Mindlin Model Research Articles

Hybrid Transformation to a Generalized Reissner–Mindlin Theory for Composite Plates

An asymptotic theory of composite plates is constructed using the variational asymptotic method. To maximize simplicity and promote efficiency of the developed model, a transformation procedure is required to establish a mathematical link between an asymptotically correct energy functional derived herein and a simpler engineering model, such as a generalized Reissner–Mindlin model. Without relaxing the warping constraints and performing “smart minimization” or optimization procedures introduced in previous work, a different approach is suggested in this paper. To eliminate all partial derivatives of the 2D generalized strains in the asymptotically correct energy functional, a hybrid transformation procedure is systematically carried out by involving modified equilibrium and compatibility equations, and solving a system of linear algebraic equations via the pseudo-inverse method. Equivalent constitutive laws for the generalized Reissner–Mindlin plate model are then estimated. Several examples as a preliminary validation are used to demonstrate the capability and accuracy of this new model.

CFD-DEM simulation of the hole cleaning process in a deviated well drilling: The effects of particle shape

We investigate the effect of particle shape on the transportation mechanism in well-drilling using a three-dimensional model that couples computational fluid dynamics (CFD) with the discrete element method (DEM). This numerical method allows us to incorporate the fluid–particle interactions (drag force, contact force, Saffman lift force, Magnus lift force, buoyancy force) using momentum exchange and the non-Newtonian behavior of the fluid. The interactions of particle−particle, particle−wall, and particle−drill pipe are taken into account with the Hertz–Mindlin model. We compare the transport of spheres with non-spherical particles (non-smooth sphere, disc, and cubic) constructed via the multi-sphere method for a range of fluid inlet velocities and drill pipe inclination angles. The simulations are carried out for laboratory-scale drilling configurations. Our results demonstrate good agreement with published experimental data. We evaluate the fluid–particle flow patterns, the particle velocities, and the particle concentration profiles. The results reveal that particle sphericity plays a major role in the fluid–solid interaction. The traditional assumption of an ideal spherical particle may cause inaccurate results.

Reprint of “Erosion prediction of liquid-particle two-phase flow in pipeline elbows via CFD–DEM coupling method”

A CFD–DEM-based liquid-particle two-phase flow erosion prediction model occurring over such pipeline connection as elbows, which takes the interaction of liquid–particle, particle–particle and particle–wall into account, is proposed in this study. The standard k−ε model is adopted for fluid turbulent flow, the standard wall functions for near-wall zone treatment, and the Hertz–Mindlin (no slip) model for particle–particle and particle–wall contact. Numerical simulations have been performed by combining CFD code ANSYS FLUENT with DEM code EDEM to predict the maximum erosion rate and location in 90°, 60° and 45° elbows with diameter 40mm. The fluid is water with inlet velocity 3m/s, and the particles are 150μm in diameter and 2650kg/m3 in density. While the maximum erosion rates of the three are found to be quite different, the maximum erosive locations have been indicated to be at or near the exit of the elbows. In addition, the effect of bend-angle on erosion, particle motion and flow field has been discussed.

Stiffness of sphere-plate contacts at MHz frequencies: dependence on normal load, oscillation amplitude, and ambient medium.

The stiffness of micron-sized sphere–plate contacts was studied by employing high frequency, tangential excitation of variable amplitude (0–20 nm). The contacts were established between glass spheres and the surface of a quartz crystal microbalance (QCM), where the resonator surface had been coated with either sputtered SiO2 or a spin-cast layer of poly(methyl methacrylate) (PMMA). The results from experiments undertaken in the dry state and in water are compared. Building on the shifts in the resonance frequency and resonance bandwidth, the instrument determines the real and the imaginary part of the contact stiffness, where the imaginary part quantifies dissipative processes. The method is closely analogous to related procedures in AFM-based metrology. The real part of the contact stiffness as a function of normal load can be fitted with the Johnson–Kendall–Roberts (JKR) model. The contact stiffness was found to increase in the presence of liquid water. This finding is tentatively explained by the rocking motion of the spheres, which couples to a squeeze flow of the water close to the contact. The loss tangent of the contact stiffness is on the order of 0.1, where the energy losses are associated with interfacial processes. At high amplitudes partial slip was found to occur. The apparent contact stiffness at large amplitude depends linearly on the amplitude, as predicted by the Cattaneo–Mindlin model. This finding is remarkable insofar, as the Cattaneo–Mindlin model assumes Coulomb friction inside the sliding region. Coulomb friction is typically viewed as a macroscopic concept, related to surface roughness. An alternative model (formulated by Savkoor), which assumes a constant frictional stress in the sliding zone independent of the normal pressure, is inconsistent with the experimental data. The apparent friction coefficients slightly increase with normal force, which can be explained by nanoroughness. In other words, contact splitting (i.e., a transport of shear stress across many small contacts, rather than a few large ones) can be exploited to reduce partial slip.

