Transverse Displacement Field Research Articles

Non-standard interface conditions in flexure of mixture unified gradient Nanobeams

Structural schemes of applicative interests in Engineering Science frequently encounter the intricate phenomenon of discontinuity. The present study intends to address the discontinuity in the flexure of elastic nanobeam by adopting an abstract variational scheme. The mixture unified gradient theory of elasticity is invoked to realize the size-effects at the ultra-small scale. The consistent form of the interface conditions, stemming from the established stationary variational principle, is meticulously set forth. The boundary-value problem of equilibrium is properly closed and the analytical solution of the transverse displacement field of the elastic nanobeam is addressed. As an alternative approach, the eigenfunction expansion method is also utilized to scrutinize the efficacy of the presented variational formulation in tackling the flexure of elastic nanobeams with discontinuity. The flexural characteristic of mixture unified gradient beams with diverse kinematic constraints is numerically illustrated and thoroughly discussed. The anticipated nanoscopic features of the characteristic length-scale parameters are confirmed. The demonstrated numerical results can advantageously serve as a benchmark for the analysis and design of pioneering ultra-sensitive nano-sensors. The established variationally consistent size-dependent framework paves the way ahead in nanomechanics and inspires further research contributing to fracture mechanics of ultra-small scale elastic beams.

Improved Finite Element Thermomechanical Analysis of Laminated Composite and Sandwich Plates Using the New Enhanced First-Order Shear Deformation Theory

This paper proposes a simple yet accurate finite element (FE) formulation for the thermomechanical analysis of laminated composites and sandwich plates. To this end, an enhanced first-order shear deformation theory including the transverse normal effect based on the mixed variational theorem (EFSDTM_TN) was employed in the FE implementation. The primary objective of the FE formulation was to systematically interconnect the displacement and transverse stress fields using the mixed variational theorem (MVT). In the MVT, the transverse stress field is derived from the efficient higher-order plate theory including the transverse normal effect (EHOPT_TN), to enhance the solution accuracy, whereas the displacement field is defined by the first-order shear deformation theory including the transverse normal effect (FSDT_TN), to amplify the numerical efficiency. Furthermore, the transverse displacement field is modified by incorporating the components of the external temperature loading, enabling the consideration of the transverse normal strain effect without introducing additional unknown variables. Based on the predefined relationships, the proposed FE formulation can extract the C0-based computational benefits of FSDT_TN, while improving the solution accuracy for thermomechanical analysis. The numerical performance of the proposed FE formulation was demonstrated by comparing the obtained solutions with those available in the literature, including 3-D exact solutions.

Machine learning-based modelling, feature importance and Shapley additive explanations analysis of variable-stiffness composite beam structures

In the current work, the displacement and inner stress state of orthotropic, composite, variable-stiffness beam structures is investigated, combining the extended Kantorovich method (EKM) with machine learning (ML) modeling and explainable artificial intelligence (AI) techniques. The complete displacement and inner stress fields of variable stiffness (VS) composite beams with a through-thickness variation of their material properties are computed for different orthotropy, material gradation and slenderness attributes. Both deep learning and tree-based machine learning architectures are considered, identifying high accuracy and low computational cost modeling architectures. Thereupon, the significance of underlying fundamental design features is assessed, classifying their importance for the first time. In particular, material orthotropy, stiffness gradation and slenderness are analyzed for different boundary conditions, elucidating their interdependence and relative significance. It is shown that reduced end kinematic constraints overall favor the importance of stiffness gradation. Differences in the importance of each design feature for the transverse displacement and inner normal and shear stress fields are identified and quantified using Shapley additive explanations (SHAP) analysis. It is found that material orthotropy is more important than stiffness gradation for the transverse displacement rather than for the normal or shear stress fields developed, for which their relative importance is inverted. What is more, evidence is provided that increased gradation and orthotropy values have opposite signed contributions to the stress fields developed at critical positions of the VS composite, while they uniquely yield equal-signed contributions to the transverse displacement field.

