Bending strength degradation of a cantilever plate with surface energy due to partial debonding at the clamped boundary

This paper investigates the bending fracture problem of a micro/nanoscale cantilever thin plate with surface energy,where the clamped boundary is partially debonded along the thickness direction.Some fundamental mechanical equations for the bending problem of micro/nanoscale plates are given by the Kirchhoff theory of thin plates,incorporating the Gurtin-Murdoch surface elasticity theory.For two typical cases of constant bending moment and uniform shear force in the debonded segment,the associated problems are reduced to two mixed boundary value problems.By solving the resulting mixed boundary value problems using the Fourier integral transform,a new type of singular integral equation with two Cauchy kernels is obtained for each case,and the exact solutions in terms of the fundamental functions are determined using the PoincareBertrand formula.Asymptotic elastic fields near the debonded tips including the bending moment,effective shear force,and bulk stress components exhibit the oscillatory singularity.The dependence relations among the singular fields,the material constants,and the plate's thickness are analyzed for partially debonded cantilever micro-plates.If surface energy is neglected,these results reduce the bending fracture of a macroscale partially debonded cantilever plate,which has not been previously reported.

Determination of Feature Line Equation for Self-adapting Incremental Press Bending

The basie idea and method about determination of the feature line equations and how to apply them to the numerical control of the press bending of panei skins were introduced. Research indicates that it is feasible to realize the self adapting incremental press bending by adopting the feature line equation. The feature line equation, which is based on the database of the status of practical processes, can be adjusted in time, and the forming precision can be improved. It is important to correctly select and reasonably predict the feature line equations to enhance the accuracy of the incremental press bending based on the feature line database and algorithm. The determination of the feature line equation settles necessary data foundation for further research on the database of self-adapting incremental press bending, and it supplies a new clue for the development of self-adapting incremental press bending.

The symplectic eigenfunction expansion theorem and its application to the plate bending equation

This paper deals with the bending problem of rectangular plates with two opposite edges simply supported. It is proved that there exists no normed symplectic orthogonal eigenfunction system for the associated infinite-dimensional Hamiltonian operator H and that the two block operators belonging to Hamiltonian operator H possess two normed symplectic orthogonal eigenfunction systems in some space. It is demonstrated by using the properties of the block operators that the above bending problem can be solved by the symplectic eigenfunction expansion theorem, thereby obtaining analytical solutions of rectangular plates with two opposite edges simply supported and the other two edges supported in any manner.

On Feasibility of Variable Separation Method Based on Hamiltonian System for a Class of Plate Bending Equations

The eigenfunction system of infinite-dimensional Hamiltonian operators appearing in the bending problem of rectangular plate with two opposites simply supported is studied. At first, the completeness of the extended eigenfunction system in the sense of Cauchy＇s principal value is proved. Then the incompleteness of the extended eigenfunction system in general sense is proved. So the completeness of the symplectic orthogonal system of the infinite-dimensional Hamiltonian operator of this kind of plate bending equation is proved. At last the general solution of the infinite dimensional Hamiltonian system is equivalent to the solution function system series expansion, so it gives to theoretical basis of the methods of separation of variables based on Hamiltonian system for this kind of equations.

BOUNDARY INTEGRAL EQUATIONS FOR THE BENDING PROBLEM OF PLATES ON TWO-PARAMETER FOUNDATION

By means of Fourier integral transformation of generalized function, the fundamental solution for the bending problem of plates on two-parameter foundation is derived in this paper, and the fundamental solution is expanded into a uniformly convergent series. On the basis of the above work, two boundary integral equations which are suitable to arbitrary shapes and arbitrary boundary conditions are established by means of the Rayleigh-Green identity. The content of the paper provides the powerful theories for the application of BEM in this problem.

