Kinematic Assumptions Research Articles

Framework of Physically Consistent Homogenization and Multiscale Modeling of Shell Structures**

A framework of physically consistent homogenization and multiscale modeling (PCHMM) for reduced-order analysis of plate/shell structures is developed in this paper. To address the inapplicability of conventional periodic boundary conditions and Hill’s condition involved in homogenization of shear-deformable shell structures, the paper proposes physically consistent boundary conditions and modified Hill’s condition for plate/shells. Unlike the PCHMM method for beams, considering the contradiction between high-order displacement fields induced by shear forces and low-order kinematic assumptions, additional constraints are applied to the plate/shell structure sectional strains during the solution of perturbation fields. The correctness and effectiveness of the proposed plate/shell PCHMM framework and method are verified by typical numerical examples. The proposed theory can also be conveniently embedded into commercial finite element software for homogenization and multiscale analysis of structures such as microscale metamaterials like lattice plates and large complex structures like aircraft fuselage sections.

Prediction of sectional collapse of thin-walled structure under pure bending by nonlinear composite beam theory

Brazier (1927) found that when one dimension of the beam cross-section was relatively smaller than the others, large in-plane displacements over the cross-section might occur, even though the strains could remain very small. Under this circumstance, the so-called Brazier effect refers to the cross-sectional ovalization, which leads to nonlinear bending buckling and collapses. This paper studies the Brazier effect by the nonlinear Variational Asymptotic Beam Sectional Analysis (VABS) theory which considers finite cross-sectional deformations. Nonlinear VABS reduces three-dimensional (3D) continuum to a one-dimensional (1D) beam analysis and a two-dimensional (2D) cross-sectional analysis featuring both geometric and material nonlinearities without unnecessary kinematic assumptions. The present theory is implemented using the finite element method (FEM) in the VABS code, a general-purpose beam cross-sectional analysis tool. An iterative method is applied to solve the finite warping field for the classical-type model using the Euler–Bernoulli beam theory. The deformation gradient tensor is directly used to deal with finite deformation, various strain definitions, and several types of material laws. Numerical examples demonstrate the capabilities of VABS to predict the sectional collapse of thin-walled structures under pure bending.

Free Vibration of Joined Spherical–Conical Panel and Shells Using the 2D-GDQ Method

The free vibration analysis of joined conical–spherical panels and shells of revolution is the primary aim of this investigation. The equations of motion are derived by considering Donnell’s kinematic assumptions and the first-order shear deformation theory (FSDT), which considers the effects of shear deformation as well as rotational inertia. The equations of motion are discretized by using the two-dimensional generalized differential quadratic (2D-GDQ) method in the [Formula: see text] and [Formula: see text] directions for conical and spherical panels and shells, respectively. Natural frequencies of joined conical–spherical shells and panels are determined for different boundary conditions. The accuracy of results has been compared with the results of the other researchers and those obtained using Abaqus and Ansys, which showed good agreement. Moreover, the effects of length, thickness, and the angle of the semi-vertex of the conical shell on the dynamic behavior of joined spherical–conical panels/shells are investigated.

Nonlinear vibrations of temperature-dependent FGM cylindrical panels subjected to instantaneous surface heating

An analysis is performed in this research to deal with the large amplitude vibrations caused by rapid surface heating in an FGM cylindrical panel. It is assumed that the properties of the panel are distributed through the thickness in terms of volume fraction of constituents. Following the first-order shear deformation theory, von Karman type of kinematics, and Donnell kinematic assumptions, definition of kinetic and strain energies of the shell are established. These obtained equations are discretized via the method of Ritz, where the shape functions are estimated by the Chebyshev polynomials. Temperature distribution within the body is also obtained using the one-dimensional heat conduction equation. This equation is discretized using the finite different method and solved iteratively following the Picard and Crank-Nicolson methods. Thermally induced force and moment resultants are evaluated and inserted into the matrix representation of equations of motion. The temporal evolution of displacement is obtained using the iterative Picard method and Newmark time marching method. Results of this study are compared with the available data in the open literature and after that novel results are given to explore the effects of geometrical, mechanical, and thermal parameters. It is highlighted that power law index, shallowness, thickness, edge supports, temperature dependency, geometrical nonlinearity and thermal boundary conditions are all important factors on the response of the structure under thermal shock. It is highlighted that thermally induced vibrations indeed exists especially in thin class of shells. Besides, thermally induced deflections may be controlled through using a proper power law index.

