Anticlastic Deformation Research Articles

Parametric estimation of Poisson's ratio for thin hinged-hinged plates

Cornu's method is an elegant calculation besieged by an impractical approach to obtain accurate estimates for Poisson's ratio. Conventionally, Cornu's method requires several components, and each component can adversely affect the accuracy of the measurement. Furthermore, Cornu's conventional method requires a long beam because beams with short length-to-width ratios cause the estimate of Poisson's ratio to diverge from the true value of Poisson's ratio. We believe that, with the right modifications, Cornu's method can become an attractive approach to obtaining precise estimates for Poisson's ratio from mode shapes. We use finite element simulations to show how to use Cornu's method to estimate Poisson's ratio from a mode shape. Our modified Cornu's method removes knife-edges and loading components for a hinged-hinged plate under steady-state excitation. Given the true value of Poisson's ratio, as a performance specification, we show that simple parametric expressions can fit estimates for Poisson's ratio for different length-to-width ratios of thin hinged-hinged plates. Additionally, we show that estimates for Poisson's ratio from higher modes align with the results from the first mode and explain our expectation for this outcome. Furthermore, our results challenge the idea that anticlastic, monoclastic, and synclastic deformation uniquely correspond to positive, zero, and negative estimates of Poisson's ratio, respectively. With the emergence of materials-by-design, we expect that this parametric technique will be able to assist in experimental qualification of thin beam and plate structures with respect to the desired value of Poisson's ratio.

Competing anticlastic and piezoelectric deformation at large deflections

In bending of a purely elastic beam or plate, it is well established that the cross-sectional shape changes character with decreasing bending radius-of-curvature and that the transition can be characterized by the Searle parameter. In a piezoelectric structure, the cross-sectional deformation is affected by the opposite anticlastic and electromechanical bending curvatures. The behavior is consequently more complicated and it is an open question how the cross-sectional shape develops with increasing bending. In this paper, analytical solutions are used to study the cross-sectional deformation of piezoelectric cantilever-actuators taking both anticlastic and electromechanical bending effects into account. We consider unimorph and bimorph actuators. In the case of electrical actuation, as for the purely mechanical case, we find that the Searle parameter is an important parameter characterizing the shape of the cross-section. A load scaling rule gives a criterion for fixed cross-section-deflection for different actuator widths. Using this scaling rule, the Searle parameter is kept unchanged. The analytical results are verified by non-linear finite element analysis using electric potential and mechanical moment as applied loads.

Analysis of geometric imperfections of spirally welded slender steel tubes

Abstract The objective of this paper is to analyze and characterize high-resolution measurements of geometric imperfections taken from a set of seven slender tapered steel tubes in order to provide insights that can improve methods for predicting buckling behavior. The seven tubes are each ~3400 mm long with diameters between 800 mm and 1100 mm, diameter to thickness ratios between 300 and 350, and taper angles between 0.67° and 0.86°. The tubes are manufactured from steel plates using an innovative spiral welding process. The geometric imperfections of these tubes are characterized with harmonic analysis of the overall imperfection measurements and with regression analysis of the measured shapes of weld depressions. The results show a consistent imperfection signature caused by the manufacturing process including distinct features attributed to both the rolling and welding processes, i.e. anticlastic deformations and weld depressions. Variability in the imperfection measurements is also analyzed and used to generate a probabilistic scheme capable of generating random fields of geometric imperfections that are consistent with the measurements considered here.

Influence of curved delamination front on R-curve of DCB specimen

This work investigated the evolution of the non-uniform distribution of Mode-I strain energy release rate (SERR) along a straight pre-crack front of double cantilever beams (DCB), caused by the anticlastic deformation and the longitudinal free-edge effects. A modified DCB specimen with a curved pre-crack front was introduced in order to alleviate this problem and produce an uniform SERR distribution at the initiation and propagation stages. The first order optimization method was used to determine the appropriate shape of the curved pre-crack front in order to guarantee an uniform SERR during the fracture process. The introduced DCB specimen configuration with the curved pre-crack front was shown to be more convenient for the estimation of a consistent SERR value throughout the fracture procedure without been affected by the visual deviation of the crack length and the non-uniformity ratio of Mode-I SEE distribution along the crack front.

