Strain-displacement Relations Research Articles

Vibration analysis of graphene-reinforced porous aluminum-based variable-walled thickness sandwich joined conical-conical panel with elastic boundary conditions using differential quadrature method

In this paper, a unified solution is proposed to investigate the vibration characteristics of the variable-walled thickness graphene-reinforced porous aluminum-based (GRPA) sandwich joined conical-conical panel (JCCP) with the arbitrary elastic support boundary conditions by using the differential quadrature method (DQM). The two surfaces of the sandwich JCCP are made of metallic aluminum and the central core layer is the GRPA. The core thickness of each conical plate varies linearly with its generatrix. Three types of the graphene distributions and two types of porosity distributions are considered along the core thickness direction. Based on the first-order shear deformation theory (FSDT), von-Karman strain displacement relationship, constitutive relationship and Hamiltonian principle, the partial differential governing equations of motion are obtained for the variable wall thickness GRPA sandwich JCCP. Using the DQM, the dynamic equation is discretized into the ordinary differential equation. The matrix of the characteristic equation is analyzed to solve the frequencies and mode shapes of the variable wall thickness GRPA sandwich JCCP. The effects of the spring stiffness, boundary conditions, graphene distributions, porosity distributions and geometric parameters on the vibration properties are studied for the variable wall thickness GRPA sandwich JCCP with two interesting elastic supported boundary conditions. At the same time, this article provides a useful approach for studying the arbitrary boundary coupled plate and shell structures with the variable wall thickness.

Calculation of natural frequencies of doubly curved laminated shells using a modified higher order zigzag theory

In this study, the analysis of free vibrations for doubly curved laminated shells has been performed with modified higher order zigzag theory (MHOZT). This approach incorporates higher order displacement kinematics and considers both transverse normal and shear strains. The model assumes that the displacement fields within the plane are a combination of a global cubicly varying field and a locally zigzag linearly varying field. On the other hand, the out-of-plane displacement field varies quadratically with shell thickness coordinates. This formulation considers extended thickness criteria, ensuring the inclusion of the ratio of thickness to the radius of curvature (z/R) in strain displacement relations. Additionally, this model guarantees that there is no transverse shear stress at the extreme surfaces of the shell and that inter-laminar shear stress at interfaces is continuous. To implement this formulation effectively, a C° finite element is used. The outcomes are compared against a three-dimensional (3D) elasticity solution, as well as other pertinent results available in the literature. The suggested model accurately predicts the vibration characteristics of laminated shells, showing good agreement with the 3D elasticity solutions.

Nonlinear motion modeling and analysis of underwater cables under complex loads based on global nodal position finite element method in Hamiltonian formulation

The nonlinear dynamic response of underwater cable is essential for underwater equipment system. This paper presents an effective global nodal position finite element method in Hamiltonian formulation to investigate the nonlinear motion of underwater cable under complex loads. The method fully considers the axial, bending, torsional and shear deformation of underwater cable. The non-linear Green-Lagrange strain-displacement relationship is applied to obtain the strain energy. Furthermore, the fourth-order circular cross-section quadrature is introduced to integrate the volume of underwater cable element after deformation. The dynamic governing equations in Hamiltonian formulation are derived and converted into the corresponding canonical equations, which are calculated by the symplectic difference algorithm. Then an initial curved pendulum system, a 180° turn towed underwater cable system and the spiral turning motion of ROV are designed to verify the presented method. The results show that compared with the existing Hamiltonian global nodal position finite element method which ignores torsional and shear deformation, the proposed method is more accurate in describing the nonlinear motion of the underwater cable with axial, bending, torsional and shear deformation.

