Kinematic Assumptions Research Articles

Transverse free vibration of resonant nanoplate mass sensors: Identification of an attached point mass

In this paper we analyse for the first time the bending vibration of a nanoplate with an attached mass using the strain gradient elasticity theory for homogeneous Lamé material, under Kirchhoff-Love’s kinematical assumptions. The exact eigenvalues of the nanoplate vibrating with an attached mass are obtained for a general case, and an approximate closed form expression is provided if the intensity of the mass is small with respect to the total mass of the nanoplate. The inverse problem of identifying a point mass attached on a simply supported rectangular nanoplate from a selected minimal set of resonant frequency data is also considered. We show that if the point mass is small, then the position of the point mass and mass size can be determined by means of closed form expressions in terms of the changes induced by the point mass on the first three resonant frequencies. The identification procedure has been tested on an extended series of numerical simulations, varying the scale parameter of the nanoplate’s material and the position and size of the point mass.

Thermodynamic formulation of a unified multi-mechanism continuum viscoplastic damage model with application to high-Cr steels

In this paper, a unified multi-mechanism continuum viscoplastic damage model is proposed to simulate the thermomechanical behavior of high-Cr steels at elevated temperatures. To represent the effects of material degradation under external loads, a total damage tensor is incorporated, which is composed of the low-cycle fatigue damage, the creep damage and the ductile damage. Within the small strain framework, the model is established through a thermodynamically consistent approach. First, some kinematic assumptions are proposed and the concept of effective stress is adopted. Then, based on a state potential with proper constitutive form, the constitutive equations can be derived from the second law of thermodynamics. By further considering the postulate of maximum dissipation, a Lagrangian functional is constructed through a regularization scheme. The stationary points of the Lagrangian functional yield the evolution equations of the dissipative variables. For the isotropic damage case, the damage tensor can be represented by a scalar damage variable and the constitutive evolution equations in the model can be simplified. To be prepared for practical applications, numerical integration algorithms are proposed to solve the constitutive evolution equations, and the material parameters in the model are identified based on the experimental data. To demonstrate the efficiency of the current model, it is applied to simulate the thermomechanical response of high-Cr steels under different loading conditions. The simulation results can fit the experimental data at a quantitative level and the damage mechanisms under the different loading conditions can be revealed. Besides that, the model is further modified to take into account the microcracks closure effect of the ductile damage. The current model would be helpful for the safety design and lifetime evaluation of high-Cr steel components in practical applications.

Node-dependent kinematic elements for the dynamic analysis of beams with piezo-patches

This article extends the use of one-dimensional elements with node-dependent kinematics to the dynamic analysis of beam structures with piezo-patches. Node-dependent kinematics allows the kinematic assumptions to be defined individually on each finite element node, leading to finite element models with variable nodal kinematics. Derived from Carrera unified formulation, node-dependent kinematics facilitates the mathematical refinement to an arbitrary order at any desirable region on the nodal level while keeping the compactness of the formulation. As an ideal approach to simulate structures with special local features, node-dependent kinematics has been employed to model piezo-patches in static cases. In this work, the application of node-dependent beam elements in dynamic problems is demonstrated. Node-dependent kinematics is applied to increase the numerical accuracy in the areas where the piezo-patches lie in through sufficiently refined models, while lower order assumptions are used elsewhere. The dissimilar constitutive relations of neighboring components are appropriately considered with layer-wise models. Both open- and short-circuit conditions are considered. The results are compared against those from literature. The numerical study shows that the adoption of node-dependent kinematics allows accurate results to be obtained at reduced computational costs.

Buckling of functionally graded graphene reinforced conical shells under external pressure in thermal environment

In the present research, buckling analysis of composite laminated conical shells reinforced with graphene sheets is investigated. Graphene sheets as reinforcements are distributed in each lamina. Volume fraction of graphene in each layer may be different which results in a piecewise functionally graded conical shell. First order shear deformation shell theory, Donnell kinematic assumptions and von Kármán type of geometrical non-linearity are used to establish the governing equations of the conical shell and the associated boundary conditions. The pre-buckling forces of the shell are obtained employing a membrane analysis. The linear stability equations are developed using the adjacent equilibrium criterion. These equations are discreted by means of the generalised differential quadratures across the shell length and Fourier expansion through the circumferential direction. An eigenvalue problem is obtained which yields the critical buckling pressure of the conical shell in thermal environment and the circumferential mode number at the onset of buckling. Comparison studies are provided for graphene reinforced and conventional composite laminated cylindrical shells and also isotropic conical shells with and without thermal environment. Afterwards parametric studies are given for buckling of functionally graded graphene reinforced composite laminated conical shells in thermal environment with different boundary conditions. It is shown that, temperature elevation decreases the critical buckling pressures of the conical shell significantly. Also buckling pressure of the shell may be enhanced through a piecewise functionally graded distribution of volume fraction of reinforcements.

