Extension Twist Research Articles

Assessment of structural mechanical effects related to torsional deformations of propellers

Lifting propellers are of increasing interest for Advanced Air Mobility. All propellers and rotors are initially twisted beams, showing significant extension–twist coupling and centrifugal twisting. Torsional deformations severely impact aerodynamic performance. This paper presents a novel approach to assess different reasons for torsional deformations. A reduced-order model runs large parameter sweeps with algebraic formulations and numerical solution procedures. Generic beams represent three different propeller types for General Aviation, Commercial Aviation, and Advanced Air Mobility. Simulations include solid and hollow cross-sections made of aluminum, steel, and carbon fiber-reinforced polymer. The investigation shows that centrifugal twisting moments depend on both the elastic and initial twist. The determination of the centrifugal twisting moment solely based on the initial twist suffers from errors exceeding 5% in some cases. The nonlinear parts of the torsional rigidity do not significantly impact the overall torsional rigidity for the investigated propeller types. The extension–twist coupling related to the initial and elastic twist in combination with tension forces significantly impacts the net cross-sectional torsional loads. While the increase in torsional stiffness due to initial twist contributes to the overall stiffness for General and Commercial Aviation propellers, its contribution to the lift propeller’s stiffness is limited. The paper closes with the presentation of approximations for each effect identified as significant. Numerical evaluations are necessary to determine each effect for inhomogeneous cross-sections made of anisotropic material.

Hygrothermally stable stacking sequence for tailoring of extension–twist coupling in composite structures

Hygrothermally stable stacking sequence for tailoring of extension–twist coupling in composite structures

Improving the extension–twist coupling performance of practically warpage-free laminates via layup hybridization

This article investigates whether the extension–twist and bend–twist coupling performance of practically warpage-free laminates can be improved by layup hybridization (carbon/epoxy–glass/epoxy hybrids) compared to non-hybrid (mono) laminates. We performed numerical layup search by simulating the thermal warpage, and the bend–twist and extension–twist behaviour of 4096 4-ply layup permutations. The upper limit of allowable manufacturing-induced warpage was based on the ISO2768 standard. The optimal hybrid and mono layups were selected to have the most significant coupling performance within the allowable warpage range. A mono layup achieved the maximum bend–twist performance (which does not require layup asymmetry), but layup hybridization increased the extension–twist performance by more than 30% in the practically warpage-free range. The numerical results were validated through 3D scanning and 3D digital image correlation–aided mechanical tests. We concluded that layup hybridization can significantly improve the extension–twist performance of realistically usable composite laminates with negligible thermal warpage.

The Effect of Non-Conservative Compressive Force on the Vibration of Rotating Composite Blades

This paper investigates the effectiveness of a resonance avoidance concept for composite rotor blades featuring extension–twist elastic coupling. The concept uses a tendon, attached to the tip of the blade, to apply a proper amount of compressive force to tune the vibration behavior of the blade actively. The tendon is simulated by applying a non-conservative axial compressive force applied to the blade tip. The main load carrying part of the structure is the composite spar box, which has an antisymmetric layup configuration. The nonlinear dynamic behavior of the composite blade is modelled by using the geometrically exact fully intrinsic beam equations. The resulting nonlinear differential equations are discretized using a time–space scheme, and the stationary and rotating frequencies of the blade are obtained. It is observed that the proposed resonance avoidance mechanism is effective for tuning the vibration behavior of composite blades. The applied compressive force can shift the frequencies and the location at which the frequency veering take place. Furthermore, the compressive force can also cause the composite blade to get unstable depending on the layup ply angle. Finally, the results, highlighting the importance of compressive force and ply angle on the dynamic behavior of composite blades, are presented and discussed.

Analytic continuation of Bethe energies and application to the thermodynamic limit of the SL(2, \u2102) non-compact spin chains

We consider the problem of analytically continuing energies computed with the Bethe ansatz, as posed by the study of non-compact integrable spin chains. By introducing an imaginary extensive twist in the Bethe equations, we show that one can expand the analytic continuation of energies in the scaling limit around another ‘pseudo-vacuum’ sitting at a negative number of Bethe roots, in the same way as around the usual pseudo-vacuum. We show that this method can be used to compute the energy levels of some states of the SL(2, ℂ) integrable spin chain in the infinite-volume limit, and as a proof of principle recover the ground-state value previously obtained in [1] (for the case of spins s = 0, overline{s} = −1) by extrapolating results in small sizes. These results represent, as far as we know, the first (partial) description of the spectrum of SL(2, ℂ) non-compact spin chains in the thermodynamic limit.

