Deformation Theory Research Articles

Large Deflection Analysis of Bimodular Functionally Graded Truncated Thin Conical Shells Under Mechanical and Thermal Loads

The purpose of this study is to analyze the large deflection problem of bimodular functionally graded truncated thin conical shells under the transverse mechanical load and non-uniform thermal load, in which two different boundary constraints of the truncated shell with two ends simply supported and fully fixed are considered. It is assumed that the temperature distribution along the thickness direction satisfies the Fourier law of heat transfer, and the material properties change exponentially along the thickness direction while different properties in tension and compression are considered. The geometric equation of the conical shell is established based on the equivalent method of curvature correction of von-Kármán deformation theory, and the analytical solution of the problem is obtained by Ritz method. Numerical simulation of bimodular functionally graded conical shells under the thermal and mechanical loads is carried out by Abaqus, and the numerical solution agrees with the theoretical solution. The results show that the introduction of bimodular functionally graded material will affect the maximum displacement and this effect has different rules under the mechanical load and thermal load. In addition, factors such as the cone apex angle and the truncated distance have a great influence on the maximum displacement and its location of the conical shell.

A Mathematical Approach to the Buckling Problem of Axially Loaded Laminated Nanocomposite Cylindrical Shells in Various Environments

In this study, the solution of the buckling problem of axially loaded laminated cylindrical shells consisting of functionally graded (FG) nanocomposites in elastic and thermal environments is presented within extended first-order shear deformation theory (FOST) for the first time. The effective material properties and thermal expansion coefficients of nanocomposites in the layers are computed using the extended rule of mixture method and molecular dynamics simulation techniques. The governing relations and equations for laminated cylindrical shells consisting of FG nanocomposites on the two-parameter elastic foundation and in thermal environments are mathematically modeled and solved to find the expression for the axial buckling load. The numerical results of the current analytical approach agree well with the existing literature results obtained using a different methodology. Finally, some new results and interpretations are provided by investigating the influences of different parameters such as elastic foundations, thermal environments, FG nanocomposite models, shear stress, and stacking sequences on the axial buckling load.

Geometry optimization of auxetic honeycomb and dynamic analysis of sandwich plate and shell panels with variable stiffness composite laminates

In the present investigation, the fiber orientations and geometric parameters of the sandwich plate and shell panels with auxetic honeycomb core and curvilinear fiber-reinforced facesheets are optimized for the maximum fundamental frequency. The dynamic analysis is carried out considering a higher-order shear deformation theory and employing a C0 eight-noded isoparametric element with nine degrees of freedom per node. The artificial bee colony algorithm is used to optimize different parameters of the auxetic core of the sandwich panel. The mechanical properties of the original material and the geometrical features of the unit cells are used to determine the mechanical properties of the auxetic core. The effects of fiber orientations, and geometric parameters like rib thickness, inclined cell angle, vertical cell rib length, and inclined cell rib length of the core on maximizing fundamental frequency parameters of the panels are investigated. It is evident from the present study that the natural frequency of the panels can be altered without changing their plan dimensions and mass by suitably altering the geometric parameters of the auxetic honeycomb core. The analysis is extended for the different ratios of the density of facesheets and auxetic core materials. The analysis is carried out for three and five-layer sandwich panels with different facesheet-auxetic core arrangements and boundary conditions. For five layered auxetic honeycomb sandwich plates and shell panels, the non-dimensional fundamental frequency is observed to be lower for sandwich with alternate facesheet-core arrangement i.e., three facesheets and two core layers, compared to that of a single core at the middle. The non-dimensional frequency parameter is also found to be sensitive to geometry parameters of auxetic core and fiber angles of the facesheets.

