Second-order Deformation Research Articles

Irrelevant and marginal deformed BMS field theories

In this study, we investigate various deformations within the framework of Bondi-van der Burg-Metzner-Sachs invariant field theory (BMSFT). Specifically, we explore the impact of Bondi-van der Burg-Metzner-Sachs (BMS) symmetry on the theory by introducing key deformations, namely, TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document}, JTμ, and TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{T\\overline{T}} $$\\end{document} deformations. In the context of generic seed theories possessing BMS symmetry, we derive the first-order correction of correlation functions using the systematic application of BMS symmetry ward identities. However, it is worth noting that higher-order corrections are intricately dependent on the specific characteristics of the seed theories. To illustrate our findings, we select the BMS free scalar and free fermion as representative seed theories. We then proceed to analytically determine the deformed action by solving the nontrivial flow equations. Additionally, we extend our analysis to include second-order deformations within these deformed theories.

Calculation of second-order bearing capacity of members subjected to axial load and biaxial bending

The second-order effects of biaxial bending members involve dual non-linear problems, namely material nonlinearity and geometric nonlinearity. These problems typically require iterative calculations and cannot be solved manually. To address this issue, this paper founded upon the strain-based resolution of stress and internal forces, deduces the formula for calculating the load-bearing capacity of the cross-section. The elastoplastic second-order deformation is then calculated at the member level, and the final result is presented in the form of a dimensionless nomogram for calculating the second-order biaxial bending members. In this paper, the limit range of section strain is determined based on the constitutive relationship of concrete and reinforcement. This range is divided into three limited strain areas, which enable the calculation of section bearing capacity and limit curvature, and subsequently the second-order bearing capacity of the member. The results are in good agreement with experimental data. The dimensionless nomogram in this paper considers the complete parabolic and rectangular of concrete stress-strain relationship and the ideal elastoplastic constitutive relationship of reinforcement and the second-order effect. It does not require iteration. It can quickly calculate the reinforcement or check the section bearing capacity by hand. It can be used to check the calculation results of engineering software and provides ideas for the design method of member subjected to axial load and biaxial bending based on seismic restoring force.

Second-order deformation behaviors of free-rotational tension after free-end torsion for rolled AZ31B magnesium alloy

The accompanying dimensional change caused by the second-order deformation behaviors severely limits the accuracy of the size and shape control of the structural components. The effect of the microstructure evolution mechanism on the second-order deformation behaviors (e.g., Swift effect, inverse Swift effect) of magnesium (Mg) alloys has not been systematically reported. Therefore, in this study, in order to promote the precision forming , the deformation mechanisms of the second-order mechanical behaviors of rolled AZ31B Mg alloy were systematically investigated by employing the specifically designed multi-degree of freedom and multi-axial force-field coupling loading mode. The free-end torsion (FET) experiments were carried out in the range of 30°–90° along the transverse direction (TD) at room temperature and the free-rotatial tension was performed after FET. The Swift effect of axial shortening was observed during FET, and the cumulative effect of misfit strain caused by tensile twinning is the main reason for the Swift effect. Meanwhile, obvious inverse Swift effect was detected in the free-rotatial tension. It is found that the external reverse residual shear stress dominates the rapid reverse rotation of the first stage in the inverse Swift effect, and the nonlinear distribution of the residual shear strain increases this reverse rotation. The competitive effect of detwinning caused by reverse rotation and prismatic slip leads to a slow reverse rotation. The prismatic slip provides the driving force for the inverse Swift effect in the positive rotation stage until the residual shear stress is completely released.

