Measurement of bending stresses in shells of arbitrary shape using the reflection moiré method

The basic equation for fringe formation in the case of reflection moiré applied to surfaces of arbitrary curvatures is derived. A practical point-by-point solution for the application of this method is introduced, and the corresponding simplified equations are given. The technique is applied to an industrial problem, the stress analysis of a shell-shaped door.     

A relook at Reissner’s theory of plates in bending

Shear deformation and higher order theories of plates in bending are (generally) based on plate element equilibrium equations derived either through variational principles or other methods. They involve coupling of flexure with torsion (torsion-type) problem and if applied vertical load is along one face of the plate, coupling even with extension problem. These coupled problems with reference to vertical deflection of plate in flexure result in artificial deflection due to torsion and increased deflection of faces of the plate due to extension. Coupling in the former case is eliminated earlier using an iterative method for analysis of thick plates in bending. The method is extended here for the analysis of associated stretching problem in flexure.     

On geometrically exact post-buckling of composite columns with interlayer slip—The partially composite elastica

This paper is focused on the geometrically exact elastic stability analysis of two interacting kinematically constrained, flexible columns. Possible applications are to partially composite or sandwich columns. A partially composite column composed of two inextensible elastically connected sub-columns is considered. Each sub-column is modeled by the Euler–Bernoulli beam theory and connected to each other via a linear constitutive law for the interlayer slip. The paper discusses the validity of parallel and translational kinematics beam assumptions with respect to the interlayer constraint. Buckling and post-buckling behavior of this structural system are studied for cantilever columns (clamped-free boundary conditions). A variational formulation is presented in order to derive relevant boundary conditions in a geometrically exact framework. The exact post-buckling behavior of this partially composite beam-column is investigated analytically and numerically. The Euler elastica problem is obtained in the case of non-composite action. The “partially composite elastica” is then treated analytically and numerically, for various values of the interaction connection parameter. An asymptotic expansion is performed to evaluate the symmetrical pitchfork bifurcation, and comparisons are made with some exact numerical results based on the numerical treatment of the non-linear boundary value problem. A boundary layer phenomenon, similar to that also observed for the linearized bending analysis of partially composite beams, is observed for large values of the connection parameter. This boundary layer phenomenon is investigated with a straightforward asymptotic expansion, that also is valid for large rotations. Finally, the paper analyses the effect of some additional imperfection eccentricities in the loading mode, that lead to some pre-bending phenomena.     

Bifurcations and exact solutions of an asymptotic rotation-Camassa–Holm equation

This paper investigates the rotation-Camassa–Holm equation, which appears in long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the earth’s rotation. The rotation-Camassa–Holm equation contains the famous Camassa–Holm equation and is a special case of the generalized Camassa–Holm equation. By using the approach of dynamical systems and singular traveling wave theory to its traveling wave system, in different parameter conditions of the five-parameter space, the bifurcations of phase portraits are studied. Some exact explicit parametric representations of the smooth solitary wave solutions, periodic wave solutions, peakons and anti-peakons, periodic peakons as well as compacton solutions are obtained.

     

Buckling of geometrically imperfect cylindrical shells — definition of a buckling load

On the basis of extensive buckling tests and analytical and numerical buckling analyses for composite cylinders it became desirable to provide a recommendation for the most reliable evaluation of stability limits for imperfect CFRP cylinders subjected to axial compression. This paper reports on different approaches including linear, non-linear and dynamic non-linear FE analysis results and discusses the related effects and potential difficulties.     

Asymptotic homogenization modeling of smart composite generally orthotropic grid-reinforced shells: Part I – theory

We develop in this paper a comprehensive micromechanical model for the analysis of thin smart composite grid-reinforced shells with an embedded periodic grid of generally orthotropic cylindrical reinforcements that may also exhibit piezoelectric properties. The original boundary value problem which characterizes the thermopiezoelastic behavior of the smart shell is decoupled via the asymptotic homogenization technique into three simpler problems the solution of which permits the determination of the effective elastic, piezoelectric and thermal expansion coefficients. The general orthotropy of the constituent materials is very important from the practical viewpoint and it renders the resulting analysis a lot more complicated. In Part II of this work the model is applied to the analysis of several practically important examples including cylindrical reinforced smart composite shells and multi-layer smart shells.     