Analysis and optimization for buckling behavior of functionally graded face layers and porous core sandwich plates resting on pasternak elastic foundation

In this paper, the first-order shear deformation theory (FSDT) is employed to derive analytical solutions for the buckling analysis of functionally graded material—porous—functionally graded material (FGM-porous-FGM) sandwich plate resting on Pasternak elastic foundation. The top and bottom layers of the plate consist of functionally graded materials that vary according to the exponential law along the thickness direction, while the core layer is composed of a porous material. The governing equations are derived by using the principle of minimum total potential energy. Based on the Navier’s solution, the displacements are obtained to satisfy the boundary conditions. The results are then compared with existing published literature showing the accuracy of the proposed solutions. Moreover, parametric studies investigate the effects of material parameters, geometric dimensions, foundation coefficients and face-core-face thickness ratios on the critical buckling load of the rectangular FGM-porous-FGM sandwich plate. In the case of uniaxial loading, the critical bucking load is almost twice for the case of biaxial loading, no matter the value of volume fraction index. For all the cases of face-core-face thickness ratio, the biaxial critical buckling load linearly depends on the porosity coefficient. The porosity coefficient is crucial when and only when the volume of the core is almost eight times bigger than the volume of the face layer. Optimization studies are then presented in order to: (i) maximize the critical buckling load and (ii) optimize the input variables corresponding to a given value of the critical buckling load. The proposed study aims to provide valuable insights into the design and application of FGM-porous-FGM sandwich plates in various engineering fields, contributing to the development of more resilient and efficient structural components.

Elementary-level intrusive coupling of machine learning for efficient mechanical analysis of variable stiffness composite laminates: a spatially-adaptive fidelity-sensitive computational framework

AbstractMechanical analysis of the complex configurations of composite laminates can be computationally prohibitive based on accurate higher-order theories, especially when the analyses involve multiple realizations corresponding to different sets of input parameters such as uncertainty quantification, optimization, reliability and sensitivity analysis. Efficient lower-order theories should not be adopted in such situations since the error accumulates with multiple realizations, leading to poor outcomes. We propose an elementary-level coupling of machine learning for efficient, yet accurate mechanical analysis of laminated composites based on finite element simulations coupled with gaussian process regression. The generic parameter space of material properties, mesh size, number of layers, and ply angle in composite laminates are accounted for forming an efficient mapping with the augmentation of lower-order theory-based elementary-level structural matrices. The computationally efficient machine learning models predict the difference in the elements of the stiffness matrix for higher-order zigzag theory (HOZT) and first-order shear deformation theory (FSDT) at the first stage. Based on such machine learning-based difference mapping, we augment the elementary stiffness matrices obtained using FSDT efficiently to the equivalent of HOZT theory without any additional computational expenses (referred to here as augmented FSDT, or aFSDT). However, it is not necessary to augment all the elements in the analysis domain which might otherwise lead to unnecessary computational expenses and loss in accuracy. To achieve an optimal level of computational efficiency and accuracy, we further propose spatially-adaptive fidelity-sensitive coupling of machine learning, only for the elements within the analysis domain where it is necessary to adopt higher-order theories. The selective augmentation strategy essentially brings in a scope of integrating physics-based insights of critical stress resultant distribution into the algorithm based on best theory diagram. Subsequently, the global structural matrices are computed exploiting such adaptive criteria containing a mixed set of elements formed using FSDT and aFSDT, which leads to an accuracy equivalent to HOZT in the mechanical analysis of composite laminates almost at the computational expense of FSDT. The proposed spatially-adaptive fidelity-sensitive scheme ensures optimal performance in terms of computational efficiency by augmenting selective elements while minimizing the loss of accuracy due to the involvement of surrogates. Detailed numerical results are presented for static, dynamic and stability characterization of composite laminates including the demonstration for variable stiffness composite configurations based on the efficient machine learning-assisted elementary-level intrusive computational framework, wherein the notion of engineering judgement is introduced concerning the trade-off between computational efficiency and required level of accuracy.