Exploring novel mechanical metamaterials: Unravelling deformation mode coupling and size effects through second-order computational homogenisation

Architected materials and mechanical metamaterials are known for their unique macroscopic properties and complex behaviour that often defy conventional continuum mechanics. Therefore, in this contribution, a recent multi-scale second-order computational homogenisation method (Dos Santos et al., 2023) is employed to explore these materials under finite strains. The approach combines a second gradient continuum theory at the macro-scale and a representative volume element (RVE) with classical first-order continuum mechanics at the micro-scale. The Method of Multi-scale Virtual Power ensures a consistent scale transition. The predictive capability and applicability of the second-order computational strategy are evaluated through coupled multi-scale numerical simulations. These simulations involve two- and three-dimensional problems, with a strong focus on the development of novel metamaterials, while also accounting for diverse loading conditions, such as tension/compression-induced undulation, bending, and compression-induced torsion. Comparisons with first-order homogenisation and Direct Numerical Simulations validate the approach. Analysis of homogenised consistent tangents reveals valuable insights into macroscopic properties. Overall, the results highlight the capability of the second-order strategy to capture significant phenomena, including second-order deformation modes, coupling deformation mechanisms, and size effects.

Initializing and accelerating Stereo-DIC computation using semi-global matching with geometric constraints

Stereo Digital Image Correlation (Stereo-DIC) has become a mainstream optical metrology technique for quantitatively analyzing full-field 3D shape, displacement, or deformation of materials and structures. Whether it is to measure 3D profile or deformation, stereo matching is essential for Stereo-DIC to reconstruct 3D point clouds from stereo images. Traditional feature-based (e.g., SIFT) methods provide initial 2D displacements for stereo matching with the aid of extracting massive features, but at the cost of expensive computational overhead. In addition, these methods preclude precise measurement of objects with steep and ridged surfaces or undergoing large rotation and/or deformation due to low feature matching accuracy of complex regions caused by perspective differences. In this paper, we propose a fast and robust stereo matching method using semi-global matching with geometric constraints (GC-SGM) for initializing and accelerating Stereo-DIC computation. For GC-SGM, an optimized semi-global matching (SGM) algorithm based on GPU acceleration is first utilized to quickly estimate dense and reliable disparity maps between the rectified stereo images. The global pixel-wise 2D correspondence between raw stereo images can be established inversely using epipolar constraints and 1D disparity information, and then converted to accurate and initial second-order deformation parameters for 2D-DIC-based sub-pixel refinement by least-squares-based surface fitting. Experimental results prove that the proposed GC-SGM enhances the matching correctness and robustness for complex objects while improving the processing speed on GPU by 3∼10 times compared with SIFT-based methods, enabling high-precision and computationally efficient 3D shape and deformation measurement.

Direct-Homogenization-Based Plate Models of Integrated Thermal Protection System for Reusable Launch Vehicles

Integrated thermal protection systems of reusable launch vehicles (RLVs) can have a corrugated core sandwich structure and experience location-dependent thickness-wise temperature gradient. The sandwich structure can be optimized depending on the location on RLV using finite element method simulations of RLV components. However, such analysis can be computationally challenging due to the disparate length scales between the RLV components and features of the sandwich structure. Two different equivalent plate models based on first-order shear and normal deformation theory, and first-order shear and second-order normal deformation theory (FSSNDT) have been utilized in this work to address this drawback. A direct homogenization technique involving unit cell and cantilever beam analysis has been developed to calibrate the equivalent plate properties for different thickness-wise temperature variations. The accuracy of the plate models has been evaluated by comparing their responses with a full-scale model for different thickness-wise temperature gradients and a uniform pressure. The comparisons clearly indicate that the FSSNDT-based plate model calibrated via direct homogenization captures both the displacements and strains in the in-plane and transverse directions accurately, and can be used to perform RLV component analysis efficiently to obtain location-specific optimal design of corrugated core sandwich structures.