CFD–DEM Model for Simulation of Non-spherical Particles in Hole Cleaning Process

During the well drilling process, particles are produced in different shapes. The shape of particles can influence the characteristics of particles transport process. The aim of this work is to analyze the effects of particle shape on the transportation mechanism. For this purpose, a three-dimensional model is prepared for simulation of particle transportation with spherical and non-spherical shapes, during deviated well drilling. The motion of particles and the non-Newtonian fluid flow are simulated via discrete element method and CFD, respectively. The two-way coupling scheme is used to incorporate the effects of fluid–particle interactions. Three different samples of non-spherical shapes are constructed using multi-sphere method. The interactions of particle–particle/wall/drill pipe are taken into account via Hertz–Mindlin model. Simulations are carried out for some laboratory-scale configurations and fair agreements with the experimental data available in the literature are established.

Erosion prediction of liquid-particle two-phase flow in pipeline elbows via CFD–DEM coupling method

A CFD–DEM-based liquid-particle two-phase flow erosion prediction model occurring over such pipeline connection as elbows, which takes the interaction of liquid–particle, particle–particle and particle–wall into account, is proposed in this study. The standard k−ε model is adopted for fluid turbulent flow, the standard wall functions for near-wall zone treatment, and the Hertz–Mindlin (no slip) model for particle–particle and particle–wall contact. Numerical simulations have been performed by combining CFD code ANSYS FLUENT with DEM code EDEM to predict the maximum erosion rate and location in 90°, 60° and 45° elbows with diameter 40mm. The fluid is water with inlet velocity 3m/s, and the particles are 150μm in diameter and 2650kg/m3 in density. While the maximum erosion rates of the three are found to be quite different, the maximum erosive locations have been indicated to be at or near the exit of the elbows. In addition, the effect of bend-angle on erosion, particle motion and flow field has been discussed.

Axisymmetric/asymmetric buckling of functionally graded circular/annular Mindlin nanoplates via nonlocal elasticity

This paper provides the axisymmetric/asymmetric buckling analysis of moderately thick circular and annular functionally graded (FG) nanoplates under uniform compressive in-plane loads. The material properties change continuously through the thickness of the FG nanoplate, which can vary according to a power-law distribution of the volume fraction of the constituents whereas Poison’s ratio is set to be constant. To reflect the small scale effects on the critical buckling loads of the Mindlin model, the nonlocal elasticity of Eringen is employed. An exact closed-form solution is used to solve the nonlocal governing equations. The effect of small scales on buckling loads for different parameters such as diameter of the FG nanoplate, boundary conditions, material properties and mode numbers are investigated in details. It is found that the modes corresponding to the lowest critical buckling loads for circular and annular FG nanoplates may be axisymmetric or asymmetric. Therefore, the asymmetric modes as well as axisymmetric ones have been considered here.

Toward lattice fractional vector calculus

An analog of fractional vector calculus for physical lattice models is suggested. We use an approach based on the models of three-dimensional lattices with long-range inter-particle interactions. The lattice analogs of fractional partial derivatives are represented by kernels of lattice long-range interactions, where the Fourier series transformations of these kernels have a power-law form with respect to wave vector components. In the continuum limit, these lattice partial derivatives give derivatives of non-integer order with respect to coordinates. In the three-dimensional description of the non-local continuum, the fractional differential operators have the form of fractional partial derivatives of the Riesz type. As examples of the applications of the suggested lattice fractional vector calculus, we give lattice models with long-range interactions for the fractional Maxwell equations of non-local continuous media and for the fractional generalization of the Mindlin and Aifantis continuum models of gradient elasticity.

A multiphysics model for magneto-electro-elastic laminates

An efficient, geometrically nonlinear, multiphysics plate model for analyzing magneto-electro-elastic composite laminates is rigorously developed by applying the variational asymptotic method. By taking advantage of the inherent small parameter characterized by the ratio of the thickness to the in-plane deformation of the plate, we systematically reduced the original multiphysically coupled three-dimensional model to a series of two-dimensional plate models. A companion one-dimensional through-the-thickness analysis provides necessary constitutive models for the plate analysis. For practical uses, we also fit the asymptotically correct second-order electromagnetic enthalpy into a generalized Reissner–Mindlin model. Three-dimensional multiphysical components such as displacement/strain/stress fields as well as the electric/magnetic potentials and fluxes are obtained through a recovery process. Results for sample problems featuring electromagnetic and elastic coupling as well as characteristically different load/boundary conditions and material configurations are compared with exact solutions to systematically investigate the prediction capability of the present model.