A Time-Dependent Finite Element Formulation for Thick Shape Memory Polymer Beams Considering Shear Effects

In this paper, employing a thermomechanical small strain constitutive model for shape memory polymers (SMP), a beam element made of SMPs is presented based on the kinematic assumptions of Timoshenko beam theory. Considering the low stiffness of SMPs, the necessity for developing a Timoshenko beam element becomes more prominent. This is due to the fact that relatively thicker beams are required in the design procedure of smart structures. Furthermore, in the design and optimization process of these structures which involves a large number of simulations, we cannot rely only on the time consuming 3D finite element analyses. In order to properly validate the developed formulations, the numeric results of the present work are compared with those of 3D finite element results of the authors, previously available in the literature. The parametric study on the material parameters, e.g., hard segment volume fracture, viscosity coefficient of different phases, and the external force applied on the structure (during the recovery stage) are conducted on the thermomechanical response of a short I-shape SMP beam. For instance, the maximum beam deflection error in one of the studied examples for the Euler–Bernoulli beam theory is 7.3%, while for the Timoshenko beam theory, is 1.5% with respect to the 3D FE solution. It is noted that for thicker or shorter beams, the error of the Euler–Bernoulli beam theory even more increases. The proposed beam element in this work could be a fast and reliable alternative tool for modeling 3D computationally expensive simulations.

Kinematic error magnitude in the single-mass inverted pendulum model of human standing posture

BackgroundMany postural control studies employ a single-mass inverted pendulum model (IPM) to represent the body during standing. However, it is not known to what degree and for what conditions the model’s kinematic assumptions are valid. Research questionOur first objective was to quantify the IPM error, corresponding to a distance change between the ankle joint and center of mass (COM) during unrestricted, natural, unperturbed standing. A second objective was to quantify the error of having the ankle joint angle represent the COM angle. MethodsEleven young participants completed five standing conditions: quiet standing with eyes open (EO) and closed (EC), voluntarily swaying forward (VSf) and backward (VSb), and freely moving (FR). The modified Helen-Hayes marker model was used to capture the body kinematics. ResultsThe COM distance changed <0.1% during EO and EC, <0.25% during VSf and VSb, and <1.5% during FR. The ankle angle moderately and positively correlated with the COM angle for EO, EC, and VSf, indicating that temporal features of the ankle angle moderately represent those of the COM angle. However, a considerable offset between the two existed, which needs to be considered when estimating the COM angle using the ankle angle. For VSb and FR, the correlation coefficients were low and/or negative, suggesting that a large error would result from using the ankle angle as an estimate of the COM angle. SignificanceInsights from this study will be critical for deciding when to use the IPM in postural control research and for interpreting associated results.

Free vibration of FG-CNT reinforced composite skew cylindrical shells using the Chebyshev-Ritz formulation

Abstract In the present research, the free vibration characteristics of a skew cylindrical panel made of functionally graded carbon nanotube reinforced composites (FG-CNTRCs) is investigated. A dual distribution of CNTs is considered across the panel thickness, namely a uniform and a nonuniform distribution. A refined rule of mixtures approach is applied to estimate the mechanical properties of the composite body, by means of the introduction of some efficiency parameters. A first order shear deformation shell theory (FSDT) is also combined with the Donnell's kinematic assumptions to determine the basic governing equations of the problem for thin-to-moderately thick shells. The governing equations are here referred to an oblique coordinate system, in order to handle any kind of boundary conditions. With the aid of the Ritz method, a system of homogeneous equations governs the eigenvalue problem, whose shape functions are built on Chebyshev polynomials. This system allows to compute the natural frequencies of the shell. A comparative evaluation of the formulation is performed to demonstrate its accuracy and efficiency. Further parametric studies are aimed at exploring the sensitivity of the response to some reinforcement parameters, as the volume fraction or the distribution of CNTs within the matrix.