Definition of a beam-like reduced order model element by means of a mixed dimensional coupling

The use of Reduced Dimensional Models (RDM) discretized like beams, plates and shell elements drastically decreases the computational cost of solving a full 3D elastic problem with a Finite Element Method (FEM). However, its kinematic assumptions are only applicable to bodies with regular sections or continuous layouts. For the correct analysis of irregular regions, it is necessary to rely on bi-dimensional or solid models that fully reproduce the geometry of the body and its behavior but have a much higher computational cost. The Mixing Dimensional Coupling (MDC) technique allows linking models discretized with elements of different topologies, allowing the possibility of considering the most cost-effective model in each region. This coupling takes place at the interface that delimits both models and relies on the equilibrium of work and reactions on its two faces. In this paper, the formulation is presented for coupling beams with laminar sections and 2D Plane-Stress (PS) models demonstrating its proper behavior. Finally, this coupling is used for defining a new beam element, the Beam-Like Reduced Order Model (BLROM), which is obtained from a Plane-Stress model of their longitudinal section.

A general element for shell analysis based on the scaled boundary finite element method

AbstractA novel technique, the scaling surface‐based Scaled Boundary Finite Element Method (SBFEM), is introduced as a method for formulating a general element for shell analysis. This displacement‐based element includes three translational degrees of freedom (DOFs) per node. Notably, only two‐dimensional discretization for one of the two parallel shell surfaces, referred to as the scaling surface, is necessary. The interpolation scheme for the scaling surface is postulated to be applicable to all surfaces parallel to it in the thickness. The derivation strictly adheres to the 3D theory of elasticity, without making additional kinematic assumptions. As a result, the displacement field along the thickness is analytically solved, and the element formulation is immune to transverse locking, membrane locking, and other issues, eliminating the need for additional remedies. Extensive investigations into the robustness and accuracy of the elements have been conducted using well‐known benchmark problems, along with additional challenging problems. Numerical examples confirm that the element formulation is free from transverse shear locking and membrane locking. Moreover, the proposed formulation is easily extendable to cases involving shell elements with varying thickness and holds the potential for extension to the nonlinear response analysis of shell structures.

Efficient large-deformation shell elements for the simulation of electromechanical coupling effects in incompressible dielectric elastomers

So-called electro-active polymers are not only capable of undergoing very large deformations, but they also exhibit different electromechanical coupling effects due to their dielectric or electrostrictive characteristics. In the present paper the sub-class of dielectric elastomers is studied; in particular, we focus on thin dielectric elastomer shells, which undergo large deformations under an applied electric field. We develop a framework for the modeling and subsequent efficient simulation of thin structures made from incompressible dielectric elastomers. A thermodynamically consistent phenomenological continuum model based on the free energy density is adapted imposing the kinematic assumptions of Kirchhoff-Love shells in combination with including the thickness deformation and the electric field as independent unknown fields. To make the formulation accessible to finite element simulations, existing non-linear elastic shell mixed finite elements are extended to the present hyperelastic electromechanically coupled formulation for dielectric elastomer shells. Computational results of the shell theory are compared to three-dimensional results validating the proposed modeling and numerical simulation methodology, and showing high accuracy already for coarse finite element discretizations of the shell.

Recent advances in stability analysis and design of 3D slopes

Slope failures in nature and engineering are typically three-dimensional (3D). The rotational failure mechanism derived from the variational limit equilibrium (LE) method shows superior performance in the stability analysis of the 3D slope. In contrast to the traditional LE methods, it avoids arbitrary kinematical and statical assumptions. Stability charts obtained by the variational LE method are used to derive explicit expression equations of the safety factor, also known as the stability equations, for both 3D reinforced and unreinforced slopes. These equations are highly accurate and can provide a convenient means to assess slope stability in practical engineering. An example of a convex reinforced slope with a turning arc is illustrated in this study to investigate the effect of the 3D effects on the required reinforcement length for design. The results indicate that the 2D method underestimates the required reinforcement length when dealing with a 3D reinforced slope problem. Furthermore, a forensic analysis of the Yeager Airport reinforced slope is conducted within the framework of the variational LE method. The required strength for stability is found to be significantly less than the allowable strength of reinforcements without considering the decrease in soil shear strength. However, the required strength greatly exceeds the allowable strength when the decrease in soil shear strength is considered. The results verify that the decrease in shear strength of the weak layer is responsible for the collapse.