Bending device and anticlastic surface measurement of solids under large deformations and displacements

Large bending of elastic bodies gives rise to significant transverse effects. Based on a recent theoretical model in the context of finite elasticity, both the longitudinal and anticlastic curvatures in bent solids under large deformation and displacement can be accurately assessed. In order to experimentally investigate the anticlastic deformation induced by large inflexion and corroborate the theoretical predictions, a properly designed mechanical bending device is here proposed. By imposing a rotation at the ends of the sample, both the longitudinal and anticlastic curvatures are measured by DIC (digital image correlation) monitoring instrumentation and compared with the theoretical results, finding good agreement. Compact analytical formulae for assessing the radii of curvature within the thickness of the sample are provided. Conversely to existing studies of the anticlastic surface induced by infinitesimal bending, the present analysis takes into account large through-to-thickness curvature variations, whose knowledg can plays a key role for a wide class of mechanical applications.

Interlaminar Stresses in Composite Laminates Subjected to Twisting Deformation

The interlaminar stress in angle-ply and cross-ply composite laminates subjected to twisting deformation are investigated. Two mechanisms of interlaminar load transfer have been developed by studying the angle-ply laminate and the cross-ply laminate subjected to uniform axial extension, thermoelastic deformation and anticlastic bending deformation. In the present, these mechanisms are investigated in laminates subjected to twisting deformation. It is shown that the mechanisms of interlaminar load transfer in twisting deformation are identical to those previously investigated, though they arise from different causes. Furthermore, a unified treatment of the mechanisms of interlaminar load transfer is presented for the angle-ply laminate and the cross-ply laminate subjected to the four aforementioned modes of deformation.

Analysis of the metal adhesively bonded double cantilever beam specimen

The double cantilever beam specimen is currently standardized for measuring the mode I fracture energy of adhesive joints. In addition, it has been increasingly employed to evaluate the adhesive traction-separation law by the direct method, which involves crack tip separation measurements. The three-dimensional finite element analyses here conducted showed that significant anticlastic deformations of the metal adherends compromise the accuracy of the direct method in the elastic domain. It was also seen that the adherend plane stress and adhesive uni-axial strain hypotheses are adequate for the typical specimen geometries. Finally, the new elastic crack length correction derived from a beam model can be used to predict accurately the initial specimen compliance, to obtain conservative fracture energy values and to gain additional insight into the adhesive fracture behaviour.

A Sagittally Bent Crystal for the Short Pulse X-ray Beamline at the Advanced Photon Source

The Short Pulse X-ray Imaging and Microscopy (SPXIM) Beamline at the Advanced Photon Source (APS) is being designed to provide a short-pulse focused beam at the sample some 60 m from the source. The second crystal in the horizontally diffracting double-crystal monochromator system is sagittally bent to focus the full vertical fan of the beam, preserving the flux of a typical undulator beamline at the APS, while providing a time-angle correlated beam on the sample with a few-picosecond pulse duration. An energy scan using this system requires changing the bent radius of the second crystal. Undesirable distortions in the sagittally bent second crystal are due to anticlastic deformations. In this paper, the analysis and design of the bent crystal is presented including an examination of the mechanical interaction between the crystal and the two pairs of rollers used to bend it. Ray tracings including the anticlastic deformations obtained from an optimized finite element model show a minimal effect on the spot size at the sample position.

Anisotropy and the optimal aspect ratio of a bent crystal

A thin bent rectangular crystal plate can be used as the second crystal in a double-crystal monochromator to focus sagittally a wide synchrotron X-ray beam. When bent in the sagittal direction, the plate also bends in the tangential direction (so-called anticlastic bending), which can adversely affect the diffracted beam. It is known that the undesirable anticlastic deformation can be minimized if the plate aspect ratio (length to width) is optimized. The present study investigates the influence of crystal material anisotropy and shows that ignoring it in the design of a sagittally bent crystal can lead to a significant underestimation of anticlastic deformation. For example, analysis shows that the optimal aspect ratio for a simply supported bent Si(220) plate ranges from 2.13 to 2.82 depending on the in-plane orientation of the cut rectangular plate. Treating silicon as isotropic with material properties in the literature leads to a plate design with an optimal aspect ratio of 2.36 and a plate with significant anticlastic bending.

