Equivalent Continuum Research Articles

Elastic moduli of covalently functionalized single layer graphene sheets

Due to their flexibility, large surface area and high specific mechanical properties, single layer graphene sheets (SLGSs) are potential candidates as filler materials for improving mechanical properties of polymers. Their effective utilization as reinforcements requires strong interfacial binding with the matrix surrounding them. The covalent functionalization of SLGSs is an effective technique to enhance this binding. However, covalent bonds introduced by a functional group usually alter the pristine structure of the SLGS that may affect its mechanical properties. Thus it is important to delineate effects of covalent functionalization on elastic moduli of an SLGS. We consider five groups of different polarities, namely, hydrogen (H), hydroxyl (OH), carboxyl (COOH), amine (NH2), and fluorine (F) as model functional groups and investigate their effects on values of Young’s modulus and the shear modulus of the SLGS. We use molecular mechanics (MM) simulations with the MM3 potential and the software TINKER to conduct the study. The pristine and the functionalized SLGSs are deformed in simple tension and simple shear, and from curves of the strain energy density of deformation vs. the axial strain and the shear strain, values of Young’s modulus and the shear modulus, respectively, are derived. These values are based on the hypothesis that the response of an SLGS is the same as that of an energetically and geometrically equivalent continuum structure of wall thickness 3.4Å. It is found that functionalization reduces the elastic moduli of the SLGSs which could be due to nearly 120% local strains induced at the functionalized sites of relaxed but unloaded SLGS, and the change in hybridization from sp2 to sp3. The decrease in the value of the modulus increases with an increase in the amount of functionalization but is essentially independent of the functionalizing agent. For 10% functionalization, Young’s modulus and the shear modulus of the SLGS are found to decrease by about 73% and 42%, respectively. However, the moduli of a fully functionalized SLGS are about the same as those of a 4% functionalized sheet. Even though the moduli of the pristine armchair and zigzag SLGSs are the same, the moduli of functionalized armchair SLGSs are about 20% less than those of the corresponding zigzag SLGSs. The work will help material scientists interested in designing graphene sheet reinforced polymeric composites.

Numerical simulation on slope stability analysis considering anisotropic properties of layered fractured rocks: a case study

Anisotropy is one of the natural properties of layered fractured rock, and it plays an important role in slope stability analysis, which is a vital problem in the geotechnical engineering. However, in the present engineering design, rock mass is simply treated as isotropic material, which fails to take into account the anisotropic properties. This work begins with formulizations of the anisotropic seepage-stress coupled model for the layered fractured rock slope stability analysis based on the equivalent continuum theory. Next, the model is applied in the numerical simulation of the southern slope of an open-pit mine in China to understand the mechanism of the existing failure strain as well as the failure mechanism of the potential landslide. The computed water table and damage zone have been compared with the field measurements and found to be in good agreement with field observations. Finally, the effective measures to prevent the slope failure and strengthen the slope stability have been suggested. The proposed model successfully applied in the case study indicated that it is much more feasible and efficient comparing with using the traditional isotropic coupled model in such layered fractured rock slopes stability problems.

A Two-Phase Flow Simulation of Discrete-Fractured Media using Mimetic Finite Difference Method

AbstractVarious conceptual models exist for numerical simulation of fluid flow in fractured porous media, such as dual-porosity model and equivalent continuum model. As a promising model, the discrete-fracture model has been received more attention in the past decade. It can be used both as a stand-alone tool as well as for the evaluation of effective parameters for the continuum models. Various numerical methods have been applied to the discrete-fracture model, including control volume finite difference, Galerkin and mixed finite element methods. All these methods have inherent limitations in accuracy and applicabilities. In this work, we developed a new numerical scheme for the discrete-fracture model by using mimetic finite difference method. The proposed numerical model is applicable in arbitrary unstructured gridcells with full-tensor permeabilities. The matrix-fracture and fracture-fracture fluxes are calculated based on powerful features of the mimetic finite difference method, while the upstream finite volume scheme is used for the approximation of the saturation equation. Several numerical tests in 2D and 3D are carried out to demonstrate the efficiency and robustness of the proposed numerical model.

