Equivalent Continuum Research Articles

FEM analysis of metal matrix nanocomposites reinforced with off-line atomistically-informed equivalent nanofillers

With ever increasing the applications of metal matrix nanocomposites (MMNCs) reinforced with carbon nanofillers, demand for accurate and effective modeling techniques in this area has been increased in recent years. To provide such a route, this study is dedicated to apply information form interfacial atomic interactions in modified continuum models implemented to analyze the mechanical behavior of these nanocomposites. In the proposed methodology, the metal matrix is considered as a continuous medium. Meanwhile, to capture the atomic-scale phenomena governing the problem, the carbon nanofiller, local metal atoms neighboring it, and the nanofiller/metal interface are analyzed using molecular dynamics (MD) simulations. Subsequently, employing the equivalent continuum modeling (ECM) concept, geometrical and mechanical characteristics of the equivalent nanofiller is obtained in each case and introduced in a finite element analysis (FEA). It is demonstrated that the atomistically informed FEA can successfully reproduce the results of full atomistic MD simulations with ignorable run-time compared to that of this method. Finally, constructing several computational cells with different metal matrices and reinforcing agents, the developed MD/FE multiscale approach is utilized to capture the influences of temperature, nanofiller geometry and orientation, its volume fraction and interfacial interactions on the axial elastic constants of these nanocomposites.

Characterization of Thickness and Thermoelastic Properties of Interphase in Polymer Nanocomposites using Multiscale Analysis

In this study, a multiscale method for solving a thermoelasticity problem for interphase in the polymeric nanocomposites is developed. Molecular dynamics simulation and finite element analysis were numerically combined to describe the geometrical boundaries and the local mechanical response of the interfacial region where the polymer networks were highly interacted with the nanoparticle surface. Also, the micrmechanical thermoelasticity equations were applied to the obtained equivalent continuum unit to compute the growth of interphase thickness according to the size of nanoparticles, as well as the thermal phase transition behavior at a wide range of temperatures. Accordingly, the equivalent continuum model obtained from the multiscale analysis provides a meaningful description of the thermoelastic behavior of interphase as well as its nanoparticle size effect on thermoelasticity at both below and above the glass transition temperature.

Vibrations of fluid-conveying inclined single-walled carbon nanotubes acted upon by a longitudinal magnetic field

This work deals with the influence of the longitudinal magnetic field on vibrations of inclined single-walled carbon nanotubes (SWCNTs) subjected to an inside fluid flow. Using an equivalent continuum structure model for the SWCNT and a plug-like model for the moving inside fluid flow, the nonlocal longitudinal and transverse equations of motion of magnetically affected SWCNTs are obtained in the context of small deformations. By application of the assumed-mode methodology, the displacements are discretized in terms of vibration mode shapes, and by exploiting generalized Newmark-β scheme, their corresponding time-dependent parameters are determined at each time. In the presence of the longitudinal magnetic field, the effects of the small-scale parameter, fluid flow velocity, and inclination angle on both longitudinal and transverse vibrations of SWCNTs are addressed. The obtained results reveal that the longitudinal magnetic field has fairly no effect on the longitudinal dynamic behavior of the nanostructure. However, maximum values of both transverse displacement and nonlocal bending moment of the fluid-conveying SWCNT would reduce as the strength of the magnetic field grows. Such a fact becomes more highlighted for high levels of the fluid flow velocity. The obtained results indicate that the longitudinal magnetic field can be exploited as an efficient way to control transverse vibrations of SWCNTs conveying fluids.