Extension twist deformation response of an auxetic cylindrical structure inspired by deformed cell ligaments

Recent advances in artificial metamaterials have allowed unique structures to deform in distinct modes not previously found in traditional material form. Auxetic structures that induce twist through deformation have been a focus of recent research. In this paper a cell-based tubular structure with pre-deformed ligaments is proposed which exhibits efficient extension induced twist (EIT) without the limitations of buckling. An analytical model of the structure is proposed using cell parameters to characterize the non-linear twist-deformation relationship. Samples of the structure were fabricated using the Ultimaker 3 platform with elastomer based TPU 95a build material and PVA support material. The analytical model and FEM were validated by tensile experiments, where fabricated samples reached 60 degrees of non-linear axial twist over 40 mm of deformation. In order to further characterize the structure’s twist response, key design parameters of the lattice structure were varied and modelled. Actuation through tensile elastic deformation has key advantages in reliability and reduced complexity. Due to these advantages, this technology has potential applications in aerospace, biomedical and robotic fields.

Inhomogeneous Gaussian free field inside the interacting arctic curve

The six-vertex model with domain-wall boundary conditions is one representative of a class of two-dimensional lattice statistical mechanics models that exhibit a phase separation known as the arctic curve phenomenon. In the thermodynamic limit, the degrees of freedom are completely frozen in a region near the boundary, while they are critically fluctuating in a central region. The arctic curve is the phase boundary that separates those two regions. Critical fluctuations inside the arctic curve have been studied extensively, both in physics and in mathematics, in free models (i.e. models that map to free fermions, or equivalently to determinantal point processes). Here we study those critical fluctuations in the interacting (i.e. not free, not determinantal) six-vertex model, and provide evidence for the following two claims:(i) the critical fluctuations are given by a Gaussian free field (GFF), as in the free case, but(ii) contrarily to the free case, the GFF is inhomogeneous, meaning that its coupling constant becomes position-dependent, .The evidence is mainly based on the numerical solution of appropriate Bethe ansatz equations with an imaginary extensive twist, and on transfer matrix computations, but the second claim is also supported by the analytic calculation of and its first two derivatives in selected points. Contrarily to the usual GFF, this inhomogeneous GFF is not defined in terms of the Green’s function of the Laplacian inside the critical domain, but instead, of the Green’s function of a generalized Laplacian parametrized by the function . Surprisingly, we also find that there is a change of regime when the anisotropy parameter Δ is smaller than , with becoming singular at one point.

Composite Blade Twist Modification by Using a Moving Mass and Stiffness Tailoring

In this paper, a new concept for morphing composite blades is proposed, and how this concept changes the twist distribution of the blade is explained. A change in the blade twist is obtained by adding a mass to the blade that produces an extra centrifugal force. This centrifugal force then may produce a moment that can change the blade twist via the extension–twist or bend–twist coupling of the composite lamination. These types of couplings are present in antisymmetrically and symmetrically laminated beams, respectively. The dynamics of the rotating composite blade is modeled by using the geometrically exact fully intrinsic beam equations. The concentrated mass is considered as a nonstructural concentrated mass that has offsets with respect to the beam reference line. The nonlinear partial differential equations are discretized by using a time–space scheme, the converged results are compared with those reported in the literature, and very good agreement is observed. It is found that, for an antisymmetric lamination, the spanwise location of the concentrated mass affects the twist, whereas in the symmetric case, the chordwise position of the concentrated mass is the source of twist change. It is also found that introducing the concentrated mass to a real blade can change the twist dramatically.

Analysis of extension–twist coupling of thick-walled composite circular tubes

Existing extension–twist coupling in composite tubes can result in undesirable deformations, which may in turn interfere with their normal performance. In this work, a quantitative factor was developed to examine the existence of such coupling in circular composite tubes using the non-classical composite beam theory. The theoretical formulation considered transverse shear and three-dimensional (3D) elastic effects in laminated walls. Therefore, the coupling factor can be used in both thin-walled and thick-walled composite tubes. The numerical values for the coupling factor calculated using the present approach were compared to those obtained from a 3D validated finite element model on 10 composite tubes with different stacking sequences. The findings suggested that the coupling factor could accurately detect existing coupling between extension and twist in composite tubes regardless of their stacking sequence. The paper also proposed a number of design guidelines that could be used in order to eliminate the extension–twist coupling in composite tubes.