Second-order homogenisation of crystal plasticity and martensitic transformation

Second-order computational homogenisation is combined with constitutive modelling of crystal plasticity and martensitic transformation to analyse the influence of strain gradients on the multi-scale response of polycrystalline materials and the kind of size effects that can be captured. The impact of the length of the Representative Volume Element (RVE) model and the size of the grains is analysed independently, resorting to parametric studies with several random realisations. It is observed that the captured size effects are due to the RVE length, while the size of the grains, which is related to the number of grains in the RVE, is mainly associated with the representativeness of the homogenised response. Macroscopic deformation states are extracted from a plate subjected to bending to guarantee the physical admissibility of the strain gradients and deformation gradients applied to the RVE. It is also found that the influence of the RVE length on the multi-scale response of polycrystals is also dependent on the relationship between the magnitude of the first and second-order deformations. The TRIP effect is also studied by performing these analyses with and without martensitic transformation.To quantify the differences between the results obtained with first- and second-order homogenisation, analytical expressions that take into consideration the macroscopic deformation measures and the RVE length are identified to approximate numerical observations. Finally, an adaptive framework for multi-scale simulation accounting for strain gradients is proposed, where these expressions can be employed in a criterion for the transition from first- to second-order homogenisation.

Comparative dynamic analysis of SCR and LWR in the coupled platform–mooring–riser system under internal solitary wave conditions

As a frequently occurring and widely distributed environmental load, an internal solitary wave (ISW) always appears with a large amplitude and may result in the instability of deepwater structures and offshore accidents. In this study, a self-developed program was established to describe the dynamic behavior of a coupled platform–mooring–riser system. The response of the platform is based on the rigid body dynamics theory, that of the risers/mooring lines is obtained by the lump mass method, and the coupled effect between them is achieved by a semi-coupled analysis. Two types of riser configurations, steel catenary risers (SCRs) and lazy-wave steel catenary risers (LWRs), in the coupled system were compared under only ISWs and combined excitation of ISWs and random waves. The numerical results were proven to be reliable through validation against previous literature. The results show that different riser configurations have little influence on the platform response or mooring top tension. Both riser configurations exhibit a second-order deformation mode along the riser length and a half-wave shape with time. Additionally, a mutation in the horizontal tension, shear force, and bending moment occurs near the ISW interface. Compared with SCRs, LWRs always produce larger motion characteristics and responses with time differences owing to their complex configuration.

Influence of Fluid Viscosity and Compressibility on Nonlinearities in Generalized Aerodynamic Forces for T-Tail Flutter

The numerical assessment of T-tail flutter requires a nonlinear description of the structural deformations when the unsteady aerodynamic forces comprise terms from lifting surface roll motion. For linear flutter, a linear deformation description of the vertical tail plane (VTP) out-of-plane bending results in a spurious stiffening proportional to the steady lift forces, which is corrected by incorporating second-order deformation terms in the equations of motion. While the effect of these nonlinear deformation components on the stiffness of the VTP out-of-plane bending mode shape is known from the literature, their impact on the aerodynamic coupling terms involved in T-tail flutter has not been studied so far, especially regarding amplitude-dependent characteristics. This term affects numerical results targeting common flutter analysis, as well as the study of amplitude-dependent dynamic aeroelastic stability phenomena, e.g., Limit Cycle Oscillations (LCOs). As LCOs might occur below the linear flutter boundary, fundamental knowledge about the structural and aerodynamic nonlinearities occurring in the dynamical system is essential. This paper gives an insight into the aerodynamic nonlinearities for representative structural deformations usually encountered in T-tail flutter mechanisms using a CFD approach in the time domain. It further outlines the impact of geometrically nonlinear deformations on the aerodynamic nonlinearities. For this, the horizontal tail plane (HTP) is considered in isolated form to exclude aerodynamic interference effects from the studies and subjected to rigid body roll and yaw motion as an approximation to the structural mode shapes. The complexity of the aerodynamics is increased successively from subsonic inviscid flow to transonic viscous flow. At a subsonic Mach number, a distinct aerodynamic nonlinearity in stiffness and damping in the aerodynamic coupling term HTP roll on yaw is shown. Geometric nonlinearities result in an almost entire cancellation of the stiffness nonlinearity and an increase in damping nonlinearity. The viscous forces result in a stiffness offset with respect to the inviscid results, but do not alter the observed nonlinearities, as well as the impact of geometric nonlinearities. At a transonic Mach number, the aerodynamic stiffness nonlinearity is amplified further and the damping nonlinearity is reduced considerably. Here, the geometrically nonlinear motion description reduces the aerodynamic stiffness nonlinearity as well, but does not cancel it.

