On the structure of continua and the mathematical properties of algebraic elastodynamic of a triclinic structural system

This paper is neither laudatory nor derogatory but it simply contrasts with what might be called elastostatic (or static topology), a proposition of the famous six equations. The extension strains and the shearing strains which were derived by A.L. Cauchy, are linearly expressed in terms of nine partial derivatives of the displacement function (u _i ,u _j ,u _h )=u(x ⁱ ,x ^j ,x ^k ) and it is impossible for the inverse proposition to sep up a system of the above six equations in expressing the nine components of matrix (∂(u _i ,u _j ,u _h )/∂(x ⁱ ,x ^j ,x ^k )). This is due to the fact that our geometrical representations of deformation at a given point are as yet incomplete^[1]. On the other hand, in more geometrical language this theorem is not true to any triangle, except orthogonal, for “squared length” in space^[2]. The purpose of this paper is to describe some mathematic laws of algebraic elastodynamics and the relationships between the above-mentioned important questions.     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅳ)-CALCULATION FOR U-SHAPED BELLOWS

This is one of the applications of Part (I), in which the angular stiffness, and the corresponding stress distributions of U-shaped bellows were discussed. The bellows was divided into protruding sections, concave sections and ring plates for the calculation that the general solution (I) with its reduced form to ring plates were used respectively, but the continuity of the surface stresses and the meridian rotations at each joint of the sections were entirely satisfied. The resent results were compared with those of the slender ring shell solution proposed earlier by the authors, the standards of the Expansion Joint Manufacturers Association (EJMA), the experiment and the finite element method. It is shown that the governing equation and the general solution (I) are very effective. Contributed by HUANG Qian Biography: ZHU Wei-ping (1962-)     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅲ)-CALCULATION FOR C-SHAPED BELLOWS

This is one of the applications of Part (I), in which the angular stiffness, the lateral stiffness and the corresponding stress distributions of C-shaped bellows were calculated. The bellows was divided into protruding sections and concave sections for the use of the general solution (I), but the continuity of the stress resultants and the deformations at each joint of the sections were entirely satisfied. The present results were compared with those of the other theories and experiments, and are also tested by the numerically integral method. It is shown that the governing equation and the general solution (I) are very effective. Contributed by HUANG Qian Biography: ZHU Wei-ping (1962-)     

ENDOCHRONIC ANALYSIS FOR COMPRESSIVE BUCKLING OF THIN－WALLED CYLINDERS IN YIELD REGION

ＥＮＤＯＣＨＲＯＮＩＣＡＮＡＬＹＳＩＳＦＯＲＣＯＭＰＲＥＳＳＩＶＥＢＵＣＫＬＩＮＧＯＦＴＨＩＮ－ＷＡＬＬＥＤＣＹＬＩＮＤＥＲＳＩＮＹＩＥＬＤＲＥＧＩＯＮＰｅｎｇＸｉａｎｇｈｅ（彭向和）ＣｈｅｎＹｕａｎｑｉａｎｇ（陈元强）ＺｅｎｇＸｉａｎｇｇｕｏ（曾祥...     

Higher approximations to the homogeneous solutions for toroid

To analyze the strength of the elastic, isotropic toroid with axisymmetrical loads, it is necessary to find the solutions for a nonhomogeneous second-order ordinary-differential equation in complex form. When the parameter = 12(1-v²) a² / (r ₀ h), which occurs in the equation, is large, the asymptotic approach is often used. Up to now, only the first approximations to its homogeneous solutions are found. In this paper the author has obtained higher approximations to the homogeneous solutions, thus, reaching the precision of linear thin shell theory.This work was supported by the National Natural Science Foundation of China.     