Geomechanical characterization, 3-D optical monitoring and numerical modeling in Kirkgecit-1 tunnel, Turkey

The geomechanical characterization of a twin-tube tunnel during exploration and excavation stages is presented and used as a base (a) to interpret the monitored long-term displacements, (b) to numerically investigate if the displacement pattern at a given section could be predicted using an equivalent continuum model and (c) to assess adequacy of the NATM recommended excavation method and support systems. The pre-excavation rock mass classification by RMR and Q systems utilizing logs of two exploration boreholes lead to reasonably similar predictions of the as-built rock mass quality.The resultant displacement magnitudes and angles are interpreted with reference to the as-built geological cross-sections of the tunnel. All displacement patterns monitored at five posts in each section are consistently effective in providing advance warning for the presence of zones of contrasting conditions despite very low magnitudes of displacements. Numerical simulations predict no imminent danger of stress-induced instability and that the twin tubes could stand unsupported. A light support system as recommended by NATM (and similarly by RMR and Q systems) is necessary to prevent structural instability, loosening and overbreaks. The numerical analysis using the core-softening approach also exemplifies the sensitivity of tunnel convergence behavior to moving position of the excavation face, and consequently supports the combined use of the displacement vector magnitudes and angles to predict the presence and to appreciate the magnitude of changes in rock mass stiffness ahead of the excavation.

Coupled seepage-elastoplastic-damage analysis of saturated porous media and its application to water conveyance tunnel

Based on the plasticity, damage and seepage theory of equivalent continuum media, a fully coupled seepage-elastoplasticity-damage model for saturated porous media is proposed, which could reproduce the evolution of tensile and compressive damage, plasticity, porosity, permeability, and observed spatial distribution of pore pressure of the surrounding rock. This presented method is validated by the in-place testing data of the water conveyance tunnel project under the Yellow River, which is part of the South–North Water Diversion Project in China, and can be used as a reference for safety monitoring of similar hydraulic tunnel.

Underwater explosion research on ship with polymer hull coating

A polymer coating applied to the wet surface of a ship’s hull can reduce some types of damage to a ship subjected to a far-field underwater explosion. Results from numerical simulations are compared with full-scale underwater explosion test data for a ship with and without polymer coating to study the effectiveness of such coatings in blast-resistant design. The polymer coating is a thin wall structure applied to the wet surface of the ship hull, while the ship is a relatively large structure. Hence, an effective approximate simulation method for the full-scale ship with polymer coating is developed. Based on the homogenization theory of polymer coating, a continuum model that takes into account the strain rate dependence is introduced. The shock response subjected to underwater explosion of the finite element ship model is investigated. The results obtained from underwater explosion tests and numerical simulation are compared and discussed. The blast-resistant property of polymer coating is elaborated. Mesh sensitivity and the validity of the polymer coating equivalent continuum model are studied.

On lateral-torsional buckling of discrete elastic systems: A nonlocal approach

The lateral-torsional buckling of a discrete repetitive elastic beam-like structure, composed of rigid links connected by bending and torsional elastic springs, is investigated herein using discrete and continuum approaches. It is shown that the governing equations of the microstructured model are equivalent to a finite difference formulation of a continuous lateral-torsional buckling problem. The discrete equations are introduced through variational arguments and solved exactly for the hinged-hinged boundary conditions using a finite difference approach. A nonlocal equivalent continuum is sought via a continualization method. It is shown that the equivalent continuum is an Eringen's based elastic nonlocal continuum, which perfectly fits the exact discrete problem. The length scale effect related to the size of the repetitive cell tends to soften the lateral-torsional buckling limit of the asymptotically local continuum. Prandtl's lateral-torsional buckling solution is a particular case, associated with an infinite number of cells. Warping can also be included into the discrete thin-walled structural model, leading to a microstructured-based nonlocal thin-walled model. Love's equations augmented by the so-called Vlasov effect (warping bimoment) are used to compare the nonlocal approach with the one arising from the continualization of the discrete equation. It is shown that, in this last case, the length scale effect may depend on the warping stiffness.

Buckling of spherical cap gridshells: A numerical and analytical study revisiting the concept of the equivalent continuum

The use of gridshell structures in architecture and structural engineering has risen in the past decade, yet fundamental research on the mechanics of such structures is lacking. The majority of gridshells built use either quadrilateral or triangular grids. In this paper, we assess the effect of the grid shape (triangular or quadrilateral), the grid spacing, and the span-to-height ratio on the failure load of spherical cap gridshells. The failure load is calculated using finite element methods and analytical methods derived from the equivalent continuum. Both a linear eigenvalue and a nonlinear analysis using geometric imperfections are performed and compared. While a triangular grid is expected to be greater in structural capacity, it is found that for shallower shells there is less of a structural advantage to using a triangular grid. Further, the triangular grid is highly sensitive to imperfections whereas a quadrilateral grid is not. The equivalent continuum defined by area equivalence provides a conservative estimate for the buckling load and multiple definitions of the equivalent continuum can be used to establish a practical design tool in the early phases of design. By understanding the fundamental mechanical behavior of gridshells, design guidelines aimed to maximize their capacity and efficiency and intended to facilitate the discussion between architect and engineer are proposed.