Two-phase numerical simulation of discrete fracture model based on multiscale mixed finite element method

Fractured reservoirs are characterized by complex fractures on multiple scales and are quite difficult to model. Numerical simulation of fractured media is usually done based on dual-porosity model and equivalent continuum model. Dual-porosity model treats matrix system and fracture system as two parallel continuous systems coupled by crossflow function. This model is only valid for reservoirs with highly developed fractures. Equivalent continuum model treats fractured media as a continuum media and equivalent absolute permeability tensors for each grid block are calculated to describe the heterogeneity of the reservoir. This model is efficient only when there exist representative element volume and the equivalent permeability are difficult to decide. Both models treat the fractured media as a simplified model and cannot describe the multiscale flows exactly, because they cannot precisely consider the diversion effect of the fractures. Although the discrete fracture network (DFN) model can provide a detailed representation of flow characteristic, traditional numerical method does not suitable for DFN. The major difficulty is the size of the computation. A tremendous amount of computer memory and CPU time are required, and this can exceed limit of today’s computer resources. Upscaling methods are generally used to reduce the computational cost. However, it is not possible to have a priori estimates of the errors that are present when complex flow processes are investigated using coarse models constructed via simplified settings. In this paper, multiscale mixed finite element method (MsMFEM) is proposed to simulate water/oil two phase flow in discrete fracture media. By combining MsMFEM with the discrete fracture model, we aim towards a numerical scheme that facilitates fractured reservoir simulation without upscaling. The MsMFEM uses a standard Darcy model to approximate pressure and fluxes on a coarse grid. The multiscale basis functions are constructed numerically by solving local differential equations on the fine-scale grid. The advantage of MsMFEM is that the basis functions capable of reflecting information about fractures within elements. Therefore, this method can capture the fine-scale effects on the coarse grid, that is, multiscale method can reduce the computational cost and keep high calculation accuracy at the same time. Traditional numerical methods generally difficult to deal with complex grid element, in this paper, mimetic finite difference (MFD) method is used to construct the multiscale basis functions due to its local conservativeness and applicability of complex grids. Compared with traditional multiscale mixed finite element methods, this method is suitable for arbitrary complex grid system. This paper introduced fundamental principles of the multiscale mixed finite element method and described the numerical scheme of discrete fracture model based on MsMFEM in detail. Then we deduced discrete fracture model computing formulate for the multiscale basis function by using mimetic finite difference method. Oversampling technique is applied to get more accurate small-scale details. IMPES scheme is used in the two-phase flow simulation. Physical experiment is used to prove the validity the multiscale method. The numerical results show that compared with traditional numerical method, the MsMFEM can represent the fine-scale flow in fracture networks exactly and meanwhile has a higher computational efficiency.

Modulus Ratio and Joint Factor Concepts to Predict Rock Mass Response

The commonly adopted rock mass classifications, namely RMR, Q and GSI, are used to estimate compressive strength and modulus of rock masses. These values have been examined as per modulus ratio concept, M (rj) , for their reliability. The design parameters adopted in some of the recent case studies based on these classifications indicate that the M (rj) values for rock masses are higher than those of the corresponding intact rocks. The joint factor, J (f), which is defined as a weakness coefficient in rock mass suggests that modulus ratio of rock mass (M (rj) ) has to be less than the modulus ratio of the corresponding intact rock (M (ri) ), on the basis of extensive experimental evidence. With joint factor, compressive strength, elastic modulus, cohesion and friction angle were estimated and applied in the analyses of a few cases. The predictions of deformations with this approach agreed well with the field measurements by adapting equivalent continuum approach. The modulus ratio concept is considered to present a unified classification for intact rocks and rock masses. Soil-rock boundary, standup time in under ground excavations and also penetration rate of TBM estimates have been linked to M (rj) .

An equivalent continuum multiscale formulation for 2D geometrical nonlinear analysis of lattice truss structure

In this work, an equivalent continuum multiscale formulation is presented for the geometrical nonlinear analysis of the structures with lattice truss materials. This formulation is established by combining the extended multiscale finite element method and the co-rotational approach. Firstly, the lattice truss unit cell is equivalent to a continuum coarse element by using a numerical constructed interpolation function in the local coordinate system. Then the tangent stiffness matrix of this coarse element is derived by employing the basic idea of the co-rotational approach in the global coordinate system. Thus, the global nonlinear equilibrium equations of the structure at the macroscopic level can be solved by using the general displacement control algorithm to capture the equilibrium path with multiple critical points. After performing all of the incremental steps and the iterative steps on the macroscopic scale, the microscopic information, such as the displacement, stress and strain, can be obtained easily by virtue of the afore-constructed numerical interpolation functions once again. In addition, several numerical examples are carried out to study the effects of the layout and size of unit cell, investigate the sensitivity of coarse-scale meshes and verify the validation and efficiency of the presented multiscale formulation.