Shell vibrations at high-frequency

The free vibration is called high-frequency when the frequency parameter is limited by the inequalities >max{R ₂ ^–2 (s)} and O(h ⁰). In this case there is only one boundary layer type of solution in the neighbourhood of any edge which is not sufficient to satisfy the two non-tangential boundary conditions to be dropped by the membrane equations at the edge, and is called non-complete.An asymptotic approach is presented in this paper, by means of which we find that there are two types of principal modes to be operative over the whole range of the shell surface, when the shell vibrates axisymmetrically at high frequency. One of the principal modes is a membrane type (¦u¦¦w¦, and the index of variation is zero) and the other is a quasi-transverse one with quick variation (¦u¦¦w¦, and the index of variation is equal to 1/2). Correspondingly, the set of frequency parameters can also be divided into two subsets, one of which corresponds to the membrane modes as their eigenvectors, while the other subset corresponds to the quasi-transverse modes with quick variation as their eigenvectors.     

Boundary and angular layer behavior in singular perturbed quasilinear systems

In this paper, using the method of differential inequalities, we study the existence of solutions and their asymptotic behavior, as 0⁺, of Dirichlt problem for second order quasilinear systems. Depending on whether the reduced solutionu(t) has or does not have a continuous first-derivative in (a, b), we study two types of asymptotic behaviour, thus leading to the phenomena of boundary and angular layers.     

Some asymptotic results concerning the buckling of a spherical shell of arbitrary thickness

For a spherical shell of arbitrary thickness which is subjected to an external hydrostatic pressure, symmetrical buckling takes place at a value of μ₁ which depends on and the mode number, where A₁ and A₂ are the undeformed inner and outer radii, and μ₁ is the ratio of the deformed inner radius to the undeformed inner radius. In the large mode number limit, we find that the dependence of μ₁ on has a boundary layer structure: it is a constant over almost the entire region of and decreases sharply from this constant value to unity as tends to unity (the thin-shell limit). Simple asymptotic expressions for the bifurcation condition are obtained. The classical result for thin shells is recovered directly from the equations of finite elasticity, and an asymptotic critical neutral curve (which envelops the neutral curves corresponding to different mode numbers) is obtained.     

Postbuckling analysis of axially-loaded laminated cylindrical shells with piezoelectric actuators

A compressive postbuckling analysis is presented for a laminated cylindrical shell with piezoelectric actuators subjected to the combined action of mechanical, electric and thermal loads. The temperature field considered is assumed to be a uniform distribution over the shell surface and through the shell thickness, and the electric field is assumed to be the transverse component E_Z only. The material properties are assumed to be independent of the temperature and the electric field. The governing equations are based on the classical shell theory with von Kármán–Donnell-type kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of hybrid laminated cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the compressive postbuckling behavior of perfect and imperfect, cross-ply laminated cylindrical thin shells with fully covered or embedded piezoelectric actuators under different sets of thermal and electric loading conditions. The effects played by temperature rise, applied voltage, shell geometric parameter, stacking sequence, as well as initial geometric imperfections are studied.     

Generalized variational principles with several arbitrary parameters and the variable substitution and multiplier method

The functional transformations of variational principles in elasticity are classified as three patterns: Ⅰ relaxation pattern, Ⅱ augmented pattern and III equivalent pattern.On the basis of pattern Ⅲ, the generalized variational principles with several arbitrary parameters are formulated and their functionals are defined. They are: the generalized principle of single variable u with several parameters, the generalized principle of two variables u, σ with several parameters, the generalized principle of two variables u, ε with several parameters, and the generalized principle of three veriables u, ε, σ with several parameters. From these principles, a series of new forms of equivalent functionals can be obtained. When the values of these parameters are properly chosen, a series of finite element models can be formulated.In this paper, the question of losing effectiveness for Lagrange multiplier method is also discussed. In order to "recover" effectiveness for multiplier method, a modified method, namely, the variable substitution and multiplier method, is proposed.     

Nonlinear stability of large k-value shallow spherical shells with a symmetrically distributed line load

In this paper, we treat the nonlinear stability problem of shallow spherical shells with large values ofk(k=12(1–v) · 2f/h,f = shell rise,h = shell thickness) under the action of uniformly distributed line load along a circle concentric with the shell boundary. Load-deflection curves are computed at successive increments of uniformly distributed line loads by using both cubic B-spline approximations and iterative techniques. Our algorithm yields fairly good convergent results for values ofk as large as 400. The limiting case in which shells are loaded along a circle of small radius has been specially investigated and the computed critical loads are compared with those obtained with central point loads by other authors.     

