Equivalent Continuum Research Articles

Wave propagation in equivalent continuums representing truss lattice materials

Stiffness scales linearly with density in stretch-dominated lattice meta-materials offering the possibility of very light yet very stiff structures. Current additive manufacturing techniques can assemble structures from lattice materials, but the design of such structures will require accurate, efficient simulation methods. Equivalent continuum models have several advantages over discrete truss models of stretch dominated lattices, including computational efficiency and ease of model construction. However, the development an equivalent model suitable for representing the dynamic response of a periodic truss in the small deformation regime is complicated by microinertial effects. This paper derives a dynamic equivalent continuum model for periodic truss structures suitable for representing long-wavelength wave propagation and verifies it against the full Bloch wave theory and detailed finite element simulations. The model must incorporate microinertial effects to accurately reproduce long wavelength characteristics of the response such as anisotropic elastic soundspeeds. The formulation presented here also improves upon previous work by preserving equilibrium at truss joints for simple lattices and by improving numerical stability by eliminating vertices in the effective yield surface.

Discrete and non-local elastica

In this paper, the buckling and post-buckling behavior of an elastic lattice system referred to as the discrete elastica problem is investigated using an equivalent non-local continuum approach. The geometrically exact post-buckling analysis of the elastic chain, also called Hencky system, is first numerically solved using the shooting method. This discrete physical model is also mathematically equivalent to a finite difference formulation of the continuum elastica. Starting from the exact difference equations of the discrete problem, a continualization method is applied for approximating the difference operators by differential ones, in order to better characterize the discrete system by an enriched continuous one. It is shown that the new continuum associated with the discrete system exactly fits the discrete elastica post-buckling problem, where the non-locality is of Eringen׳s type (also called stress gradient non-local model). An asymptotic expansion is performed for both the discrete and the non-local continuum models, in order to approximate the post-buckling branches of the discrete system. Some numerical investigations show the efficiency of the non-local approach, especially for capturing the scale effects inherent to the cell size of the lattice model.

Analysis of failure loads and optimal design of composite lattice cylinder under axial compression

Application of large-scale composite lattice cylinders in aeronautic and astronautic engineering is an effective way to reduce structure mass and achieve high mass efficiency. In this paper, structural stiffness and critical axial force of the lattice cylinder are analyzed theoretically by equivalent continuum method. For the lattice cylinder under axial compression, there are four failure modes, which are global buckling, out-of-plane strut buckling, in-plane strut buckling and strength failure. Curves of load capacity and failure maps related to the variation of non-dimensional variables are plotted respectively. According to the failure maps, failure of the large-scale lattice cylinder is generally dominated by global buckling due to relatively small strut thickness of the lattice cylinder. Based on finite element simulations, influence of four key geometrical parameters on stiffness and critical buckling force of the large-scale lattice cylinder under axial compression is discussed. Strut thickness and number of oblique strut rows have relatively great effects on the stiffness and critical buckling force, while number of horizontal strut rows should maintain a relatively small value in order to achieve high mass efficiency. Aimed to minimize mass of a specific Kagome lattice cylinder in practical engineering, a multi-parameter optimization model is built in MATLAB Optimization Toolbox™ and successfully applied to optimize the four geometrical parameters of the large-scale lattice cylinder under axial compression.

On the applicability of geomechanical models for carbonate rock masses interested by karst processes