Buckling response of CNT based hybrid FG plates using finite element method and machine learning method

In this study, a C0 finite element model (FEM) based on modified third-order shear deformation (MTSDT) theory in conjunction with a deep neural network (DNN), extreme gradient boosting (XGBoost), and random forest (RF) models are implemented to predict the critical buckling load of a carbon nanotube (CNT)-reinforced hybrid functionally graded (FG) plate structure Cubic and quadratic variations of the thickness terms are adopted in the longitudinal and transverse displacement fields, respectively. A zero transverse shear condition is assumed for the outermost surfaces of the plate. A C0 FEM is developed and coded in a MATLAB environment. The MTSDT-based FEM is validated with previous literature and found suitable for finding new results. The DNN model is coded in Python and XGBoost and the RF models are coded in RStudio. Using the random value of the elastic properties of the material as an input parameter and the buckling strength as an output parameter, a regression problem is expressed using the given machine learning method to predict the buckling response of the CNT-reinforced hybrid FG plates.

Spin and orbital metallic magnetism in rhombohedral trilayer graphene

We provide a complete theoretical interpretation of the metallic broken spin/valley symmetry states recently discovered in ABC trilayer graphene (ABC) perturbed by a large transverse displacement field. Our conclusions combine insights from ABC trilayer graphene electronic structure models and mean field theory, and are guided by precise magneto-oscillation Fermi-surface-area measurements. We conclude that the physics of ABC trilayer graphene is shaped by the principle of momentum-space condensation, which favors Fermi surface reconstructions enabled by broken spin/valley flavor symmetries when the single-particle bands imply thin annular Fermi seas. We find one large outer Fermi surface enclosed majority-flavor states and one or more small inner hole-like Fermi surfaces enclosed minority-flavor states that are primarily responsible for nematic order. The smaller surfaces can rotate along a ring of van-Hove singularities or reconstruct into multiple Fermi surfaces with little cost in energy. We propose that the latter property is responsible for the quantum oscillation frequency fractionalization seen experimentally in some regions of the carrier-density/displacement-field phase diagram.

Application of UPV-Instrument in Health Monitoring of Indian Rail Section Using AE Technique

In present work, buckling analysis of laminated composite plates is carried out using recently proposed higher-order zigzag theory (HOZT). Third order variation of in-plane displacement field is taken across the thickness of the plate. For predicting the behavior efficiently for thick plates, quadratic variation of transverse displacement field is assumed for core layer and constant for face layers. The present theory satisfies inter-laminar transverse shear stress continuity condition at interface along with zero value at top and bottom surface of the plate. In present study, nine-noded finite element having eleven degrees of freedom per node is used. Present model is free from requirement of any kind of penalty function or C-1 condition.

Buckling Analysis of Laminated Composite Plates under Thermal Conditions

In present work, buckling analysis of laminated composite plates is carried out using recently proposed higher-order zigzag theory (HOZT). Third order variation of in-plane displacement field is taken across the thickness of the plate. For predicting the behavior efficiently for thick plates, quadratic variation of transverse displacement field is assumed for core layer and constant for face layers. The present theory satisfies inter-laminar transverse shear stress continuity condition at interface along with zero value at top and bottom surface of the plate. In present study, nine-noded finite element having eleven degrees of freedom per node is used. Present model is free from requirement of any kind of penalty function or C-1 condition.

A cell-based smoothed radial point interpolation method applied to kinematic limit analysis of thin plates

The cell-based smoothed radial point interpolation method in conjunction with a second order cone programming is proposed for kinematic limit analysis of rigid-perfectly plastic thin plates in this paper. The transverse displacement field is interpolated by the radial point interpolation method (RPIM) and no rotational degrees of freedom are involved. The gradient smoothing technique is utilized to construct smoothed curvature field in every smoothing domain and there is no need to compute the second derivatives of shape functions. The rotational boundary conditions are satisfied in the process of curvature smoothing and the translational boundary conditions can be directly enforced without any special treatment. The limit analysis problem of thin plates is formulated by minimizing the dissipation power subject to a set of equality constraints and this minimization problem can be expressed easily as a standard second order cone programming. It is testified from the computational results that the proposed procedure can provide reasonable and satisfactory upper bound limit load multipliers for thin plates.