Numerical Analysis of Plate-Load Test Results on Fractured Rocks Using an Equivalent-Continuum Model: Case Study of the Bakhtiary Dam Site

AbstractThe plate-load test (PLT) is widely performed at large dam sites to determine the rock mass deformability. Analytical relations based on the theory of elasticity, as suggested by the International Society for Rock Mechanics and the American Society for Testing and Materials test standards, are commonly used to interpret the test results and calculate the rock mass deformation modulus. However, because of inconsistency of the test gallery geometry with the half-space medium assumption embedded in the analytical relations, a systematic error occurs in the calculation of the deformation modulus. Furthermore, when the tests are conducted on jointed rock masses, the test results show a considerable scatter, mainly due to the anisotropy of the deformability of jointed rock masses. In this paper, numerical modeling is used to interpret the results of a series of PLTs conducted at the Bakhtiary dam site in southwest Iran. An equivalent continuum model, called JointedRock, is used to back-calculate the sti...

Vibration analysis of periodic cellular solids based on an effective couple-stress continuum model

Cellular solids are usually treated as homogeneous continuums with effective properties. Nevertheless, these mechanical properties depend strongly on the ratio of the specimen size to the cell size. These size effects may be accounted for according to preliminary static analysis of effective continuums based on couple-stress theory. In this paper an effective dynamic continuum model, based on couple-stress theory, is proposed to analyze the behavior of free vibrations of periodic cellular solids. In this continuum model, the effective mechanical constants of the effective continuum are deduced by an equivalent energy method. The cellular solid structure is then replaced with the equivalent couple-stress continuum with same overall dimension and shape. Moreover, the finite element formulation of the couple-stress continuum for the generalized eigenvalue analysis is developed to implement the free vibration analysis. The eigenfrequencies of the effective continuum are then obtained via the shear beam theory or the finite element method. A conventional finite element analysis by discretizing each cell of the cellular solids is also carried out to serve as an exact solution. Several structural cases are calculated to demonstrate the accuracy and effectiveness of the proposed continuum model. Good agreement on structural eigenfrequencies between the effective continuum solutions and the exact solutions shows that the proposed continuum model can accurately simulate the dynamic behavior of the cellular solids.

Elasto-Dynamic Behavior of a Two-Dimensional Square Lattice With Entrained Fluid II: Microstructural and Homogenized Models

This paper presents a detailed study of the pressure waves and effective mechanical properties of a closed-cell cellular solid with entrained fluid. Plane-harmonic-waves are analyzed in a periodic square with a finite-element model of a representative-volume element, which explicitly considers fluid-structure interactions, structural deformations, and the fluid dynamics of entrained fluid. The wall, cavity, and coupled-system resonance frequencies are identified as key parameters that describe the propagation characteristics. A tube-piston model based on computed microstructural deformations allows us to determine the effective stiffness tensor of an equivalent continuum at the macroscale. The analysis of dispersion surfaces indicates a single isotropic pressure mode for frequencies below resonance of the lattice walls, unlike Biot's theory which predicts two pressure modes. Shear modes are instead strongly anisotropic for all values of relative density ρ* describing both cellular ρ*<0.3 and porous solids ρ*≥0.3. The dependence of the pressure wave phase velocity on the relative density is analyzed for varying properties of the entrained fluid. Depending on the relative density and mass coupling of the solid and fluid phases, the microstructural deformations can be of three types: bending, through-the-thickness, and the combination of the two. For heavy and stiff entrained fluid, the bending regime is confined to extremely small values of relative density, whereas for light fluid such as a gas, deformations are of the bending-type for ρ*<0.1. Through-the-thickness deformations appear only for the heavy entrained fluid for large values of ρ*.

A nonlocal meshless solution for flexural vibrations of double-walled carbon nanotubes

The true understanding of free vibration of double-walled carbon nanotubes (DWCNTs) plays a vital role in optimal design and dynamic control of such nanostructures. This paper is aimed to examine free flexural vibration of lengthy DWCNTs with arbitrary boundary conditions in the framework of nonlocal elasticity theory. The DWCNTs are embedded in an elastic medium and are subjected to initially axial forces. Equivalent continuum structures associated with the innermost and outermost tubes of the DWCNT are considered. The transverse and rotational interactions of the DWCNT with the surrounding elastic medium are also taken into account. The generalized equations of motion of lengthy DWCNTs are established based on the nonlocal Rayleigh beam theory. Seeking an analytical solution to the developed equations, particularly in their general form, is a very problematic task. As an alternative solution, an efficient numerical scheme is proposed. The effects of slenderness ratio, small-scale parameter, lateral and rotational stiffness of the surrounding matrix, and initially axial force on the first five natural frequencies of DWCNTs under different boundary conditions are comprehensively scrutinized.

