Kinematic Assumptions Research Articles

Multi-objective shape optimization for axially functionally graded microbeams

Nonuniform microbeams made of functionally graded materials (FGMs) have been studied extensively in literature to predict their mechanical and thermal behavior, those demonstrated that each of material variation, non-uniformity and micro-scale effects have significant influences on the static stability sand dynamic behavior. Therefore, this research exploited the multi-objective shape optimization method to optimize the beam shape and its volume fraction distribution in order to maximize the critical buckling loads and fundamental frequencies while minimizing the mass and cost of the FG microbeam, for the first time. Modified continuum model based on both Euler-Bernoulli beam theory as kinematic assumptions and constitutive equation of modified couple stress theory, is developed to derive equilibrium equations (in static analysis) and equations of motion (in dynamic analysis) of axially FGMs nonuniform microbeam. To control the variation of height and width along the beam length, three different shape functions are proposed in the analysis. The multiobjective particle swarm optimization (MOPSO) is adopted to get the Pareto optimal solutions. In addition to the FGM power index, the shape functions types and parameters are considered as the design variables. Several optimization problems are studied to demonstrate the multi-objective optimal shape design of axially functionally graded microbeams.

Enriched beam finite element models with torsion and shear warping for the analysis of thin-walled structures

This paper presents three beam Finite Element (FE) formulations developed for the analysis of thin-walled structures. These account for out-of-plane cross-section warping by removing the classical rigid body cross-section hypothesis and capture the interaction of axial/bending stress components with shear and torsion.The beam FE models rely on different kinematic assumptions to describe out-of-plane cross-section deformations. Indeed, warping displacement field is interpolated in the element volume according to different approaches, with increasing level of accuracy and detail. First two models adopt a coarse warping description, where warping displacement field is defined as the linear combination of assumed warping profiles and unknown kinematic parameters. In the first model, these are considered as equal to the generalized cross-section torsional curvature and shear strains and a classical displacement-based formulation is adopted to derive the element governing equations. In the second model, warping parameters are assumed as independent kinematic quantities and a mixed approach is considered to derive the FE formulation. Third model, also relying on a mixed formulation, independently interpolates warping by introducing additional degrees of freedom on the cross-section plane, thus, resulting in a richer description of the out-of-plane deformations. This latter is also adopted to propose a numerical procedure for the warping profile evaluation of thin-walled beams subjected to torsional and shear forces, for general cross-section geometry.The efficiency and accuracy of the proposed FE formulations are validated by simulating the response of thin-walled structures under torsion and coupled torsion/shear actions and the influence of the kinematic assumptions characterizing each formulation is discussed.

Numerical evaluation of beam models based on the modified couple stress theory

An error in the beam model proposed by Lam, Yang, Chong, Wang, & Tong (2003) for the modified couple stress theory is found and corrected, and then the corrected model is verified by the finite element analysis. In addition, various beam models based on the modified couple stress theory are classified according to their characteristics, and evaluated numerically by the finite element analysis using 2D and 3D continuum elements through elastic cantilever beam problems. The results show that the corrected beam model of Lam et al. is the best fit for plane strain beam problems, while none of the beam models are appropriate for general beams of similar width and thickness dimensions. The reason is confirmed by the energy term analysis which reveals that the kinematic assumptions applied to the beam models are inappropriate when the beam shape is general.

Nonlinear analysis of thermal-mechanical coupling bending of FGP infinite length cylindrical panels based on PNS and NSGT