Study of anticlastic deformation in a silicon crystal for channeling experiments

Anticlastic deformation (AD) is an established mechanical property of isotropic solid bodies, which has been recently used to steer particle beams through channeling in Si crystals. An analysis of AD in an anisotropic material, such as Si, has been worked out with particular emphasis to the cases used in channeling experiments. Both a theoretical model and finite element simulations were developed and compared to experimental data achieved by optical profilometry on bent Si crystals. A quantifier of the extent of AD, namely the ratio between primary and secondary curvature radii, has been found to be orientation dependent and determined analytically. The realistic case of crystal bending by a mechanical holder has been studied for applications. We addressed the case of the holder with a crystal at the energy currently under operation in the CERN super proton synchrotron and its possible extension to higher energy cases such as for the large hadron collider. Anisotropy-driven torsion of the crystal was investigated, determining a limitation to the portion of the crystal suitable for channeling. The geometry of the jaws clamping the crystal was considered too.

Determination of stress distribution in III–V single crystal layers for heterogeneous integration applications

AbstractDouble crystal X‐ray diffraction imaging and a variable temperature stage are employed to determine the stress distribution in heterogeneous wafer bonded layers though the superposition of images produced at different rocking curve angles. The stress distribution in InP layers transferred to a silicon substrate at room temperature exhibits an anticlastic deformation, with different regions of the wafer experiencing different signs of curvature. Measurements at elevated temperatures (≤125 °C) reveals that differences in thermal expansion coefficients dominate the stress and that interfacial particulates introduce very high local stress gradients that increase with increased temperature. For thinned GaAs substrates (100 μm) bonded using patterned metal interlayers to a separate GaAs substrate at ≈200 °C, residual stresses are produced at room temperature due to local stress points from metallization contacts and vias and the complex stress patterns can be observed using the diffraction imaging technique. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Thermoelastic relaxation in elastic structures, with applications to thin plates

A new result enables direct calculation of thermoelastic damping in vibrating elastic solids. The mechanism for energy loss is thermal diffusion caused by inhomogeneous deformation, flexure in thin plates. The general result is combined with the Kirchhoff assumption to obtain a new equation for the flexural vibration of thin plates incorporating thermoelastic loss as a damping term. The thermal relaxation loss is inhomogeneous and depends upon the local state of vibrating flexure, specifically, the principal curvatures at a given point on the plate. Thermal loss is zero at points where the principal curvatures are equal and opposite, that is, saddle shaped or pure anticlastic deformation. Conversely, loss is maximal at points where the curvatures are equal, that is, synclastic or spherical flexure. The influence of modal curvature on the thermoelastic damping is described through a modal participation factor. The effect of transverse thermal diffusion on plane wave propagation is also examined. It is shown that transverse diffusion effects are always small provided the plate thickness is far greater than the thermal phonon mean free path, a requirement for the validity of the classical theory of heat transport. These results generalize Zener's theory of thermoelastic loss in beams and are useful in predicting mode widths in micro- and nano-electromechanics systems oscillators.

A cryogenically cooled, sagittal focusing scanning monochromator for ESRF

A double crystal monochromator for fast scans over large energy ranges has been designed, constructed, and tested at ESRF. The instrument is based on a drive unit operated in air that rotates a large wheel sitting in UHV and supporting the two crystal assemblies. The link between the drive and the wheel is realized by a differentially pumped feedthrough. In order to withstand the full power of typical ESRF undulators without noticeable deformation, the first crystal is cooled by liquid nitrogen via pipes located in the hollow core of the differentially pumped feedthrough. The second crystal mount is based on an elastically deformable structure driven by piezoelectric and pneumatic actuators for precise angular adjustments. In its basic version the relative distance between the first and second crystal is kept constant (channel cut type of design), but a nearly fixed beam at the focal plane can be obtained, as the Bragg angle is changed, by the partial compensation of two different effects: the vertical displacement of the exit beam typical of channel cut designs and its divergence with respect to the incoming one due to the difference in lattice spacing between the first cooled crystal and the second one at room temperature. Horizontal focusing of the beam is achieved by sagittal bending of the second crystal. The major drawbacks of crystal focusing, the necessity of changing the radius of curvature while scanning, and the losses in throughput due to the anticlastic deformation experienced at high energy are circumvented by an appropriate choice of the crystal geometry and orientation.