Viscoelastic modelling and analysis of two-dimensional woven CNT-based multiscale fibre reinforced composite material system

Carbon nanotube (CNT) has fostered research as a promising nanomaterial for a variety of applications due to its exceptional mechanical, optical, and electrical characteristics. The present article proposes a novel and comprehensive micromechanical framework to assess the viscoelastic properties of a multiscale CNT-reinforced two-dimensional (2D) woven hybrid composite. It also focuses on demonstrating the utilisation of the proposed micromechanics in the dynamic analysis of shell structure. First, the detailed constructional attributes of the proposed trans-scale composite material system are described in detail. Then, according to the nature of the constructional feature, mathematical modelling of each constituent phase or building block’s material properties is established to evaluate the homogenised viscoelastic properties of the proposed composite material system. To highlight the novelty of this study, the viscoelastic characteristics of the modified matrix are developed using the micromechanics method of Mori–Tanaka (MT) in combination with the weak viscoelastic interphase (WI) theory. In the entire micromechanical framework, the CNTs are considered to be randomly oriented. The strength of the material (SOM) approach is used to establish mathematical frameworks for the viscoelastic characteristics of yarns, whereas the unit cell method (UCM) is used to determine the viscoelastic properties of the representative unit cell (RUC). Different numerical results have been obtained by varying the CNT composition, interface conditions, agglomeration, carbon fibre volume percentage, excitation frequency, and temperature. The influences of geometrical parameters like yarn thickness, width, and the gap length to yarn width ratio on the viscoelasticity of such composite material systems are also explored. The current study also addresses the issue of resultant anisotropic viscoelastic properties due to the use of dissimilar yarn thickness. The results of this micromechanical analysis provide valuable insights into the viscoelastic properties of the proposed composite material system and suggest its potential applications in vibration damping. To demonstrate the application of developed novel micromechanics in vibration analysis, as one of the main contributions, comprehensive numerical experiments are conducted on a shell panel. The results show a significant reduction in vibration amplitudes compared to traditional composite materials in the frequency response and transient response analyses. To focus on the aspect of micromechanical behaviour on dynamic response and for the purpose of brevity, only linear strain displacement relationships are considered for dynamic analysis. These insights could inform future research and development in the field of composite materials.

Finite element model updating and damage detection using strain-based modal data in the Bayesian framework

This paper proposes a novel methodology for Bayesian model updating and damage detection of a structural system using strain-based modal data. The strain-based modal data are assumed to consist of the posterior statistics of natural frequencies and strain mode shapes corresponding to measured strains from a classically damped system. The dynamic strain-based equation of the motion is formulated utilizing the energy functional of the structural system and the strain displacement relation. The state variables in the considered strain-based formulation consist of the strains and the elemental rigid body displacements. The methodology is developed by incorporating strain-based formulation into the Bayesian framework for Finite Element (FE) model updating. Besides, additional uncertain parameters in the form of system strain-based modal parameters are introduced in the eigenvalue equation to avoid mode-matching. The model reduction technique is utilized to eliminate the rigid body displacements and the strains that are not measured. Metropolis-Hastings (MH) within Gibbs sampling is proposed to simulate samples from the posterior Probability Density Function (PDF) of the model parameters of the FE model. The effectiveness of the proposed methodology is demonstrated using simulated examples. An experimental study is also conducted to validate the proposed methodology.

Full-field deformation reconstruction of beams using the inverse Finite Element Method: Application to thin-walled structures

This work presents a methodology for the real-time structural deformation reconstruction of solid or thin-walled prismatic beams using discrete strains. Based on the 1D inverse Finite Element Method (iFEM), the approach combines the beam kinematics of Timoshenko theory, strain–displacement relations, and the finite element discretisation framework to reconstruct the full-field deformations of beams with any general cross-section. Although applicable for any general class of beams, the present work investigates its experimental application specifically to conventionally and additively manufactured thin-walled beams discretised using various low- and high-order inverse beam elements. The results demonstrate the method’s accuracy and robustness, albeit influenced by the element discretisation scheme and the number of strain sensors used.

A new set of explicit solutions for postbuckling behaviour of anisotropic shear deformable laminated cylindrical shells with general imperfection distribution and thermal loading

Thermal postbuckling analysis is presented for shear-deformable anisotropic laminated cylindrical shell with general imperfection distribution under several typical temperature fields are analyzed, including uniform temperature, tent-like linear distribution, axial and circumferential quadratic parabolic temperature fields over the shell surface and through the shell thickness. On this basis, the general temperature distribution can be obtained based on the three-dimensional steady-state heat transfer equation. At the same time, based on the high order shear deformation theory including anisotropic coupling effect, geometrically nonlinear large deflection displacement-strain relationship, the general initial geometric imperfections obtained by measurement and analytical means are introduced, and the thermal buckling equilibrium differential equation of anisotropic shear deformable laminated cylindrical shell with initial imperfections is established. A two-step perturbation-Galerkin integral method is used to obtain explicit solutions for the path development direction of post-thermal buckling equilibrium. The results of parameter analysis show that the material properties and temperature dependence, temperature distribution, lay-up and geometric parameters such as radius-to-thickness ratio have important effects on the thermal buckling load and postbuckling equilibrium path. The present method provides an effective theoretical prediction and analysis method for the design of the carrying capacity of submarine pipelines.