Two-phase elastic axisymmetric nanoplates

In the present work, the two-phase integral theory of elasticity developed in Barretta et al. (Phys E 97:13–30, 2018) for nano-beams is generalized to model two-dimensional nano-continua. Notably, a well-posed mixture local/stress-driven nonlocal elasticity is proposed to accurately predict size effects in Kirchhoff axisymmetric nanoplates. The key idea is to express the elastic radial curvature as a convex combination of local and nonlocal integral responses, that is a coherent choice motivated by virtue of the plate axisymmetry. The relevant structural problem is shown to be governed by a set of integro-differential equations, whose solution is computationally onerous. Thus, Helmholtz’s averaging kernel is advantageously adopted, since it enables explicit inversion of the integral constitutive law by virtue of an equivalence property. Specifically, the elastostatic problem of axisymmetry nanoplates is equivalently formulated in a differential form whose solution in terms of transverse displacement field is governed by nonlocal and mixture parameters. A parametric study is performed for case studies of applicative interest, and numerical solutions are finally provided and discussed. The presented methodology can be adopted to design and optimization of plate-based nano-electro-mechanical-systems (NEMS).

A sub-laminate based higher-order model for bending of laminated beams containing multiple delaminations

The paper presents a new sub-laminate based higher-order laminated model (HOLM) for modelling the bending response of laminated composite and sandwich beams with and without delaminations of any configuration. The variation of in-plane and transverse displacement fields through the thickness of a sub-laminate is assumed cubic and quadratic, respectively. Such variation involves specifying displacement parameters at the top and bottom surfaces of sub-laminates that provide significant benefit in conveniently modelling perfect and delaminated multilayered beams with any desired level of accuracy. The option exists to therefore trade-off between prediction and computational efficiency by adapting a suitable sub-laminate modelling scheme. A C0 continuous beam element with quadratic in-plane variation of displacements is developed for implementation of the proposed laminate model. The versatility and accuracy of the model is demonstrated by via a number of numerical examples of composite and sandwich beams, covering a range of variables such as boundary conditions, loading, and delamination configuration. A number of the newly developed results reported in this paper will serve as baseline data for delaminated structures.

New refined higher-order shear deformation theories for functionally graded plates conforming to graded variations of material properties

This paper proposes new refined higher-order shear deformation theories (RHSDTs) for functionally graded (FG) plates conforming to graded variations of material properties. The theoretical formulation starts with initial displacement descriptions based on general higher-order theories, determines in-plane stress fields using geometrical and constitutive relations, and further achieves transverse shear stress fields though equilibrium equations. By imposing tangential stress-free conditions and defining new variables, the transverse shear stress expressions without differential variables are obtained. Then, the uniform and quadratic distributions of the transverse displacement field are considered, respectively, and the final displacement expressions of the present 2D and quasi-3D RHSDTs are further formulated by integrating the associated shear strain expressions. This novel formulation approach first introduces gradient parameters of FG materials into the shear strain shape functions. Compared with traditional HSDTs, the present refined theories fully incorporate the influences of the graded variations of material properties and can better predict mechanical responses for FG plates. Moreover, the governing equations and associated analytical solutions are presented. Numerical examples demonstrate high accuracy of the present RHSDTs and reflect their obvious accuracy advantages over traditional HSDTs especially when predicting the transverse shear stresses for FG plates with arbitrary gradients. Furthermore, it is also validated that the present RHSDTs with more shear expansion terms have higher accuracy for the prediction of transverse shear stress fields.

Analysis on excavation stability of tunnel portal with different design methods

The stability of tunnel portal has its particularity. Due to the influence of natural or human factors, it is easy to produce cracking, landslide, collapse and other diseases. Taking the No.5 highway tunnel of F3 bid in Georgia as the background, using the finite element analysis software MIDAS-GTS to establish a three-dimensional model to simulate the tunnel construction process. The tunnel portal adopts two design schemes of micro pile wall support and bolt mesh shotcrete support. Through the analysis and comparison of the transverse and vertical displacement and stress field characteristics of surrounding rock caused by two schemes of tunnel excavation, the influence of tunnel excavation on the stability of tunnel portal is studied. The numerical results show that the maximum transverse and longitudinal displacements of surrounding rock are 3.55mm and 8.73mm respectively, and the maximum tensile stress is 1.91mpa. Compared with the micro pile wall support design, the bolt mesh shotcrete support scheme is more feasible. The conclusion has a certain guiding role for the construction of highway tunnels in mountain areas.