Novel shear deformation model for moderately thick and deep laminated composite conoidal shell

This article presents a novel mathematical model for moderately thick and deep laminated composite conoidal shell. The zero transverse shear stress at top and bottom of conoidal shell conditions is applied. Novelty in the present formulation is the inclusion of curvature effect in displacement field and cross curvature effect in strain field. This present model is suitable for deep and moderately thick conoidal shell. The peculiarity in the conoidal shell is that due to its complex geometry, its peak value of transverse deflection is not at its center like other shells. The C¹ continuity requirement associated with the present model has been suitably circumvented. A nine-node curved quadratic isoparametric element with seven nodal unknowns per node is used in finite element formulation of the proposed mathematical model. The present model results are compared with experimental, elasticity, and numerical results available in the literature. This is the first effort to solve the problem of moderately thick and deep laminated composite conoidal shell using parabolic transverse shear strain deformation across the thickness of conoidal shell. Many new numerical problems are solved for the static study of moderately thick and deep laminated composite conoidal shell considering 10 different practical boundary conditions, four types of loadings, six different hl/hh (minimum rise/maximum rise) ratios, and four different laminations.     

A new kind of mixed elements for shells of revolution subjected to arbitrary loads

In this paper a generalized variational principle with two-field variables is derived from the Reissner principle of elasticity in the curvilinear coordinates of a revolution shell, based on which, a new kind of mixed elements with independent transverse rotations is formulated for revolution shells subjected to harmonic external loads. The resultant-stress interpolations are carefully selected so that the shear part of the element stiffness contains the Kirchhoff hypothesis for thin shells and element stiffness matrices have correct ranks. The elements are free from shear locking and spurious kinematic modes. Numerical examples show that the new elements have good generality and high accuracy for thin and moderately-thick revolution shells.     

Natural frequencies of rotating functionally graded cylindrical shells

Love’s first approximation theory is used to analyze the natural frequencies of rotating functionally graded cylindrical shells.To verify the validity of the present method,the natural frequencies of the simply supported non-rotating isotropic cylindrical shell and the functionally graded cylindrical shell are compared with available published results.Good agreement is obtained.The effects of the power law index,the wave numbers along the x-and θ-directions,and the thickness-to-radius ratio on the natural frequencies of the simply supported rotating functionally graded cylindrical shell are investigated by several numerical examples.It is found that the fundamental frequencies of the backward waves increase with the increasing rotating speed,the fundamental frequencies of the forward waves decrease with the increasing rotating speed,and the forward and backward waves frequencies increase with the increasing thickness-to-radius ratio.     

The stress-strain fields at crack tip in axially cracked cylindrical shells and the calculation of stress intensity factors

A perturbation solution for stress-strain fields (including modes I, II, III) at crack tip in axially cracked cylindrical shells is given. The analysis, using 10th-order differential equations which take the transverse shear deformations into account, involves perturbation in a curvature parameter λ², (λ²=[12(1-v ²)]^1/2 a ²/Rh). Stress intensity factors for finite size cylindrical shells under bending and internal pressure loading are evaluated. A good accuracy can be obtained without using fine meshes in a region near the crack tip. Besides, the influence of the finite size and the shearing stiffness on bulging factors, which are commonly used in engineering, are analyzed.     

