Edge-compression fixture for buckling studies of corrugated board panels

A test fixture, developed for evaluating the preand postbuckling response of simply supported, nearly flat, rectangular corrugated board panels subjected to edge compression is evaluated. The test fixture enables loading of panels into the postbuckling regime until collapse. The shadowmoiré method verified that buckling in the first mode occurred, and that there was symmetry of the adge-boundary conditions. Through an iterative regression model, experimental curves of load versus out-of-plane displacement for isotropic panels were fitted to an equation governing the nonlinear postbuckling response. This method provides the critical buckling load, a postbuckling parameter and the amplitude of initial imperfection of the panel. Comparison with analytical results revealed that simply supported boundary conditions were closely achieved. Examination of compressively loaded corrugated board panels showed that collapse occurred due to compressive failures of the facings in the highly stressed edge regions without severe influence from stress concentrations at load introduction and edge supports.     

Compressive strength of edge-loaded corrugated board panels

Postbuckling strength of simply supported corrugated board panels subjected to edge compressive loading has been studied experimentally using a specially developed test fixture. Although the load versus out-of-plane displacement response was highly sensitive to the presence of initial imperfections in the panels, the collapse loads did not vary much, which is attributed to the stable postbuckling behavior of the plates. Thin plates collapsed at nearly twice the buckling load, while thick panels collapsed at loads below the elastic critical buckling load. Local buckling of the facing on the concave side of the buckled plate was observed at load levels close to the collapse load. The plate collapse was triggered by compressive failure of the facings that initiated at the unloaded edges. A simplified design analysis was derived based on approximate postbuckling analysis and compared with an existing design formula for corrugated board panels and boxes.     

Large strain elasto-plastic model of paper and corrugated board

An anisotropic elasto-plastic constitutive model of paper material is presented. It is formulated in a spatial setting in which anisotropic properties are accounted for by use of structural variables. A multiplicative split of the deformation gradient is employed to introduce plasticity. A similar approach is used to model the plastic deformation of the substructure. The yield surface adopted is based on the Tsai–Wu failure criterion, used previously to model failure of paper material. A non-associated plasticity theory is employed to calibrate the model to experimental data. It turns out that a multi-axial loading situation is needed to calibrate the model and here a biaxial tension test is audited. The model was implemented into a finite element environment and the creasing process of a corrugated board panel is investigated.     

Discrete buckling model for corrugated beam

A methodology for the analysis of the mechanical behaviour of beams is presented, which relies on the lemma of stroboscopy, a tool from Non-Standard Analysis. This problem traces back to the conventional Euler's column load. Starting from a discrete rigid rod-torsional spring model of the beam, stroboscopy enables us to build a continuous model for the variation of the angular position along the beam. This position is characterised as a solution of a differential equation. A perturbation analysis of this equation, leads to buckling condition, in particular a sufficient buckling conditions for corrugated beams.     

Creep and thermal cycling fixture design for metal matrix composites

A creep-thermal cycling system with a constant stress cam has been designed to be inexpensive and easily fabricated in order to identify the creep and thermal-cycling damage mechanisms in metal-matrix composites (MMCs) with high ductility. The current system can be modified to be a stress thermal cycling (or fatigue) system.     

Calculations for circular arc type corrugated membrane

Using Chien Wei-zang’s general Solutions of axial symmetrical ring shells, we calculate the stress and displacement of circular are type corrugated membrane under axial loads in this paper.According to the results of calculation, we plot the usable graphs for engineering.Scientists and lechnicians can keep these graphs for easy reference.     

Mechanics of corrugated surfaces

Experimental studies of the surface stress of solids typically work with surfaces that are not perfectly planar. The experiment then probes an effectively averaged surface stress. The evolution of the surface morphology, for instance during film growth or reconstruction, is also affected by the surface stress acting on a corrugated surface. Here, we analyze the mechanics of rough surfaces in a continuum framework. In a generalization of the approach of Weissmüller and Duan [2008. Phys. Rev. Lett. 101, 146102] to solids with anisotropic elasticity, anisotropic surface stress and anisotropic roughness, we focus on the effectively averaged surface stress that determines the mean compensating stress in the bulk. Important concepts are the projection of out-of-plane stresses at inclined segments of a surface into the macroscopic surface plane, and the transverse coupling between the out-of-plane and in-plane components of the surface-induced stress in the bulk. We show that the coupling of the surface stress at a corrugated surface into a planar substrate depends on the geometry of the corrugation exclusively through the surface orientation distribution function. Special geometries are inspected with an eye on illustrating the impact of anisotropic elasticity as well as geometric anisotropy, which both feed into the anisotropy of the effective surface stress.     