Nonlinear Vibration of Saturated Porous Functionally Graded Plate in Thermal Environment

In this work we studied investigates the nonlinear vibrations of a rectangular saturated porous functionally graded (FG) plate on elastic foundations in thermal environment. The mechanical properties of the saturated porous material vary smoothly with the thickness in three different distributions of porosity including uniform, symmetrically irregular, and asymmetrically irregular. The basic equations are employed by the Reddy’s higher order shear deformation theory, incorporating the geometrically nonlinear von Kármán strain-displacement relationship, stress-strain relations based on the elastic theory for porous materials by Biot, and an analytical solution obtained through the Galerkin method and Airy’s stress function for the simply supported plate. The influence of the geometrical and material parameters, elastic foundations and temperature increment on the nonlinear vibrations of the saturated porous FG plate were specifically evaluated through numerical investigations. 
 
 

Displacement Reconstruction Based on Physics-Informed DeepONet Regularizing Geometric Differential Equations of Beam or Plate

Physics-informed DeepONet (PI_DeepONet) is utilized for the reconstruction task of structural displacement based on measured strain. For beam and plate structures, the PI_DeepONet is built by regularizing the strain–displacement relation and boundary conditions, referred to as geometric differential equations (GDEs) in this paper, and the training datasets are constructed by modeling strain functions with mean-zero Gaussian random fields. For the GDEs with more than one Neumann boundary condition, an algorithm is proposed to balance the interplay between different loss terms. The algorithm updates the weight of each loss term adaptively using the back-propagated gradient statistics during the training process. The trained network essentially serves as a solution operator of GDEs, which directly maps the strain function to the displacement function. We demonstrate the application of the proposed method in the displacement reconstruction of Euler–Bernoulli beams and Kirchhoff plates, without any paired strain–displacement observations. The PI_DeepONet exhibits remarkable precision in the displacement reconstruction, with the reconstructed results achieving a close proximity, surpassing 99%, to the finite element calculations.

Anti-galloping analysis of iced quad bundle conductor based on compound damping cables

Based on the strain displacement relationship of spatial curved beam theory, a nonlinear dynamic equation for a hybrid model of iced bundle conductor galloping analysis with three translational degrees of freedom and one rotational degree of freedom was established. The element independence is numerically studied, and the impact of modal truncation on the galloping response is analyzed to verify the accuracy of the model. In addition, the compound damping cable is applied to transmission lines for the first time to prevent galloping, and a corresponding nonlinear vibration control finite element equation for the iced quad bundle conductor with a compound damping cable structure under wind load is developed. The study also examines the influence of various parameters on galloping amplitude. The results show that the hybrid model accurately predicts the galloping response of transmission lines. The compound damping cable can effectively suppress the galloping of the quad bundle conductor, and the vibration reduction effect reaches more than 85%. Increasing the installation height of the compound damping cable improves its damping effect. At the same installation height, when the horizontal installation position is close to the mid-span of the conductor, the damping effect first increases and then decreases, suggesting an optimal installation position exists. Moreover, properly increasing the stiffness of the primary cable and reducing the stiffness of the return spring can improve the vibration reduction effect of the compound damping cable. Additionally, if the damping coefficient and the mass of the primary cable are properly increased, the vibration reduction effect of the compound damping cable will be better.