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ɛ; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum.

Large deflection geometrically nonlinear analysis of carbon nanotube-reinforced functionally graded cylindrical panels

A large deflection geometrically nonlinear behavior of carbon nanotube-reinforced functionally graded (CNTR-FG) cylindrical panels under uniform point transverse mechanical loading is studied. The analysis is carried out using the kp-Ritz method with kernel particle function is employed to construct the shape functions for the two-dimensional displacement approximations. Based on the first-order shear deformation shell theory, nonlinear governing equations are developed with geometric nonlinearity taking the form of von Kármán strains. It is assumed that carbon nanotubes are uniaxially aligned in the axial direction and are functionally graded in thickness direction of the cylindrical panels. The effective material properties of resulting CNTR-FG panels are estimated by employing an equivalent continuum model based on the Eshelby–Mori–Tanaka approach. A stabilized conforming nodal integration scheme is employed to evaluate the system bending stiffness and the membrane as well as shear terms are calculated by the direct nodal integration method to eliminate shear locking, for a very thin cylindrical panel. Several numerical example problems are examined to reveal the influences of volume fraction of carbon nanotubes, span angle, edge-to-radius ratio and thickness on nonlinear responses of the CNTR-FG panels. Moreover, effects of different boundary conditions and distribution type of carbon nanotubes are also investigated.

Damping in Periodic Structures: A Continuum Modeling Approach

A homogenized modeling technique is developed in this paper to account for viscous damping properties in beam-like lattice structures of repeated patterns. Necessary assumptions regarding the local free deformations, shear deformation beam theory, and compatibility conditions are made to obtain an equivalent continuum model of a three-dimensional lattice. A dissipated energy equivalence approach is then used to relate the energy dissipation for a general case of damping matrix in a lattice element to an equivalent proportionally damped model. As a result, a continuum model with Kelvin–Voigt damping is obtained. Damped bending natural frequencies, frequency response functions, and damping ratios are found using this method and compared to the results of a finite-element analysis for several structures for the purpose of validation.

Static and dynamic of carbon nanotube reinforced functionally graded cylindrical panels

The analysis of flexural strength and free vibration of carbon nanotube reinforced composite cylindrical panels is carried out. Four types of distributions of uniaxially aligned reinforcements are considered, i.e. uniform and three kinds of functionally graded distributions of carbon nanotubes along thickness direction of the panels. Material properties of nanocomposite panels are estimated by employing an equivalent continuum model based on the Eshelby–Mori–Tanaka approach. The governing equations are developed based on the first-order shear deformation shell theory. Detailed parametric studies have been carried out to reveal the influences of volume fraction of carbon nanotubes, edge-to-radius ratio and thickness on flexural strength and free vibration responses of the panels. In addition, effects of different boundary conditions and types of distributions of carbon nanotubes are examined.

Equivalent upscaled hydro-mechanical properties of a damaged and fractured claystone around a gallery (Meuse/Haute-Marne Underground Research Laboratory)

Abstract In this work, we present calculations and analyses of equivalent continuum (upscaled) coefficients describing the damaged, fissured and fractured claystone around an underground gallery. We focus here on mechanical and coupled hydro-mechanical properties of the damaged claystone (the upscaled Darcy permeability of the same claystone was studied in a previous paper focused on hydraulics without mechanical deformations). Concerning the geometric structure of the damaged clay stone around the cylindrical excavation, we use a hybrid 3D geometric model of fissuring and fracturing, comprising (a) a set of 10 000 statistical fissures with radially inhomogeneous statistics (size, thickness and density increasing near the wall), and (b) a deterministic set of large curved ‘chevrons’ fractures, periodically spaced along the axis of the drift according to a 3D chevron pattern. The hydro-mechanical coefficients calculated here are second- and fourth-rank tensors, which are displayed using ellipsoids. For simplicity, we also calculate equivalent isotropic coefficients extracted from these tensors: Young's modulus ( E ), bulk modulus ( K ), Lamé shear modulus (μ), Poisson's ratio (ν), Biot coefficient ( B , stress–pressure coupling) and Biot modulus ( M , pressure–fluid production coupling). All of these coefficients are affected by the degree of damage and fracturing, which increases near the wall of the gallery. Both 3D and ‘2D transverse’ distributions are analysed, on grids of 3D cubic voxels and 2D pixels, respectively. Global coefficients upscaled over the entire damaged and fractured zone are also analysed. Other types of averages are presented, for example, upscaled values over a cylindrical annular shell at various radial distances from the gallery wall. The relation to the degree of fracturing is discussed, including for instance the effect of fracturing on bulk and shear stiffnesses, and on the hydro-mechanical coupling coefficients of the damaged claystone.