A homogenized model for the nonlinear analysis of masonry columns in compression

Abstract A homogenized model based on simplified kinematic assumptions for the analysis of masonry columns subjected to axial forces is presented. A Unit Cell (UC) characterized by different arrangements of clay bricks and mortar joints is modeled and analyzed. The kinematic unknown fields are approximated by cubic interpolation functions and the compatible strain fields are accordingly derived. Stresses in bricks and mortar are evaluated on the basis of a damage constitutive law, relying on a modified Willam-Warnke yield criterion and an exponential evolution for the damage variable. Equilibrium equations are consistently deduced via the virtual displacement principle. A numerical solution procedure is proposed and described in detail, using the collocation technique to solve the nonlinear evolution problem of damage variables in masonry constituents. Numerical applications are presented to validate the proposed model. The response of UCs characterized by different geometrical textures is numerically studied and the results are compared with those obtained by well assessed nonlinear nonlocal finite element (FE) modeling approaches. Finally, a parametric investigation on the effects of constituents mechanical properties on the overall UC response is performed and a comparison with experimental evidences is illustrated.

Effect of rainfall spatial distribution and duration on minimum spatial resolution of rainfall data for accurate surface runoff prediction

The effect of rainfall spatial distribution and duration on determining rainfall spatial resolution for runoff discharge prediction and equilibrium time estimation was studied. We simulated hydrographs and equilibrium times of storms which have the same spatial averaged rainfall intensity with wide range of spatial distributions and durations under stationary storm conditions on an idealized plane. The equilibrium times under various rainfall conditions were estimated with an approximate solution based on the kinematic wave assumption. The runoff hydrographs were simulated using a kinematic wave runoff model. The approximate and computed results were used to investigate the relationship between the spatial–temporal rainfall characteristics and rainfall spatial resolution. From the simulated results, it was found that (i) A required rainfall spatial resolution to achieve threshold accuracy depended on rainfall spatial distribution. For the simulated cases, the critical rainfall spatial resolution was higher as the slope of rainfall spatial distribution was steeper. (ii) The rainfall duration, as well as rainfall spatial distribution, was an important factor for determining a required rainfall spatial resolution for accurate flood prediction. The required rainfall spatial resolution to achieve threshold prediction accuracy was higher as the rainfall duration was shorter. (iii) The rainfall spatial distribution and duration which were traditionally neglected need to be considered in lumped runoff modelling or measurement as an important factor. Although limited to the ideal test cases, the maximum errors due to the low rainfall spatial resolution for the equilibrium time and peak discharge were about 25% and 35%, respectively.

Shear deformable hybrid finite element formulation for buckling analysis of composite columns

The objective of this study is to develop a shear deformable hybrid finite element formulation for the flexural buckling analysis of fiber-reinforced laminate composite columns with doubly symmetric cross sections. The hybrid finite element formulation is developed by using the Hellinger-Reissner functional which is obtained by introducing the conditions of compatibility as auxiliary conditions to the potential energy functional. The shear deformation effects due to bending are included by equilibrating shear stress. In 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 beam theory. The agreement with Engesser formulation for flexural buckling analysis of columns with shear-weak cross sections shows the applicability and accuracy of the current hybrid finite element method for composite structural elements. The applicability of the developed method herein to sandwich and built-up columns are also illustrated.

Buckling of Multilayer Graphene Sheets Subjected to Axial Compression Based on a Continuum Mechanics Model

Buckling of multilayer graphene sheets (MLGSs) subjected to an axial compressive load in plane-strain condition is studied. Closed-form solutions for buckling load of MLGSs are obtained based on a continuum model for MLGSs. Two different kinematic assumptions, which lead to MLGS beam, which was recently proposed by the authors, and the Euler beam, are used to obtain the buckling loads. The obtained solutions yield significantly different buckling loads when the axial length is small. To validate obtained results, molecular dynamics (MD) simulations are conducted, and they show that the MLGS beam model well captures the buckling load of MLGSs. The buckling solution of MLGS beam model provides two interesting facts. First, the buckling load of MLGSs coincides with the Euler buckling load when the length is large. Second, when the number of layers is large, the buckling strain converges to a finite value, and could be expressed as a linear combination of the buckling strain of single-layer graphene and the ratio between the shear rigidity of interlayer and the tensile rigidity of graphene layer. We validate the asymptotic behavior of buckling strain through MD simulations and show that buckling occurs even when the overall thickness is larger than the axial length. Finally, we present a diagram that contains buckling strain of MLGSs according to the boundary conditions, the number of layers, and the axial length.