Computation of the effective nonlinear mechanical response of lattice materials considering geometrical nonlinearities

The asymptotic homogenization technique is presently developed in the framework of geometrical nonlinearities to derive the large strains effective elastic response of network materials viewed as repetitive beam networks. This works extends the small strains homogenization method developed with special emphasis on textile structures in Goda et al. (J Mech Phys Solids 61(12):2537---2565, 2013). A systematic methodology is established, allowing the prediction of the overall mechanical properties of these structures in the nonlinear regime, reflecting the influence of the geometrical and mechanical micro-parameters of the network structure on the overall response of the chosen equivalent continuum. Internal scale effects of the initially discrete structure are captured by the consideration of a micropolar effective continuum model. Applications to the large strain response of 3D hexagonal lattices and dry textiles exemplify the powerfulness of the proposed method. The effective mechanical responses obtained for different loadings are validated by FE simulations performed over a representative unit cell.

절리간극의 변화가 이차원 DFN 시스템의 수리적 특성에 미치는 영향

본 연구는 절리의 수리간극의 변화가 절리암반의 수리상수에 미치는 영향을 평가하기 위하여 등가유로관 연결구조에 기반을 둔 이차원 DFN(discrete fracture network) 유체유동 해석 프로그램 코드를 개발하고 수리간극 변화를 고려한 수치실험을 수행하였다. 수치실험에 사용된 이차원 DFN 시스템은 두 절리군을 사용하여 절리의 방향성, 빈도 및 길이분포를 고정하고 절리의 수리간극을 절리군별로 서로 다른 일정한 값을 갖는 경우와 수리간극이 확률분포 특성을 갖는 경우를 고려하였다. 추계론적으로 생성한 총 216개의 <TEX>$20m{\times}20m$</TEX> DFN 블록에 대하여 블록수리상수가 산정되었다. 수리간극의 변동성은 이차원 DFN 시스템의 이방성 및 등가연속체 취급 가능성에 유의미한 영향을 미치는 것으로 평가되었다. A computer program code was developed to estimate hydraulic behavior of the 2-D connected pipe network system, and implemented to evaluate the effect of joint aperture on hydraulic parameters of fractured rock masses through numerical experiments. A total of 216 stochastic 2-D DFN(discrete fracture network) blocks of <TEX>$20m{\times}20m$</TEX> were prepared using two joint sets with fixed input parameters of joint orientation, frequency and size distribution. Two different cases of joint aperture variation are considered in this study. The hydraulic parameters were estimated for generated 2-D DFN blocks. The hydraulic anisotropy and the chance for equivalent continuum behavior of the DFN system were found to depend on the variability of joint aperture.

Overall thermomechanical properties of layered materials for energy devices applications

This paper is concerned with the analysis of effective thermomechanical properties of multi-layered materials of interest for solid oxide fuel cells (SOFC) and lithium ions batteries fabrication. The recently developed asymptotic homogenization procedure is applied in order to express the overall thermoelastic constants of the first order equivalent continuum in terms of microfluctuations functions, and these functions are obtained by the solution of the corresponding recursive cell problems. The effects of thermal stresses on periodic multi-layered thermoelastic composite reproducing the characteristics of solid oxide fuel cells (SOFC-like) are studied assuming periodic body forces and heat sources, and the solution derived by means of the asymptotic homogenization approach is compared with the results obtained by finite elements analysis of the associate heterogeneous material.