Rock mass classification and geomechanical models have a particular importance for carbonate rocks, due to their peculiar fabric, variability of the main features, and scarce availability of experimental data. Carbonates are particularly sensitive to syn-depositional and post-depositional diagenesis, including dissolution and karstification processes, cementation, recrystallisation, dolomitisation and replacement by other minerals. At the same time, as most of sedimentary rocks, they are typically stratified, laminated, folded, faulted and fractured. The strength and deformability of carbonate rock masses are, therefore, significantly affected by the discontinuities, as well as by their pattern and orientation with respect to the in situ stresses. Further, discontinuities generally cause a distribution of stresses in the rock mass remarkably different from those determined by the classical elastic or elasto-plastic theories for homogeneous continua. Goal of this work is the description of the difficulties in elaborating geomechanical models to depict the stress–strain behavior of karstified carbonate rock masses. Due to such difficulties, a high degree of uncertainty is also present in the selection of the most proper approach, the discontinuum one or the equivalent continuum, and in the numerical model to be used within a specific engineering application as well. The high uncertainty might cause wrong assessments as concerns the geological hazards, the design costs, and the most proper remediation works. Even though recent developments in the application of numerical modeling methods allow to simulate quite well several types of jointed rock masses, as concerns carbonate rock masses many problems in representing their complex geometry in the simulation models still remain, due to peculiarity of the structural elements, and the presence of karst features. In the common practice, the improper use of the geomechanical models comes from a superficial geological study, or from the lack of reliable geological and structural data that, as a consequence, bring to erroneous evaluations of the influence of the geological-structural features on the in situ stress state and the stress–strain rock mass behavior.

A nonlinear mechanical model for the fatigue life of thin-film carbon nanotube supercapacitors

The effects of cyclic loading on the mechanical performance and fatigue life of a novel carbon nanotube supercapacitor are investigated. The highly flexible supercapacitor is a monolithic, pre-fabricated, fully functional film made of a nanostructured free-standing layer in which ions are stored within two vertically aligned multi-walled carbon nanotube (MWCNs) electrodes that are monolithically interspaced by a solution of microcrystalline cellulose in a room temperature ionic liquid electrolyte. To study the cyclic mechanical response of such nanostructured multilayer composite, an original framework is adopted by combining the equivalent continuum approach of Eshelby–Mory–Tanaka and a Weibull-like approach for the evolution of debonding carbon nanotubes electrodes. One- and three-layer models of the supercapacitor are proposed. Cyclic tests are numerically carried out in strain control. A fatigue life limit is determined by considering a confidence interval for the number of cycles corresponding to the states at which the effective elastic modulus of the partially debonded nanostructured portion of the supercapacitor is reduced by a percentage between 20% and 30%. The simulated cyclic tests yield Wholer-type fatigue curves showing the fatigue life limit as the maximum number of cycles N for each strain amplitude.The sensitivity of the fatigue life with respect to meaningful parameters such as the interfacial strength between the MWCNs and cellulose is investigated. Frequency-response functions of the multilayer nanostructured composite are further computed as function of the strain amplitude during cyclic tests.

On large deformation granular strain measures for generating stress–strain relations based upon three-dimensional discrete element simulations

Generating stress–strain relations based upon three-dimensional discrete element simulations, for hierarchical multiscale constitutive modeling of granular materials at finite strain, requires measures of stress and strain in the reference and current configurations. The Cauchy stress tensor calculated from interparticle forces and branch vectors resulting from discrete element simulations, is well-established, but the various finite strain definitions existing in the literature are not as well studied or accepted. Thus, the paper develops seven finite strain measures in three-dimensions (Cartesian coordinates assumed): two are calculated in the reference configuration, and five are calculated in the current configuration. In the process, a time-integrated deformation gradient is formulated, with which the Cauchy stress tensor can be mapped to either the Kirchhoff stress, first Piola–Kirchhoff stress, or second Piola–Kirchhoff stress. The difficulty in calculating strain measures for granular materials is the discrete particulate nature of the materials. In the paper, strain measures are motivated by the equivalent continuum method, but are extended to finite strain in three-dimensions. Simulations are conducted to test the finite strain measures, and it is found that these strain measures are independent of rigid body rotation of a particle assembly. Granular Cauchy stress is also calculated for these discrete element simulations to be able to plot stress–strain curves in the current configuration. Finally, two numerical examples with large deformation are simulated: (1) cavity expansion, and (2) pile penetration; and their results are analyzed.

Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua

In this paper, we revisit the capability of numerical approaches such as finite difference methods and finite element methods, in approximating exact one-dimensional continuous eigenvalue problems (such as lateral vibrations of a string, the axial or the torsional vibrations of a bar, and the buckling of elastic columns). The numerical methods analysed in this paper are converted into difference equations. Following a continualization procedure or the method of differential approximation, the difference operators are then expanded in differential operators via Taylor expansion or Pade approximants. Analogies between the finite numerical approaches and some equivalent enriched continuum are shown, using this continualization procedure. The finite difference methods (first-order or higher-order finite difference methods) are shown to behave as integral-based nonlocal media (or stress gradient media), while the finite element method is found to behave as gradient elasticity media (or strain gradient media). The length scale identification of each equivalent enriched continuum strongly depends on the order of the numerical method considered. For the finite difference methods, the length scale identification of the equivalent nonlocal medium depends on the static versus dynamic analysis, whereas this length scale appears to be independent of inertia effects for the finite element method. Some comparisons between the exact discrete eigenvalue problems and the approximated continuous ones show the efficiency of the continualization procedure. An equivalent enriched Rayleigh quotient can be defined for each numerical method: the integral-based nonlocal method gives a lower bound solution to the exact eigenvalue multiplier, whereas the gradient elasticity method furnishes an upper bound solution.

Multiscale homogenization model for thermoelastic behavior of epoxy-based composites with polydisperse SiC nanoparticles

A multiscale statistical homogenization method based on a finite element analysis is proposed to predict the embedded filler size-dependent thermoelastic properties of nanoparticulate polymer composites. A molecular dynamics simulation is used to predict the particle size-dependent elastic constants and coefficient of thermal expansion (CTE) of nanocomposites. Because of the densified interphase zones formed in the vicinity of nanoparticles, and their relative dominance according to particle size, the thermoelastic properties are more prominently increased by smaller nanoparticles. The equivalent continuum microstructure of a nanocomposite is modeled as a three-phase periodic unit cell consisting of the nanoparticle, matrix, and effective interphase zone. An inverse numerical scheme is proposed to predict the thermoelastic properties of the effective interphase zone from the known properties of nanocomposites. In order to account for more realistic variation of the nanoparticle radius and spatial distribution in nanocomposites, statistical homogenization is performed by assigning randomness to the particle size using a beta distribution and to the spatial distribution through larger many-particle embedded representative volume elements (RVEs). Compared with the result from a regular homogenization method using a mono-particle RVE, the mean value of the elastic moduli increases, while that of the CTE decreases in the many-particle RVEs whose mean particle radius corresponds to that of the mono-particle RVE at the same volume fraction of nanoparticles.

Anisotropic characteristics of jointed rock mass: A case study at Shirengou iron ore mine in China

Rock mass is characterized by the existence of distributed joints whose properties and geometry strongly affect the mechanical behavior of jointed rock masses. A finite element model considering the anisotropic characteristics of fractured rock mass was proposed which could deal with a wide variety of joint distribution in rock mass and then applied in Shirengou iron ore mine in Tangshan, China. First, the scale effects and anisotropy were investigated by using multi-scale discrete fracture network models under uniaxial compression tests. Then, the principal direction of elasticity was found and used in the constitutive law of the equivalent continuum model. Finally, the deformation and failure behavior were studied and verified through site-specific microseismic data. It is found that the stress and damage zone are influenced by joint orientation. This proposed model can efficiently study the effects of rock joints on rock mass behavior and thus contribute to a more reasonable explanation on the dominant effect of the joint sets on deformation and failure of rock mass.

An equivalent plate model for corrugated-core sandwich panels

This paper suggests an equivalent plate model to analyze the mechanical behavior of corrugated-core sandwich panels under tensile and bending loads. A homogenization-based theory based on the equivalent energy method is used to obtain the stiffness matrices of corrugated cores of a sandwich panel. Equivalent continuum layers with orthotropic elastic constants corresponding to the membrane and bending stiffness terms of the corrugated layers are determined by using classical lamination theory (CLT). The importance of this work is that an equivalent plate model for corrugated-core sandwich panels can be easily obtained by combining the equivalent energy method and the CLT. The proposed equivalent model is verified by numerical simulations of sandwich panels with sinusoidal and trapezoidal corrugated cores.

