Equivalent Continuum Research Articles

On the vibration of postbuckled functionally graded carbon nanotube-reinforced composite annular plates

This paper studies the free vibration charachterstics of post-buckled functionally graded nanocomposite annular plates reinforced by single-walled carbon nanotubes (SWCNTs). The analysis is performed by employing a generalized differenitail quadrature (GDQ)-type numerical technique and psedue arc-length continuation scheme. The SWCNT reinforcement is considered to be either uniformly distributed (UD) or functionally graded (FG) in the thickness direction. The material properties of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates are estimated using an equivalent continuum model based on the modified rule of mixture. The vibration problem is formulated on the basis of the first-order shear deformation theory for moderately thick laminated plates and von Karman geometric nonlinearity. By employing Hamilton’s principle and a variational approach, the governing equations and the associated boundary conditions (BCs) are derived which are then discretized via the GDQ method. The postbuckling characteristics of FG-CNTRC annular plates are investigated by plotting the equilibrium postbuckling path as the load-deflection curves. Thereafter, the free vibration behavior of FG-CNTRC annular plates in pre- and post-buckled states is examined. Effects of different parameters including type of BCs, CNT volume fraction, outer radius-to-thickness ratio and inner-to-outer radius ratio are investigated in detail.

Deformation of atomic models and their equivalent continuum counterparts using Eringen’s two-phase local/nonlocal model

The aim of this contribution is to formulate equivalent continuum finite element model for two-dimensional atomic arrays under plane-stress condition, based on Eringen’s two phase local/nonlocal model. The interaction between the atoms is modelled using translational and rotational linear elastic springs including both nearest and second nearest neighbor relations. Explicit relations between those set of springs and material properties of associated continuum model is looked for by means of equivalency of potential energy stored in atomic bonds and strain energy of continuum. Possibility of reducing computational burden of full atomic models by equivalent continuum models is discussed. This study may be regarded as the first step in composing a partitioned-domain multiscale model; with possibly smoother transition between coarse and fine scales due to the ability of nonlocal continuum model in incorporating long-range interactions.

An efficient multiscale homogenization modeling approach to describe hyperelastic behavior of polymer nanocomposites

The mechanical characterization of the interphase zone has attracted considerable attention in the fields of mechanical engineering and material design. Especially, the constitutive modeling for the description of the nonlinear mechanical behaviors of the interphase zone within the finite strain is essential for the material design of the polymer nanocomposites (e.g., the thermo-mechanical design, the fracture toughness design, and the fatigue life design). In this study, we propose an efficient multiscale homogenization modeling approach to describe the hyperelastic behavior of the polymer nanocomposites. This is the first attempt to achieve the hyperelastic constitutive modeling of the interphase zone by the multiscale framework, which is based on the full-atomistic molecular dynamics simulations and the multiscale homogenization analysis. The role of interfacial interactions between the nanoparticle and the polymer matrix on the nonlinear mechanical behaviors of polymer nanocomposites is characterized within the finite strain range by the proposed multiscale framework. To overcome the numerical inefficiencies induced by the excessively large number of iterations, the proper orthogonal decomposition method is merged into the proposed multiscale framework. The equivalent continuum models of the polymer nanocomposites are verified by the full atomistic molecular dynamics simulations.

Shallow unloading deformation analysis on Baihetan super-high arch dam foundation

The Baihetan super-high arch dam is the largest hydropower station in the world under construction. Columnar jointed basalt (CJB) is widely distributed at the Baihetan dam foundation, and the many closely spaced joints pose potential risks on the unloading quality of the foundation surface. In this study, typical unloading cracking and relaxation features of CJB based on field survey and the acoustic wave test are revealed during foundation excavation. Considering the large-scale joints in category-I CJB, an equivalent continuum constitutive model was proposed for describing anisotropic deformation and the unloading relaxation modes and the mechanism of columnar joints are further analysed. Both monitoring and numerical results show that: 1) the unloading behaviours include shallow relaxation of columnar joints and combined rebound deformation of columnar joints and shear belts; 2) the relaxation range of columnar joints mainly occurs at the shallow foundation surface with the maximum depth of 3~4 m, and the relaxation degree is obviously exaggerated owing to the sliding of shear belts; 3) the unloading relaxation mode at the shallow foundation mainly belongs to rebound deformation and relaxation of oral joints. The tensile failure occurs in steep joints, and both tensile and shear failure occur in gentle joints. The proposed model can be applied to effectively simulate the unloading relaxation processes of dam foundation excavation.