Impact of layer thickness and well orientation on caprock integrity for geologic carbon storage

Economic feasibility of geologic carbon storage demands sustaining large storage rates without damaging caprock seals. Reactivation of pre-existing or newly formed fractures may provide a leakage pathway across caprock layers. In this study, we apply an equivalent continuum approach within a finite element framework to model the fluid-pressure-induced reactivation of pre-existing fractures within the caprock, during high-rate injection of super-critical CO2 into a brine-saturated reservoir in a hypothetical system, using realistic geomechanical and fluid properties. We investigate the impact of reservoir to caprock layer thickness, wellbore orientation, and injection rate on overall performance of the system with respect to caprock failure and leakage. We find that vertical wells result in locally higher reservoir pressures relative to horizontal injection wells for the same injection rate, with high pressure inducing caprock leakage along reactivated opening-mode fractures in the caprock. After prolonged injection, leakage along reactivated fractures in the caprock is always higher for vertical than horizontal injection wells. Furthermore, we find that low ratios of reservoir to caprock thickness favor high excess pressure and thus fracture reactivation in the caprock. Injection into thick reservoir units thus lowers the risk associated with CO2 leakage.

The calculation method based on the equivalent continuum for the fracture initiation pressure of fracturing of coalbed methane well

Coal rock is a type of fractured rock masses in which lots of cleats, fractures and weak planes develop, and these structures make the coal rock mechanics properties largely different from those of conventional sandstone rock. Previously established calculation models for the fracture initiation pressure of fracturing of conventional reservoirs cause large deviations when used to calculate the fracture initiation pressure of coalbed methane fracturing. In order to accurately calculate the fracture initiation pressure of coalbed methane fracturing, a new calculation model needs to be established while taking coal rock mechanical properties into consideration. In this paper, first, we carry out the macroscopic measurements and scanning electron microscopy (SEM) observation experiments on coal rock samples to describe the cleats distribution characteristics of coal rock samples. Second, we have carried out tests on rock mechanical parameters of different coring directions to obtain the anisotropy of coal rock mechanical parameters. Third, a calculating model has been established to determine the elastic modulus and Poisson’s ratio of coal rock based on the equivalent continuum mechanical methods, and then the model gets validated using the results of mechanical parameters tests. Finally, we have established the calculation models for borehole wall stress and fracture initiation pressure of hydraulic fracturing, which is based on the calculation model for equivalent coal rock mechanical parameters, to analyze the influence of mechanics properties of coal cleats on initiation pressure. The research result provides a new kind of research idea and approach for the calculation of fracture initiation pressure of coal hydraulic fracturing.

Investigation of hydraulic parameters of a weathered mylonite fault from field pumping tests: A case study

This paper presents a case history where the hydraulic parameters of a weathered mylonite fault located in the way of shield tunneling for the Xie-Zhong section on Metro Line No. 7 in Guangzhou City were determined using a series of field pumping tests. The weathered mylonite fault contains fissure groundwater, which would result in a high potential of water ingress during shield tunneling within the weathered mylonite strata. Therefore, the first stage of pumping tests involving five pumping wells was initially carried out. To verify the validity of the hydraulic parameters derived, the second stage of pumping tests involving four pumping wells was carried out. In addition, both an analytical method that involves a well-flow calculation for a fissure-confined aquifer for non-orthogonal anisotropic rock mass and a numerical simulation based on the theory of the equivalent continuum porous media were used to derive the hydraulic parameters based on the field pumping test data. The analytical method can provide the values of hydraulic conductivity only in the horizontal plane and is not able to capture the hydraulic characteristics of standalone stratum of the multi-aquifer system. In the numerical model, the weathered mylonite fault due to the nature of non-uniform mylonite strata distribution is further divided into four sub-regions with different hydraulic conductivities, according to the calculated influence radius of groundwater drawdown or geological formation. This leads to estimates of hydraulic parameters that are closer to real values, and enhances our understanding of non-uniform fissure development.

Rock Mass Characterization by High-Resolution Sonic and GSI Borehole Logging

We investigate the relationships between the in situ P-wave velocity (Vp) of rock masses, measured by borehole acoustic logging, and their Geological Strength Index (GSI), to support a reliable assessment of equivalent continuum rock mass properties at depth. We quantified both Vp and GSI in three deep boreholes drilled in a crystalline core complex of the central Italian Alps. The boreholes were driven up to 400 m in depth and provided high-quality drill cores in gneiss, schist and metasedimentary rocks with variable lithology. Geological and geomechanical logging was carried out for over 800 m of cores, and acoustic logging was performed for more than 600 m of borehole length. High-resolution core logging in terms of GSI was obtained using an original quantitative approach. Candidate empirical correlation functions linking Vp and GSI were tested by a two-step statistical analysis of the experimental dataset, including outlier removal and nonlinear regression analysis. We propose a sigmoid Vp–GSI equation valid over a depth range between 100 and 400 m. This accounts for extremely variable lithological, weathering and rock mass damage conditions, complementing existing shallow-depth approaches and showing potential for practical applications in different engineering settings.