Symplectically convex and symplectically star-shaped curves: a variational problem

In this article, we propose a generalization of the 2-dimensional notions of convexity resp. being star-shaped to symplectic vector spaces. We call such curves symplectically convex resp. symplectically star-shaped. After presenting some basic results, we study a family of variational problems for symplectically convex and symplectically star-shaped curves which is motivated by the affine isoperimetric inequality. These variational problems can be reduced back to two dimensions. For a range of the family parameter, extremal points of the variational problem are rigid: they are multiply traversed conics. For all family parameters, we determine when non-trivial first- and second-order deformations of conics exist. In the last section, we present some conjectures and questions and two galleries created with the help of a Mathematica applet by Gil Bor.

In-plane-dominated vibration characteristics of piezoelectric thick circular plates based on higher-order plate theories

ABSTRACT Numerous engineering applications exist for the piezoelectric effect, which results from the electromechanical coupling between electrical and mechanical fields. In-plane vibrations of piezoelectric plates’ resonance frequencies and associated mode shapes have been thoroughly investigated. However, analytical solutions for in-plane-dominated vibrations of thick piezoelectric circular plates are limited. In this paper, higher-order plate theories for the in-plane-dominated vibration characteristics of piezoelectric circular thick plates under fully clamped and completely free boundary conditions are presented. The resonant frequencies and associated mode shapes were investigated based on two higher-order plate theories: second-order shear deformation plate theory and third-order shear deformation plate theory, as well as simplified third-order linear piezoelectric theory. Hamilton's principle was applied to derive equations of motion and boundary conditions. In the theoretical analysis, the resonant frequencies, associated mode shapes and distribution of electric displacements for various radius-to-thickness ratios were calculated. The numerical results obtained by the finite element method were compared with those obtained from theoretical analysis. Excellent agreement was found between the theoretical and numerical results for the thick piezoelectric circular plates.

Dynamic modeling and fuzzy compensation sliding mode control for flexible manipulator servo system

The flexible manipulator servo system considering the second-order modal and two-dimensional deformation is too complicated, which is not conducive to real-time control. In order to establish dynamic equations that can be controlled in real-time, it is necessary to analyze each nonlinear factor. Based on the results of dynamic analysis, a model simplification method is proposed that ignores the second-order modal deformation. What's more, an improved fuzzy compensation sliding mode control (FCSMC) strategy is proposed to control the angular velocity of the servo system. The friction torque and the ignoring nonlinear factors are compensated by the fuzzy systems. The improved FCSMC strategy can reduce the chattering phenomenon and the angular velocity errors. Simulation experiments and physical control experiments prove the effectiveness of the improved FCSMC strategy. The results show that the improved FCSMC strategy has smaller angular velocity errors. Therefore, the strategy can effectively improve the movement accuracy of flexible manipulators, which is conducive to the precise end-effectors.

Second–order models for the bending analysis of thin and moderately thick circular cylindrical shells

The suitability of employing the second-order shear deformation theory to static bending problems of thin and moderately thick isotropic circular cylindrical shells was investigated. Two variant forms of the polynomial second-order displacement models were considered. Both models account for quadratic expansions of the surface displacements along the shell thickness, although the second model (SSODM) was augmented by the initial curvature term. The equilibrium equations were derived by use of the principle of virtual work. Navier analytical solutions were obtained under simply supported boundary conditions. The results of the displacements and stresses revealed that the theory formulated on the SSODM provides a good depiction of the bending response of thin and moderately thick shells and are in close agreement with those of the first and higher-order shear deformation theories (FSDT; HSDT). The ability of the theory formulated on the first model (FSODM) to predict adequate values of displacements and stresses in thin shells was found to be significantly affected by changes in length to radius of curvature (l/a) ratios.