A systematic methodology for port-Hamiltonian modeling of multidimensional flexible linear mechanical systems

This article introduces a novel systematic methodology for modeling a class of multidimensional linear mechanical systems that directly allows to obtain their infinite-dimensional port-Hamiltonian representation. While the approach is tailored to systems governed by specific kinematic assumptions, it encompasses a wide range of models found in current literature, including ℓ-dimensional elasticity models (where ℓ = 1, 2, 3), vibrating strings, torsion in circular bars, classical beam and plate models, among others. The methodology involves formulating the displacement field using primary generalized coordinates via a linear algebraic relation. The non-zero components of the strain tensor are then calculated and expressed using secondary generalized coordinates, enabling the characterization of the skew-adjoint differential operator associated with the port-Hamiltonian representation. By applying Hamilton's principle and employing a specially developed integration by parts formula for the considered class of differential operators, the port-Hamiltonian model is directly obtained, along with the definition of boundary inputs and outputs. To illustrate the methodology, the plate modeling process based on Reddy's third-order shear deformation theory is presented as an example. To the best of our knowledge, this is the first time that a port-Hamiltonian representation of this system is presented in the literature.

New Accomplishments on the Equivalence of the First-Order Displacement-Based Zigzag Theories through a Unified Formulation

The paper presents a critical review and new accomplishments on the equivalence of the first-order displacement-based zigzag theories for laminated composite and sandwich structures. Zigzag theories (ZZTs) have widely spread among researchers over the last few decades thanks to their accuracy in predicting the response of multilayered composite and sandwich structures while retaining the simplicity of their underlying equivalent single-layer (ESL) theory. The displacement field consists of two main contributions: the global one, able to describe the overall structural behaviour, and the local layer-wise one that considers the transverse shear continuity at the layer interfaces that describe the “zigzag” displacement pattern typical of multilayered structures. In the framework of displacement-based linear ZZTs, various assumptions have been made on the local contribution, and different theories have been deduced. This paper aims to provide a unified formulation for first-order ZZTs, highlighting some common aspects and underlying equivalencies with existing formulations. The mathematical demonstrations and the numerical examples prove the equivalence of the approaches to characterising local zigzag enrichment. Finally, it is demonstrated that the kinematic assumptions are the discriminants of the ZZTs’ accuracy.

Frequency Regulation in Nanocomposite Sandwich Joined Conical-Conical Shells with Open Cell Foam Partially Supported by Pasternak Elastic Foundation Using GDQ

This study investigated the natural vibration of a sandwich conical-conical shell that is partially supported by Winkler–Pasternak foundations. The sandwich system comprises an open-cell foam (OCF) core and laminated composite face sheets reinforced with graphene platelets using functionally graded models (FG-GPLRC). To account for through-the-thickness shear deformations and rotary inertias, the first-order theory of shells is combined with Donnell-type kinematic assumptions. Hamilton’s principle is utilized to establish the general motion equations and related boundary and continuity conditions. The resulting system of equations is discretized using the semi-analytical generalized differential quadrature (GDQ) approach. An eigenvalue problem is formulated to analyze the corresponding mode shapes and vibration frequencies for the shell ends and continuity criteria under various boundary conditions. Parametric experiments are conducted for sandwich conical-conical shells partially supported by Winkler–Pasternak foundations. This study examined the impact of multiple factors on the frequency of a joined shell structure, including the total thickness, FG-GPL face sheet thickness, various boundary conditions, the length of each cone segment, and the Winkler and shear elastic foundation. Notably, this research also analyzed the effects of a partial elastic foundation on the structure’s frequency for the first time.

A general kinematic model for lubricated ball bearings based on the minimum energy hypothesis

Accurately predicting the performance of lubricated rolling elements is critical for certain key bearing applications. A widely employed simulation technique is quasi-static models, which although computationally efficient to implement, lack accuracy in predicting the kinematics of bearing components. This problem arises from their construction under kinematic assumptions, instead of solving the kinematic based on real frictional interactions. This paper presents a quasi-static model based on the minimum energy hypothesis for lubricated contacts, which determines the kinematic of the ball considering the effects of the frictional interactions governed by the elastohydrodynamic lubrication mechanism. The influence of kinematic variables on the sliding patterns and power loss distribution of the contact are analysed. The results are compared with other published quasi-static models, revealing substantial differences. The influence of the kinematic hypotheses on power losses are also studied at a bearing component scale. Finally, the effect of the governing lubrication mechanism of the contact is discussed, concluding that the limiting shear stress of the lubricant and the centrifugal forces have a significant effect on the kinematic of the ball.