The present paper investigates the nonlinear static bending behavior of infinite length cylindrical panels made of functionally graded porous (FGP) materials. The shell with clamped edges is subjected to uniform temperature rise and transverse pressure loading. Thermomechanical properties of the shell with even distributed porosities are temperature-dependent and are graded along the thickness. The principle of virtual displacement and nonlocal strain gradient theory (NSGT) are employed to derive the equilibrium equations of the shallow shell resting on nonlinear elastic foundation. The concept of physical neutral surface (PNS) and higher-order shear deformation theory are also included into the formulation. Using the uncoupled thermoelasticity and Donnell kinematic assumptions, three differential equations of the shell under thermomechanical loading are established. The nonlinear system of governing equations is solved for the shell with infinite length which is clamped on both straight edges and free at other curved edges. The two-step perturbation technique and Galerkin procedure are employed to derive analytical solutions for the thermal, mechanical, and thermomechanical responses of cylindrical panels. The important parameters governing the bending behavior of shells under thermal-mechanical coupling load are identified and discussed. Parametric studies are given to show the influences of nonlocal/length scale parameter, porosity coefficient, foundation stiffness, geometrical parameter and functionally graded pattern. It is shown that the geometrical parameters have an important role on the bending behavior of cylindrical panels. Also, the temperature dependence of material properties results in higher deflection of the heated shells.

On nonlinear vibration and snap-through buckling of long FG porous cylindrical panels using nonlocal strain gradient theory

In the current research, nonlinear vibration and snap-buckling analysis of long cylindrical panels made of functionally graded (FG) porous materials are studied. The case of cylindrical panels with shallow curvature and long length is analyzed using an approximate nonlocal strain gradient model. Non-homogeneous properties of the cylindrical panel with uniform distributed porosities are graded across the thickness. The interaction of the cylindrical panel with a nonlinear elastic foundation is also included into the formulation. The mathematical formulations are expressed based on the high-order shear deformation theory and the Donnell kinematic assumptions. Three nonlinear motion equations of the cylindrical panel are established by employing the Hamilton principle. These coupled partial differential equations are analytically solved for the case of long cylindrical panels which is immovable pinned on both straight edges. The two-step perturbation technique and Galerkin’s method are implemented to extract the nonlinear free vibration and snap-buckling characteristics of the shell. The natural frequencies and load-deflection curves are first validated for the cases of long plates and shells. Afterwards, novel parametric studies are developed to show the effects of nonlocal and length scale parameters, power law index, porosity coefficient, foundation components, and the geometrical parameters of the shell.

On coupled FEM-BEM simulation of the magneto-mechanical behavior of single-crystalline Ni-Mn-Ga alloys

In this paper, a coupled FEM-BEM method is applied to simulate the magneto-mechanical behavior of a single-crystalline Ni-Mn-Ga sample. First, based on some kinematic and constitutive assumptions, the governing system for modeling the response of the Ni-Mn-Ga sample is formulated, which is composed of the magnetic and mechanical field equations, the evolution laws of the internal variables and the twin interface movement criteria. To solve the governing system, an iterative numerical algorithm is proposed. Especially, a coupled FEM-BEM method is adopted in the algorithm to tackle the magnetic field equation. By using this method, one doesn’t need to consider the distribution of the demagnetization field in the surrounding space, thus the number of elements can be reduced significantly. To show the efficiency of the numerical algorithm, some typical examples are studied. It can be found that the numerical results obtained from the coupled FEM-BEM method and the conventional FEM method are in good agreement with each other. The numerical algorithm proposed in the current work would be helpful for the design of novel smart devices made of Ni-Mn-Ga alloys.

A specific 3D shell approach for textile composite reinforcements under large deformation

The deformation of textile composite reinforcements is strongly conditioned by their fibrous composition. Standard plate and shell theories are based on kinematic assumptions that are not verified for textile reinforcements. A 3D shell approach specific to fibrous reinforcements is proposed. It is based on two specificities: the inextensibility of the fibres and the possible slippage between the fibres. The approach is developed in a continuum-based shell element. The form of the virtual work reflects the specificities of the deformation of the fibrous reinforcements. It takes into account the tensile and bending stiffness of the fibres. Friction between fibres is taken into account in a simple way in connection with bending. The present approach is based on the actual physics of the deformation of the textile reinforcements. It makes it possible to simulate the 3D deformations of textile reinforcements and provides displacements and strains for all points in the fabric thickness and the proper rotations of the material normal.