Reissner–Mindlin plate model with uncertain input data

A Reissner–Mindlin model of a plate resting on unilateral rigid piers and a unilateral elastic foundation is considered. Since the material coefficients of the orthotropic plate, stiffness of the foundation, and the lateral loading are uncertain, a method of the worst scenario (anti-optimization) is employed to find maximal values of some quantity of interest.The state problem is formulated in terms of a variational inequality with a monotone operator. Using mixed-interpolated finite elements, approximations are proposed for the state problem and for the worst scenario problem. The solvability of the problems and a convergence of approximations is proved.

Analysis of the discontinuous Petrov–Galerkin method with optimal test functions for the Reissner–Mindlin plate bending model

We analyze the discontinuous Petrov–Galerkin (DPG) method with optimal test functions when applied to solve the Reissner–Mindlin model of plate bending. We prove that the hybrid variational formulation underlying the DPG method is well-posed (stable) with a thickness-dependent constant in a norm encompassing the L2-norms of the bending moment, the shear force, the transverse deflection and the rotation vector. We then construct a numerical solution scheme based on quadrilateral scalar and vector finite elements of degree p. We show that for affine meshes the discretization inherits the stability of the continuous formulation provided that the optimal test functions are approximated by polynomials of degree p+3. We prove a theoretical error estimate in terms of the mesh size h and polynomial degree p and demonstrate numerical convergence on affine as well as non-affine mesh sequences.

DEM simulation and physical testing of rice seed impact against a grain loss sensor

A triaxial ellipsoidal particle model was established according to the physical properties of rice seed, and its impact behaviour against a grain loss sensor was simulated using the discrete element method (DEM). The contact criterion was developed directly by solving the intersection equations, and the contact forces were calculated according to elastic–plastic and Mindlin models. It was shown that the seeds may perform translational and rotational motion in a 3D space after the impact. With the influences of particle shape, orientation and angle of incidence, three typical impact processes were found: single impact, multiple impacts in a short-time, and continuous impacts. Two important parameters for the design of loss sensors are the maximum normal impact force Fn max and the force rise-time tr. Simulations showed that an increase in particle ellipticity strongly enlarged the differences in Fn max. As the ellipticities increased from unity to 2, the defined force ratio η decreased from 100% to about 40%, and this value decreased to less than 20% when ellipticities continuously increased from 2 to 4. Tangential velocity led to an asymmetric variation of η. tr was generally distributed between 12 and 54 μs. In laboratory tests, rice seeds were allowed to free fall onto a loss sensor from a height of 320 mm. Results indicated that the peak output voltage was fluctuated in 1.5–4.5 V, and the rise-time was in 14–48 μs.

Scaling and hierarchies of wave motion in solids

AbstractThe concept of wave hierarchies in the Whitham's sense is generalized to hierarchies of second order wave operators. Based on Mindlin's model of microstructured solids, the scaling procedure is described and the corresponding hierarchical equation derived which includes two wave operators. It is shown that waves in the Cosserat' medium are described by a similar hierarchical equation. These results are generalized to a multiscale case (a scale within a scale) and to nonlinear media. It is shown also how to construct hierarchies for waves in elastic ferroelectrics. The results obtained by Scott for hierarchies in thermoelasticity are presented in the similar framework. Finally, the cases with first order wave operators are described.

Zeroth-order shear deformation micro-mechanical model for composite plates with in-plane heterogeneity

This article introduces a new model for investigating the mechanical behavior of heterogeneous plates, which are composed of periodically-repeated microstructures along the in-plane directions. We first formulate the original three-dimensional problem in an intrinsic form for implementation into a single unified formulation and application to geometrically nonlinear problem. Taking advantage of smallness of the plate thickness-to-length parameter and heterogeneity and performing homogenization along dimensional reduction simultaneously, the variational asymptotic method is used to rigorously construct an effective zeroth-order plate model, which is similar to a generalized Reissner–Mindlin model (the first-order shear deformation model) capable of capturing the transverse shear deformations, but still carries out the zeroth-order approximation which can maximize simplicity and promote efficiency. This present approach is incorporated into a commercial analysis package for the purpose of dealing with realistic and complex geometries and constituent materials at the microscopic level. A few examples available in literature are used to demonstrate the consistence and efficiency of this new model, especially for the structures, in which the effects of transverse shear deformations are significant.

Contact parameter estimation for DEM simulation of iron ore pellet handling

The appropriate choice of contact parameters is of great importance, particularly when simulations using the discrete element method (DEM) are run with the aim of capturing quantitative information on collision events that occur among the individual particles, as what happens during handling of iron ore pellets. The paper analyzes the use of tests conducted with single pellets for estimating material and contact parameters that are required in DEM simulations using the no-slip Hertz–Mindlin model. Tests included impact load cell, tribometer, drop tests and a simple test in which the rolling angle of individual pellets is measured. It is concluded that, provided particle shapes are described appropriately, DEM simulations provide reasonably realistic predictions of pellet motion in assemblies from contact parameters measured on the basis of single-particle tests. In this case the physical significance of the contact parameters is maintained. However, if pellets are described in a simplistic way as spheres, then the DEM simulation of their flow no longer matches the experimental results, requiring fitting some of the contact parameters in order to match the observed global flow, so that much of the physical meaning of the parameters is lost.