Embedded Discrete Fracture Modeling as a Method to Upscale Permeability for Fractured Reservoirs

Fractured reservoirs are distributed widely over the world, and describing fluid flow in fractures is an important and challenging topic in research. Discrete fracture modeling (DFM) and equivalent continuum modeling are two principal methods used to model fluid flow through fractured rocks. In this paper, a novel method, embedded discrete fracture modeling (EDFM), is developed to compute equivalent permeability in fractured reservoirs. This paper begins with an introduction on EDFM. Then, the paper describes an upscaling procedure to calculate equivalent permeability. Following this, the paper carries out a series of simulations to compare the computation cost between DFM and EDFM. In addition, the method is verified by embedded discrete fracture modeling and fine grid methods, and grid-block and multiphase flow are studied to prove the feasibility of the method. Finally, the upscaling procedure is applied to a three-dimensional case in order to study performance for a gas injection problem. This study is the first to use embedded discrete fracture modeling to compute equivalent permeability for fractured reservoirs. This paper also provides a detailed comparison and discussion on embedded discrete fracture modeling and discrete fracture modeling in the context of equivalent permeability computation with a single-phase model. Most importantly, this study addresses whether this novel method can be used in multiphase flow in a reservoir with fractures.

Numerical simulation of hydro-mechanical coupling in fractured vuggy porous media using the equivalent continuum model and embedded discrete fracture model

In this study, an efficient numerical model is proposed to simulate the hydro-mechanical coupling in the fractured vuggy porous media containing multi-scale fractures and vugs. Specifically, an improved Equivalent Continuum Model (ECM), which is obtained through the asymptotic homogenization method, is applied to model micro fractures and vugs, while macro fractures are modeled explicitly by using the Embedded Discrete Fracture Model (EDFM) and non-matching grids. As an important feature of the explicit representment, the effects of fillings and fluid pressure on preventing macro fractures from closing can be considered. After that, the new mixed space discretization (i.e., Mimetic Finite Difference (MFD) method for flow and stabilized eXtended Finite Element Method (XFEM) for geomechanics) and fully coupled methods are applied to solve the proposed model. The mixed space discretization can deal with complex heterogeneous properties (e.g., full tensor permeability), and yields the benefits of local mass conservation and numerical stability in space. Finally, we demonstrate the accuracy and application of the proposed model to capture the coupled hydro-mechanical impacts of multiscale fractures and vugs on fluid flow in fractured porous media.

Strength Anisotropy of Rock with Crossing Joints: Results of Physical and Numerical Modeling with Gypsum Models

The geometric and mechanical characteristics of multiple intersecting joints can govern the strength anisotropy behavior of a rock mass. Laboratory uniaxial compression tests with artificial rock-like material (gypsum) were conducted to investigate the strength anisotropy behavior of jointed specimens. Three special kinds of gypsum were used to ensure that the handmade joints have a user-defined-strength for all samples. Specimens with one or two crossing joints covering more than 20 angle configurations and two different property sets were prepared and tested. The strength anisotropy behaviors of specimens with constant joint angles (90°, 80°, 60°, 45°, and 30°) were investigated, and the failure mechanisms were assessed through the damage pattern of the colored gypsum. The details of the design scheme and figures of the evaluated experimental results are presented in this paper. A new equivalent continuum model, called the multi-joint model, is developed for jointed rock masses that contain up to three arbitrary persistent joint sets. The Mohr–Coulomb yield criterion is used to check failure of the intact rock and the joints. The multi-joint model is implemented into the finite difference method code FLAC and compared with the distinct element method code UDEC. The experimental results are used to verify the developed multi-joint constitutive model and to investigate the behavior of jointed specimen. Experimental observations agree well with the simulation results and analytical solutions.