A peridynamic Reissner‐Mindlin shell theory

SummaryIn this work, we have developed a state‐based peridynamics theory for nonlinear Reissener‐Mindlin shells to model and predict large deformation of shell structures with thick wall. The nonlocal peridynamic theory of solids offers an integral formulation that is an alternative to traditional local continuum mechanics models based on partial differential equations. This formulation is applicable for solving the material failure problems involved in discontinuous displacement fields. The governing equations of the state‐based peridynamic shell theory are derived based on the nonlocal balance laws by adopting the kinematic assumption of the Reissner and Mindlin plate and shell theories. In the numerical calculations, the stress points are employed to ensure the numerical stability. Several numerical examples are conducted to validate the nonlocal structure mechanics model and to verify the accuracy as well as the convergence of the proposed shell theory.

Multi-objective genetic algorithm optimization of composite sandwich plates using a higher-order theory

The main purpose of this paper is multi-objective optimization of soft-core composite sandwich plates using a known accurate high-order sandwich plate theory. In this three-layer theory, high-order kinematic assumptions are dedicated to each face sheet and the core layer. This theory considers the transverse flexibility of the soft core and satisfies transverse shear stresses continuity conditions and zero transverse shear stresses conditions at the upper and lower surfaces of the plate. The governing equations for bending and buckling analyses are derived using Hamilton’s principle, and their analytical solutions are presented for cross-ply plates. Two multi-objective optimization problems consist of weight/deflection optimization and weight/buckling load optimization of the unidirectional and cross-ply sandwich plates being studied. The thicknesses of the core and the face sheet layers are set as problems variables. The genetic algorithm is employed to find the optimal solutions for the problems, and the results are presented in form of the Pareto front. The accuracy of the present modeling and optimization method is evaluated for special case of buckling load maximization of laminated composite plates. For each problem, the optimization process is continued for more than 200 generations, but the results don’t change after 100 generations and the optimized results are obtained. The results confirm that there is no significant difference between the optimal solutions of unidirectional and cross-ply sandwich plates in both optimization problems. The process statistics show that cross-ply layup optimization takes about 25% more time than the unidirectional layup.

Autonomous Vehicles: Vehicle Parameter Estimation Using Variational Bayes and Kinematics

On-board sensory systems in autonomous vehicles make it possible to acquire information about the vehicle itself and about its relevant surroundings. With this information the vehicle actuators are able to follow the corresponding control commands and behave accordingly. Localization is thus a critical feature in autonomous driving to define trajectories to follow and enable maneuvers. Localization approaches using sensor data are mainly based on Bayes filters. Whitebox models that are used to this end use kinematics and vehicle parameters, such as wheel radii, to interfere the vehicle’s movement. As a consequence, faulty vehicle parameters lead to poor localization results. On the other hand, blackbox models use motion data to model vehicle behavior without relying on vehicle parameters. Due to their high non-linearity, blackbox approaches outperform whitebox models but faulty behaviour such as overfitting is hardly identifiable without intensive experiments. In this paper, we extend blackbox models using kinematics, by inferring vehicle parameters and then transforming blackbox models into whitebox models. The probabilistic perspective of vehicle movement is extended using random variables representing vehicle parameters. We validated our approach, acquiring and analyzing simulated noisy movement data from mobile robots and vehicles. Results show that it is possible to estimate vehicle parameters with few kinematic assumptions.

A unified formulation for composite quasi-2D finite elements based on global-local superposition

A unified theory to formulate a family of quasi-2D composite finite elements that capture through-the-thickness effects is proposed. The formulation is based on the superposition of global and local displacement fields and includes thermal effects, as a response to the claim of the scientific community. First are presented the fundamentals of the unified formulation: the kinematic assumptions; the imposition of continuity of displacements and transverse stresses, and of non-homogeneous boundary conditions; and the mathematical manipulations that render its numerical efficiency. Then, the new family of elements is assessed in problems involving different boundary conditions. Results are compared to analytical and highly refined 2D finite elements solutions, providing a benchmark of the elements capabilities. Mesh convergence is assessed, together with the influence of the global and local displacement fields. The numerical efficiency of the formulation is attested and an expansion for a 3D formulation is outlined.

