Flexural Anisotropy Research Articles

Bending and twisting rigidities of 2D materials

A novel torus-based surface parametrization is proposed and used to evaluate the bending and twisting rigidities of 2D materials of arbitrary symmetries. A generalized constitutive model is used to capture the effects of simultaneous bending and twisting curvatures on the strain energy density of a 2D material. The flexural rigidities are evaluated by subjecting the 2D material specimens to different curvature states, including simultaneous bending and twisting, evaluating the corresponding strain energy densities and curve fitting the constitutive model to the data. The proposed methodology is independent of the method used to evaluate the strain energy of the 2D materials, e.g., density functional theory or classical interatomic potentials. The flexural rigidities of graphene, molybdenum disulfide, and black phosphorus are evaluated and compared to those published in the literature. The directional dependence of the flexural rigidities is investigated. Dimensionless coefficients are presented for quantifying the bidirectional bending and bending-twisting coupling effects in 2D materials. Lastly, this study includes an analysis of the flexural anisotropy and flexure-thickness distension coefficients of black phosphorus.

Flexural rigidity of hawkmoth antennae depends on the bending direction

To probe its environment, the flying insect controllably flexes, twists, and maneuvers its antennae by coupling mechanical deformations with the sensory output. We question how the materials properties of insect antennae could influence their performance. A comparative study was conducted on four hawkmoth species: Manduca sexta, Ceratomia catalpae, Manduca quinquemaculata, and Xylophanes tersa. The morphology of the antennae of three hawkmoths that hover while feeding and one putatively non-nectar-feeding hawkmoth (Ceratomia catalpa) do not fundamentally differ, and all the antennae are comb-like (i.e., pectinate), markedly in males but weakly in females. Applying different weights to the free end of extracted cantilevered antennae, we discovered anisotropy in flexural rigidity when the antenna is forced to bend dorsally versus ventrally. The flexural rigidity of male antennae was less than that of females. Compared with the hawkmoths that hover while feeding, Ceratomia catalpae has almost two orders of magnitude lower flexural rigidity. Tensile tests showed that the stiffness of male and female antennae is almost the same. Therefore, the differences in flexural rigidity are explained by the distinct shapes of the antennal pectination. Like bristles in a comb, the pectinations provide extra rigidity to the antenna. We discuss the biological implications of these discoveries in relation to the flight habits of hawkmoths. Flexural anisotropy of antennae is expected in other groups of insects, but the targeted outcome may differ. Our work offers promising new applications of shaped fibers as mechanical sensors. Statement of significanceInsect antennae are blood-filled, segmented fibers with muscles in the two basal segments. The long terminal segment is muscle-free but can be flexed. Our comparative analysis of mechanical properties of hawkmoth antennae revealed a new feature: antenna resistance to bending depends on the bending direction. Our discovery replaces the conventional textbook scenario considering hawkmoth antennae as rigid rods. We showed that the pectinate antennae of hawkmoths behave as a comb in which the bristles resist bending when they come together. This anisotropy of flexural resistance offers a new mode of environmental sensing that has never been explored. The principles we found apply to other insects with non-axisymmetric antennae. Our work offers new applications for shaped fibers that could be designed to sense the flows.

Influence of shear direction on the buckling of CFRP composite perforated plate

The shear buckling performance of CFRP composite perforated plates taking due account of the influence of the flexural anisotropy is examined in this paper using the finite element analysis procedure. Typical laminated composite lay-up causes many complexities but can easily be dealt with in the numerical simulations. The results from the finite element models are shown to agree favourably with the buckling curves from the independent simulations using finite strip method of analysis. Results in the paper clearly show that care should be taken seriously for the case of composite structural members having opening in their webs. Depending on the nature of the applied shear loading, openings, and varied symmetric angle ply laminates, the buckling behavior and capabilities have also been altered.

Flexural anisotropy of rift-sawn softwood boards induced by the end-grain orientation

Inspired by the use of rift-sawn softwood board for covering curved surfaces in Kokerabuki, a traditional Japanese roofing method, we investigated the flexural anisotropy of wood caused by its end-grain orientation. We measured the flexural displacement of softwood species, Chamaecyparis obtusa and Cryptomeria japonica, and hardwood species, Populus suaveolens and Cerasus serrulata. For the softwood species, this was approximately five times longer for the rift-sawn specimens than for the other grain patterns. Using the replica method to measure the softwood tracheid deformation with different flexural displacements, we confirmed the different deformation mechanisms of the tracheid in the flat- and quarter-sawn specimens, and rift-sawn specimens. In the flat- and quarter-sawn woods, on-axis loading was generated, in which the stress was concentrated on the radial and tangential cell walls parallel to the direction of tension or compression. By contrast, in the rift-sawn wood, off-axis loading was generated, in which the stress was evenly distributed throughout the corner cell walls without the wall directly resisting the tensile and compressive forces. We also concluded that the tapered shape of the tracheid walls contributes to the excellent flexibility of rift-sawn softwood.