Bending and Stretching an Elastic Strip Around a Narrow Cylindrical Drum

An analysis is made of the deflected forms assumed by an elastic strip when it is bent and stretched around a rigid drum. A summary is first given of the method of analysis used and the results obtained in a previous study where the drum width is equal to or greater than that of the strip. This work is then extended to the case where the strip width exceeds that of the drum. Deflected forms and contact regions are delineated for a strip-width/drum-width ratio of two and for various values of two parameters: anticlastic deformation and tension.

The Bending and Stretching of a Uniform Elastic Strip Around a Cylindrical Drum

The investigation covers a flat strip bent into a circular ring and a cylindrical drum expanding inside the ring. The resulting anticlastic deformations of the strip and the contact patterns are representative of a flat strip being bent around a rigid, frictionless, cylindrical drum and then put under tension. The analysis shows that three contact patterns are possible and their respective regions are delineated in terms of the geometric and tension parameters. Expressions for anticlastic deformations and some numerical solutions are also presented.

Optimization study of anticlastic curvature of cylindrically bent strips with variable thickness

This paper presents a nonlinear extension of a previous investigation. Tapering the edges on the concave side of a longitudinally bent strip according to a power-law is studied aiming to alleviate the anticlastic deformations of the strip. Conjugate gradient method is employed to calculate the specific dimensions of power-law taperings which will reduce the peak-to-trough deflection of the cross-section of a bent strip to a minimum. Optimum tapering width and thickness ratios and thickness variation parameter are presented in graphic and tabular form. The influence of Poisson's ratio on the tapering configuration is discussed. The results show that power-law tapering can reduce the peak-to-trough deflection to less than 1 per cent of the thickness of the strip.

A Generalized Analysis of Conically Bent Plates by Large-Deflection Theory

This paper presents a general method of solution for the problem which concerns bending plates into cones. The plates considered are those whose thickness obeys the power-law. The large-deflection theory of von Ka´rma´n for plates is employed and the distortion of the cross-section of the bent plate is obtained in terms of generalized hypergeometric functions. Numerical examples are given and results are presented in graphic form. It is shown that by tapering the plate according to a negative power-law, the cross-sectional distortions can be reduced considerably. Also discussed is an analogy between the anticlastic deformation of a plate bent into a cone and the lateral deflection of a conical shell subjected to normal pressure.

Hypergeometric series solutions of some anticlastic deformation problems

Diese Arbeit befasst sich mit der allgemeinen Behandlung antiklastischer Durchbiegungsprobleme. Von Karmans Gleichung fur grosse Durchbiegungen und antiklastische Deformation wurden mit Hilfe der Frobeniusschen Methode gelost. Fur Platten, deren Dicke proportional einer beliebigen Potenz des Abstandes vom Plattenrand ist, wurden Losungen in Form allgemeiner hypergeometrischer Funktionen ausgearbeitet. Numerische Resultate fur antiklastische Deformation werden fur verschiedene Plattendicken prasentiert.

Moiré study of anticlastic deformations of strips with tapered edges

When a thin elastic strip is bent, anticlastic deforming of the cross section takes place, and the edges move away from the center of curvature. This effect can have serious consequences in several applications. However, it has been found that the magnitudes of the deformations can be very greatly reduced if the concave edges of the bent strip are tapered. The proportions of the tapers have already been worked out theoretically so as to optimize the reduction in anticlastic deformation in any given strip bent to a known radius of curvature. The main purpose of the present paper is to verify the theoretical deformations for optimally tapered strips by experiments using the moire method.

An optimum study of the anticlastic deformations of strips with tapered edges

If the edges on the concave side of an elastically bent strip are tapered, the anticlastic deformations of the cross sections can be greatly reduced. This paper concerns the calculation of the specific dimensions of the tapers which will reduce the deformations of the cross sections to a minimum when the strip is bent to a prescribed radius. The optimum tapers are obtained by a Newton-Raphson iteration process, together with a parametric study. The results are presented in tabular form for Poisson's ratio values of 0.3–0.5. It is shown that the maximum relative displacements of the bent strips can be reduced by as much as 60–95 per cent by suitably tapered concave edges.