Shear-deformable hybrid finite-element formulation for lateral-torsional buckling analysis of composite thin-walled members

A shear deformable hybrid finite element formulation is developed for the lateral-torsional buckling analysis of fibre-reinforced composite thin-walled members with open cross-section. The method is developed by using the Hellinger–Reissner functional. Comparison to the displacement-based formulations the current hybrid formulation has the advantage of incorporating the shear deformation effects easily by using the strain energy of the shear stress field without modifying the basic kinematic assumptions of the thin-walled beam theory. Numerical results are validated through comparisons with results based on other formulations presented in the literature. Examples illustrate the effects of shear deformations and stacking sequence of the composite layers in predicting bucking loads.

Coupled vibration analysis for equivalent dynamic model of the space antenna truss

A novel equivalent dynamic model is developed for coupled vibration analysis of the space antenna truss to enhance the design capacity of vibration controllers. Based on energy equivalence principle, the space antenna truss with two important features of rigid joints and complicated configuration is equivalent to a spatial anisotropy Timoshenko beam model. According to the kinematic assumptions, strain and kinetic energy expressions of the spatial periodic element can be obtained in accordance with displacement components at its center. Hamilton's principle is carried out to formulate the governing partial differential equations of motion for the equivalent beam model (EBM) which is divided into two sets of PDEs. Each set of PDEs includes three degrees of freedom and describes bending-torsion and bending-extension couplings respectively, due to the asymmetry of the antenna truss. An exact analytical method is developed to solve the two sets of coupled motion equations. The natural characteristics of the EBM are shown to be in excellent agreement with those of the finite element method, which demonstrates that the proposed EBM can provide a satisfactory accuracy for the antenna truss. In addition, since the mode shapes of the EBM are expressed as analytical functions of the spatial coordinate, such an approach may lead the investigations of dynamic property and the design of vibration control law to become convenient for the space antenna truss.

A thermodynamically consistent non-linear mathematical model for thermoviscoelastic plates/shells with finite deformation and finite strain based on classical continuum mechanics

This paper presents a kinematic assumption free and thermodynamically consistent non-linear formulation incorporating finite strain and finite deformation for thermoviscoelastic plates/shells based on the conservation and balance laws of the classical continuum mechanics (CCM) in R3 (see Surana and Mathi, (2020) for linear theory). The conservation and balance laws in Lagrangian description with finite strain measure and the conjugate stress measure in R3 are considered. The conjugate pairs in the second law of thermodynamics (SLT), the consideration of additional physics and the principle of equipresence are utilized to determine the constitutive variables and their argument tensors. The constitutive theory for the contravariant second Piola–Kirchhoff stress tensor is derived using Green’s strain tensor and its convected time derivatives of up to order n as its argument tensors using representation theorem with complete basis (i.e. using integrity). The convected time derivatives of the Green’s strain tensor up to order n provide ordered rate constitutive theory for the dissipation mechanism that is naturally non-linear due to Green’s strain measure. The constitutive theory for heat vector derived using representation theorem and integrity is also a non-linear constitutive theory. Simplified linear forms of these constitutive theories are also presented. The solution methods for the mathematical model for the BVPs as well as the IVPs using p-version hierarchical higher degree and higher order finite element method are presented. Due to dissipation, the energy equation is integral part of the mathematical model that accounts for rate of mechanical work resulting in rate of entropy production, hence heat.The plate/shell geometry (flat or curved) is described by its middle surface containing the nodal vectors (at each of the nine nodes), the ends of which define bottom and top surfaces of the plate/shell (Surana and Mathi, 2020). The geometry is mapped into ξηζ natural coordinate space in a two unit cube. The p-version hierarchical local approximations in ξηζ are constructed to describe the deformation of the plate/shell middle surface as well as its faces that is controlled by p-levels in ξ,η and ζ directions (element local approximation). The formulation presented here is accurate for very thin as well as very thick plate/shells and for small as well as finite deformation and finite strains. Non-linear constitutive theories based on integrity allow more complex constitutive behavior in term of strains as well as strain rates, hence dissipation mechanism. Dissipation mechanism is described by an ordered rate theory based on SLT and additional physics. The formulation presented here always ensures thermodynamic equilibrium during the evolution as it is derived using the conservation and balance laws of CCM in R3. The formulation always preserves three dimensional nature of the deformation physics regardless of the plate/shell thickness and is free of locking problems and shear correction factors that plague most of the currently used plate/shell formulations.