Invertibility of Infinite-Dimensional Hamiltonian Operators and Its Application to Plate Bending Equation

The results of invertibility and spectrum for some different classes of infinite-dimensional Hayniltonian operators, after a brief classification by domains. are given. By the above results, the associated infinite-dimensional Hamiltonian operator with simple supported rectangular plate is proved to be invertible. Furthermore, by a certain compactness, we find that the spectrum of this operator consists only of isolated eigenvalues with finite geometric multiplicity, which will play a significant role in finding the analytical and numerical solution based on Hamiltonian system for a class of plate bending equations.

EQUIVALENT BOUNDARY INTEGRAL EQUATIONS WITH INDIRECT UNKNOWNS FOR THIN ELASTIC PLATE BENDING THEORY

Equivalent Boundary Integral Equations (EBIE) with indirect unknowns for thin elastic plate bending theory, which is equivalent to the original boundary value problem, is established rigorously by mathematical technique of non-analytic continuation and is fully proved by means of the variational principle. The previous three kinds of boundary integral equations with indirect unknowns are discussed thoroughly and it is shown that all previous results are not EBIE.

On Applications of Generalized Functions in the Discontinuous Beam Bending Differential Equations

This paper discusses the mathematical modeling for the mechanics of solid using the distribution theory of Schwartz to the beam bending differential Equations. This problem is solved by the use of generalized functions, among which is the well known Dirac delta function. The governing differential Equation is Euler-Bernoulli beams with jump discontinuities on displacements and rotations. Also, the governing differential Equations of a Timoshenko beam with jump discontinuities in slope, deflection, flexural stiffness, and shear stiffness are obtained in the space of generalized functions. The operator of one of the governing differential Equations changes so that for both Equations the Dirac Delta function and its first distributional derivative appear in the new force terms as we present the same in a Euler-Bernoulli beam. Examples are provided to illustrate the abstract theory. This research is useful to Mechanical Engineering, Ocean Engineering, Civil Engineering, and Aerospace Engineering.

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.

A SOLVING METHOD FOR A SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS WITH AN APPLICATION TO THE BENDING PROBLEM OF A THICK PLATE

A theorem of solving a system of linear non-homogeneous differential equations through integrating and adding its basic solutions is put forward and proved, the mathematical role and physical nature of the theorem is interpreted briefly. As an example, the theorem is applied to solve the problem of thermo-force bending of a thick plate.

METHOD TO CALCULATE BENDING CENTER AND STRESS INTENSITY FACTORS OF CRACKED CYLINDER UNDER SAINT_VENANT BENDING

Using the single crack solution and the regular solution of plane harmonic function, the problem of Saint_Venant bending of a cracked cylinder by a transverse force was reduced to solving two sets of integral equations and its general solution was then obtained. Based on the obtained solution, a method to calculate the bending center and the stress intensity factors of the cracked cylinger whose cross_section is not thin_walled, but of small torsion rigidity is proposed. Some numerical examples are given.

NONLINEAR BENDING OF SIMPLY SUPPORTED RECTANGULAR SANDWICH PLATES

The fundamental equations and boundary conditions of the nonlinear bending theory for a rectangular sandwich plate with a soft core were derived by using the method of calculus of variations. Then the nonlinear bending for a simply supported rectangular sandwich plate under the uniform lateral load was investigated by using the perturbation method. As a result, a quite accurate analytic solution was obtained.

Bending analysis and control of rolled plate during snake hot rolling

In order to study the bending behavior of aluminum alloy 7050 thick plate during snake hot rolling, several coupled thermo-mechanical finite element(FE) models were established. Effects of different initial thicknesses, pass reductions, speed ratios and offset distances on the bending value of the plate were analyzed. ‘Quasi smooth plate' and optimum offset distance were defined and quasi smooth plate could be acquired by adjusting offset distance, and then bending control equation was fitted. The results show that bending value of the plate as well as the extent of the increase grows with the increase of pass reduction and decrease of initial thickness; the bending value firstly increases and then keeps steady with the ascending speed ratio; the bending value can be reduced by enlarging the offset distance. The optimum offset distance varies for different rolling parameters and it is augmented with the increase of pass reduction and speed ratio and the decrease of initial thickness. A proper offset distance for different rolling parameters can be calculated by the bending control equation and this equation can be a guidance to acquire a quasi smooth plate. The FEM results agree well with experimental results.

