Showing papers on "Displacement field published in 2019"

New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations

Abstract: The aim of this work is to establish a two dimensional (2D) and quasi three dimensional (quasi-3D) shear deformation theories, which can model the free vibration of FG plates resting on elastic foundations using a new shear strain shape function. The proposed theories have a novel displacement field which includes undetermined integral terms and contains fewer unknowns with taking into account the effects of both transverse shear and thickness stretching. The mechanical properties of the plates are assumed to vary through the thickness according to a power law distribution in terms of the volume fractions of the constituents. The elastic foundation parameters are introduced in the present formulation by following the Pasternak (two-parameters) mathematical model. Hamilton's principle is employed to determine the equations of motion. The closed form solutions are derived by using Navier's method and then fundamental frequencies are obtained by solving the results of eigenvalue problems. The efficiency of the proposed theory is ascertained by comparing the results of numerical examples with the different 2D, 3D and quasi-3D solutions found in literature.

Thermomechanical analysis of antisymmetric laminated reinforced composite plates using a new four variable trigonometric refined plate theory

Abstract: The thermo-mechanical bending behavior of the antisymmetric cross-ply laminates is examined using a new simple four variable trigonometric plate theory The proposed theory utilizes a novel displacement field which introduces undetermined integral terms and needs only four variables The validity of the present model is proved by comparison with solutions available in the literature

Static analysis of laminated reinforced composite plates using a simple first-order shear deformation theory

Abstract: This paper aims to present an analytical model to predict the static analysis of laminated reinforced composite plates subjected to sinusoidal and uniform loads by using a simple first-order shear deformation theory (SFSDT). The most important aspect of the present theory is that unlike the conventional FSDT, the proposed model contains only four unknown variables. This is due to the fact that the inplane displacement field is selected according to an undetermined integral component in order to reduce the number of unknowns. The governing differential equations are derived by employing the static version of principle of virtual work and solved by applying Navier\'s solution procedure. The non-dimensional displacements and stresses of simply supported antisymmetric cross-ply and angle-ply laminated plates are presented and compared with the exact 3D solutions and those computed using other plate theories to demonstrate the accuracy and efficiency of the present theory. It is found from these comparisons that the numerical results provided by the present model are in close agreement with those obtained by using the conventional FSDT.

Dense Tactile Force Estimation using GelSlim and inverse FEM

Abstract: In this paper, we present a new version of tactile sensor GelSlim 2.0 with the capability to estimate the contact force distribution in real time. The sensor is vision-based and uses an array of markers to track deformations on a gel pad due to contact. A new hardware design makes the sensor more rugged, parametrically adjusTable AND Improves illumination. leveraging the sensor’s increased functionality, we propose to use inverse finite element method (ifem), a numerical method to reconstruct the contact force distribution based on marker displacements. the sensor is able to provide force distribution of contact with high spatial density. experiments and comparison with ground truth show that the reconstructed force distribution is physically reasonable with good accuracy.A sequence of Kendama manipulations with corresponding displacement field (yellow) and force field (red). Video can be found on Youtube: https://youtu.be/hWw9A0ZBZuU

Thermomechanical bending study for functionally graded sandwich plates using a simple quasi-3D shear deformation theory

Abstract: In this article, a simple quasi-3D shear deformation theory is employed for thermo-mechanical bending analysis of functionally graded material (FGM) sandwich plates. The displacement field is defined using only 5 variables as the first order shear deformation theory (FSDT). Unlike the other high order shear deformation theories (HSDTs), the present formulation considers a new kinematic which includes undetermined integral variables. The governing equations are determined based on the principle of virtual work and then they are solved via Navier method. Analytical solutions are proposed to provide the deflections and stresses of simply supported FGM sandwich structures. Comparative examples are presented to demonstrate the accuracy of the present theory. The effects of gradient index, geometrical parameters and thermal load on thermo-mechanical bending response of the FG sandwich plates are examined.

Failure characteristics of rock-like materials with single flaws under uniaxial compression

Abstract: A uniaxial compression test was conducted with a servo loading apparatus to study the failure of a rock-like specimen with a pre-existing single flaw. The evolution of cracks was monitored with digital image correlation technology and simulated with the expanded distinct element method based on the strain strength criterion. The concentration and evolution of the principal strain field were found to be consistent with the initiation, propagation, and coalescence of cracks. As the inclination angle increased, the position of the maximum principal strain concentration changed from within the flaw to the flaw tips, and the distribution of the horizontal displacement field changed from symmetric to antisymmetric. The initiation stress and peak strength were affected by the inclination angle; they were minimum when the inclination angle was 60°. As the inclination angle increased, the failure mode of the specimens transformed from mostly tensile failure to mostly shear failure.