Asymptotical construction of a Reissner-like model for multilayer functionally graded magneto-electro-elastic plates

A magneto-electro-elastic Reissner–Mindlin model is developed for heterogeneous multilayer laminates made of functionally graded magneto-electro-elastic material using the variational asymptotical method. This model is applicable to laminates without prescribed electric and magnetic potential though the thickness. Taking advantage of the smallness of the plate thickness, we asymptotically decouple the original 3D magneto-electro-elastic problem into a 1D through-the-thickness analysis and a 2D plate analysis, and both are fully-coupled magneto-electro-elastic analyses. Interface continuity conditions are explicitly derived and solved to obtain the multilayer solutions. Various examples have been used to demonstrate the accuracy and application of this model.

On the influence of internal degrees of freedom on dispersion in microstructured solids

In the present paper a model describing wave propagation in the nonlinear dispersive media with microstructure is investigated. The model is based on the continuum approach following Mindlin's and Eringer's earlier theories which model a microstructure as a deformable cells in a macrostructure assuming that the deformation gradient is small. A generalized version of the Mindlin model called the Mindlin–Engelbrecht–Pastrone model (MEP) is used. The MEP model is solved numerically using the pseudospectral method and localized initial conditions together with periodic boundary conditions. The main focus of the study is on clarifying the influence of internal degrees of freedom of a microstructure on solutions.

Asymptotical construction of a fully coupled, Reissner–Mindlin model for piezoelectric and piezomagnetic laminates

The variational asymptotic method is used to construct a fully coupled Reissner–Mindlin model for piezoelectric and piezomagnetic laminates with some surfaces parallel to the reference surface coated with electrodes and magnetism. Taking advantage of the smallness of the plate thickness, we asymptotically split the original 3D electromagneto-mechanical problem into a 1D through-the-thickness analysis and a 2D plate analysis, and both are fully-coupled multiphysics analyses. The through-the-thickness analysis serves as a link between the original 3D analysis and the plate analysis by providing a constitutive model for the plate analysis and recovering the 3D field variables in terms of global responses calculated by the plate analysis. The present theory is implemented into the computer program VAPAS (Variational Asymptotic Plate and Shell Analysis). A numerical example of three-layer sandwich plate has been used to validate the present model.

Robust C0 high‐order plate finite element for thin to very thick structures: mechanical and thermo‐mechanical analysis

SUMMARYThis paper presents a new C0 eight‐node quadrilateral finite element (FE) for geometrically linear elastic plates. This finite element aims at modeling both thin and thick plates without any pathologies of the classical plate finite elements (shear and Poisson or thickness locking, spurious modes, etc). A C1 FE was previously developed by the first author based on the kinematics proposed by Touratier. This new FE can be viewed as an evolution towards three directions: (1) use of only C0 FE approximations; (2) modeling of thick to thin structures; and (3) capability in multifield problems. The transverse normal stress is included allowing use of the three‐dimensional constitutive law. The element performances are evaluated on some standard plate tests, and comparisons are given with exact three‐dimensional solutions for plates under mechanical and thermal loads. Comparisons are made with other plate models using C1 and semi‐C1 FE approximations as well as with an eight node C0 FE based on the Reissner–Mindlin model. All results indicate that the present element is highly insensitive to mesh distortion, has very fast convergence properties and gives accurate results for displacements and stresses. Copyright © 2012 John Wiley & Sons, Ltd.

Mitigating the dead-layer effect in nanocapacitors using graded dielectric films

As ideal candidates for next-generation energy storage devices, nanocapacitors are predicted to exhibit very high capacitance according to classical theory. However, the actual capacitance of nanocapacitors is dramatically lower than expected. This is attributed to the so-called ‘dead-layer’ effect associated with the flexoelectricity of dielectric films and the incomplete screening of metal electrodes. In this paper, a way to mitigate this negative effect is demonstrated by using graded dielectric films instead of homogeneous films. The enhancements due to grading dielectric films were obtained by using perturbation theory to solve the governing equations with boundary conditions in Mindlin's model of parallel-plate capacitors. We have shown that by grading both the relative permittivity and the elastic constant, we can obtain enhancement of almost 27% in capacitance for the 2.7 nm SrTiO3 dielectric film. In addition, the impact of various dielectric film properties on the overall capacitance was investigated.