Stress-strain solutions for circumferentially corrugated elliptic cylindrical shells

The paper presents an approach to solve the boundary-value stress-strain problem for circumferentially corrugated elliptic cylindrical shells. The approach employs splines to approximate the solution and the stable discrete-orthogonalization method to solve the resulting one-dimensional problem. The results are presented as plots and a table __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 70–78, September 2006.     

Viscous flow between two rotating nonconcentric cylinders for small eccentricity

The flow of a viscous and incompressible fluid between two rotating nonconcentric cylinders is investigated. An approximate solution of the Navier-Stokes equations is obtained by a perturbation method for the case of small eccentricity. A second solution of the basic flow is obtained by imposing the additional geometric restriction of small gap between the two cylinders and employing the asymptotic expansion of Bessel functions by Meissel's series. This second solution is also obtained by formulating a small gap boundary value problem. The transverse velocity profiles are presented for the case when the eccentricity and gap are small and the outer cylinder is stationary.     

Effective characteristics of corrugated plates

Corrugated plates are widely used in modern constructions and structures, because they, in contrast to plane plates, possess greater rigidity. In many cases, such a plate can be modeled by a homogeneous anisotropic plate with certain effective flexural and tensional rigidities. Depending on the geometry of corrugations and their location, the equivalent homogeneous plate can also have rigidities of mutual influence. These rigidities allow one to take into account the influence of bending moments on the strain in the midplane and, conversely, the influence of longitudinal strains on the plate bending [1]. The behavior of the corrugated plate under the action of a load normal to the midsurface is described by equations of the theory of flexible plates with initial deflection. These equations form a coupled system of nonlinear partial differential equations with variable coefficients [2]. The dependence of the coefficients on the coordinates is determined by the corrugation geometry. In the case of a plate with periodic corrugation, the coefficients significantly vary within one typical element and depend on the values of local variables determined in each of the typical elements. There is a connection between the local and global variables, and therefore, the functions of local coordinates are simultaneously functions of global coordinates, which are sometimes called rapidly oscillating functions [3].One of the methods for solving the equations with rapidly oscillating coefficients is the asymptotic method of small geometric parameter. The standard procedure of this method usually includes preparatory stages. At the first stage, as a rule, a rectangular periodicity cell is distinguished to be a typical element. At the second stage, the scale of global coordinates is changed so that the rectangular structure periodicity cells became square cells of size l × l. The third stage consists in passing to the dimensionless global coordinates relative to the plate characteristic dimension L. As a result, the dependence between the new local variables and the new scaled dimensionless variables is such that the factor 1/α, where α=l/L ? 1 is a small geometric parameter, appears in differentiating any function of the local coordinate with respect to the global coordinate. After this, the solution of the problem in new coordinates is sought as an asymptotic expansion in the small geometric parameter [1], [4–10].We note that, in the small geometric parameter method, the asymptotic series simultaneously have the form of expansions in the gradients of functions depending only on the global coordinates. This averaging procedure can be applied to linear and nonlinear boundary value problems for differential equations with variable periodic coefficients for which the periodicity cell can be affinely transformed into the periodicity cube. In the case of an arbitrary dependence of the coefficients on the coordinates (including periodic dependence), another averaging technique can be used in linear problems. This technique is based on the possibility of the integral representation of the solution of the original problem for the linear equation with variables coefficients in terms of the solution of the same problem for an equation of the same type but with constant coefficients [11–13]. The integral representation implies that the solution of the original problem can be represented in the form of the series in the gradients of the solution of the problem for the equation with constant coefficients [13].The aim of the present paper is to develop methods for calculating effective characteristics of corrugated plates. To this end, we first write out the equilibrium equations for a flexible anisotropic plate, which is inhomogeneous in the thickness direction and in the horizontal projection, with an initial deflection. We write these equations in matrix form, which allows one to significantly reduce the length of the expressions and to simplify further calculations. After this, we average the initial matrix equations with variable coefficients. The averaging procedure implies the statement of problems such that, after solving them, we can calculate the desired effective characteristics. By way of example, we consider the case of a corrugated plate made of a homogeneous isotropic material whose corrugations are hexagonal in the horizontal projection. In this case, we obtain approximate expressions for the components of the effective tensors of flexural rigidity and longitudinal compliance and expressions for the effective plate thickness.     