Scaling Methodology Applied to Buckling of Sandwich Composite Cylindrical Shells

Studying buckling behavior of large shell structures through full-scale test articles can be complex and expensive. Therefore, reduced-scale structures are often preferred for investigating buckling behavior. However, designing reduced-scale structures that are representative of the full-scale structure can be difficult. An analytical scaling methodology for compression-loaded sandwich composite cylindrical shells based on the nondimensionalization of the buckling equations is presented herein. The methodology was used to develop scaled configurations that show similar buckling responses to the full-scale baseline configuration. Finite element analysis results showed that both a baseline and a scaled configuration buckled similarly, when the nondimensional stiffness, defined as the ratio between the nondimensional load and nondimensional displacement, was matched between the different scale models. Limitations of the methodology are discussed and are believed to be a result of neglecting the flexural anisotropy and the transverse shear compliance. A preliminary material failure assessment for the different scales is also considered.

The buckling of CFRP composite plates in compression and shear and thin-walled composite tubes in torsion – The effects of bend-twist coupling and the applied shear direction on buckling performance

The influence of flexural anisotropy or bend-twist coupling on the buckling performance of CFRP composite plates in compression and shear and thin-walled CFRP composite tubes in torsion is examined in this paper using the finite strip method of analysis. The finite strip formulation employed in the analysis procedure is readily able to deal with the many complexities associated with typical laminated composite construction. Lay-up configurations resulting in membrane anisotropy, flexural anisotropy and membrane-flexural coupling are all easily dealt with. The strip perturbation or buckling displacement fields are postulated to vary sinusoidally along the strip length and algebraicly across the strip width and any combination of linearly varying bi-axial tension or compression coupled with in-plane shear loading on the strip can be accommodated. The paper gives an in-depth understanding of the complex buckling mechanics associated with flexural anisotropy in polymer composite construction and highlights the importance of realising the significance of the applied in-plane shear or torque direction on buckling performance. Markedly different stability levels are shown to be in existence for the load cases of positive and negative shear in composite plates and for the clockwise and anticlockwise torsional loading of thin-walled composite tubes. This situation is, of course, not realised in isotropic metal construction or in orthotropic composite construction since such material systems are devoid of the effects of bend-twist coupling.

Instability Paths in the Kirchhoff–Plateau Problem

The Kirchhoff–Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a soap film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. Adopting a variational approach, we define an energy associated with shape deformations of the system and then derive general equilibrium and (linear) stability conditions by considering the first and second variations of the energy functional. We analyze in detail the transition to instability of flat circular configurations, which are ground states for the system in the absence of surface tension, when the latter is progressively increased. Such a theoretical study is particularly useful here, since the many different perturbations that can lead to instability make it challenging to perform an exhaustive experimental investigation. We generalize previous results, since we allow the filament to possess a curved intrinsic shape and also to display anisotropic flexural properties (as happens when the cross section of the filament is noncircular). This is accomplished by using a rod energy which is familiar from the modeling of DNA filaments. We find that the presence of intrinsic curvature is necessary to obtain a first buckling mode which is not purely tangent to the spanning surface. We also elucidate the role of twisting buckling modes, which become relevant in the presence of flexural anisotropy.

Some approaches to buckling analysis of flexurally anisotropic composite plates, subjected to combined in-plane loading

Buckling of flexurally anisotropic composite plates under combined loading is considered. The purpose of the paper is to develop approaches for predicting buckling loads for the plates, basing on buckling analysis results for separate loadings. Theoretical aspects of the problem are explored. The derivatives of the buckling interaction curves are analytically derived. The finite predicting formulae are proposed. The Galerkin-type numerical method for combined loading buckling analysis is developed. The method is based on buckling modes of separate loadings. Numerical examples are discussed. Convergence of the method is shown. Influence of the flexural anisotropy on the buckling behavior is demonstrated.