Impact of mechanical stratification on the structural style of the Lublin Basin, SE Poland: results of seismic interpretation and implications for quantification of deformation within the frontal parts of thin-skinned fold-and-thrust belts

We demonstrate how lithological and mechanical stratification of Ediacaran–Carboniferous sedimentary package governs strain partitioning in the Lublin Basin (LB) which was incorporated in the marginal portion of the Variscan fold-and-thrust belt. Based on the geometry of seismic reflectors, the pre-Permian–Mesozoic sedimentary sequence was subdivided into two structural complexes differing in structural style. The lower one reveals forelandward-vergent imbrication, while the upper one comprises fold train, second-order deformations, and multiple local detachments. Lithological composition of the upper structural complex controlled geometry, kinematics, and position of compressional deformations in stratigraphic profile. System of foreland-vergent thrusts which links lower and upper detachment developed due to efficiency of simple shear operating in heterogeneous clastic-carbonate-evaporitic strata of the Lower–Upper Devonian age. Internal homogeneity promoted the formation of conjugate sets of thrusts in Silurian shales and Upper Devonian limestones. Structural seismic interpretation combined with sequential restoration revealed localised thickening of Devonian strata and up to 5% difference in length of Devonian horizons. This mismatch is interpreted as a manifestation of distributed shortening, including layer-parallel shortening (LPS), which operated before or synchronously to the initiation of folding. The amount of distributed strain is comparable with numbers obtained in external parts of other fold-and-thrust belts. The outcomes derived from this study may act as a benchmark for studying variability in a structural style of multilayered sequences which were incorporated in the external portion of other fold-and-thrust belts.

RESEARCH ON THE PREDICTION OF DEFORMATION PROPERTIES BASED ON NEURAL NETWORK

The direct analysis method based on deformation properties needs to predict the lateral deformation of the steel frame-support structure in advance, but it takes time and labor to use both finite element numerical simulation and experimental research. In this paper, the BP neural network receives the parameters of the steel frame-support structure, such as the aspect ratio, the second-order effect parameters, the slenderness ratio of frame columns, the rigidity ratio of beam-column bus and the deformation limit, with the lateral displacement of the column as the output layer, the second-order lateral deformation properties of the steel frame-support structure can be quickly obtained. Comparing the prediction results of BP neural network with those of the existing fitting formulas, the absolute difference and the relative error ratio between them are very small, which shows that the accuracy of the second-order column roof migration using the BP neural network prediction is high. The BP neural network can be used to judge the applicability of the direct analysis method based on the deformation properties of the steel frame-support structure under various parameters.

Slender RC Columns Strengthened with a Novel Hybrid Strengthening System of External Longitudinal and Transverse FRPs

In this study, the performance of slender circular concrete columns strengthened with a novel hybrid system of longitudinally bonded prefabricated fiber-reinforced polymer (FRP) laminates and transverse FRP wrapping is investigated. The novelty of the hybrid system is to improve the load-carrying capacity of slender steel-reinforced concrete (RC) columns under eccentric axial compression by providing high modulus longitudinal carbon FRP (CFRP) laminates through enhancing the flexural stiffness of the slender column and to laterally support the longitudinal laminates by FRP wraps to prevent debonding and local buckling. A total of 6 large-scale circular slender RC columns with a diameter of 260 mm and a length of 3,048 mm were tested under combined axial and flexural loads. The results showed that, for the strengthening of the slender columns, the hybrid system is a more effective strengthening system than wrapping controlling second-order deformations due to the slenderness effect and enhancing the load-bearing capacity. Also, the performance of the system was further investigated using an analytical-numerical model considering the second-order deformations of the slender columns. The model considered nonlinearity in material and confinement effects plus the geometrical nonlinearity via an iterative second-order analysis. The model was verified against experimental data from the current study (hybrid system) and an independent study (wrapping system) and showed a good agreement with the test results. Then, a comprehensive parametric study was conducted to study the effect of various parameters including slenderness ratio, load eccentricity, longitudinal and transverse FRP reinforcement ratios, concrete strength, and column diameter on the performance of slender RC columns strengthened with the hybrid system. It was found that the hybrid strengthening system was more effective for RC columns with high slenderness ratios, high load eccentricities, and low concrete strength.