Nonlocal-Strain-Gradient-Based Anisotropic Elastic Shell Model for Vibrational Analysis of Single-Walled Carbon Nanotubes

In this study, a new anisotropic elastic shell model with a nonlocal strain gradient is developed to investigate the vibrations of simply supported single-walled carbon nanotubes (SWCNTs). The Sanders–Koiter shell theory is used to obtain strain–displacement relationships. Eringen’s nonlocal elasticity and Mindlin’s strain gradient theories are adopted to derive the constitutive equations, where the anisotropic elasticity constants are expressed via Chang’s molecular mechanics model. An analytical method is used to solve the equations of motion and to obtain the natural frequencies of SWCNTs. First, the anisotropic elastic shell model without size effects is validated through comparison with the results of molecular dynamics simulations reported in the literature. Then, the effects of the nonlocal and material parameters on the natural frequencies of SWCNTs with different geometries and wavenumbers are analyzed. From the numerical simulations, it is confirmed that the natural frequencies decrease as the nonlocal parameter increases, while they increase as the material parameter increases. As new results, the reduction in natural frequencies with increasing SWCNT radius and the increase in natural frequencies with increasing wavenumber are both amplified as the material parameter increases, while they are both attenuated as the nonlocal parameter increases.

Nonlinear vibrations of porous-hyperelastic cylindrical shell under harmonic force using harmonic balance and pseudo-arc length continuation methods

The resonant responses are investigated for the porous-hyperelastic Mooney–Rivlin cylindrical shell subjected to a radial harmonic excitation. Considering the higher-order shear deformation theory (HSDT), fourth-order strain-displacement relations are derived, which include the radial geometric imperfections varying along the thickness direction. Using the porous-hyperelastic Mooney–Rivlin constitution relation and Lagrange equation, the differential governing equations of motion are obtained for the porous-hyperelastic cylindrical shells. The resonant conditions are presented and the accuracy of the mathematical models is verified. The harmonic balance and pseudo-arc length continuation methods are used to obtain the amplitude-frequency and forced-amplitude curves. The stability of the solutions is analyzed by Floquet theory. The effects of the external excitation amplitudes, structure parameters and porosity infill parameters on the linear frequencies, amplitude-frequency responses and force-amplitude responses are discussed for the imperfect porous-hyperelastic cylindrical shells. The results show that the linear frequencies of the porous-hyperelastic cylindrical shells increase obviously as the uniform and sigmoid function porosity parameters reach the certain values. The increase of the structure parameters enhances the response amplitudes of the first-order mode and minified response amplitudes of the second-order mode. The decreasing porosity ratios weaken the softening nonlinear behaviors of the porous-hyperelastic cylindrical shells. With the changes of the external excitation amplitudes and structure parameters, the motion of the porous-hyperelastic cylindrical shell indicates that the synchronous vibrations occur with the period and chaotic vibrations alternately.

Harmonic in-plane patch loading induced nonlinear dynamic instability of unidirectional and bidirectional stiffened composite plates

A finite element framework is established to study the nonlinear dynamic instability of stiffened plates subjected to in-plane harmonic patch loading. An eight-noded isoparametric degenerated shell element and a three-noded curved beam element are used to model the skin and the stiffener, respectively. The Green–Lagrange strain displacement relationship is adopted to formulate the system matrices for the plate and the stiffener, adopting the total Lagrangian approach. The Bolotin method is adopted to trace the boundaries of linear dynamic instability region. The Incremental Harmonic Balance (IHB) method is used to investigate the nonlinear instability behavior of stiffened plates. The effect of varying loading patches along with the number of stiffeners in unidirectional ( x-directional and y-directional) and bidirectional ( x as well as y-directional) stiffened plates on the nonlinear frequency response is studied. It is observed that the cross-stiffened plates with the higher number of stiffeners are dynamically more stable than the unidirectional stiffened plates. Moreover, the stiffened plates with smaller loading patches are dynamically more stable than those with larger loading patches.

Nonlinear wave propagation in graphene incorporating second strain gradient theory

The exploration of graphene has attracted extensive interest owing to its significant structural and mechanical properties. In this research, we numerically investigate wave propagation in defect-free single-layer graphene, considering its geometrically nonlinear behavior through second-strain gradient elasticity. To capture the geometric nonlinearity, firstly, the nonlinear strain–displacement relations are introduced. The governing equation and associated boundary conditions are derived using Hamilton’s principle. Then, the weak form, including the element matrices, is established. The eigenvalue problem is solved for 2D wave propagation through periodic structures theory. Finally, the dynamical properties such as band structures, mode shapes, energy flow, and wave beaming effects are analyzed. The numerical results reveal that the geometric nonlinearity through the second strain gradient influences the wave propagation characteristics in graphene. The findings are significant and contribute to the understanding of graphene’s dynamic response, with implications for the engineering applications of graphene-based nanostructures.