Generalization of the Vlasov theory for lateral torsional buckling analysis of built-up monosymmetric assemblies

The present study provides a formal generalization of the elastic lateral torsional buckling solution based on the Vlasov theory, initially intended for monolithic sections, to extend its applicability to any number of cross-sections connected together to form a monosymmetric assembly. The theory applies the well-known Vlasov kinematic assumptions to each of the components and augments them by assuming no relative slip between the components, as would be the case when components are bolted or welded. The formulation naturally leads to general expressions for effective sectional properties for the assembly (e.g., location of the shear centre, warping constant, monosymmetry parameter, and the Saint-Venant torsional constant). By adopting the effective sectional property definitions arising from the theory, the resulting lateral torsional buckling variational principle for the assembly is cast in a form analogous to that of monolithic sections. Alternative techniques are proposed to estimate the effective warping constant and monosymmetry parameter for assemblies. The solution shows that assemblies of sections connected through plug welds have lower critical moments than those continuously welded at the edges, while assemblies with intermittent welds have intermediate critical moment values. It is shown that treating the assembly as a monolithic section yields higher critical moments than assemblies with plug welds. The theory is used to quantify the critical moments in a number of built-up applications. Comparisons with the predictions of shell finite element model in Abaqus and S-Frame demonstrate the validity of the formulation.

Theoretical analysis of the effects of radial acceleration on centrifuge modeling of slope instability

In a centrifuged model, the centrifugal acceleration increases radially outward from the axis of rotation. The variations in the magnitude and direction of the centrifugal acceleration will inevitably result in a distortion of the stress field of the model compared with its desired prototype counterpart. This distortion alone is generally considered to be slight and negligible, but the extent of its impact on the accuracy of the centrifuge modeling of the soil slope instability is yet unknown. To answer this question, analytical solutions for a generally shaped critical slip surface and its normal stress distribution are derived for a plane-strain model slope by using a variational limiting equilibrium approach, without a need for arbitrary kinematic and static assumptions. Their validities are verified by comparing the extreme cases of the proposed solutions with available analytical solutions for the slope instability under the Earth's gravity condition. The proposed solutions are also validated by a well-documented centrifuge study of the slope instability with slip surface measurements, and reasonably good agreement is obtained. A parametric study is carried out by using the proposed solutions. The computed results for the model slopes are compared with those for the corresponding prototype slopes. The comparison shows that when the cross section of the model slope is radially oriented to the axis of rotation, the variation in the magnitude of the centrifugal acceleration alone has a negligible influence on the slope instability. When the cross section of the model slope is tangentially oriented to the circumference of the rotation of the centrifuge, however, the radial acceleration is shown to have a potentially important influence on the similitude of the slope instability. An optimum location of a model slope in a model container for the minimal effect of the radial acceleration can be determined by using the proposed solutions.

Nonlinear bending analysis of size-dependent FG porous microtubes in thermal environment based on modified couple stress theory

This study deals with the nonlinear bending analysis of elastic tubes made of functionally graded (FG) porous material based on modified couple stress theory. Immovable simply supported and clamped boundary conditions are assumed for the FG porous microtubes resting on nonlinear elastic foundation. Transverse pressure load is applied uniformly on the upper surface of the microtube which is exposed to uniform temperature field. Temperature-dependent material properties of the microtube with uniformly distributed porosity are graded across the radius of the cross-section. The higher-order shear deformation theory of circular beams is presented to approximate the displacement field. The von Kármán type of kinematic assumptions is utilized for nonlinear strain-displacement relations. The uncoupled thermoelasticity theory is used to derive the constitutive equations. With the establishment of the virtual displacement principle, nonlinear differential equations governing the equilibrium position of the microtube are extracted. These equilibrium equations are analytically solved by means of the two-step perturbation technique and Galerkin procedure. Parametric investigations are performed to demonstrate the size-dependent nonlinear bending responses of an FG porous microtube on nonlinear elastic foundation in thermal environment.