Transition from progressive buckling to global bending of circular shells under axial impact––Part II: Theoretical analysis

This is the second paper of a two-part paper investigating the complex phenomenon of the dynamic transition from progressive buckling to global bending collapse of a long circular cylindrical shell subjected to an axial impact. The paper focuses on the theoretical analysis of the phenomenon. The two-phase concept for the deformation of ‘Type II’ structures is employed to explore the influence of the loading parameters and the material and geometrical characteristics of a shell on the critical length that marks the transition between the two collapse modes. Simple models for the initial compressive phase in the case of global bending and for the development of a single axisymmetric wrinkle in a circular shell, are used to analyse some numerical results presented in Part I of this study [International Journal of Solids and Structures 41 (2004) 1565].     

An asymmetric theory of nonlocal elasticity—Part 1. Quasicontinuum theory

In this article, an asymmetric theory of nonlocal elasticity is developed on the basis of three dimensional atomic lattice model, the Galileo invariance for constitutive equations and by use of Fourier transformation of generalized function and energy method. It is shown that nonlocal characteristic functions (or constitutive parameters of internal elastic energy) can be explicitly expressed in terms of interacting forces connecting atoms, and the general model of nonlocal theory with rotation effects is asymmetric. Both symmetric stress and anti-symmetric stress is a nonlocal function of strain and local rotation for anisotropic materials. For isotropic materials, symmetric stress is only a nonlocal function of strain, while antisymmetric stress is only a nonlocal function of local rotation.     

Localized bulging in an inflated cylindrical tube of arbitrary thickness – the effect of bending stiffness

We study localized bulging of a cylindrical hyperelastic tube of arbitrary thickness when it is subjected to the combined action of inflation and axial extension. It is shown that with the internal pressure P and resultant axial force F viewed as functions of the azimuthal stretch on the inner surface and the axial stretch, the bifurcation condition for the initiation of a localized bulge is that the Jacobian of the vector function $(P,F)$ should vanish. This is established using the dynamical systems theory by first computing the eigenvalues of a certain eigenvalue problem governing incremental deformations, and then deriving the bifurcation condition explicitly. The bifurcation condition is valid for all loading conditions, and in the special case of fixed resultant axial force it gives the expected result that the initiation pressure for localized bulging is precisely the maximum pressure in uniform inflation. It is shown that even if localized bulging cannot take place when the axial force is fixed, it is still possible if the axial stretch is fixed instead. The explicit bifurcation condition also provides a means to quantify precisely the effect of bending stiffness on the initiation pressure. It is shown that the (approximate) membrane theory gives good predictions for the initiation pressure, with a relative error less than 5%, for thickness/radius ratios up to 0.67. A two-term asymptotic bifurcation condition for localized bulging that incorporates the effect of bending stiffness is proposed, and is shown to be capable of giving extremely accurate predictions for the initiation pressure for thickness/radius ratios up to as large as 1.2.     

Stochastic response of an axially loaded composite Timoshenko beam exhibiting bending–torsion coupling

A spectral finite element method is proposed to investigate the stochastic response of an axially loaded composite Timoshenko beam with solid or thin-walled closed section exhibiting bending–torsion materially coupling under the stochastic excitations with stationary and ergodic properties. The effects of axial force, shear deformation (SD) and rotary inertia (RI) as well as bending–torsion coupling are considered in the present study. First, the damped general governing differential equations of motion of an axially loaded composite Timoshenko beam are derived. Then, the spectral finite element formulation is developed in the frequency domain using the dynamic shape functions based on the exact solutions of the governing equations in undamped free vibration, which is used to compute the mean square displacement response of axially loaded composite Timoshenko beams. Finally, the proposed method is illustrated by its application to a specific example to investigate the effects of bending–torsion coupling, axial force, SD and RI on the stochastic response of the composite beam.     

The closed-form motion equation of redundant actuation parallel robot with joint friction: an application of the Udwadia–Kalaba approach

Nonlinear Dynamics - The closed-form motion equation of the redundant actuation parallel robot with joint friction is established by an extended application of the Udwadia–Kalaba modeling....     

Transition from progressive buckling to global bending of circular shells under axial impact––Part I: Experimental and numerical observations

The influence of impact velocity and material characteristics on the dynamic buckling response of circular shells subjected to axial impact loads is studied. It is shown experimentally that the critical buckling length, which marks the transition between progressive and global buckling of aluminium alloy circular tubes, is significantly influenced by the axial impact velocity. A finite element analysis is undertaken to further explore the effects of material yield stress, strain hardening and strain rate sensitivity on the transition phenomenon. It is observed that circular tubes made of ductile alloys with a high yield stress and low strain hardening characteristics have a better performance as energy absorbers than tubes made of alloys with a low yield stress and high strain hardening characteristics. Theoretical analysis of some particular features of the dynamic buckling transition is presented in Part II [International Journal of Solids and Structures (2004)].     