Investigation on Heat Extraction Performance of Fractured Geothermal Reservoir Using Coupled Thermal-Hydraulic-Mechanical Model Based on Equivalent Continuum Method

Natural fractures and artificial fractures in a tight rock matrix of an enhanced geothermal system make flow and heat transfer become seriously anisotropic. In this paper, a field-scale fractured heterogeneous geothermal reservoir model is built to study the heat transfer process. Based on an equivalent continuum method and local thermal non-equilibrium model, an equivalent permeability tensor is mapped from discrete fractures and a coupled thermal-hydraulic-mechanical mathematical model is adopted in which logarithmic stress sensitivity model is used to couple effective stress and permeability. From numerical simulation results at different injection rates, the contour results show the heterogeneity of flow, heat transfer and stress sensitivity are dominated by fractures distribution. Temperature contours reveal that the heat convection between water and rock in a fracture is more intense than the heat conduction between rock under different temperatures. The predicted power generation of a geothermal plant reveals the adverse effect on heat conversion efficiency, which is caused by the temperature drop at high injection rates.

Evaluation of Equivalent Bending Stiffness by Simplified Theoretical Solution for an FRP–aluminum Deck–truss Structure

A hybrid Fiber-Reinforced Polymer (FRP)–aluminum spatial deck–truss structure that incorporates an innovative structural form and advanced extrusion-type FRP profiles has the advantages of a lightweight, high-bearing capacity, and faster installation. In this study, a design-oriented investigation was conducted with a simplified theoretical solution to conveniently evaluate the equivalent bending stiffness of the above-mentioned unique structure. The simplified theoretical solution was derived using the equivalent continuum beam method based on the homogenization concept and shearing equivalence principle. The theoretical prediction enabled a direct format and simple calculation process convenient for engineers in terms of the calculation and design. The theoretical solution was experimentally and numerically calibrated to ensure that the formulae have the satisfactory accuracy. The accuracy of the predicted bending stiffness exceeded 90%. The derivation procedures and formulae further indicated that the FRP web diagonals played a key role in resisting the global bending stiffness of the unique structure, and thus, the shear deformation should be considered in the structural design in terms of stiffness-driven. When compared with the conventional calculation method for obtaining the bending stiffness of steel solid-web beams and planar trusses, the proposed method was more accurate and applicable for the unique hybrid structures.

A New Three-Dimensional Numerical Model Based on the Equivalent Continuum Method to Simulate Hydraulic Fracture Propagation in an Underground Coal Mine

Hydraulic fracturing has been proven to be the most efficient way to improve the permeability of coal seams. In this work, the hydraulic fracture propagation in an underground coal mine was numerically investigated. A new numerical approach was developed based on the equivalent continuum methodology to model the hydro-mechanical behavior of multiple fractures in three dimensions. It was solved using a hybrid combination of the embedded element method (EEM) and the finite volume method (FVM) using an iterative coupling schema. The FVM was employed to calculate the pressure field, while the EEM was used to track the displacement discontinuity caused by fractures. A fracture constitutive model was implemented to describe the aperture variation, shear slippage, and shear dilation for both contact and open fractures, as well as for contact and open criteria. To verify the developed model, two benchmark examples were presented. Then, the developed model was used to numerically investigate hydraulic fracture propagation in Datong underground coal mine in Songzao in Chongqing. According to the numerical study, it was found that (1) a fracture network created by a hydraulic fracturing operation in a coal mine is more complex than the ideal cross-cutting-shape, H-shape, T-shape, and Z-shape patterns; (2) the orientation of the minimum principal stress controlled the main propagation direction; (3) the complexity of the fracture pattern is controlled by the geological structure, the in situ stress and the injection rate.