Nonlinear free vibrational behavior of temperature-dependent two-directional functionally graded truncated cone-like shells in thermal environment

This article investigates nonlinear free vibrations of temperature-dependent two-directional functionally graded (TDTDFG) cone-like shells in thermal environment. The structure material is a combination of ceramic and metal with temperature-dependent mechanical characteristics varying continuously in the thickness and longitudinal directions. Initially, the thermomechanical dynamic equations of the system are modeled based on the first-order shear deformation theory (FSDT) and the strain-displacement relations of von Kármán. Afterward, the final governing equation for the structure’s transverse motion is obtained using the Galerkin discretization method. Finally, the governing equation is solved using improved Lindstedt-Poincaré method as an analytical technique, resulting in an equation for calculating the nonlinear frequencies of the shell. To validate the research’s results, the system frequencies are calculated under different conditions and compared with the results of previous studies. After ensuring the accuracy of the results, a parametric study is performed to evaluate the effects of various parameters on the linear and nonlinear frequencies of the TDTDFG cone-like shell in thermal environment. The results of this study have demonstrated that gradient indices, geometric ratios, and thermal environment have a significant influence on the nonlinear vibration behavior of structures.

Tension buckling and postbuckling of nanocomposite laminated plates with in-plane negative Poisson’s ratio

Abstract Mechanical metamaterials with negative Poisson’s ratio (NPR) have emerged as a novel class of engineering material, and have attracted increasing attention in various engineering sectors. Most studies available on the buckling problem of laminated plates with positive or NPR are those under uniaxial compression. Here, we report that the buckling phenomenon may occur for auxetic nanocomposite laminated plates under uniaxial tension when the unloaded edges of the plates are immovable. Two types of nanocomposites are considered, including graphene/Cu and carbon nanotube/Cu composites. Governing equations of the auxetic nanocomposite laminated plates are formulated based on the framework of Reddy’s higher-order shear deformation theory. In modeling, the von Kármán nonlinear strain–displacement relationship, temperature-dependent material properties, thermal effects, and the plate–substrate interaction are considered. The explicit analytical solutions for postbuckling of auxetic nanocomposite laminated plates subjected to uniaxial tension are obtained for the first time by employing a two-step perturbation approach. Numerical investigations are performed for tension buckling and postbuckling behaviors of auxetic nanocomposite laminated rectangular plates with in-plane NPR rested on an elastic substrate under temperature environments.

A novel uncertainty quantification method for determining deformations and reliabilities of stochastic laminated composite plates with geometric nonlinearity

Stochastic finite element method (SFEM) for uncertainty quantification is widely applied for analysis of structures with intrinsic randomness. For determining the geometrically nonlinear deformations of laminated composite plates with random fields, the existing intrusive SFEMs have the limitations of low applicability, insufficient accuracy, or low efficiency. To this end, this paper proposes a novel and efficient non-intrusive SFEM incorporating direct probability integral method to achieve probability density functions (PDFs) of stochastic responses and reliabilities of laminated composite plates with geometric nonlinearity. Firstly, the von Kármán strain-displacement relation based on the third-order shear deformation theory is employed to model the geometric nonlinearity of laminated plates. The random field is discretized via Karhunen–Loève expansion. Secondly, the probability density integral equation (PDIE) is derived from the new perspective of probability conservation. The proposed non-intrusive SFEM decouples the equilibrium equation and PDIE to compute the response PDFs and reliabilities of uncertain laminated composite plates in a unified way. Moreover, the criterion which can judge the applicability of geometrically nonlinear theory is suggested for performing uncertainty quantification. Finally, comparisons of the results in terms of Monte Carlo simulation and the literature demonstrate the high accuracy and efficiency of the proposed method. For stochastic laminated plates, the response statistical moments vary nonlinearly with linear increase of load amplitude due to geometric nonlinearity, the deflection variability increases and structural reliability decreases with the increase in variability and correlation length of random field, and the stacking angle significantly affects the stochastic nonlinear deflections and reliabilities.