On the robustness of estimates of mechanical anisotropy in the continental lithosphere: A North American case study and global reanalysis

Lithospheric strength variations both influence and are influenced by many tectonic processes, including orogenesis and rifting cycles. The long, complex, and highly anisotropic histories of the continental lithosphere might lead to a natural expectation of widespread mechanical anisotropy. Anisotropy in the coherence between topography and gravity anomalies is indeed often observed, but whether it corresponds to an elastic thickness that is anisotropic remains in question. If coherence is used to estimate flexural strength of the lithosphere, the null-hypothesis of elastic isotropy can only be rejected when there is significant anisotropy in both the coherence and the elastic strengths derived from it, and if interference from anisotropy in the data themselves can be plausibly excluded. We consider coherence estimates made using multitaper and wavelet methods, from which estimates of effective elastic thickness are derived. We develop a series of statistical and geophysical tests for anisotropy, and specifically evaluate the potential for spurious results with synthetically generated data. Our primary case study, the North American continent, does not exhibit meaningful anisotropy in its mechanical strength. Similarly, a global reanalysis of continental gravity and topography using multitaper methods produces only scant evidence for lithospheric flexural anisotropy.

Electrical and flexural anisotropy in freeze tape cast stainless steel porous substrates

Engineered metal foams have strong potential in applications such as fuel cell electrodes, sensors, variable springs, filtering media, and compositionally graded composite structures. In this study, the variation in mechanical and electrical properties was characterized in engineered porous metal foams with aligned porosity to ascertain the degree of anisotropy that can be induced in these substrates. Porous substrates were prepared by freeze tape casting of powdered ferritic stainless steel. After directional solidification of the slurry, the solvent was sublimed from ~ 1 mm thick tapes, yielding porous green metal compacts which were sintered in a protective atmosphere. The resulting disks exhibit long range ordered acicular pores with substantial anisotropy in the mechanical and electrical properties that is related to the cross sectional pore morphology and connectivity. DC conductivity testing revealed up to 61% variation depending on direction of measurement relative to the alignment of pores. Also, an 89% variation in flexural rigidity in relation to pore orientation was observed in identical disks.

Optimization of Long Anisotropic Laminated Fiber Composite Panels with T-Shaped Stiffeners

A method to optimize long anisotropic laminated fiber composite panels with T-shaped stiffeners is presented. The technique splits the optimization problem into two steps. At the first step, composite optimization is performed using mathematical programming in which the skin and the stiffeners are characterized by lamination parameters accounting for their membrane and flexural anisotropy. Skin and stiffener laminates are assumed to be symmetric or midplane symmetric laminates with 0-, 90-, 45-, or -45- deg ply angles. The stiffened panel consists of a series ofskin-stiffener assemblies or superstiffeners. Each superstiffener is further idealized as a group of flat laminated plates that are rigidly connected. The stiffened panel is subjected to a combined loading under strength, buckling, and practical-design constraints. At the second step, the actual skin and stiffener layups are obtained using a genetic algorithm and considering the ease of manufacture. This approach offers the advantage of introducing numerical analysis methods such as the finite element method at the first step, without significant increases in processing time. Furthermore, modeling the laminate anisotropy enables the designer to explore and potentially use elastic tailoring in a beneficial manner.

Flap–torsion coupling in sandwich beams and filled box sections

Slender aerodynamic structures such as helicopter rotor blades are routinely idealised as 1D beams with associated cross-sectional stiffness properties, and often exhibit structural couplings. A strength of materials based analytical model is presented for determining the cross-sectional stiffness that couples flap bending and torsion (a coupling that is of particular interest in helicopter rotor blade design), by considering strains of individual substructures of the cross-section. The section may be square or rectangular in nature and be foam-filled or hollow. The top and bottom skins may exhibit anisotropic elastic behaviour through non-symmetry and both membrane and flexural anisotropy. This allows a more detailed parametric investigation of this cross-sectional property than is possible with existing analytical models. The analytical model is validated by comparison with a finite element model of the cross-section.