Damage constitutive model of single flaw sandstone under freeze-thaw and load

The deterioration of mechanical properties subjected to freeze-thaw action is a determining factor for natural rock masses in cold region engineering. Taking four cases of flaw sandstone as the research object, freeze-thaw cycles experiments and triaxial compression tests were conducted to analyze the mechanical properties after 0, 20, 40 and 60 freeze-thaw cycles. The damage evolution equation of freeze-thaw sandstone under load is established based on the Lemaitre strain equivalent principle and continuum damage mechanics theory. In this regard, the damage model was developed to predict the degradation of triaxial compression strength for single flaw sandstone samples, it takes into account pre-existing cracks, confining pressure, freeze-thaw action and load. The results show that the sandstone damage consists of factors mentioned above, the coupled damage variable acquired from the proposed method well agree with the actual condition, and the damage constitutive model validations are given through comparisons between the calculated and the measured.

Relaxed micromorphic model of transient wave propagation in anisotropic band-gap metastructures

In this paper, we show that the transient waveforms arising from several localised pulses in a micro-structured material can be reproduced by a corresponding generalised continuum of the relaxed micromorphic type. Specifically, we compare the dynamic response of a bounded micro-structured material to that of bounded continua with special kinematic properties: (i) the relaxed micromorphic continuum and (ii) an equivalent Cauchy linear elastic continuum. We show that, while the Cauchy theory is able to describe the overall behaviour of the metastructure only at low frequencies, the relaxed micromorphic model goes far beyond by giving a correct description of the pulse propagation in the frequency band-gap and at frequencies intersecting the optical branches. In addition, we observe a computational time reduction associated with the use of the relaxed micromorphic continuum, compared to the sensible computational time needed to perform a transient computation in a micro-structured domain.

Performance of New Austrian Tunneling Method (NATM) in Weak Rock Case Study

The decline in the over ground utilizable space and increment in development of metro structures, cut and cover structures are winding up fairly difficult to conceptualize and build. In this examination, a nonlinear two dimensional limited component investigation was completed to show the New Austrian Tunneling Method (NATM) burrow developed in frail shake utilizing the business limited component with joint programming PHASE 2.The validity of the numerical modeling procedure performed by the author was checked by making back-analysis for an actual case study of Strengen Tunnel which is one of the biggest expressways in western Austria. A comprehensive parametric study was performed on a hypothetical circle tunnel. Two dimensional numerical simulations with the finite element with joint software PHASE 2 have been performed to ground behaviour with the results of the numerical analysis are presented and discussed for recommendations for future work. In general the tangential stress at side wall and crown obtained from finite element with joints are nearly equal or higher than the closed form solution and equivalent continuum.

Meticulous simulation and evaluation of seepage control effects at right bank dam foundation and underground powerhouse of Jinping I hydropower station

This case study presents a systematic performance assessment of the seepage control system at the right bank abutment of Jinping I Hydropower Station. By utilizing the in-situ measurements of seepage field, the characteristics of groundwater flow behaviors in the dam foundation and around the underground powerhouse were analyzed. Based on an equivalent continuum method, an anisotropic seepage tensor model of the fractured rock mass in the dam site area was established. Considering the features of topography, geological settings and hydraulic structures, a three-dimensional finite element mesh with complex seepage control structures was generated. A numerical method combining a substructure technique, a variational inequality formulation of Signorini’s type and an adaptive penalized Heaviside function (short for SVA method) is adopted to simulate the three-dimensional seepage field at the dam site. On this basis, the distribution of seepage field during the operation period was predicted, and the performance of the seepage control system was evaluated. Besides, the influence of critically oriented fractures along the direction of NWW on the seepage control effect was also discussed. The methodology presented in this study is useful for understanding the groundwater flow behavior and the performance of the seepage control system in hydraulic engineering.