Underwater explosion approximate method research on ship with polymer coating

Battle damage survivability is the prime design criterion of naval ships. Composites and hybrid structures such as polymer coating attract much attention to increase the ship survivability. Aiming to develop the parameter optimization design of the coating and to assess the underwater explosion shock environment and the blast-resistant effect of polymer coating efficiently, an approximate numerical method was performed in this preliminary study. An equivalent continuum model based on the homogenization theory was introduced to simulate the dynamic behavior of the coating. The test data of full-scale underwater explosion tests and the numerical simulation results were investigated. The blast-resistant property of the polymer coating was elaborated. The core strength had an important influence on the ship dynamic response: under low-intensity underwater explosion shock load, “soft” core had greater blast-resistant effect than the other ones, while under high-intensity underwater explosion shock load, “ordinary” core and “hard” core had better blast-resistant effect than the “soft” core.

Numerical modeling of two-phase fluid flow in deformable fractured porous media using the extended finite element method and an equivalent continuum model

In the present paper, a numerical model is developed based on a combination of the extended finite element method and an equivalent continuum model to simulate the two-phase fluid flow through fractured porous media containing fractures with multiple length scales. The governing equations involve the linear momentum balance equation and the flow continuity equation for each fluid phase. The extended finite element method allows for an explicit and accurate representation of cracks by enriching the standard finite element approximation of the field variables with appropriate enrichment functions, and captures the mass transfer between the fracture and the matrix. Due to the high computational cost of X-FEM, this technique is only used to model large fractures. The pre-existing short fractures, which are distributed randomly in the porous medium, contribute to the increase of the effective permeability tensor and are modeled with an equivalent continuum model. Finally, the robustness of the proposed computational model is demonstrated through several numerical examples, and the effects of crack orientation, capillary pressure function, solid skeleton deformation, and existence of short cracks on the pattern of fluid flow are investigated. It is shown that the developed model provides a correct prediction of flow pattern for different crack configurations.

An efficient embedded discrete fracture model based on mimetic finite difference method

Various numerical models exist for numerical simulation of fluid flow in fractured porous media, such as dual-porosity model, discrete fracture model and equivalent continuum model. As a promising model, the embedded discrete fracture model is a powerful tool for fractured porous media with complex fracture distribution, because it incorporates the effect of each fracture explicitly without requiring the simulation mesh to conform to the fracture geometry. Moreover, it does not need mesh refinement near fractures and offers computationally-efficient simulations compared to other discrete fracture models. In this paper, the Mimetic Finite Difference method and Finite Volume Method are used to improve the numerical solution of the embedded discrete fracture model, the improved method can deal with permeability tensor and can be used to simulate fractured reservoir with complex geometrical shape, which fails to be solved by the primal method based on the finite difference method. Several numerical simulations and physical experiment demonstrate the applicability of the proposed method for studying flow processes in fractured porous media.

Vibration of a carbyne nanomechanical mass sensor with surface effect

This paper presents a comprehensive model to investigate the influence of surface elasticity and residual surface tension on the natural frequency of flexural vibrations of nanomechanical mass sensor using a carbyne resonator. Carbyne is modeled as an equivalent continuum circular cross-section Timoshenko nanobeam including rotary inertia and shear deformation effects. Surface stress and surface elasticity are presented via the Young–Laplace equation. The analytical solution is presented and verified with molecular dynamics solution. The results show that the carbyne resonator can measure a very small mass with weight below 10−3 zg. The effects of surface elasticity, residual surface tension, carbyne length, and mass position on the fundamental frequencies are illustrated. This study is helpful for characterizing the mechanical behavior of high-precision measurement devices such as chemical and biological sensor.