Numerical analysis of nonlinear free and forced vibrations of buckled curved beams resting on nonlinear elastic foundations

This paper presents a novel numerical procedure to predict nonlinear free and steady state forced vibrations of clamped–clamped curved beam in the vicinity of postbuckling configuration. Nonlinear Euler–Bernoulli kinematics assumptions including mid-plane stretching are proposed to exhibit a large deformation but a small strain of von Kármán. To simulate the interaction of beam with the surrounding elastic medium, nonlinear elastic foundation with cubic nonlinearity and shearing layer are employed. The nonlinear integro-differential equation that governs the buckling of beam is discretized using the differential-integral quadrature method (DIQM) and then is solved using Newton’s method. The problem of linear vibration is discretized using DIQM and then is solved as a linear eigenvalue problem. Afterwards, a single-mode Galerkin discretization is used to reduce the nonlinear governing equation into a time-varying Duffing equation. The Spectral differentiation matrix operators are exploited to discretize the Duffing equation. The discretized Duffing equation is a nonlinear eigenvalue problem which is directly solved using pseudo arc length continuation method. Results obtained by the proposed numerical solution are compared with analytical solutions available in the literature and good agreement is obtained. Parametric studies are carried out to show the effects of applied axial load, imperfection and nonlinear elastic foundations on the natural frequency as well as forced damped vibration behavior of the beam. The above mention effects play very important role on the dynamic behavior of buckled curved beam.

Free vibration of joined conical–cylindrical–conical shells

The present research considers the free vibration characteristics of a joined shell system that consists of three segments. The joined shell system contains two conical shells at the ends and a cylindrical shell at the middle. All shell elements are made from isotropic homogeneous material. The shell elements are unified in thickness. With the aid of the first-order shear deformation shell theory and the Donnell type of kinematic assumptions, the equations of motion of a conical shell and the associated boundary conditions are obtained. These equations are valid for each segment. The obtained equations are then discreted using the generalised differential quadratures (GDQ) method. Applying the intersection continuity conditions for displacements, rotations, forces, and moments between two adjacent shells, and also boundary conditions at the ends of the joined shell system, a set of homogeneous equations is obtained, which governs the free vibration motion of the joined shell. Comparisons are made with the available data in the open literature for the case of thin conical–cylindrical–conical shells with special types of geometry or boundary conditions. Afterwards, numerical results are provided for moderately thick shells with different geometrical and boundary conditions.

Application of the FOSDT to the solution of buckling problem of FGM sandwich conical shells under hydrostatic pressure

Abstract In this study, the buckling problem of functionally graded materials (FGMs) sandwich truncated conical shells (STCSs) under hydrostatic pressure was solved using the modified form of first-order shear deformation theory (FOSDT). The FGM properties of the constituents are graded in the thickness direction in accordance with a power-law distribution. The basis equations of FGM sandwich truncated conical shells (FGMSTCSs) are derived from the Donnell kinematics assumptions. Using the Galerkin method to the partial differential governing equations, the formula for the dimensionless hydrostatic buckling pressure (DHBP) in the framework of the FOSDT is obtained to demonstrate the buckling response of FGMSTCSs. A numerical analysis is performed for various types and geometry of FGMSTCSs under FOSDT and the classical shell theory (CST).

Modified porosity model in analysis of functionally graded porous nanobeams

This work studies the mechanical bending and vibration of functionally graded nanobeams using finite elements according to Euler beam theory. We implement a modified porosity model that represents porosity and Young’s modulus in an implicit form, where the density is assumed as a function of the porosity parameter, while Young’s modulus is assumed as a ratio of the mass density with porosity to that without porosity. The effect of nano-scale is described by the nonlocal continuum theory by adding the length scale into the constitutive equations as a material parameter comprising information about nanoscopic forces and its interactions. The material gradation of constituents is described by a power function through the thickness of nanobeam. The beam is simply supported and is assumed to be thin, and hence, the kinematic assumptions of Euler–Bernoulli beam theory are held. The mathematical model is solved numerically using the finite-element method. Numerical results show that the increase of porosity, material graduation, and nano-scale parameters tend to decrease the bending resistance as well as the fundamental frequencies of the nanobeam.