Buckling of long compression-loaded anisotropic plates restrained against inplane lateral and shear deformations

An approach for synthesizing buckling results and behavior for thin balanced and unbalanced symmetric laminates that are subjected to uniform axial compression loads and elastically restrained against inplane expansion, contraction, and shear deformation is presented. This approach uses a nondimensional analysis for infinitely long, flexurally anisotropic plates (coupling between bending and twisting) that are subjected to combined mechanical loads and is based on nondimensional parameters. In addition, nondimensional loading parameters are derived that account for the effects of the elastic inplane deformation restraints, membrane orthotropy, and membrane anisotropy on the induced prebuckling stress state. The loading parameters are used to determine buckling coefficients that include the effects of flexural orthotropy and flexural anisotropy. Many results are presented, for some selected laminates, that are intended to facilitate a structural designer’s transition to the use of the generic buckling-design curves that are presented and discussed in the paper. Several buckling response curves are presented that provide physical insight into the behavior for combined loads, in addition to providing useful design data. An example is presented that demonstrates the use of the generic design curves, which are applicable to a wide range of laminate constructions. The analysis approach and generic results indicate the effects and characteristics of laminate orthotropy and anisotropy in a very general and unifying manner.

Isostatic response of the Australian lithosphere: Estimation of effective elastic thickness and anisotropy using multitaper spectral analysis

Gravity and topography provide important insights regarding the degree and mechanisms of isostatic compensation. The azimuthally isotropic coherence function between the Bouguer gravity anomaly and topography evolves from high to low for increasing wavenumber, a diagnostic that can be predicted for a variety of lithospheric loading models and used in inversions for flexural rigidity thereof. In this study we investigate the isostatic response of continental Australia. We consider the effects of directionally anisotropic plate strength on the coherence. The anisotropic coherence function is calculated for regions of Australia that have distinctive geological and geophysical properties. The coherence estimation is performed by the Thomson multiple‐Slepian‐taper spectral analysis method extended to two‐dimensional fields. Our analysis reveals the existence of flexural anisotropy in central Australia, indicative of a weaker N‐S direction of lower Te. This observation is consistent with the suggestion that the parallel faults in that area act to make the lithosphere weaker in the direction perpendicular to them. It can also be related to the N–S direction of maximum stress and possibly the presence of E–W running zones weakened due to differential sediment burial rates. We also demonstrate that the multitaper method has distinct advantages for computing the isotropic coherence function. The ability to make many independent estimates of the isostatic response that are minimally affected by spectral leakage results in a coherence that is more robust than with modified periodogram methods, particularly at low wavenumbers. Our analysis elucidates the reasons for discrepancies in previous estimates of effective elastic thickness Te of the Australian lithosphere. In isotropic inversions for Te, we obtain values that are as much as a factor of 2 less than those obtained in standard inversions of the periodogram coherence using Bouguer gravity and topography but greater than those obtained by inversions that utilize free‐air rather than Bouguer gravity and ignore the presence of subsurface loads. However, owing to the low spectral power of the Australian topography, the uncertainty on any estimate of Te is substantial.

Stress‐induced azimuthal anisotropy in borehole flexural waves

A nonlinear elastic model for acoustic waves in a stressed medium is used to calculate tectonic stress‐induced changes in borehole flexural dispersions. Our theoretical analysis shows that a horizontal uniaxial stress in the formation causes a crossover in flexural dispersions for the radial polarization aligned parallel and normal to the stress direction. This crossover in flexural dispersions is caused by stress‐induced radial heterogeneities in acoustic wave velocities that are different in the two principal stress directions. Other sources of borehole flexural anisotropy caused by finely layered dipping beds, aligned fractures, or microstructures found in shales, exhibit neither such radial heterogeneities nor flexural dispersion crossovers. Consequently, a crossover in flexural dispersion can be used as an indicator of stress‐induced anistropy. In this situation, the fast shear polarization direction coincides with the far‐field uniaxial stress direction. The analysis also yields an expression for the largest shear stress parameter in terms of the fast and slow seismic shear‐wave velocities with shear polarization parallel and perpendicular to the far‐field stress direction.

Flexural anisotropy of glass-fibre-reinforced thermoplastics injection mouldings

The anisotropy of flexural stiffness has been examined at low strain using ASTM bar and 3-mm and 6-mm thick edge-gated disc injection mouldings for two glass-fibre-reinforced thermoplastics (GFTP). Experimentally determined values were obtained under three-point bending and compared with theoretical prediction: which took account of the fibre orientation and length distributions in the mouldings. The flexural stiffness of the discs was also studied as a function of direction of test. The prediction of the resulting anisotropies in flexure compares well with experiment, but these anisotropies are markedly different from those measured previously in tension. It is concluded that it is necessary to have a knowledge of the fibre distribution of a GFTP moulding in order to predict successfully its mechanical properties, or to interpret measured data.