A non-local plasticity model of stimulated volume evolution during hydraulic fracturing

Hydraulic fracturing in naturally fractured rocks often leads to the creation of a stimulated zone in which the target rock formation is deformed and fractured by the reactivation and shear dilation of natural fractures and the plastic deformation, damaging, and fracturing of the bulk. In this paper, we present a novel mathematical model with the goal of simulating the evolution of the stimulated volume during hydraulic fracturing. This was achieved by introducing an equivalent continuum non-local poro-elastic-plastic zone of enhanced permeability for the stimulated region, characterized by an internal length scale. The non-local plastic constitutive behavior of the rock, combined with the classical Biot’s poroelastic theory, was implemented using a new implicit C0 non-local finite element method. A predictor-corrector return algorithm for the non-local plasticity model was formulated as an extension of the classical plasticity algorithm. To improve the performance of the iterative solution scheme, a consistent algorithmic stiffness tangent modulus was developed. First, the elastic-plastic constitutive behavior of the proposed methodology is verified using the standard non-porous biaxial compression test with strain softening behavior. Next, it is verified that the poro-elastic-plastic model correctly simulates the evolution of the stimulated zone and the subsequent change in the flow and fluid pressure for several hydraulic fracturing examples under various far-field in-situ stress conditions. Lastly, the non-local poro-elastic-plastic model is shown to be mesh-independent and capable of capturing a wide range of complex fracturing behavior.

Identification of equivalent couple-stress continuum models for planar random fibrous media

The purpose of this paper is to develop a homogeneous, couple-stress continuum model as a representation of 2D random fiber networks in the small deformation regime. The couple-stress substitution continuum is calibrated based on the response of a network model (window of analysis, WOA) subjected to prescribed kinematic boundary conditions applied on part of the WOA boundary, while the free surface boundary conditions are applied on complementary surfaces. Each fiber in the network is considered as a Timoshenko beam and the cross-links between fibers are modeled as welded joints in which the relative angles between the crossing beams remain constant during deformation, and hence they transmit moments along and between crossing fibers. The effective elastic constants of the couple-stress continuum are deduced by an equivalent strain energy method, and the characteristic lengths are identified from the resulting homogenized moduli. The competition between the affine (ADR) and non-affine (NADR) deformation regimes is shown to be quantified by the bending length, a scalar quantity that measures the relative importance of fiber bending in comparison with fiber stretch. The scaling laws of the effective moduli versus the bending length, network density and window size are determined in the affine and non-affine deformation regimes. The motivation of the adopted couple-stress substitution continuum is brought by comparing the identified effective non-classical properties with the mechanical properties predicted by FE simulations performed over the fully resolved fibrous network.

Upscaled Anisotropic Methane Hydrate Critical State Model for Turbidite Hydrate‐Bearing Sediments at East Nankai Trough

AbstractTurbidite formation is a common feature of natural hydrate‐bearing sediments that has been observed and reported at several hydrate exploration sites. It is therefore important to incorporate this anisotropic geological feature into the constitutive modeling when evaluating the geomechanical risks involved during hydrate‐based gas production. To date, a number of constitutive models have been proposed to capture the isotropic geomechanical behavior of homogeneous hydrate‐bearing sediments. Since the turbidite formation contains soil layers at a scale much smaller than the size of the numerical element used for reservoir scale simulations, it is necessary to upscale the geomechanical behavior of a layered system to an equivalent anisotropic continuum model by adopting some homogenization techniques. In this study, an anisotropic methane hydrate critical state model is developed by modifying the original isotropic version of Uchida et al. (2012, https://doi.org/10.1029/2011JB008661). The calibration methodology of the anisotropic model parameters for a given set of hydrate heterogeneity and the turbidite formation at the Eastern Nankai Trough is proposed and demonstrated. The upscaled parameters are calibrated by curve fitting the numerically simulated stress‐strain curves of the layered system with the original isotropic constitutive model at the layered scale. Forty‐two sets of model parameters are calibrated from different site element models of this site. They are used to develop empirical correlations between the model parameters and the site input properties within the turbidite formation. This paper presents the details of the new anisotropic constitutive model and the performance of the proposed upscaling procedure for the Eastern Nankai Trough case.