Numerical and experimental approach on vibration characterization of non-uniform hybrid sandwich plates

In this paper, the free vibrational response of various non-uniform composite sandwich plates with different hybrid honeycomb core types and varied weight percentages of multi-walled carbon nanotube (MWCNT) is analyzed experimentally and numerically. The governing differential equations of motion of the various honeycomb non-uniform composite sandwich plates are derived using higher order shear deformation theory and numerically solved. The various non-uniform fiber-reinforced composite face plate configurations were fabricated using the vacuum-assisted hand lay-up technique with the ply-dropping off at different domains to achieve different taper configurations. The honeycomb core was fabricated using a corrugated die, and hand lay-up method to yield various configurations. In the corrugated honeycomb core, apart from the standard hexagonal shape, composite laminated strips were longitudinally reinforced between the corrugated shape continuously and intermediately to increase the strength and damping property of the structure. The various weight percentages of R-COOH functionalized MWCNT were distributed homogeneously into epoxy using the titanium probe-assisted ultra-sonication method. Material properties such as Young’s modulus and Poison’s ratio, were evaluated for the face plate using ASTM E1876. An alternative dynamic approach was carried out to obtain the shear moduli along the corrugated direction (Gxz ) and joining direction (Gyz ) of the various core models with and without MWCNT reinforcement. An experimental and numerical investigation was performed on the various prototypes of non-uniform sandwich plates with various hybrid honeycomb core material to obtain natural frequencies and loss factors under various boundary conditions. The effect of MWCNT reinforcement in face sheet and various honeycomb core material, aspect ratio at various boundary conditions on the free vibration and transverse vibrational responses of the structure was also studied. It was found that the MWCNT reinforcement in the various honeycomb core materials increases the shear modulus considerably, thus enhancing the overall dynamic behavior of the various non-uniform honeycomb composite sandwich plates.

Deformation theory and energy mechanism of cyclic dynamic mechanical damage for granite in the diversion tunnel under cyclic loading-unloading

Deformation theory and energy mechanism of cyclic dynamic mechanical damage for granite in the diversion tunnel under cyclic loading-unloading

A force-based beam element model based on the modified higher-order shear deformation theory for accurate analysis of FG beams

A force-based beam element model based on the modified higher-order shear deformation theory for accurate analysis of FG beams

Spectral method for bending analysis of rectangular functionally graded plates

This study investigates the bending behavior of functionally graded rectangular plates using the spectral collocation method to offer precise modeling. We apply the First-order Shear Deformation Theory (FSDT) to accurately capture the effects of shear deformation in our plate behavior models. The effectiveness of the proposed method is demonstrated by a comparison of the results obtained with those of literature.

Stability and bifurcation analysis of simply supported rectangular nanoplate embedded on visco-Pasternak foundation based on first-order shear deformation theory

Vibrations of simply supported nanoplates embedded on visco-Pasternak foundation and stressed by in-plane periodic forces are investigated. The governing size-dependent equations employ the first-order shear deformation theory, nonlinear von Kármán strains, and the nonlocal elasticity theory. Reducing the governing system is carried out by the Bubnov-Galerkin method, which is based on two-mode model, thus the resulting system contains two ordinary differential equations with size- and time-dependent coefficients. Analysing the linearized system, the stability of the nanoplate structure is studied depending on the foundation parameters, force parameters and small-scale parameter. Nonlinear dynamics approaches are employed to determine the nature of vibration after the loss of stability. Time history, Poincare section, Lyapunov exponent are analysed for chosen values of excitation parameters, visco-Pasternak foundation parameters to indicate small-scale effects and nonlinear effects. The novelty of the work consists of the combined application of the nonlocal theory, first-order shear deformation theory, Floquet theory, methods of nonlinear dynamics and obtaining new results by a numerical experiment.