Absolute Nodal Coordinate Formulation-Based Shape-Sensing Approach for Large Deformation: Three-Dimensional Beam

In recent years, fiber-optic shape sensing, which means reconstructing the deformation state of structures from strain information measured by fiber-optic sensors, has attracted interest from many fields. Most of the existing shape-sensing research for flexible three-dimensional (3D) slender structures is based on simple strain-curvature transformation, relying on standardized substrates with a circular cross section and particular forms of fiber layouts. To develop a more general shape-sensing method that can be applied to a wider range of practical engineering conditions, in this paper, the strain–displacement relationship of 3D slender structures under the effect of multiple deformation modes coupling is described by introducing absolute nodal coordinate formulation element, and the deformation reconstruction problem is reorganized into a nonlinear optimization problem that can be applied to large deformation and accommodates different cross-sectional shapes and optical fiber layouts. Due to the rapid increase in the calculation difficulties of nonlinear optimization problems with the number of variables, an element-by-element solving strategy is adopted, and nodal degrees of freedom that have less influence on the overall shape of the structure are merged. In addition, in order to address the issue of local strain anomalies caused by unmodeled factors such as section warping, the pointwise matching between the theoretical strains and measured strains is relaxed to an average matching in subregions to capture the overall deformation, improving the robustness and computational efficiency of the solution process. The accuracy and computational performance of the proposed method are verified through numerical simulation and experiment.

Honeycomb structure filling morphing wing trailing edge: Design strategy, deformation feedback, and active control

Abstract Pursuing highly efficient aerodynamic efficiency in aircraft has driven the development of morphing wing technology. However, there are still limitations to morphing wing technology, including adaptation of load and deformation, and deformation monitoring and control. This work introduces an intelligent trailing edge structure that balances deformation and load-bearing and achieves deformation monitoring and active control. Firstly, we employ a honeycomb structure for non-uniform filling of the trailing edge. The filling method is obtained through inverse design using a genetic algorithm based on neural networks, allowing the device to undergo continuous deformation while meeting load-bearing requirements. The bending deformation of the wing is achieved using shape memory alloy (SMA) wire. Additionally, we design and fabricate a metal-based multichannel flexible sensor, and based on beam bending theory, we establish the strain–displacement relationship. These sensors are affixed to the trailing edge surface, enabling real-time monitoring and active control of trailing edge deformation. Building an experimental platform to test this system, the results show that the sensors can accurately give feedback on the degree of wing deformation, and the error of active deformation control technology is less than 4%. This provides a new method for the deformation feedback control closed-loop system of intelligent variant wings.

Nonlinear isogeometric analysis of axially functionally graded graphene platelet-reinforced composite curved beams

The linear and nonlinear isogeometric finite element models of an axially functionally graded graphene platelet-reinforced composite (AFG-GPLRC) curved beam are established within the framework of the third-order shear deformation beam theory (TSDT) and von-Kármán’s nonlinear geometric relation. The AFG-GPLRC curved beams can be seen as composite structures in which the graphene platelets (GPLs) are continuously distributed in the matrix along the length direction of the curved beam according to different patterns. The modified Halpin-Tsai parallel model and the rule of mixture are implemented to predict the effective Young’s modulus and mass density as well as Poisson’s ratio, respectively. Hamilton's principle, TSDT, and von-Kármán’s strain-displacement relation are combined to derive the governing partial differential equation of motion and corresponding boundary conditions. Furthermore, the Non-Uniform Rational B-splines (NURBS)-based isogeometric analysis (IGA) approach together with a direct iterative technique are utilized to solve the nonlinear governing equation. The accuracy and efficiency of the proposed IGA framework are confirmed by comparing corresponding numerical solutions with other available results. The parametric investigations, such as the curved beam’s geometric parameters, boundary conditions, and GPL’s distribution patterns, on the nonlinear bending and vibration responses of the AFG-GPLRC curved beams are carried out by some illustrative examples.