Finite element continuum modeling of vibrationally-fluidized granular flows

Discrete element modeling of granular flows is effective, but requires large numerical models. In an effort to reduce computational effort, this paper presents a finite element (FE) continuum model of the vibrationally fluidized granular flow produced by a tub vibratory finisher. The constitutive equations governing the continuum model were calibrated using the discrete element method (DEM). A constant stiffness was used for the normal contact of the polyurethane tub wall and the spherical steel media, and an effective coefficient of Coulomb friction was obtained by studying the variation of the shear and normal forces at different points along the tub wall using DEM simulations. The DEM predictions of the ratio of the tangential to normal contact forces showed that rolling of the media on the tub wall was much more common than sliding, and the average effective coefficient of friction was relatively uniform over the wall. Average values of the equivalent elastic and plastic properties of the media were used in the continuum model, neglecting the small local variations that existed within the vibrationally-fluidized bed. The bulk flow behavior of the equivalent continuum media was then studied using both Lagrangian and Eulerian FE formulations. The bulk flow velocities predicted by the Lagrangian approach were in good agreement with those obtained using DEM simulations over a wide range of tub wall amplitudes. The qualitative trend of the local impact velocity distribution at various locations in the tub predicted by the DEM agreed with the trend predicted by the continuum model using the shear rate as a measure of the granular temperature and hence local velocity. The FE simulations required approximately 8 times less computing time than the equivalent DEM.

Vibration analysis of a carbyne-based resonator in nano-mechanical mass sensors

Carbyne is a chain of C atoms held together by double or alternating single and triple chemical bonds, and has twice the tensile stiffness of carbon nanotubes (CNTs) and graphene sheets (GSs). In this study, we propose a nano-mechanical mass sensor using a tensioned carbyne resonator. The carbyne resonator is modeled as an equivalent continuum circular cross section beam with diameter 0.772 Å, Young’s modulus 32.71 TPa, shear modulus 11.8 TPa, Poisson’s ratio 0.386 and density 32.21 g cm−3. We analyze the resonant frequency of the proposed sensor carrying with a concentrated mass based on the Timoshenko beam theory and verify the theoretical approach using Rayleigh’s energy method and molecular dynamics simulation. The results show that the proposed mass sensor can measure a tiny mass with weight below 10−5 zg, and provide much higher sensitivity than CNTs- and GSs- based nano-mechanical mass sensors. In addition, the effects of carbyne length, mass position and tensile load on the frequency shift are also analyzed in detail, and it is preferred to use shorter carbyne and higher tensile load in the proposed mass sensor.

Numerical analysis of stability for mined-out area in multi-field coupling

There were differences between real boundary and blast hole controlling boundary of irregular mined-out area in underground metal mines. There were errors in numerical analysis of stability for goaf, if it was analyzed as regular 3D mined-out area and the influence of coupling stress-seepage-disturbance was not considered adequately. Taking a lead zinc mine as the background, the model was built by the coupling of Surpac and Midas-Gts based on the goaf model precisely measured by CMS. According to seepage stress fundamental equations based on the equivalent continuum mechanical and the theory about equivalent load of dynamic disturbance in deep-hole blasting, the stability of mined-out area under multi-field coupling of stress-seepage-dynamic disturbance was numerically analyzed. The results show that it is more consistent between the numerical analysis model based on the real model of irregular 3D shape goaf and the real situation, which could faithfully reappear the change rule of stress-strain about the surrounding rock under synthetic action of blasting dynamic loading and the seepage pressure. The mined-out area multi-field coupling formed by blasting excavation is stable. Based on combination of the advantages of the CMS, Surpac and Midas-Gts, and fully consideration of the effects of multi-field coupling, the accurate and effective way could be provided for numerical analysis of stability for mined-out area.