On infinitesimal and finite deformations in shape memory alloys

Two material models for shape memory alloys are developed and are evaluated by comparing and contrasting their predictions in three standard problems involving martensitic transformations. Both models are based on generalized plasticity and comprise a von Mises type of expression for the loading surfaces, a linear evolution law for the material martensite fraction, and a hyperelastic constitutive equation. The first model is an infinitesimal one, based on the usual additive decomposition of the small strain tensor into elastic and inelastic (transformation-induced) parts, while the second is a finite one based on the consistent use of the “physical” (intrinsic material) metric concept. This study reveals that in the first and second problem—uniaxial tension and simple shear—and under small and moderate levels of strain, both models predict almost identical response, while for higher levels of strain the models still predict comparable response, even though their basic kinematical assumptions differ vastly. The third problem, comprising infinitesimal shear with finite rotation, is considered next. In this case, it is shown that while the finite model yields the physically correct response, the infinitesimal model yields completely erroneous results.

Computational certification under limited experiments

SummaryA computational certification framework under limited experimental data is developed. By this approach, a high‐fidelity model (HFM) is first calibrated to limited experimental data. Subsequently, the HFM is employed to train a low‐fidelity model (LFM). Finally, the calibrated LFM is utilized for component analysis. The rational for utilizing HFM in the initial stage stems from the fact that constitutive laws of individual microphases in HFM are rather simple so that the number of material parameters that needs to be identified is less than in the LFM. The added complexity of material models in LFM is necessary to compensate for simplified kinematical assumptions made in LFM and for smearing discrete defect structure. The first‐order computational homogenization model, which resolves microstructural details including the structure of defects, is selected as the HFM, whereas the reduced‐order homogenization is selected as the LFM. Certification framework illustration, verification, and validation are conducted for ceramic matrix composite material system comprised of the 8‐harness weave architecture. Blind validation is performed against experimental data to validate the proposed computational certification framework.

Isogeometric large amplitude free vibration of graphene reinforced laminated plates in thermal environment using NURBS formulation

Present study proposes a nonlinear formulation to study the large amplitude free vibration of composite laminated plates reinforced by graphene sheets. Volume fraction of graphene sheets as reinforcements may be different in the layers. Material properties of the constituents are assumed to be temperature dependent. Properties of the composite media are obtained using a refined Halpin–Tsai approach where the auxiliary parameters are included into the formulation to capture the size dependency of the nanocomposite media. With the aid of third order shear deformation plate theory and the von Kármán type of kinematic assumptions, the basic governing equations of the plate are established. Afterwards, a non-uniform rational B-spline (NURBS) based isogeometric finite element method is used to study the large amplitude free vibration response of the graphene reinforced composite plates in thermal environment. Large amplitude frequencies as a function of centre point deflection are provided for different functionally graded patterns, aspect ratios, side to thickness ratios and boundary conditions.

Local buckling of laminated composite beams based on different plate theories

AbstractThis paper presents an approximate approach for the local buckling analysis of prismatic composite laminated beams under uniaxial compression wherein the segments of the beams (i.e. flanges and webs) are assumed to be moderately thick so that advanced plate theories that transcend the restrictions of classical laminated plate theory (CLPT) need to be employed. We present a novel approximate analysis method based on the so‐called discrete plate approach during which the segment of interest is separated from the beam and is subjected to rotational restraints at the cutting edges. The analysis itself employs rather simple shape functions for the local buckling modes in conjunction with a Rayleigh‐type approach using the principle of minimum elastic potential of the buckled segment wherein we use the kinematic assumptions of Reddy's third‐order shear deformation theory (TSDT). The results are compared to comparative computations based on CLPT and first‐order shear deformation theory (FSDT), and a good agreement is found especially between FSDT and TSDT which lends credibility to the present approach, however neither employing the strict limitations of CLPT imposed by Kirchhoff's classical kinematic assumptions, nor requiring shear correction factors as they are generally required in the framework of FSDT and which is still a topic of on‐going research. (© 2017 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis

A nonlinear two-dimensional (2D) continuum with a latent internal structure is introduced as a coarse model of a plane network of beams which, in turn, is assumed as a model of a pantographic structure made up by two families of equispaced beams, superimposed and connected by pivots. The deformation measures of the beams of the network and that of the 2D body are introduced and the former are expressed in terms of the latter by making some kinematical assumptions. The expressions for the strain and kinetic energy densities of the network are then introduced and given in terms of the kinematic quantities of the 2D continuum. To account for the modelling abilities of the 2D continuum in the linear range, the eigenmode and eigenfrequencies of a given specimen are determined. The buckling and post-buckling behaviour of the same specimen, subjected to two different loading conditions are analysed as tests in the nonlinear range. The problems have been solved numerically by means of the COMSOL Multiphysics finite element software.