Advanced Dynamic Thermal Vibration of Thick Composited FGM Cylindrical Shells with Fully Homogeneous Equation by Using TSDT and Nonlinear Varied Shear Coefficient

A numerical method using advanced nonlinear shear is used to study the thermal vibration of functionally graded material (FGM) thick circular cylindrical shells. The third-order shear deformation theory (TSDT) of displacements is applied and the equations are derived of the motion of cylindrical shells and the expression of the advanced nonlinear varied shear factor. The expressions of stiffness of thick composited two-layer FGM circular cylindrical shells with sinusoidal rising temperature are applied. The partial differential equation (PDE) in dynamic equilibrium of thick FGM circular cylindrical shells is derived with respect to shear rotations and displacements under terms of thermal–mechanical loads and density inertia terms. Important parametric effects of the advanced nonlinear varied shear factor, power law index, and temperature on the stress and displacement of thick FGM circular cylindrical shells are studied. Additionally, the advanced nonlinear varied shear factor effect is included and studied for a vibrating frequency using a fully homogeneous equation.

Behavior of Functionally Graded Porous Plate in Bending with Smoothed Element

The bending analysis of functionally graded porous (FGP) plates using a four-node quadrilateral element connected to the C0-type of Reddy's third-order shear deformation theory and cell-based smoothed strains is presented in this paper. Reddy's theory surely uses the advantages and desirable properties of third-order shear deformation theory. Moreover, FGP plates with advanced material properties are changed from the bottom to the top surface, respectively. Numerical results and comparisons with other reference solutions indicate the accuracy and efficiency of the current element in the analysis of FGP plates.

Free flexural vibration of cracked carbon nanotube-reinforced composite plates resting on elastic foundations using XFEM

This paper presents detailed investigations on the free flexural vibration of carbon nanotube-reinforced composite plates with cracks resting on elastic foundations. The higher-order shear deformation theory is implemented in an extended finite element framework to derive the vibration equations of cracked plates. Uniform and graded distribution of carbon nanotubes (CNTs) in the thickness direction of plates are considered. The effective material properties are computed using the Halpin Tsai model and the rule of mixture. The crack in the plate is detected using the level set method. The enrichment of the primary variable is done using enrichment functions to address the crack discontinuity issue. Detailed parametric studies have been performed to explore the effect of cracks, CNT’s volume fraction, gradation pattern of CNTs, thickness ratio of the plate, and the elastic foundations on the frequencies. Comparative results are presented by solving a few numerical examples using in-house MATLAB code to validate and ensure the accuracy and robustness of the proposed model.

A deformation of Master-Space and inertia effects within the theory of Master Space-Teleparallel Supergravity

In the framework of the theory of Master space-Teleparallel Supergravity (MS gp-TSG) (Ter-Kazarian, 2024b), having the gauge translation group in tangent bundle, in present article we address the theory of a general deformation of the flat MSp induced by external force exerted on a particle, subject to certain rules. Our idea is that the universality of gravitation and inertia attribute to the single mechanism of origin from geometry but having a different nature.

A new shear deformation micro-beam model based on a nonlocal elasticity theory

In this article, a nonlocal refined shear deformation beam theory is proposed to analyze the free vibration behavior of FG nanobeams. The formulation is based on the nonlocal differential constitutive equations of Eringen that accounts for small scale effects. Material properties vary continuously through thickness according to a power law distribution. The study is performed within the framework of a new parametric shear deformation theory which provides parabolic transverse shear strains across the thickness direction and hence, does not need shear correction factor. Moreover, zero-traction boundary conditions on the top and bottom surfaces of the nanobeam are satisfied rigorously. Equations of motion of FG nanobeam are derived from both the nonlocal differential constitutive relations of Eringen and Hamilton’s principle. The system of differential equations is solved using the Navier solution technique. The parametric study is conducted to examine the effects of volume fraction index, length scale parameter and length ratio on the free frequencies of vibration of FG thin and thick nanobeams.

Dynamic analysis of carbon nanotube-reinforced composite plates using reduced order isogeometric model

In this work, a reduced order isogeometric model is proposed to analyze the dynamic behavior of carbon nanotube-reinforced composite plates. In which, the mechanical properties of the material are functionallygraded through the plate thickness employing four distributions of carbon nanotubes. A third shear deformation theory is employed to represent the displacement along with the plate thickness, whilst a non-uniform rational B-splines surface is utilized to approximate the displacement in the plate plane. The dynamic responses at important degrees of freedom are resolved by the Newmark method instead of dealing with all degrees of freedom as those of the full model. Accordingly, a reduced order model based on the second-order Neumann series expansion is utilized to build the isogeometric analysis. Several examples are tested to illustrate the ability of the suggested paradigm. Obtained outcomes are compared with those of other works and full model to prove the reliability of the reduced Isogeometric analysis.