Study on Optimization of Anti-Seepage and Drainage System for High Concrete Sand-Gravel Dam

The design height of a concrete sand-gravel dam is up to 251 meters, which has strong unloading rock masses with great depth on the left bank. According to the topographic and hydro geologic conditions of the dam site, a three-dimensional finite element model for seepage calculation of the dam is established. Based on the equivalent continuum method, the effect of anti-seepage and drainage system within run-time with normal storage level is studied, meanwhile, the seepage elements are obtained, including the seepage of the groundwater, the discharge of the dam, the seepage gradient of the dam body and the impervious curtain and so on, thus the conclusion that the anti-seepage and drainage system is reasonable is verified. What’s more, the influence of the different length and depth of the anti-seepage curtain to the distribution of groundwater seepage as well as the effect of "L" type drainage in the dam body are analyzed in order that the suggestion of the optimization of anti-seepage and drainage system is put forward for the high concrete sand-gravel dam.

Use of the equivalent continuum approach to model the behavior of a rock mass containing an interlayer shear weakness zone in an underground cavern excavation

An interlayer shear weakness zone (ISWZ) is a weak zonal geotechnical system of variable thickness that occurs between different rock strata (e.g., tuff and basalt). At the site of the future Baihetan hydropower station, Sichuan Province, China, because of the relatively poor ISWZ mechanical properties, the overall stability of the underground powerhouse is potentially at risk. In this study, to evaluate the effects of ISWZs on the stability of the future underground powerhouse by means of three-dimensional continuum modeling (3-D continuum modeling), the concept of a virtual rock mass composed of ISWZ and host rock is proposed. An equivalent continuum approach, including a rock–soil composite material (RSCM) model, is elaborated, with corresponding expressions for the input parameters. Comparisons were made between the predictions from the RSCM model, the results obtained by an analytic method, and existing data from physical model tests. The comparison showed that all three types of information showed good consistency in terms of failure mode and strength. This indicates the suitability of the RSCM model for describing the behavior of a rock mass containing discontinuities. Furthermore, comparison between the predictions of the proposed equivalent continuum approach, the joint element approach, and the solid element approach for a deformation of a test tunnel section containing an ISWZ show that the results produced by the first two approaches are similar, but much smaller than that using the third approach. Further comparison of the actual state of the ISWZ-containing rock mass in the test tunnel section confirmed the applicability of the proposed equivalent continuum approach to prediction of deformation of the rock masses containing ISWZs at the future Baihetan underground powerhouse site.

Active vibration control of nanotube structures under a moving nanoparticle based on the nonlocal continuum theories

The active vibration control of nanotubes under action of a moving nanoscale particle is carried out using nonlocal elasticity theory of Eringen. The nanotube is simulated by an equivalent continuum structure under simply supported boundary conditions within the framework of nonlocal Rayleigh as well as Timoshenko beam theory. A Dirac-delta function is applied to model the location of the moving mass along the nanotube as well as its inertial effects. In the following a linear classical optimal control algorithm with a time varying gain matrix and displacement-velocity feedback is applied to vibration suppression in nanotube structure. The effects of the moving nanoparticle velocity, slenderness ratio of nanotube and small scale effect parameter on the dynamic deflection are studied in some detail for both the Rayleigh and Timoshenko theories. Finally the proficiency of the control algorithm in suppressing the response of the nanostructure under the effect of moving nanoparticle with different number of controlled modes and control forces is studied. The results of the present work can be used as a benchmark in future studies of nanomachines and nanosystems.

Continuous and discrete models for masonry like material: A critical comparative study