Electric-Field-Tunable Valley Zeeman Effect in Bilayer Graphene Heterostructures: Realization of the Spin-Orbit Valve Effect.

We report the discovery of electric-field-induced transition from a topologically trivial to a topologically nontrivial band structure in an atomically sharp heterostructure of bilayer graphene (BLG) and single-layer WSe_{2} per the theoretical predictions of Gmitra and Fabian [Phys. Rev. Lett. 119, 146401 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.146401]. Through detailed studies of the quantum correction to the conductance in the BLG, we establish that the band-structure evolution arises from an interplay between proximity-induced strong spin-orbit interaction (SOI) and the layer polarizability in BLG. The low-energy carriers in the BLG experience an effective valley Zeeman SOI that is completely gate tunable to the extent that it can be switched on or off by applying a transverse displacement field or can be controllably transferred between the valence and the conduction band. We demonstrate that this results in the evolution from weak localization to weak antilocalization at a constant electronic density as the net displacement field is tuned from a positive to a negative value with a concomitant SOI-induced splitting of the low-energy bands of the BLG near the K(K^{'}) valley, which is a unique signature of the theoretically predicted spin-orbit valve effect. Our analysis shows that quantum correction to the Drude conductance in Dirac materials with strong induced SOI can only be explained satisfactorily by a theory that accounts for the SOI-induced spin splitting of the BLG low-energy bands. Our results demonstrate the potential for achieving highly tunable devices based on the valley Zeeman effect in dual-gated two-dimensional materials.

Experimental methodology to assess the dynamic equivalent stiffness properties of elliptical orthotropic plates

This paper deals with the dynamic characterisation of plate structures with elliptical orthotropic stiffness properties, using an equivalent thin plate theory using a wave fitting approach. The method consists in projecting the experimentally determined transverse displacement field of a plate on an analytical Green’s function of an elliptical orthotropic plate based on Hankel’s functions. The error between the projected and measured fields is then minimized, varying the characteristics of the function until an optimal fit is reached. The thus obtained characteristics are the two flexural rigidities defining the elliptical orthotropy of the plate, and the orthotropy angle. This fitting procedure is applied at each frequency, enabling the determination of frequency dependent dynamic material properties. The method is applied to a honeycomb sandwich panel to validate the proposed fitting approach. The identified flexural rigidities are compared to the estimations obtained by means of an analytical model and the IWC (Inhomogeneous Wave Correlation) method assuming three different type of plate characteristics (anisotropic, orthotropic and elliptical orthotropic). For the elliptical orthotropic assumption, consistent results are observed between the methods and the model over a large frequency range (from 1 to 50 kHz).

Comparative study on the bending of sandwich FGM beams made up of different material variation laws using refined layerwise theory

Comparative study is carried out in present work on sandwich functionally graded beams made up of different material property variation laws. For study, power, exponential and sigmoidal laws are used. A fourth-order zigzag theory is used for the analysis. Both in-plane and transverse displacement fields are considered to predict the behavior of thick beams more efficiently. A 3-noded 1-D finite element having 8 degrees of freedom per node is used during analysis. The present model satisfies inter-laminar transverse stress continuity conditions at interfaces along with zero value at the top and bottom surfaces of the beam for transverse shear stresses. The current model is free from the requirement of any kind of penalty or C-1 conditions and hence is computationally efficient. Present results are validated with those available in the literature. Results for exponential and sigmoidal law are new results in present work, which will serve as a benchmark for future studies. Results for stresses and deflection are presented in form of tables. For some cases, stress variation across the thickness of beam are also reported. A modified form of exponential and sigmoidal law named Type-E1 and Type-S2 are also presented in which central core is made up of FGM phase. Among all the material variation laws, sandwich FGM beam having ceramic face sheets and exponentially varying FGM core (C-Type-E1) is found to perform best among all the cases studied.