Nonlocal Effect on Buckling of Triangular Nano-composite Plates

In the present study, small scale effect on critical buckling loads of triangular nano- composite plates under uniform in-plane compression is studied. Since at nano-scale the structure of the plate is discrete, the size dependent nonlocal elasticity theory is employed to develop an equivalent continuum plate model for this nanostructure incorporating the changes in its mechanical behavior. The two-parameter, Winkler-Pasternak, elastic medium is used for precisely modelling the behavior of matrix surrounding the nano-plate. The governing stability equations are then derived using the classical plate theory and the principle of virtual work for a perfect uniform triangular nano-plate composite. The well-known numerical Galerkin method in conjunction with the areal coordinates system is used as the basis for the solution. The solution procedure views the entire nano-composite plate as a single super-continuum element. Effects of nonlocal parameter, length, aspect ratio, mode number, anisotropy, edge supports and elastic medium on the buckling loads are rigorously investigated. All of these parameters are seen to have significant effect on the stability of nano-composite plate. It is shown that the results depend obviously on the non-locality of buckled nano-composite plate, especially at very small dimensions, small aspect ratios, higher mode numbers, higher anisotropy degree, stiffer edge supports and softer elastic mediums. Also, it is seen that the classical continuum mechanics overestimates the buckling results which can lead to deficient design and analysis of these widely used nanostructures.

Simulating Biofilm Deformation and Detachment with the Immersed Boundary Method

AbstractWe apply the immersed boundary (or IB) method to simulate deformation and detachment of a periodic array of wall-bounded biofilm colonies in response to a linear shear flow. The biofilm material is represented as a network of Hookean springs that are placed along the edges of a triangulation of the biofilm region. The interfacial shear stress, lift and drag forces acting on the biofilm colony are computed by using fluid stress jump method developed by Williams, Fauci and Gaver [Disc. Con-tin. Dyn. Sys. B11(2):519–540, 2009], with a modified version of their exclusion filter. Our detachment criterion is based on the novel concept of an averaged equivalent continuum stress tensor defined at each IB point in the biofilm which is then used to determine a corresponding von Mises yield stress; wherever this yield stress exceeds a given critical threshold the connections to that node are severed, thereby signalling the onset of a detachment event. In order to capture the deformation and detachment behaviour of a biofilm colony at different stages of growth, we consider a family of four biofilm shapes with varying aspect ratio. For each aspect ratio, we varied the spacing between colonies to investigate role of spatial clustering in offering protection against detachment. Our numerical simulations focus on the behaviour of weak biofilms (with relatively low yield stress threshold) and investigate features of the fluid-structure interaction such as locations of maximum shear and increased drag. The most important conclusions of this work are: (a) reducing the spacing between colonies reduces drag by from 50 to 100% and alters the interfacial shear stress profile, suggesting that even weak biofilms may be able to grow into tall structures because of the protection they gain from spatial proximity with other colonies; (b) the commonly employed detachment strategy in biofilm models based only on interfacial shear stress can lead to incorrect or inaccurate results when applied to the study of shear induced detachment of weak biofilms. Our detachment strategy based on equivalent continuum stresses provides a unified and consistent IB framework that handles both sloughing and erosion modes of biofilm detachment, and is consistent with strategies employed in many other continuum based biofilm models.

Using an equivalent continuum model for 3D dynamic analysis of nanocomposite plates

Most of the early studies on plates vibration are focused on two-dimensional 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 3-D elasticity solution for free vibration analysis of continuously graded carbon nanotube-reinforced (CGCNTR) rectangular plates resting on two-parameter elastic foundations. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. In this study, 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). The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The formulations are based on the three-dimensional elasticity theory. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the present approach. The novelty of the present work is to exploit Eshelby-Mori-Tanaka approach in order to reveal the impacts of the volume fractions of oriented CNTs, different CNTs distributions, various coefficients of foundation and different combinations of free, simply supported and clamped boundary conditions on the vibrational characteristics of CGCNTR rectangular plates. The new results can be used as benchmark solutions for future researches.