Verification and Validation of finite element models for laminated timber structures using solid, solid-beam and solid-shell approaches

This paper presents a numerical approach to assess efficiently the vibration performance of adhesive-free engineered wood products assembled through compressed wood dowels. Predictive finite element models are obtained by applying the Verification and Validation methodology. The models are first developed using solid elements. Then, solid-beam and solid-shell approaches based on standard solid elements are developed with first-order or higher-order theories. Beam or shell kinematic assumptions are applied throughout the cross-section of the dowels and through the thickness of the layers. From a numerical point of view, a modification of the system of algebraic equations, based on the concept of independent and dependent nodes, is developed. Dependent nodes are eliminated, resulting in a significant reduction in the number of degrees of freedom and floating-point operations. The methodology is evaluated for the calculation of frequencies and mode shapes of adhesive-free laminated timber beams and timber panels with a mix of solid elements, solid-shell and solid-beam approaches. The study highlights the efficiency of the proposed modelling approach in terms of quality of results and model size reduction.

Asymptotically consistent and computationally efficient modeling of short-ranged molecular interactions between curved slender fibers undergoing large 3D deformations

This article proposes a novel computational modeling approach for short-ranged molecular interactions between curved slender fibers undergoing large 3D deformations, and gives a detailed overview how it fits into the framework of existing fiber or beam interaction models, either considering microscale molecular or macroscale contact effects. The direct evaluation of a molecular interaction potential between two general bodies in 3D space would require to integrate molecule densities over two 3D volumes, leading to a sixfold integral to be solved numerically. By exploiting the short-range nature of the considered class of interaction potentials as well as the fundamental kinematic assumption of undeformable fiber cross-sections, as typically applied in mechanical beam theories, a recently derived, closed-form analytical solution is applied for the interaction potential between a given section of the first fiber (slave beam) and the entire second fiber (master beam), whose geometry is linearly expanded at the point with smallest distance to the given slave beam section. This novel approach based on a pre-defined section–beam interaction potential (SBIP) requires only one single integration step along the slave beam length to be performed numerically. In addition to significant gains in computational efficiency, the total beam–beam interaction potential resulting from this approach is shown to exhibit an asymptotically consistent angular and distance scaling behavior. Critically for the numerical solution scheme, a regularization of the interaction potential in the zero-separation limit as well as the finite element discretization of the interacting fibers, modeled by the geometrically exact beam theory, are presented. In addition to elementary two-fiber systems, carefully chosen to verify accuracy and asymptotic consistence of the proposed SBIP approach, a potential practical application in form of adhesive nanofiber-grafted surfaces is studied. Involving a large number of helicoidal fibers undergoing large 3D deformations, arbitrary mutual fiber orientations as well as frequent local fiber pull-off and snap-into-contact events, this example demonstrates the robustness and computational efficiency of the new approach.

A Multiple Scale Approach to Nonlinear Vibrations and Stability of Doubly Curved Shells/Panels Under External and Parametric Excitations

This study focuses on nonlinear vibrations and stability of doubly curved shells/panels. The thin-walled shell structure is assumed to be acted upon by external loads as well as parametric excitation. Using the first-order shear deformation Fülgge theory assumptions, governing equations of motion are derived for a general doubly curved geometry. Nonlinear components which are compatible with the Von-Karman kinematic assumptions are retained in the formulation and the rest of nonlinear components are neglected due to the fact that strains are assumed to be small and rotations to be moderate. The formulation can be readily reduced to any geometry including cylindrical shells and spherical panels, as benchmark examples in this study. Employing the Galerkin discretization approach and applying a modification to the Volmir assumption, a reduced order form of the original system is determined, which is shown to be a great representative of the original system with much more computational advantages. Using the method of multiple scales, the reduced order nonlinear governing equations are solved to discover the primary resonance, forced vibration characteristics and stability margins of thin-walled shells/panels under the action of parametric excitation. Several interesting findings on nonlinear behavior of cylindrical shells and spherical panels (as benchmark geometries) are presented while giving a demonstration of the accuracy and computational advantages of the proposed model.

