffness of non-homogeneous cylindricallyorthotrpic circular plate varing exponentially with radius r is obtained by using thebending theory of a simple beamThe aim of this paper is to verify the effect of radius on the materal properties According to the flat stress-strain relation the values of material properties E E andwhich are the functions of radius r are obtained.Compared with the experimentalvalues the analytical values of the material properties are in essential agreement withtem.

NONLINEAR BENDINGS FOR THE ORTHOTROPIC RECTANGULAR THIN PLATES UNDER VARIOUS SUPPORTS

In this paper, the nonlinear bandings for the orthotropic rectangular thin plates under various supports are studied.The uniformly valid asymptotic solutions of the displacement ? and stress function φ are derived by the perturbation offered in [1].

THE EVALUATION OF STRESS INTENSITY FACTORS OF PLANE CRACK FOR ORTHOTROPIC PLATE WITH EQUAL PARAMETER BY F2LFEM

In this paper, the evaluation of stress intensity factor of plane crack problems for orthotropic plate of equal-parameter is investigated using a fractal two-level finite element method （F2LFEM）. The general solution of an orthotropic crack problem is obtained by assimilating the problem with isotropic crack problem, and is employed as the global interpolation function in F2LFEM. In the neighborhood of crack tip of the crack plate, the fractal geometry concept is introduced to achieve the similar meshes having similarity ratio less than one and generate an infinitesimal mesh so that the relationship between the stiffness matrices of two adjacent layers is equal. A large number of degrees of freedom around the crack tip are transformed to a small set of generalized coordinates. Numerical examples show that this method is efficient and accurate in evaluating the stress intensity factor （SIF）.

APPROXIMATE SOLUTION FOR THIN PLATES CONSTRAINED AT ARBITRARY POINTS

The energy variational formula based on the principle of minimum potential energy is proposed for the plates constrained at arbitrary points. As an instance, the orthotropic large deflection rectangular thin plates with four free edges and transverse displacement constraints under uniform transverse load are discussed. The generalized Fourier series are used as the trial functions of the transverse displacement and the stress function to establish the essential equations, which are linearized by means of the incremental method of load and displacement constraint. In the end of the paper, several computational results are compared with the former literature. Moreover, one typical example is demonstrated through advanced experimental technique. The result shows the accuracy is satisfied well.

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULA RPLATE（I）

In this paper, ihe probleins of nonlinear unsymmeirucal bending for cylindricallyorthotropic circular plale are sludied by using “ the method of two-variabie”￣[1], and theuniformly valid asympiotic soluiions of Nth-order .lor ε_1 and Mth-order for ε_2 areobiained

THE PROBLEMS OF NONLINEAR BENDING FOR ORTHOTROPIC RECTANGULAR PLATE WITH FOUR CLAMPED EDGES

In this paper,under the non-uniformtransverse load,the problems of nonlinear bending for orthotropic rectangular plate are studied by using'the method of twovariable'[1]and 'the method of mixing perturbation'[2].The uniformly valid asymptotic solutions of Nth-order for ε1 and Mth-order for ε2 for ortholropic rectangular plale with four clamped edges are oblained.

THE PROBLEM OF THE NON-LINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULAR THIN PLATE WITH VARIABLE THICKNESS

To begin with, in this paper, the governing equations of the problem of the non-linear unsymmetrical bending for cylindrically orthotropic circular thin plate with variable thickness are derived. By using 'the method of two-variable' and introducing four small parameters, the problem of the non-linear unsymmetrical bending for cylindrically orthotropic circular thin plate with linear variable thickness are studied, and the uniformly valid asymptotic solution of Nth-order for epsilon(1) and Mth-order for epsilon(2) are obtained.

BASIC EQUATIONS OF THE PROBLEM OF THE NONLINEAR UNSYMMETRICAL BENDING FOR ORTHOTROPIC RECTANGULAR THIN PLATE WITH VARIABLE THICKNESS

Under the case of ignoring the body forces and considering components caused by diversion of membrane in vertical direction (z-direction),the constitutive equations of the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with variable thickness are given;then introducing the dimensionless variables and three small parameters,the dimensionaless governing equations of the deflection function and stress function are given.

UNIFORMLY VALID ASYMPTOTIC SOLUTIONS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR ORTHOTROPIC RECTANGULAR THIN PLATE OF FOUR CLAMPED EDGES WITH VARIABLE THICKNESS

By using “the method of modified two-variable”,“the method of mixing perturbation” and introducing four small parameters,the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with linear variable thickness is studied.And the uniformly valid asymptotic solution of Nth-order for ε_1 and Mth-order for ε_2 of the deflection functions and stress function are obtained.

ON J-INTEGRALS NEAR MODELS Ⅰ,Ⅱ CRACK TIPS IN THE PLATE OF ORTHOTROPIC COMPOSITE MATERIAL

In this paper, we prove the independence of the path of the imegrals near model i II track tips in the plate of the orthotropic composite material. Then we derive the computing formulae of the J-integrals at the cases of and by using a complex variable method and redueing J-imegrals to compley farmThe J-integral computational formulae derived this paper have certain reference value for the theoretical researches and the experimental verifications in the plane fracture for composite material.

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULAR PLATE（II）

In this Paper we study the recursive equations under the recursive boundary conditions for W_(nm),nm, v_(nm) and ψ_(nm)(n=0.1.2...N:m=1.2...M. which derivedby the two-variable method ￣[3] in the preceding paper￣[1]. We then solve these problems by using the method of regular perturbation￣[2]. and the uniformly valid asymptotic solution is obtained. Lastly we consider a particular example i. e the bending problems of the axisymmetrical circular plate by using the mixed perurbation method and compare our results with the exact solution found in Ret [ 5 ]. They are similarly coincided.

Bending problems of Non-Homogeneous Cylindrical Orthotropic Circular Plate

In this paper ,the bending problem of the non-homogeneous cylindrical orthotropiccircular plate is described.A general solution for the bending of circular plate underuniformly distributed transverse load is solved.and the exact solution of such circularplate with clamped edges is obtained.

Free vibration of thermo-elastic microplate based on spatiotemporal fractional-order derivatives with nonlocal characteristic length and time

A fractional-order thermo-elastic model taking into account the small-scale effects of the thermo-elastic coupled behavior is developed to study the free vibration of a higher-order shear microplate.The nonlocal strain gradient theory is modified with the introduction of the fractional-order derivatives and the nonlocal characteristic length.The Fourier heat conduction is replaced by the non-Fourier heat conduction with the introduction of the fractional order and the memory characteristic time.Numerical calculations are performed to analyze the effects of the nonlocal strain gradient parameters,the spatiotemporal fractional order,the nonlocal characteristic length,and the memory characteristic time on the natural frequencies,the vibration attenuation,and the phase shift between the temperature field and the displacement field.The numerical results show that the new thermo-elastic model with the spatiotemporal fractional order can provide more exquisite descriptions of the thermo-elastic behavior at a small scale.