Deformation and reverse snapping of a circular shallow shell under uniform edge tension

In this paper we study the deformation and stability of a shallow shell under uniform edge tension, both theoretically and experimentally. Von Karman’s plate model is adopted to formulate the equations of motion. For a shell with axisymmetrical initial shape, the equilibrium positions can be classified into axisymmetrical and unsymmetrical solutions. While there may exist both stable and unstable axisymmetrical solutions, all the unsymmetrical solutions are unstable. Since the unsymmetrical solutions will not affect the stability of the axisymmetrical solutions, it is concluded that for quasi-static analysis, there is no need to include unsymmetrical assumed modes in the calculation. If the shell is initially in the unstrained configuration, it will only be flattened smoothly when the edge tension is applied. No snap-through buckling is possible in this case. On the other hand, if the shell is initially in the strained position, it will be snapped back to the stable position on the other side of the base plane when the edge tension reaches a critical value. Experiment is conducted on several free brass shells of different initial heights to verify the theoretical predictions. Generally speaking, for the range of initial height H < 10 the experimental measurements of the deformation and the reverse snapping load agree well with theoretical predictions.     

Stability of Axially Compressed Noncircular Cylindrical Shells Consisting of Panels of Constant Curvature

The stability problem is solved for an axially compressed cylindrical shell. Its cross section is formed by circular arcs of radius r with ends supported on a closed circle of radius R. The solution is based on the Flügge equations of the classic theory of deep cylindrical shells. It is shown that the critical axial load for shells of medium length and appropriately chosen cross-sectional profile can be increased by a factor of R/r approximately, compared with the circular shell. The shells length affects considerably the efficiency of noncircular shells of this type. This design model allows us to find out how the local properties of the shell and its stiffness are related     

Mathematical Justification of the Obstacle Problem in the Case of a Shallow Shell

This paper deals with the asymptotic formulation and justification of a mechanical model for a shallow shell in frictionless unilateral contact with an obstacle. The first three parts of the paper concern the formulation of the equilibrium problem. Special attention is paid to the contact conditions, which are usual within two or three dimensional elasticity, but which are not so usual in shell theories. Lastly the limit problem is formulated in the main part of the paper and a convergence result is presented. Two points are worth stressing here. First, we point out that unlike classical bilateral shell models justifications, the functional framework of the present analysis involves cones. Secondly, while the cones result from a positivity condition on the boundary as long as the thickness parameter is finite, leading to a Signorini problem in the Sobolev space H ¹, the cone results from a positivity condition in the domain, giving rise to a so-called obstacle problem in the Sobolev space H ² at the limit.      

Second-Order Shell Kinematics Implied by Rotation Constraint-Equation

The paper presents a general methodology of introducing the shell-type variables which is based on the rotation constraint-equation (RC-equation). The RC-equation is proven to be equivalent to the polar decomposition of the deformation gradient formula, and the rotations which it yields are interpreted in terms of rotations of vectors of an ortho-normal basis. The deformation function and rotations are assumed as polynomials of the thickness coordinate ζ, and in this form used in the RC-equation. Solving this equation, we can express the coefficients of the quadratic deformation function in terms of the following shell-type variables: (a) the mid-surface position x ₀, (b) the constant rotation Q ₀, (c) the rotation vector ψ ^* for the ζ-dependent rotations, and (d) the normal components U ₃₃ ⁰ and U ₃₃ ¹ of the right stretching tensor. This new methodology (i) ensures that all shell kinematical variables are consistent with the RC-equation, which is justified on 3D grounds, (ii) provides a general framework from which various Reissner-type hypotheses can be obtained by suitable assumptions. As an example, two generalized Reissner hypotheses are derived: one with two normal stretches, and the other with the in-plane twist and the bubble-like warping parameters. This revised version was published online in June 2006 with corrections to the Cover Date.     

On buckling of cantilever rectangular plates under symmetrical edge loading

The stability of cantilever rectangular plates under the symmetrical edge loading will be studied in this paper by lite varialional calculus. We are going to find out the minimum critical loading for cantilever rectangular plates subjected to various edge loadings symmetrically on a pair of opposite free edges. We’ll discuss the least critical loadings when the buckling of rectangular plates acted on bv a pair of concentrated forces, uniformly distributed loads, locally uniform distributed loads, distributed loads in the form of triangle and a pair of concentrated couples occur respectively.