Ensemble learning methods for the mechanical behavior prediction of tri-directional functionally graded plates

This paper aims to enhance computational performance for behavior prediction of tri-directional functionallygraded plates using ensemble learning methods such as random forest, extreme gradient boosting, and lightgradient boosting machine. Furthermore, the effectiveness of these methods is verified by comparing theirresults with those of artificial neural networks. The present investigation focuses on the buckling problem of tridirectional functionally graded plates. In this study, data pairs consisting of input and output data are generated using a combination of isogeometric analysis and generalized shear deformation theory to ensure the accuracy of the dataset. The input data in this case are eighteen control points used to characterize material distribution; the output data are total ceramic volume fraction and non-dimensional buckling load. Based on this dataset, the effect of hyperparameters in machine learning models on accuracy and computational cost is investigated to determine models with optimal hyperparameters, referred to as optimal models. The performance of the optimal models in predicting plate behavior is compared to each other. Furthermore, in terms of computational time and accuracy, the light gradient boosting machine model gives the best results compared to the others.

Analytical modeling contribution of the vibration dynamics of FGM plates placed on elastic foundations

The present article uses the non-polynomial refined shear deformation theory (NP-RSDT) with four unknowns in order to determine the natural frequencies of functionally graded material (FGM) plates that are made with advanced materials without integrating a shear correction factor. These plates rest on elastic foundations. This study aims to analyze the effects of an elastic system, which is supposed to be represented by the two Pasternak and Winkler parameters on the dynamics of free vibrations of the FGM plates, while taking into account the fact that the Winkler springs have a variable modulus while the Pasternak layer is considered as a shear layer with a constant modulus. For the purpose of demonstrating the accuracy of the current theory used in this work, various numerical investigations which were conducted on the free vibrations of FGM plates resting on elastic foundations are presented. Additionally, the effects of varying various parameters, such as the elastic foundation parameters, power law index, aspect ratio, and plate geometry are also investigated. The results obtained with this method are then compared with those obtained with other methods reported in the literature. Once the current method was validated, we proceeded in the same field by carrying out a study on the free vibrations of FGM plates simply supported and resting on an elastic system, while considering that the Winkler parameters are variable. The results obtained are displayed through tables and graphs. They are then discussed.

Some computational results on Koszul–Vinberg cochain complexes

AbstractAn affine connection is said to be flat if its curvature tensor vanishes identically. Koszul–Vinberg (KV for abbreviation) cohomology has been invoked to study the deformation theory of flat and torsion-free affine connections on tangent bundle. In this Note, we compute explicitly the differentials of various specific KV cochains, and study their relation to classical objects in information geometry, including deformations associated with projective and dual-projective transformations of a flat and torsion-free affine connection. As an application, we also give a simple yet non-trivial example of a left-symmetric algebra of which the second cohomology group does not vanish.

Mechanical behavior of porous functionally graded plates

In this study, the effect of porosity distribution pattern on the bending and free vibration analysis of porous FG plates is studied by using sinusoidal shear deformation theory. This theory account for sinusoidal variation of transverse shear strain through the depth of the plate and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The material properties of the plate and the porosities within the plate are considered to vary continuously through the thickness direction according to the volume fraction of constituents defined by the modified rule of the mixture, this includes porosity volume fraction with four different types of porosity distribution over the cross-section. The governing partial differential equation of motion for the bending and free vibration analysis is obtained using sinusoidal shear deformation theory. An analytical solution is presented for the governing equation. Results of the presented solution are compared and validated by the available results in the literature. Moreover, the effects of material and porosity distribution and geometrical parameters on bending and free vibrational properties are investigated.