Effect of Intrinsic Twist and Orthotropy on Extension–Twist–Inflation Coupling in Compressible Circular Tubes

We present effects of intrinsic twist and material orthotropy on extension–twist–inflation coupling in circular tubes about their stress-free state. Simple analytical expressions for coupling stiffnesses corresponding to extension–twist, twist–inflation and extension–inflation couplings are obtained. We show that the sign of the extension–twist coupling stiffness, which governs initial overwinding/unwinding in tubes during their extension, is not just dependent on the tube’s intrinsic twist but also on two other parameters: ratio of the Young’s moduli in the lateral surface of orthotropic tube and the excess of the Poisson’s ratio from an isotropy condition. By tuning these two parameters, one can generate the counter-intuitive overwinding as reported earlier in the case of DNA. Similarly, we show that even with positive Poisson’s ratio, an intrinsically twisted tube could inflate on being stretched. We also present a scheme to obtain all the relevant stiffnesses of chiral single-walled carbon nanotubes from a “one-atom unit cell” calculation. These stiffnesses, when plotted vs. the nanotube’s chirality, exhibit interesting periodicity which has its origin in the 6-fold symmetry of graphene. This trend is captured in the continuum model for a nanotube when it is assumed to be comprised of three families of parallel fibers on its lateral surface.

Dynamic behavior of a rotating delaminated composite beam including rotary inertia and shear deformation effects

A finite element (FE) model is developed to study the free vibration of a rotating laminated composite beam with a single delamination. The rotary inertia and shear deformation effects, as well as the bending–extension, bending–twist and extension–twist coupling terms are taken into account in the FE model. Comparison between the numerical results of the present model and the results published in the literature verifies the validity of the present model. Furthermore, the effects of various parameters, such as delamination size and location, fiber orientation, hub radius, material anisotropy and rotating speed, on the vibration of the beam are studied in detail. These results provide useful information in the study of the free vibration of rotating delaminated composite beams.

Natural frequencies of axial–torsional coupled motion in springs and composite bars

The natural frequencies corresponding to axial–torsional (extension–twist) coupled motion of a helical spring, or the corresponding motion induced through material coupling in a composite bar, are considered using an equivalent continuum approach. Closed form solution of the governing differential equations leads either to an exact dynamic stiffness matrix or to a number of exact relationships between the natural frequencies corresponding to coupled and uncoupled motion. The latter relationships both guarantee that the Wittrick–Williams root finding algorithm can still be used to converge on any required natural frequency, despite any lack of reciprocity arising from differential coupling, and for the case of symmetric material coupling coefficients, enable their value to be determined precisely from experimental results. A number of examples are then given to confirm the accuracy of the proposed theory and to indicate its range of application.

Post-buckling of internal-pressure-loaded laminated cylindrical shells surrounded by an elastic medium

This paper presents a study on the post-buckling response of an anisotropic laminated cylindrical shell of finite length embedded in a large outer elastic medium and subjected to internal pressure in thermal environments. The surrounding elastic medium is modelled as a tensionless Pasternak foundation reacting in compression only. The governing equations are based on higher-order shear deformation shell theory with von Kármán–Donnell kinematic non-linearity and including extension–twist, extension–flexural, and flexural–twist couplings. The thermal effects are also included, and the material properties are assumed to be temperature dependent. Non-linear prebuckling deformations and the initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the post-buckling response of the shells, and an iterative scheme is developed to obtain numerical results without using any assumption concerning the shape of the contact region between the shell and the elastic medium. Numerical illustrations concern the buckling and post-buckling response of cross-ply and symmetric angle-ply laminated shells surrounded by an elastic medium of tensionless foundation of the Pasternak type, from which results for conventional elastic foundations are obtained as comparators. The results reveal that unilateral constraints have a significant effect on the post-buckling response of shells subjected to internal pressure in thermal environments when the foundation stiffness is sufficiently large.

Micromechanical modelling of wood fibre composites

A concentric cylinder model for an N-phase composite with orthotropic properties of constituents was previously presented by the authors. With only minor modifications the model allows for including also free hygroexpansion terms in the elastic stress–strain relationship in order to deal with orthotropic phase swelling. Thus the effect of wood fibre ultrastructure and cell wall hygroelastic properties on wood fibre composite hygroexpansion may be analysed.Multiscale modelling was performed to calculate the hygroexpansion coefficients of both the fibre cell wall and the aligned wood fibre composite.Furthermore, the fibre's helical structure leads to an extension–twist coupling and thus a free fibre will deform axially and also rotate upon loading in longitudinal fibre direction. Within the composite, however, the fibre rotation will be restricted. Therefore, the decision was to compare the composite performance in the two extreme cases(i) free rotation(ii) no rotation of the fibre in the composite.

Buckling and postbuckling of anisotropic laminated cylindrical shells under combined axial compression and torsion

Abstract A postbuckling analysis is presented for an anisotropic laminated cylindrical shell of finite length subjected to combined loading of axial compression and torsion. The governing equations are based on classical shell theory with von Karman–Donnell-type of kinematic nonlinearity and including the extension–twist, extension–flexural and flexural–twist couplings. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, anisotropic laminated cylindrical shells for different values of load-proportional parameters. The results show that the postbuckling characteristics depend significantly upon the load-proportional parameter. The results reveal that in combined loading cases the postbuckling equilibrium path is unstable and the shell structure is imperfection-sensitive.