An investigation of models for elastic ribbons: Simulations & experiments

Understanding the feature-rich buckling-dominated behavior of thin elastic ribbons is ripe with opportunities for fundamental studies exploring the nexus between geometry and mechanics, and for conceiving of engineering applications that exploit geometric nonlinearity as a functioning principle. Predictive mechanical models play an instrumental role to this end. As a direct consequence of their physical appearance, ribbons are usually modeled either as one-dimensional rods having wide cross sections, or as narrow two-dimensional plates/shells. These models employ drastically different kinematic assumptions, which in turn play a decisive role in their predictive capabilities. Here, we critically examine three modeling approaches for elastic ribbons using detailed measurements of their complex three-dimensional deformations realized in quasistatic experiments with annulus-shaped ribbons. We find that simple and practically realizable ribbon deformations contradict assumptions underlying strain-displacement relationships in nonlinear rod and von Kármán plate models. These observations do not point at shortcomings of the theories themselves, but highlight fallacies in their application to modeling ribbon-like structures that are capable of undergoing large displacements and rotations. We identify and validate, seemingly for the first time, the 1-director Cosserat plate theory as a model for elastic ribbons over a useful range of loading conditions. In the process, we demonstrate annular ribbons to be prototypical systems for studying the mechanics of elastic ribbons. Annular ribbons exhibit a tunable degree of geometric nonlinearity in response to simple displacement and rotation boundary conditions— a feature that we exploit here for highlighting the consequences of kinematic assumptions underlying different ribbon models. We additionally provide experimental evidence for the existence of multiple stable equilibria, bifurcation phenomena correlated with the number of zero crossings in the mean curvature, and localization of energy, thus making annular ribbons interesting mechanical systems to study in their own right.

Geometrically exact thin-walled beam including warping formulated on the special Euclidean group [formula omitted

Based on a formulation on the special Euclidean group SE(3), a geometrically exact thin-walled beam with an arbitrary open cross-section is proposed to deal with the finite deformation and rotation issues. The beam strains are based on a kinematic assumption where warping deformation and Wagner effects are included such that the nonlinear behavior of a thin-walled beam is predicted accurately, particular under large torsion. To reduce the nonlinearity of rigid motion, static and dynamic equations are derived in the SE(3) framework based on the local frame approach. As the value of the iteration matrix, including the Jacobian matrix of inertial and internal forces, is invariable under arbitrary rigid motion, the number of updates required during the computation process decreases sharply, which drastically improves the computational efficiency. Furthermore, the isogeometric analysis (IGA) based on the non-uniform rational B-splines (NURBS) basis functions, which promotes the integration of computer-aided design (CAD) and computer-aided engineering (CAE), is adopted to interpolate the displacement, rotation, and warping fields separately. The interpolated strain measures satisfy the objectivity by removing the rigid motion of the reference point. To obtain the symmetric Jacobian matrix of internal forces, the linearization operation is conducted based on the previously converged configuration. A Lie group SE(3) extension of the generalized-α time integration method is utilized to solve the equations of motion for thin-walled beams. Finally, the proposed formulation is successfully tested and validated in several static and dynamic numerical examples.

Stress analyses of viscoelastic three-dimensional beam-like structures with low- and high-order one-dimensional finite elements

The mechanical responses of viscoelastic structures are evaluated by using finite beam elements based on different kinematical assumptions. The governing equations related to the structural theories are obtained with the Carrera Unified Formulation. The viscoelastic constitutive law is expressed in an integral form, which relates the viscous moduli with the strain rate. The solution of the governing equations is obtained in the Laplace domain and then transformed in the time domain through a numerical inversion procedure. The numerical simulations are performed on slender, squat, isotropic, and orthotropic beams in order to examine the actual differences between the results obtained with low-order theories and the refined models. The current results are validated with classical and higher-order solutions available in the literature. Moreover, comparisons between the equivalent-single layer and the layer-wise approach are reported and discussed.