A new three-node element for bending, free vibration and buckling analysis of composite laminated beams based on FSDT theory

In this paper, a new three-node element with three degrees of freedom per node is proposed for bending, free vibration and buckling analysis of laminated beams. The element’s formulation is based on the first-order shear deformation theory (FSDT). For this aim, transverse displacement and rotation field of the element are selected from fifth and fourth order, respectively. Moreover, the shear strain is varied as quadratic function throughout the element. It is worth noting that the quadratic function can be used for axial displacement field. By employing equilibrium equation, curvature and shear strain relations of laminated beam with FSDT theory, the exact and explicit shape functions of the displacement fields are obtained. By utilizing the obtained shape functions, the explicit form of the stiffness matrix is calculated for the element. On the other hand, by using the governing equation of the free vibration and buckling of the beam, the explicit form of the translation and rotary mass matrices, and geometric stiffness matrix of the element are obtained. It should be mentioned, the proposed element is free of shear locking. Finally, several numerical tests fulfill to assess the robustness of the developed element. For this purpose, bending, free vibration and buckling analysis of laminated beams with different boundary conditions and aspect ratios are performed. The results of the numerical tests demonstrate high accuracy and efficiency of the proposed element for free vibration and buckling analysis of laminated beams.

Novel higher-order zigzag theory for analysis of laminated sandwich beams

In the present work, a new higher-order zigzag theory is proposed for the analysis of laminated sandwich beams under static and free vibration conditions. Fourth-order in-plane and transverse displacement fields are chosen along with linear unit Heaviside step function. The present theory satisfies interlaminar transverse stress continuity conditions along with zero value at the top and bottom surface for transverse shear stresses. The proposed approach is also free from any kind of C-1 or penalty requirements. A three-noded one-dimensional finite element having eight degrees of freedom per node is used during analysis. The efficiency of the proposed model is carried out by comparing the present results with those available based on elasticity solutions and zigzag theories in the literature. New results are also reported in the present work, which will serve as a benchmark for future studies. The influence of boundary condition on the nature of stress distribution across the length of beam and frequencies of the beam with different end conditions is also carried out. A comparative study has also been carried out between symmetric and unsymmetric laminated sandwich beam.

Bending analysis of functionally graded sandwich plates using HOZT including transverse displacement effects

In the present work, a novel higher-order zigzag theory is proposed for the bending analysis of sandwich functionally graded plates. Zigzag effects are incorporated with the help of a linear Heaviside step function. The present theory satisfies zero transverse shear stress conditions at the bottom and top surface of the plate along with continuity condition at interfaces for both transverse shear and transverse normal stresses. The proposed model is free from any kind of C-1 or penalty function requirements. A nine-noded isoparametric finite element having 12 degrees of freedom per node is used during analysis. Since the present theory incorporates the transverse displacement field along with continuity conditions and hence able to predict the behavior of thick sandwich plates more efficiently.

Analysis of non-skew and skew laminated composite and sandwich plates under hygro-thermo-mechanical conditions including transverse stress variations

A new fourth-order in-plane displacement field based refined higher-order zigzag theory is proposed for analysis of laminated sandwich plate (for both skew and non-skew shaped) subjected to hygro-thermal conditions. In order to predict behavior of thick laminated sandwich plates, third-order transverse displacement field is taken. Zig-zag effects are introduced using linear unit Heaviside step function. The theory satisfies zeros transverse normal and shear stress condition at the bottom and top surface of the plate along with continuity condition at interface. The proposed model is free from any kind of C-1 or penalty function requirements. Nine-noded isoparametric finite element having twelve degrees of freedom per node is used during analysis. Since, the present theory incorporates transverse displacement field along with continuity conditions, is able to predict the behavior of thick sandwich plates more efficiently. In literature no results are present for skew laminated composite and sandwich plates therefore, present results for skew plates are entirely new and will serve as benchmark for future studies.

Free vibration analysis of laminated sandwich plates under thermal loading

In present work, free vibration analysis of laminated sandwich plates is carried out under thermal loading. C-0 finite element based higher-order zigzag theory is used during analysis. Third-order variation of in-plane displacement field is assumed while for transverse displacement field, quadratic variation is used for core region and is taken constant for the face layers. Present model is free from any kind of penalty requirements. Also, the proposed model satisfies transverse shear stress continuity condition at interface and same is zero at top and bottom faces. Present results are compared with those available in literature. New results for sandwich plates are proposed in the present work which will serve as benchmark for future studies.