Adaptive modeling and shape control of laminated plates using piezoelectric actuators

Abstract Adaptive modeling and shape control of laminate plates with piezoelectric actuators is discussed in this report. The influence of piezoelectric actuators and position sensors is studied for shape control under varying unknown loads. A finite element formulation is developed for modeling the dynamic and static response of laminated plates containing discrete piezoelectric ceramics subjected to both mechanical and electrical loadings. The numerical results on the shape control show that the stresses induced by piezoceramic may significantly affect the mechanical behavior of the plate. Actuators are modeled as additional plies completely integrated into substrate laminate. The purpose of this paper is to present the effectiveness of using piezoelectric elements to control the flexural composite plates by inducing internal stresses. Free-end deflection of a composite plate with different location of mechanical load was studied analytically and experimentally. Composite plates with bending–twist and extension–twist couplings were presented. Piezoceramic structure may be designed to tune the deformation of composite plates. The modeling and validation results show the reliability of the piezoactuation for the shape control of laminated plates.

Modifications to the Cauchy–Born rule: Applications in the deformation of single-walled carbon nanotubes

This paper presents a study of the Cauchy–Born (CB) rule as applied to the deformation analysis of single-walled carbon nanotubes (SWNTs) that are modeled as 2-dimensional manifolds. The C–C bond vectors in the SWNT are assumed to deform according to the local deformation gradient as per the CB rule or a modified version thereof. Aspects of the CB rule related to spatial inhomogeneity of the deformation gradient at the atomic scale are investigated in the context of a specific class of extension–twist deformation problems. Analytic expressions are derived for the deformed bond lengths using the standard CB rule as well as modified versions of the standard CB rule. Since the deformation map is conveniently prescribed in this work, it is possible to compare the performance of these deformation rules with the exact solution (i.e. the exact analytic expression for the deformed bond vectors) given directly by the deformation map. This approach provides insights into the CB rule and its possible modifications for use in more complicated deformations where an explicit deformation map is not available. Specifically, it is concluded that in the case of inhomogeneous deformations at the atomic scale for which the CB rule is only approximate (as demonstrated in Section 1 of this paper), the mean value theorem in calculus can be used as a guide to modify the CB rule and construct a more rigorous and accurate atomistic–continuum connection. The deformed bond lengths are used to formulate an enriched continuum hyperelastic strain energy density function based on interatomic potentials (the multi-body Tersoff–Brenner [Tersoff, J., 1988. New empirical approach for the structure and energy of covalent systems. Phys. Rev. B 37, 6991–7000; Brenner, D.W., 1990. Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys. Rev. B 42, 9458–9471] empirical interatomic potential for carbon-carbon bonds is used in this work). The deformation map (and hence the deformation gradient, the bond vectors and the continuum strain energy density) contains certain parameters, some of which are imposed and others determined as a result of energy minimization in the standard variational formulation. Numerical results for kinematic coupling and binding energy per atom are presented in the case of imposed extension and twist deformations on representative chiral, zig-zag and armchair nanotubes using the CB rule and its modifications. These results are compared with the exact solution based on the deformation map which serves as a basis for evaluating the efficacy of these deformation rules. The ideas presented in this paper can also be directly extended to other lattices.

Optimum stacking sequences for extension–twist coupled composites subject to thermal stresses

Optimum stacking sequences for extension–twist coupled composites under thermal stresses are determined. An optimization technique is utilized to determine the optimum stacking sequences. An alternative method is developed based on the identification of a parameter controlling the optimum stacking sequence. Comparison of associated extension–twist coupling values demonstrates their accuracy. It is observed that as the number of plies increases, the optimum stacking sequence follows an asymptotic behavior and that an angle ply system does not represent an optimum stacking sequence. The results are validated through manufacturing of candidate laminates.

Analysis of cord-composite plates using a higher order formulation

Abstract A two-dimensional (2D) higher order formulation that takes into account the extension–twist coupling, and bending behaviour of the cord, is presented for the static and dynamic analysis of cord-composite plates. The cords are represented by an equivalent geometrical surface having the mechanical characteristics including bending property and the coupling between twist–extension. The equations of motion and boundary conditions are formulated using the principle of virtual works. Static examples are done to show when a high order model should be used.

Observations on Electromigration and Improved Filament Life of Microminiature Lamps

Preliminary results of life tests on microminiature lamps and observations on dc notch formation show that longer lamp life can be achieved by improving the vacuum during exhaust. Experimental results show that early burn-out of filaments heated by dc is preceded by extensive twist and distortion of the filaments causing localized over-heating. Filament instability results from the thermal stresses developed by extensive notch formation due to differential rates of electromigration of tungsten atoms on the filament surfaces along various crystallographic planes. By improving the vacuum, notching and twisting of the filament are minimized with a subsequent seven-fold increase in filament life.