Abstract The aim of this paper is to present a critical comparative review of different models that may be adopted for modelling the mechanical behaviour of masonry, with particular attention to microstructured models. Several continuous and discrete models are discussed. Such models are based on the following assumptions: i) the structure is composed of rigid blocks; ii) the mortar is modelled as an elastic material or an elastic interface. The rigid block hypothesis is particularly suitable for historical masonry, in which stone blocks may be assumed as rigid bodies. For this type of masonry, mortar thickness is negligible if compared with block size, hence it can be modelled as an interface. Masonry-like materials may be modelled taking into account their heterogeneity by adopting a heterogeneous Finite Element Model (FEM) or a Discrete Element Model (DEM). The former seems to be more representative of masonry, but it is computationally onerous and results interpretation may be difficult; the latter is limited to rigid block assumption and mortar joints modelled as interfaces. For this reason, continuous equivalent models may be suitable to investigate masonry behaviour. Continuum equivalent models provide, in an analytical form, constitutive functions, but Cauchy model may be not suitable to describe masonry behaviour due to not negligible size of heterogeneity (block size) with respect to masonry panel size. For this reason, micropolar equivalent continuum may be adopted. By reference to the existing literature, a simple and effective DEM is adopted, in which masonry is modelled as a ‘skeleton’ having a behaviour depending on forces and moments transferred between blocks through the interfaces (mortar joints). Moreover for the micropolar equivalent continuum, an ad hoc enriched homogenised FEM is formulated by means of triangular elements. The proposed numerical models represent two possible simple approaches for solving heterogeneous problems. Such models are developed both by means of fast numerical routines and do not require specific computer codes, whereas the heterogeneous FEM may be studied by adopting a traditional FE code. DEM and heterogeneous FEM are adopted to verify reliability and application field of Cauchy and micropolar continua. Moreover, sensitivity of micropolar model to the Representative Elementary Volume (REV) chosen is discussed. For these purposes, ad hoc FE models are adopted, with constitutive functions obtained from an identification procedure (both for Cauchy and micropolar continua). An extensive comparison between DEM, heterogeneous FEM and equivalent homogenous FEM is presented in some meaningful cases, taking into account also the effect of heterogeneity size on models behaviour.

Modeling Approaches for Masonry Structures

Different scale approaches, micromechanical, multiscale and macromechanical or phenomenological, are presented to study the structural response of masonry elements. First, a micromechanical model is introduced and the masonry is considered to be a heterogeneous material, made of mortar and bricks joined by interfaces, where the mortarbrick decohesion mechanisms occur. To this end, a special interface model combining damage and friction is proposed. Then, two multiscale procedures are presented, that consider regular arrangements of bricks and mortar, modeled by nonlinear constitutive laws which account for damage and friction effects. A homogenization technique is developed to derive two different equivalent continuum models at the macro-level, a micropolar Cosserat continuum and a nonlocal Cauchy model. Finally, a macromechanical model, based on the adoption of a classical No-Tension Material (NTM) model, and on the presence of irreversible crushing strains, is proposed. A zero tensile strength is assumed, thus fracture strains arise when the stress is zero. Moreover, an elastoplastic model is considered for the material response in compression. Numerical applications are performed on a masonry arch and two masonry panels, by adopting the three approaches presented. Comparisons with experimental outcomes, published elsewhere, are performed.

Bending stiffness of a graphene sheet

The paper proposes a discrete mechanical model of monolayer graphene. A relation between parameters of the model and elastic characteristics of its equivalent continuum is derived by comparing the energy of small strains on micro- and macroscales. The relation allows one to determine the microscale interaction parameters from experimental data and, knowing the microscale parameters, to determine the mechanical properties of graphene. The main aim of the work is to estimate the bending stiffness of a graphene sheet. The proposed discrete model provides an analytical dependence of the graphene sheet bending stiffness on the microscale interaction parameters.

The influence of nanoparticle size on the mechanical properties of polymer nanocomposites and the associated interphase region: A multiscale approach

The characterization of the effective mechanical properties and geometry of the interphase in a heterogeneous material is a key issue for the design of polymer nanocomposites due to the dominant effect of the interphase on the overall behavior of the composites. In this study, molecular dynamics (MD) simulation and finite element (FE) analysis were integrated to develop a mechanics-based multiscale approach that can derive both the global stiffness and the local load transfer on the filler surface of the particulate nanocomposites. The unknown mechanical response and geometrical boundaries of the interphase (polymer networks adsorbed on the particle surface) are numerically obtained from a continuum model through the matching of homogenization and deformation energy to a full atomic molecular model. The equivalent continuum models given from the present multiscale method successfully represent the virial local stresses at both the interphase and matrix regions of the full-atomic model, as well as the particle size dependent stiffness of the nanocomposites. The proposed method is used to characterize the internal mechanical behavior of the intermediate media in terms of the nanoparticle size, and the nanophysics of the intermediate media are discussed in detail.