Nonlinear mechanics of sandwich plates: Layerwise third-order thickness and shear deformation theory

The nonlinear mechanics of sandwich plates is studied using a layerwise third-order thickness and shear deformation theory. In particular, the two face sheets are modelled using a second-order shear deformation theory (well-justified in case of a zero-shear stress only at one outer surface) and the core is modelled with a third-order thickness and transverse shear deformation theory. After introducing continuity of displacements at the interfaces between the core and the face sheets, 16 independent kinematic parameters are retained. An independent parameter is kept to describe the thickness deformation, which allows for introduction of the corresponding boundary condition; this represents a significant novel contribution. Geometric nonlinearity in all the kinematic parameters is introduced. Numerical results are presented for deflection of a sandwich plate under pressure. A thin core was considered, since it is numerically more critical, and a wide range of core stiffnesses was studied in order to verify the validity of the proposed model till the limit case of two plates joined by a core of negligible stiffness (i.e., two independent plates). Numerical results are compared to those obtained by a commercial finite element (FE) program with three-dimensional solid elements till the limit when the FE code fails for large deformations of the extremely weak core.

Wave dispersion characteristics of graphene reinforced nanocomposite curved viscoelastic panels

This paper tries to investigate the wave phenomenon in nanoshell made of graphene nanoplatelets (GNPs) reinforced nanocomposites. The second-order shear deformation theory in curvilinear coordinate is utilized to develop the doubly-curved shell as a continuous structure and the general nonlocal strain gradient theory is adopted to calculate nonlocality and strain gradient size-dependency. The effective material properties are estimated through the Halpin–Tsai micromechanical model and a modified rule of mixture. A virtual work of Hamilton statement is employed to obtain the governing motion equations and then a harmonic solution based on an analytical procedure is developed to find the wave response. Afterwards, a parametric study is performed to analyze the effects of the linear spring and damper coefficients, small-scale parameters, different curves, radius to thickness ratio, GNP's weight fraction as well as its number of layers on the phase velocity response of the GNPs reinforced doubly curved nanoshell with various shape panels resting on a visco-Winkler medium.

T overline{T} -flow effects on torus partition functions

In this paper, we investigate the partition functions of conformal field theories (CFTs) with the T overline{T} deformation on a torus in terms of the perturbative QFT approach. In Lagrangian path integral formalism, the first- and second-order deformations to the partition functions of 2D free bosons, free Dirac fermions, and free Majorana fermions on a torus are obtained. The corresponding Lagrangian counterterms in these theories are also discussed. The first two orders of the deformed partition functions and the first-order vacuum expectation value (VEV) of the first quantum KdV charge obtained by the perturbative QFT approach are consistent with results obtained by the Hamiltonian formalism in literature.

Stability and Hamiltonian BRST-invariant deformations in Podolsky's generalized electrodynamics

We study the problem of stability in Podolsky's generalized electrodynamics by constructing a series of 2-parametric bounded conserved quantities. In this way, we show that the 00-component of the energy-momentum tensors could be positive definite and therefore the higher derivative system is considered to be stable. Afterwards, we derive the consistent interactions in Podolsky's theory within the framework of Hamiltonian BRST-invariant deformation procedure. The key ingredients in our analysis are the local BRST-cohomology which plays a crucial role in the determination of the first-order deformation as well as the Jacobi identity that will greatly simplify the calculations for us. We assert that in our discussions, the second-order deformation and the other higher order deformations of the BRST charge naturally turn out to be zero while the third-order as well as the corresponding higher order BRST-invariant Hamiltonian deformations also vanish completely. Moreover, we evaluate the path integral of the higher derivative constrained system before and after deformation process following the standard BRST quantization method with appropriate gauge-fixing fermions.