Influence of equivalent continuum model based on the Eshelby-Mori-Tanaka scheme on the vibrational response of elastically supported thick continuously graded carbon nanotube-reinforced annular plates

Most of the early studies on plates vibration were focused on two-dimensional (2D) theories; these theories reduce the dimensions of problems from three to two by introducing some assumptions in mathematical modeling leading to simpler expressions and derivation of solutions. However, these simplifications inherently bring errors and therefore may lead to unreliable results for relatively thick plates. The main objective of this research paper is to present 3D elasticity solution for free vibration analysis of 2D continuously graded carbon nanotube-reinforced (CGCNTR) annular plates resting on a two-parameter elastic foundation. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness and radial directions. Various material profiles along the thickness and radial directions are illustrated using the 2D power-law distribution. An equivalent continuum model based on the Eshelby-Mori-Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight carbon nanotubes (CNTs). A semi-analytical approach composed of 2D differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. The convergence of the method is demonstrated and comparisons are made between the present results and results reported by well-known references and have confirmed accuracy and efficiency of the present approach. The novelty of the present work is to exploit Eshelby-Mori-Tanaka approach to reveal the impacts of the volume fractions of oriented CNTs and different CNTs distributions on the vibrational characteristics of annular plates which are assumed to be graded in one (or more) direction(s). POLYM. COMPOS., 35:1644–1661, 2014. © 2013 Society of Plastics Engineers

In-plane elastic moduli of covalently functionalized single-wall carbon nanotubes

Effective utilization of single-wall carbon nanotubes (SWCNTs) as reinforcements in composites necessitates their strong interfacial bonding with the surrounding matrix. The covalent functionalization of SWCNTs is an effective method to enhance this bonding. However, covalent bonds introduced by a functional group may alter the pristine structure of the SWCNT and affect its mechanical properties. Thus it is important to delineate effects of covalent functionalization on elastic properties of a SWCNT. We study here effects of covalent functionalization on Young’s modulus, Poisson’s ratio and shear modulus of a SWCNT in the graphitic plane. We consider hydrogen (–H), hydroxyl (–OH), carboxyl (–COOH), and amine (–NH2) as model functional groups in this work. We use molecular mechanics (MM) simulations with the MM3 potential and the software TINKER to analyze simple tension and torsional deformations of pristine and functionalized SWCNTs. As is commonly assumed, we hypothesize that the response of a SWCNT to these deformations is the same as that of an energetically and geometrically equivalent continuum cylinder of wall thickness 3.4Å. From curves of the strain energy density of deformation versus the axial strain and the shear strain for each functionalization group, values of Young’s modulus and the shear modulus, respectively, are deduced. From results of the tension tests on a pristine and a SWCNT fully functionalized with hydrogen (–H), Poisson’s ratio is computed. It is found that for each functional group studied and 20% functionalization, Young’s modulus and the shear modulus decrease by about 34% and 43%, respectively, and Poisson’s ratio of the functionalized SWCNT is more than that of the pristine SWCNT. These results should help in determining mechanical properties of SWCNT reinforced nanocomposites by using a micromechanics approach.

Numerical modeling of timber with knots: the progressively damaged lattice approach vs. the equivalent damaged continuum

Abstract Two new numerical models for simulating structural timber with knots are presented. The first one, the lattice model (LM), was built from an arrangement of linear truss elements, meanwhile, the second one, the continuum model (CM), was constructed of nonquadratic quadrilateral elements. Both models shared similar characteristics, which allows for an objective comparison between the two approaches. Common characteristics are as follows: they (1) possess identical geometry and node locations, (2) consider the fiber deviation of wood based on the flow-grain theory, (3) account for the longitudinal and transverse plasticity, and (4) permit tensile damage. Experimental four-point bending tests were performed to compare failure prediction, failure location, deflection, and qualitative failure pattern. Considerable construction and calibration work were required for the LM approach in comparison with the CM approach. However, the LM acceptably predicted the behavior of wood at the macroscale, and previous research documented satisfactory predictions for small scales. This suggests the lattice paradigm to be a valid approach for multiscalar simulations of wood. Moreover, complex mechanical principles (e.g., plasticity and damage) were found to be easier to incorporate into lattice elements. This can lead to deeper material understanding and innovation, as limitations for multiscale modeling may be more computational than theoretically constrained.