The exact solution of Stokes＇ second problem including start-up process with fractional element

The start-up process of Stokes＇ second problem of a viscoelastic material with fractional element is studied. The fluid above an infinite flat plane is set in motion by a sudden acceleration of the plate to steady oscillation. Exact solutions are obtained by using Laplace transform and Fourier transform. It is found that the relationship between the first peak value and the one of equal-amplitude oscillations depends on the distance from the plate. The amplitude decreases for increasing frequency and increasing distance.     

The compressible Coanda wall jet—an experimental study of jet structure and breakaway

An underexpanded jet issuing from a convergent slot and blowing over a surface of convex streamwise curvature was studied experimentally. The jet was confined between side walls, with the slot aspect ratio varying between 40 and 6, but tests showed that in the area of interest close to the slot the flow was effectively two-dimensional. The ratio of slot width to the radius of curvature of the downstream surface varied between 0.05 and 0.33. The main techniques used were Schlieren and shadowgraph to show the jet structure, and surface flow visualization which revealed areas of separation and reattachment. Surface static pressures were also measured on the curved surface. The curved jet proved to have a shock cell structure similar to that of a plane jet. However, the cell structure disappeared more rapidly as the outer shear layer grew more quickly due to the destabilizing effect of the curvature on the turbulence in the shear layer. Even at modest upstream jet pressures, a separated region on the Coanda surface became evident. This region was characterized by a stagnant constant pressure part followed by a region of strongly reversed flow before reattachment took place. The separation was caused by the compression at the end of the first shock cell, with reattachment taking place where expansion in the second cell started. The separated region grew rapidly as the upstream pressure was increased, until, finally, reattachment failed to occur and the jet suddenly broke away from the surface. This work is related to studies of the Coanda flare, where the jet is axisymmetric. The high level of turbulence causes rapid entrainment of air and so gives us clean combustion. However there should be more general application to devices that use the Coanda effect, varying from fluidic devices to blown jet flaps on wings.     

The plane elasticity problem of an isotropic wedge under normal and shear distributed loading––application in the case of a multi-material problem

The problem of a multi-material composite wedge under a normal and shear loading at its external faces is considered with a variable separable solution. The stress and displacement fields are determined using the equilibrium conditions for forces and moments and the appropriate Airy stress function. The infinite isotropic wedge under shear and normal distributed loading along its external faces is examined for different values of the order n of the radial coordinate r. The proposed solution is applied to the elastostatic problem of a composite isotropic k-materials infinite wedge under distributed loading along its external faces. Applications are made in the case of the two-materials composite wedge under linearly distributed loading along its external faces and in the case of a three-materials composite wedge under a parabolically distributed loading along its external faces.     

Anisotropic hardening – numerical application of a cubic yield theory and consideration of variable r-values for sheet metal

The first part of this paper contains a polynomial yield condition of third order connected with evolution equations for material tensors of higher orders. They are formulated by formal generalisation of an approach by Danilov. The second part presents a possibility of taking into account the rotation of the yield surface as a result of a variable planar anisotropy (r-value) in sheet metal. This is done by an extension of the evolution equations, based on a quadratic yield function. The corresponding deformation law and the set of evolution equations are numerically integrated for selected loading paths in the subspaces σ₁,σ₂ and σ,τ. Some of the experimentally observed effects, such as the increasing curvature of the yield locus curve in the loading direction or the specific rotation of the yield surface, are correctly reproduced.     

Multiple time scales of fluvial processes — theory and applications

Fluvial processes comprise water flow, sediment transport and bed evolution, which normally feature distinct time scales. The time scales of sediment transport and bed deformation relative to the flow essentially measure how fast sediment transport adapts to capacity region in line with local flow scenario and the bed deforms in comparison with the flow, which literally dictates if a capacity based and/or decoupled model is justified. This paper synthesizes the recently developed multiscale theory for sediment-laden flows over erodible bed, with bed load and suspended load transport, respectively. It is unravelled that bed load transport can adapt to capacity sufficiently rapidly even under highly unsteady flows and thus a capacity model is mostly applicable, whereas a non-capacity model is critical for suspended sediment because of the lower rate of adaptation to capacity. Physically coupled modelling is critical for fluvial processes characterized by rapid bed variation. Applications are outlined on very active bed load sediment transported by flash floods and landslide dam break floods.