Anisotropic evolution of viscous strain in soft biological materials

We propose a model for anisotropic viscoelastic biological materials that can handle large deformations, based on the kinematic assumption that the reinforcing fibre structure undergoes affine deformation with the underlying matrix. A generalized orientation tensor approach is used to account for the dispersion of the fibres. Moreover, we consider a strain energy function that features both an elastic and an overstress component, corresponding to distinct natural states. As a consequence of this choice, the remodelled state is not necessarily stress-free, and the material does not completely relax the stress. Notably, we consider that viscous remodelling also alters the fibre distribution, leading to a dependence of the overstress energy on the remodelled orientation tensor. An anisotropic evolution equation for the viscous strain is then derived, which has five distinct characteristic times if a single fibre family is considered and requires no additional assumptions on the viscous spin. To implement the model, we prove that the evolution of the viscous strain can be recast in a variational form by an Onsager variational principle. Finally, we discuss the algorithm used for the simulations and show numerical examples that serve as benchmark test cases for viscoelastic materials.

Assumed strain methods in micromechanics, laminate composite voxels and level sets

AbstractThis work deals with the composite voxel method, which—in its original form—furnishes voxels containing more than one material with a surrogate material law accounting for the heterogeneity in the voxel. We show that the laminate composite voxel technique naturally arises as an assumed strain method, that is, the general framework introduced by Simo‐Rifai, for a specific choice of enhanced strain field. As a consequence, laminate composite voxels may be regarded as a kinematic assumption within a discretization scheme rather than a part of material modeling, as suggested originally. We discuss how to seamlessly integrate composite voxels into the framework of a level‐set description of the microstructure, in particular the accurate and efficient computation of normals and cut‐volume fractions. In contrast to more traditional strategies based on subvoxelizations, the introduced method avoids systematic errors when computing composite voxel properties. We demonstrate the applicability of the developed technology for a number of relevant computational examples.

Efficient analysis on buckling of FG-CNT reinforced composite joined conical–cylindrical laminated shells based on GDQ method under multiple loading conditions

This work presents an efficient analysis of buckling for functionally graded carbon nanotube reinforced composite (FG-CNTRC) joined conical–cylindrical laminated shells with lateral pressure and axial compressive load considering thermal effect. The governing equilibrium equation of the joined conical–cylindrical shell (CO-CYL) is determined using the first-order shear deformation theory (FSDT), Donnell kinematic assumptions, von-Kármán geometrical nonlinearity, and considering thermal effect. A semi-analytical solution, generalized differential quadrature (GDQ) method, and artificial springs are applied during the solution procedure. The comparison studies have demonstrated a remarkable 97% reduction in running time compared to the finite element method (FEM).

Variable kinematics finite plate elements for the buckling analysis of sandwich composite panels

This paper extends sublaminate-based variable kinematics plate finite elements towards the buckling analysis of composite structures. Robust locking-free 4-node and 8-node interpolations are used to build the finite element matrices in terms of fundamental nuclei, invariant with respect to the employed kinematic model of the composite plate. Geometric nonlinearities are accounted for in the von Kármán sense. The classical linearized stability analysis is mainly conducted for sandwich panels, for which different kinematic assumptions are employed for the skins and the core. Global buckling of the sandwich panel as well as short-wavelength wrinkling of the skins are investigated by referring to a variety of case studies, including homogeneous and laminated skins as well as isotropic and orthotropic cores. Convergence studies are performed to establish the minimum number of elements for the local instabilities to be grasped. The proposed computational framework requires a simple 2D mesh and is capable of providing quasi-3D response patterns. This is as demonstrated by numerous case studies that address the transition from global to local buckling, the onset of wrinkling in flat sandwich panels with anisotropic skins under various loading conditions as well as the local face sheet instability occurring in sandwich panels working in bending. It is concluded that he proposed FEM-based tool is computationally efficient and can be advantageously employed in pre-sizing design phases without resorting to full 3D models.

Vibration Bandgap Analysis of Periodic Composite Microplates using Isogeometric Approach

For periodic composite microplates, the microstructure dependent effect has great influence on the elastic wave vibration band gaps. In this paper, an effective isogeometric analysis method based on the modified couple stress theory (MCST) is proposed to calculate the vibration band gaps in periodic composite microplates. The MCST is employed to capture the microstructure effect of vibration band gaps in periodic composite microplates. The high order continuity requirement of MCST can be easily satisfied by applying the NURBS based isogeometric analysis approach. The elasto-dynamics and discrete equations of isogeometric analysis are obtained under the Mindlin kinematics assumptions. The present approach is validated by comparison with the analytical solution. And the influence of microstructure effect, volume fraction and unit cell length on the band gaps are investigated in details.