A state space solution for the bending problem of thick laminated piezoelectric open cylindrical shells

Based on the theories of three-dimensional elasticity and piezoelectricity, and by assuming appropriate boundary functions, we established a state equation of piezoelectric cylindrical shells. By using the transfer matrix method, we presented an analytical solution that satisfies all the arbitrary boundary conditions at boundary edges, as well as on upper and bottom surfaces. Our solution takes into account all the independent elastic and piezoelectric constants for a piezoelectric orthotropy, and satisfies continuity conditions between plies of the laminates. The principle of the present method and corresponding results can be widely used in many engineering fields and be applied to assess the effectiveness of various approximate and numerical models.

METHOD TO CALCULATE BENDING CENTER AND STRESS INTENSITY FACTORS OF CRACKED CYLINDER UNDER SAINT-VENANT BENDING

Using the single crack solution and the regular solution elf plane harmonic function, the problem of Saint-Venant bending of a cracked cylinder by a transverse force was reduced to solving two sets of integral equations and its general solution was then obtained. Based on the obtained solution, a method to calculate the bending center and the stress intensity factors of the cracked cylinger whose cross-section is not thin-walled, but of small torsion rigidity is proposed. Some numerical examples are given.

AXISYMMETRIC BENDING FOR THICK LAMINATED CIRCULAR PLATE UNDER A CONCENTRATED LOAD

Based upon the fundamental equations of three dimensional elasticity, the state equation for axisymmetric bending of laminated transversely isotropic circular plate is established and the concentrated force on plate surface is expanded into Fourier_Bessel's series, therefore, an analytical solution for the problem is presented. Every fundamental equation of three dimensional elasticity can be exactly satisfied by the solution and all the independent elastic constants can be taken into account fully, furthermore, the continuity conditions between plies can also be satisfied.

THE SYMMETRICAL BENDING OF AN ELASTIC CIRCULAR PLATE SUPPORTED AT KINTERNAL POINTS

This paper treats the symmetrical bending of a uniformly loaded circular plate supported at k internal points. The boundary displacement and slope are expanded in Fourier seriesr. The method proposed by [6] is applied. As both the governing differential equation and boundary conditions are satisfied exactly, we therefore obtain the analytic expression of the transverse deflectionul equation of the circular plate. This is an easy and effective methed.

NONLINEAR BENDING OF SIMPLY SUPPORTED SYMMETRIC LAMINATED CROSS-PLY RECTANGULAR PLATES

Based on the von Karman-type theory of plates, nonlinear bending problems of simply supported symmetric laminated cross-ply rectangular plates under the combined action of pressure and inplane load are investigated in this paper. The solution which satisfies the governing equations and boundary conditions is obtained by using the double Fourier series method.

NONSINGULAR KERNEL BOUNDARY ELEMENT METHOD FOR THIN-PLATE BENDING PROBLEMS

In this paper, the nonsingular fundamental solutions were obtained from Fourier series under some given conditions. These solutions can be taken as the kernels of integral equation. So a new boundary element method was presented, with which all kinds of thin-plate bending problems can be solved, even with complicated loadings and sinuous boundaries. The calculation is much simpler and more accurate.

NON-SYMMETRICAL BENDING PROBLEMS OF INFINITE ANNULAR PLATES SUPPORTED ON INNER EDGE

For non-asymmetrical bending problems of elastic annular plates, the exact solutions are not fond. To bending problems of infinite annular plate with two different boundary conditions, based on the boundary integral formula,the natural boundary integral equation for the boundary value problems of the biharmonic equation and the condition of bending moment in infinity,bending solutions under non-symmetrical loads are gained by the Fourier series and convolution formulae. The formula for the solutions has nicer convergence velocity and high computational accuracy, and the calculating process is simpler. Solutions of the given examples are compared with the finite element method. The textual solutions of moments near the loads are better than the finite element method to the fact that near the concentrative loads the inners forces trend to infinite.