The rigidity of corrugated diaphragms

Summary When the corrugated diaphragm is replaced by a fictitious flat plate of similar properties it is possible to derive a linear differential equation for the deflection. The coefficients of this equation, however, vary in a complicated way and its solution for the pressure-loaded diaphragm is only given for thick and for thin sheets separately. For thick sheets the profile of the corrugation appears to be inessential, whereas for thin sheets it is necessary to distinguish between trapezoidal, triangular and arc-shaped corrugations. By an obvious device the results for thick and for thin sheets are fitted together, so that the deflection can also be determined for the intermediate range of medium sheet thickness. The final results of the present calculation are compared with measurements carried out by others and are found to be in satisfactory agreement with the experiments.It is to be remarked that, compared on the basis of small deflections, the introduction of corrugations into the sheet leads to a considerable increase of rigidity of the diaphragm. The prevailing assertion that the flat plate is more rigid than the corrugated diaphragm holds only for large deflections, because of the non-linearity between the load and the deflection of the flat plate.     

Morphing of curved corrugated shells

Thin sheet materials of low bending stiffness but high membrane stiffness are often corrugated in order to achieve improvements of several orders of magnitude in bending stiffness with only minimal increases in weight and cost. If these corrugated sheets are initially curved along the corrugations, much of this stiffness gain is lost. In return, the sheets are then capable of significant elastic changes in shape overall, including large changes in overall Gaussian curvature. These shape changes are described here by non-linear and coupled kinematical relationships, which are verified against experiment and finite-element simulations. It is found that gross simplifications can be made about the large displacement behaviour of such shells without a loss of accuracy.     

Equivalent models of corrugated panels

The design of corrugated panels has wide application in engineering. For example corrugated panels are often used in roof structures in civil engineering. More recently corrugated laminates have been suggested as a good solution for morphing aircraft skins due to their extremely anisotropic behaviour. The optimal design of these structures requires simple models of the panels or skins that may be incorporated into multi-disciplinary system models. Thus equivalent material models are required that retain the dependence on the geometric parameters of the corrugated skins or panels. An homogenisation-based analytical model, which could be used for any corrugation shape, is suggested in this paper. This method is based on a simplified geometry for a unit-cell and the stiffness properties of original sheet. This paper outlines such a modelling strategy, gives explicit expressions to calculate the equivalent material properties, and demonstrates the performance of the approach using two popular corrugation shapes.     

Design and validation of a new fixture for the shear testing of cellular solids

The design and validation of a new fixture for the shear testing of cellular solids are presented. The fixture is an extended version of a picture-frame shear fixture (EPF) and is suited for comparatively thick rectangular block specimens. The stress state in the specimen is analyzed using a detailed finite element model. The finite element model is based on a 3D CAD model and incorporates friction in the revolute joints. Using specimens with low stiffness, a nearly pure and uniform shear stress state is induced in the specimen. A correction factor for the shear stress is derived which takes into account the friction in the joints and the nonuniformity of the shear stress distribution in the gauge section. The shear response of the polymer foam Rohacell^® 200 WF is determined in order to demonstrate the capabilities of the EPF. The strain state is analyzed by means of digital image correlation and is detected to be very pure and uniform on the specimen’s surface, as predicted by the numerical analysis. The shear modulus obtained with the EPF is in good agreement with the calculated shear modulus derived from tensile tests on the same material. In addition, there is only little scatter of the strength values over the tested specimens which further confirms the accuracy of the new fixture.     

CONSTITUTIVE RELATION AND STIFFNESS DEGRADATION OF CRACKED ANISOTROPIC COMPOSITE LAMINATES(II)─DETERMINATION OF RESOLVED STIFFNESS AND INVESTIGATION OF STIFFNESS DEGRADATION

The study on properly degradation of damaged,.onlj,osile laminates is extendedto anisotropic laminates with matrix cracking. In (II) of the paper, a solution forpartitioned stiffness is given to complete the constitutive relations developed in (I) Thestiffness degradation in (θ_m/90_n)_s cracked laminates is calculated and the results arediscussed.     

Nonlinear vibration of circular corrugated plates

In this paper, first by using Hamilton principle, we derive the variational equation of circular corrugated plates. Taking the central maximum amplitude of circular corrugated plates as the perturbation parameter and adopting the perturbation variational method, in the first-order approximation, we obtain the natural frequency of linear vibration of circular corrugated plates and then the nonlinear natural frequency of the corrugated plates. By comparing with the linear results, the attempt of this paper is proved feasible.This paper was presented in the 2nd National Conference on Vibration (Xi'An 1984)     

Bending of corrugated thick rectangular plates

An approximate analytical solution describing the bending of hinged corrugated thick plates is obtained using the second variant of the boundary shape perturbation method and taking into account the first three approximations. The effect of the shape of the boundary surfaces in the zone of maximum external load on the magnitude and nonlinear variation of the displacements and stresses throughout the thickness of a corrugated thick plate depending on the corrugation amplitude and spatial frequency is analyzed. The results are compared to the exact solution for a flat plate