Transient response of oscillated carbon nanotubes with an internal and external damping

Abstract: The present works aims at modeling a viscoelastic nanobeam with simple boundary conditions at the two ends with the introduction of the Kelvin-Voigt viscoelasticity in a nonlocal strain gradient theory. The nanobeam lies on the visco-Pasternak matrix in which three characteristic parameters have a prominent role. A refined Timoshenko beam theory is here applied, which is only based on one unknown variable, in accordance with the Euler-Bernoulli theory, whereas the Hamilton's principle is applied to derive the equations of motion. These are, in turn, solved for a carbon nanotube with some fixed material properties. An analytical method has been used to discretize the equations in the displacement field and time, while computing the time-response of the system. For validation purposes, the results based on the proposed formulation are successfully compared to several references. A final parametric investigation focuses on the sensitivity of the time-response of a nanotube under simple boundary conditions, to different parameters such as the length scale, the viscoelasticity coefficients or the nonlocal parameter.

Modeling the dynamic and quasi-static compression-shear failure of brittle materials by explicit phase field method

Abstract: The phase field method is a very effective method to simulate arbitrary crack propagation, branching, convergence and complex crack networks. However, most of the current phase-field models mainly focus on tensile fracture problems, which is not suitable for rock-like materials subjected to compression and shear loads. In this paper, we derive the driving force of phase field evolution based on Mohr–Coulomb criterion for rock and other materials with shear frictional characteristics and develop a three-dimensional explicit parallel phase field model. In spatial integration, the standard finite element method is used to discretize the displacement field and the phase field. For the time update, the explicit central difference scheme and the forward difference scheme are used to discretize the displacement field and the phase field respectively. These time integration methods are implemented in parallel, which can tackle the problem of the low computational efficiency of the phase field method to a certain extent. Then, three typical benchmark examples of dynamic crack propagation and branching are given to verify the correctness and efficiency of the explicit phase field model. At last, the failure processes of rock-like materials under quasi-static compression load are studied. The simulation results can well capture the compression-shear failure mode of rock-like materials.

On the Nonlinear Vibrations of Polymer Nanocomposite Rectangular Plates Reinforced by Graphene Nanoplatelets: A Unified Higher-Order Shear Deformable Model

Abstract: This paper presents a unified higher-order shear deformable plate model to numerically examine the nonlinear vibration behavior of thick and moderately thick polymer nanocomposite rectangular plates reinforced by graphene platelets (GPLs). Four distribution patterns of graphene nanoplatelet nanofillers across the plate thickness are considered. The effective material properties of graphene platelet-reinforced polymer (GPL-RP) nanocomposite plate are approximately calculated by employing the modified Halpin–Tsai model and rule of mixture. Using a generalized displacement field, a unified mathematical formulation is derived based on Hamilton’s principle in conjunction with von Karman geometrical nonlinearity. By selecting appropriate shape functions, the proposed unified nonlinear plate model can be reduced to that on the basis of Mindlin, Reddy, parabolic, trigonometric and exponential shear deformation plate theories. The investigation of nonlinear vibration behavior is performed by employing a multistep numerical solution approach. In this regard, the discretization process is done through the generalized differential quadrature method. Then, the discretized governing equations are solved by employing the numerical-based Galerkin technique, periodic time differential operators and pseudo-arc length continuation algorithm. A detailed parametric study is carried out to examine the effect of GPL distribution pattern, weight fraction, geometry of GPL nanofillers and boundary constraints on the nonlinear vibration characteristics of the GPL-RP nanocomposite rectangular plates.

Micromorphic computational homogenization for mechanical metamaterials with patterning fluctuation fields

Abstract: This paper presents a homogenization framework for elastomeric metamaterials exhibiting long-range correlated fluctuation fields. Based on full-scale numerical simulations on a class of such materials, an ansatz is proposed that allows to decompose the kinematics into three parts, i.e. a smooth mean displacement field, a long-range correlated fluctuating field, and a local microfluctuation part. With this decomposition, a homogenized solution is defined by ensemble averaging the solutions obtained from a family of translated microstructural realizations. Minimizing the resulting homogenized energy, a micromorphic continuum emerges in terms of the average displacement and the amplitude of the patterning long-range microstructural fluctuation fields. Since full integration of the ensemble averaged global energy (and hence also the corresponding Euler–Lagrange equations) is computationally prohibitive, a more efficient approximative computational framework is developed. The framework relies on local energy density approximations in the neighbourhood of the considered Gauss integration points, while taking into account the smoothness properties of the effective fields and periodicity of the microfluctuation pattern. Finally, the implementation of the proposed methodology is briefly outlined and its performance is demonstrated by comparing its predictions against full scale simulations of a representative example.

Isogeometric analysis for nonlinear planar pantographic lattice: discrete and continuum models

Abstract: Pantographic sheets are metamaterials constituted by two interconnected layers of straight fibers. One of the great features of these structures is that they are extremely elastically compliant toward large nonlinear deformations. To model pantographic lattices, Kirchhoff rods based on Euler–Bernoulli assumptions can be used. Otherwise, if the fibers are sufficiently dense, homogenization of the microstructure results in a two-dimensional second-order gradient continuum model. The discrete and continuum models have in common the fact that their energy terms depend on the second-order derivatives of the displacement field, such that the classical finite element method cannot be directly employed. We propose instead the use of the isogeometric analysis where the basis functions are endowed with a higher inter-element continuity, allowing the rotation-free discretization of these problems. The discrete and continuum models are compared to existing benchmarks and are found in excellent agreement, validating the proposed approach.

Influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates

Abstract: Present study aims to investigate the influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates. Reddy's higher order shear deformation theory has been employed to model the displacement field. Variational approach has been used to derive the governing differential equations. Effect of material uncertainties in the formulation have been incorporated using first order perturbation technique (FOPT). An efficient stochastic finite element formulation (SFEM) have been used for the calculation of first and second order statics of natural frequency and transverse deflection. Validation of the results have been performed with the help of available literature and separately developed Monte Carlo formulation (MCS) algorithm. A large number of examples have been solved to quantify the effect of uncertainties on the vibration and bending characteristics of functionally graded skew sandwich plates.

Augmented Lagrangian Digital Image Correlation

Abstract: Digital image correlation (DIC) is a powerful experimental technique for measuring full-field displacement and strain. The basic idea of the method is to compare images of an object decorated with a speckle pattern before and after deformation, and thereby to compute the displacement and strain fields. Local subset DIC and finite element-based global DIC are two widely used image matching methods. However there are some drawbacks to these methods. In local subset DIC, the computed displacement field may not be compatible, and the deformation gradient may be noisy, especially when the subset size is small. Global DIC incorporates displacement compatibility, but can be computationally expensive. In this paper, we propose a new method, the augmented-Lagrangian digital image correlation (ALDIC), that combines the advantages of both the local (fast) and global (compatible) methods. We demonstrate that ALDIC has higher accuracy and behaves more robustly compared to both local subset DIC and global DIC.

On existence and uniqueness of weak solutions for linear pantographic beam lattices models

Abstract: In this paper, we discuss well-posedness of the boundary-value problems arising in some “gradient-incomplete” strain-gradient elasticity models, which appear in the study of homogenized models for a large class of metamaterials whose microstructures can be regarded as beam lattices constrained with internal pivots. We use the attribute “gradient-incomplete” strain-gradient elasticity for a model in which the considered strain energy density depends on displacements and only on some specific partial derivatives among those constituting displacements first and second gradients. So, unlike to the models of strain-gradient elasticity considered up-to-now, the strain energy density which we consider here is in a sense degenerated, since it does not contain the full set of second derivatives of the displacement field. Such mathematical problem was motivated by a recently introduced new class of metamaterials (whose microstructure is constituted by the so-called pantographic beam lattices) and by woven fabrics. Indeed, as from the physical point of view such materials are strongly anisotropic, it is not surprising that the mathematical models to be introduced must reflect such property also by considering an expression for deformation energy involving only some among the higher partial derivatives of displacement fields. As a consequence, the differential operators considered here, in the framework of introduced models, are neither elliptic nor strong elliptic as, in general, they belong to the class so-called hypoelliptic operators. Following (Eremeyev et al. in J Elast 132:175–196, 2018. https://doi.org/10.1007/s10659-017-9660-3 ) we present well-posedness results in the case of the boundary-value problems for small (linearized) spatial deformations of pantographic sheets, i.e., 2D continua, when deforming in 3D space. In order to prove the existence and uniqueness of weak solutions, we introduce a class of subsets of anisotropic Sobolev’s space defined as the energy space E relative to specifically assigned boundary conditions. As introduced by Sergey M. Nikolskii, an anisotropic Sobolev space consists of functions having different differential properties in different coordinate directions.

Optimization problems for a viscoelastic frictional contact problem with unilateral constraints

Abstract: We consider a mathematical model which describes the contact between a viscoelastic body and a rigid-deformable foundation with memory effects. We derive a variational formulation of the model which is in the form of a history-dependent variational inequality for the displacement field. Then we prove the existence of a unique weak solution to the problem. We also study the continuous dependence of the solution with respect to the data and prove two convergence results, under different assumptions on the data. The proofs are based on arguments of lower semicontinuity, pseudomonotonicity, and compactness. Finally, we use our convergence results in the study of several optimization problems associated to the viscoelastic contact model.

Vibration and buckling behaviours of thin-walled composite and functionally graded sandwich I-beams

Abstract: The paper proposes a Ritz-type solution for free vibration and buckling analysis thin-walled composite and functionally graded sandwich I-beams. The variation of material through the thickness of functionally graded beams follows the power-law distribution. The displacement field is based on the first-order shear deformation theory, which can reduce to non-shear deformable one. The governing equations of motion are derived from Lagrange's equations. Ritz method is used to obtain the natural frequencies and critical buckling loads of thin-walled beams for both non-shear deformable and shear deformable theory. Numerical results are compared to those from previous works and investigate the effects of fiber angle, material distribution, span-to-height's ratio, and shear deformation on the critical buckling loads and natural frequencies of thin-walled I-beams for various boundary conditions.