3D modelling of the Excavation Damaged Zone using a Marked Point Process technique

The Excavation Damage Zone (EDZ) is a key aspect in the assessment of engineering feasibility and long-term safety of deep geological repositories for radioactive waste. The EDZ represents a possible release path for radionuclides dissolved in the porewater and for gases from corrosion and degradation processes which needs to be addressed quantitatively in Safety Assessment (SA). The EDZ is often represented by a discrete fracture network, whose complexity precludes its implementation in conventional SA models. To overcome this problem, two-dimensional abstraction methodologies have been developed that replace the complex geometry of the discrete fracture network by a hydraulically equivalent continuum model. Although such simple 2D models are amenable to SA modelling, they cannot capture the inherent three-dimensional and stochastic nature of the phenomena associated with the EDZ. In this paper we present a methodology based on Marked Point Processes to infer 3D stochastic continuum models of the EDZ from 2D discrete characterizations of the fracture network. The methodology is illustrated with the calibration of a multi-rate injection experiment in Mont Terri (Switzerland).

Experimental and Numerical Investigations of Influence on Overland Flow and Water Infiltration by Fracture Networks in Soil

Preferential flow is common in clay or expansive clay soils, involving water bypassing a large portion of the soil matrix. Dye tracer experiment and numerical modeling are used to simulate the surface runoff and subsurface preferential flow patterns influenced by the soil fracture network of a relatively steep hillslope system (slope angle equals to 10 degrees). The result of the experiments indicates that part of the water is infiltrated through cracks, leading to the delay of the initial runoff-yielding time and reduction of the discharge of the surface runoff. The soil water flow is mainly in the matrix when the intensity of precipitation is low. With the increasing of precipitation, soil water movement may become in the form of preferential flow through cracks. In addition, the nonuniformity of soil water infiltration and the depth of the average water infiltration increase as the precipitation intensity increases. To this end, the complete coupling model was established by using the surface-matrix-crack (SMC) model to simulate water flow within discrete fracture as well as to simulate water flow in the soil matrix based on the concept of dual permeability using the traditional Richards’ equation. In this model, the “cubic law” of fluid motion in cracks within smooth parallel plates and the two-dimensional diffusion wave approximation to Saint-Venant equations with momentum term ignored (two-dimensional shallow water equations) were used. The model divides soil water infiltration into two forms and uses the overall method to calculate the exchange of water between the crack networks and matrix regions as well as the exchange water between surface runoff and infiltration water. Results indicate that the SMC model has better performance compared with the traditional equivalent continuum model when those models are used to simulate the surface runoff movement and the soil water movement in the presence of cracks.

Vibrational analysis of sandwich sectorial plates with functionally graded sheets reinforced by aggregated carbon nanotube

In the present work, by considering the agglomeration effect of single-walled carbon nanotubes, free vibration characteristics of functionally graded nanocomposite sandwich sectorial plates are presented. The volume fractions of randomly oriented agglomerated single-walled carbon nanotubes are assumed to be graded in the thickness direction. To determine the effect of carbon nanotube agglomeration on the elastic properties of carbon nanotube-reinforced composites, a two-parameter micromechanical model of agglomeration is employed. In this research work, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach is considered to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented straight carbon nanotubes. The two-dimensional generalized differential quadrature method as an efficient and accurate numerical tool is used to discretize the equations of motion and to implement the various boundary conditions. The proposed sectorial plates are simply supported at radial edges, while all possible combinations of free, simply supported, and clamped boundary conditions are applied to the other two circular edges. The benefit of using the considered power-law distribution is to illustrate and present useful results arising from symmetric and asymmetric profiles. The effects of agglomeration, geometrical, and material parameters together with the boundary conditions on the frequency parameters of the sandwich functionally graded nanocomposite plates are investigated. It is shown that the natural frequencies of structure are seriously affected by the influence of carbon nanotubes agglomeration. This study serves as a benchmark for assessing the validity of numerical methods or two-dimensional theories used to analyze the sandwich sectorial plates.