Equivalent Continuum Model Research Articles

Enhanced high-frequency continualization scheme for inertial beam-lattice metamaterials

A multifield continualization technique is introduced that offers a thermodynamically consistent description of the constitutive and dispersive properties of beam-lattice inertial metamaterials with periodic microstructures. The balance equations governing the mechanics of the discrete Lagrangian system are appropriately handled using an innovative continualization scheme to derive an equivalent integral-type continuum model. Based on formal Taylor series expansion of the integral kernels or the corresponding pseudo-differential functions incorporating shift operators and appropriate pseudo-differential downscaling laws, the proposed multifield enhanced continualization scheme allows the derivation of a gradient-type continuum model of given rank and equivalent to lattices. Two different resolution techniques are proposed. Firstly, the corresponding infinite-order average differential equations are tackled using a perturbative approach to describe the forced Bloch wave propagation in the metamaterial. Secondly, higher-order continuum models are employed through proper differential equation truncation to characterize the dispersive properties of the metamaterial in both high- and low-frequency regimes. Moreover, an energetically consistent generalized equivalent Micropolar continua, with non-local inertial terms, are here identified. The multifield continualization procedure is applied to two-dimensional periodic microstructures with tetrachiral, hexachiral, and hexa-tetrachiral topologies. Illustrative examples highlight the ability of the equivalent continuum model to accurately describe the effective constitutive properties of inertial metamaterials with periodic microstructures and to define a dynamic response consistent with the discrete Lagrangian model, validated and tested through virtual experimental verification under free and forced wave conditions.

A New Simplified Discrete Fracture Model for Shearing of Intersecting Fractures and Faults

Shearing of fractures and faults is important because it can result in permeability change or even induce seismicity—both are keys for efficient and safe energy recovery and storage in Earth systems. Quantitative analysis of shearing of intersecting fractures and faults is challenging because it can involve dynamic frictional contacts that are complicated by deformation of the rock matrix. To predict the shearing of intersecting fractures/faults, we attempt to answer the question of how intersections impact the shearing of a fracture network and whether we can simplify the description as compared to classical discrete fracture network (DFN) models. To answer these questions, we conducted a series of numerical simulations on scenarios for variable numbers of intersecting fractures. All these examples yield consistent results: the results of using DFNs are consistent with those of using hypothetical major paths. This leads to a new model, which we name simplified discrete fracture network model, to analyze shearing of intersecting fractures/faults using major path(s). We found that the intersections of fractures do not fundamentally change the shearing of two intersecting fractures if the intersecting angles are small. Furthermore, increasing the number of fractures/faults may relax the stress as more fractures/faults become available for shearing and distributing the stress. The simplified DFN model, which can capture efficiently the shearing behavior of each major paths from a large number of intersecting fractures/faults, will be a promising conceptual model that is complementary to existing equivalent continuum and discrete fracture models to analyze shearing of intersecting fractures/faults.

Transverse-inertia-induced wave dispersion in plate–shell chains with a variable Poisson ratio

One-dimensional shell chain structures can cause stress wave dispersion, and the dispersion ability is mainly affected by the apparent elastic Poisson ratio of the unit cell composing the chain. In this study, a novel plate–shell chain structure with a variable Poisson ratio is proposed by combining bent plates and arc shells (BPAS). The dispersive wave propagation features in the BPAS chains under striker impact are studied theoretically, numerically, and experimentally. An equivalent continuum model of the unit cell under compression in a linear-elastic deformation regime is established, which contains two material parameters, i.e., the apparent elastic modulus and the apparent elastic Poisson ratio. It is found that the unit cell can reach a relatively large Poisson ratio by adjusting its structural parameters. A continuum-based wave propagation model considering transverse inertia is developed, and an explicit analytical solution is obtained to represent the dispersion wave propagation behavior within a linear-elastic deformation regime. The model can reasonably predict the force and velocity evolution features of the chains and is verified by the numerical/experimental tests. The results show that the BPAS chain with a large Poisson ratio exhibits superior wave dispersion performances. The underlying mechanism is that more impact energy is dispersed by the transverse movement of the cells in the BPAS chain. Increasing the aspect ratio of the unit cell can improve the wave dispersion ability of the BPAS chains, achieving a high attenuation of the stress wave and is helpful for impact mitigation.

Modeling density-driven flow and solute transport in heterogeneous reservoirs with micro/macro fractures

In this paper, a numerical model is presented to simulate the density-driven flow in heterogeneous porous media with micro- and macro-fractures. The micro-fractures (like fissures) with relatively small fracture lengths compared to the reservoir are modeled using an equivalent continuum model. The macro-fractures (like faults) with large fracture lengths are modeled using the extended finite element method (XFEM). Governing equations are based on the mass conservation equation for both the matrix and fracture domains together with the proper constitutive laws for the fluid velocity, dispersive velocity of solute, and equation of state for density of fluid phase. Several parameter studies are performed to investigate the effects of fracture characteristics on the solute spread in the medium. It is demonstrated that in the absence of macro-fractures, the influence of micro-fractures’ aperture on the solute spread is fully substantial and predictable. However, in the presence of macro-fractures, there is a region where the macro-fracture performs like a vacuum diminishing the influence of nearby micro-fractures. Finally, a real hydrological problem of the Gödöllő Hills reservoir located in Hungary with five different layers and eight faults, which is full of saline water is numerically modeled to investigate the effect of pressure gradient, anisotropy, and density difference on the solute transport. It is illustrated that in the top layers of the Gödöllő Hills reservoir, the gravitational force and vertical permeability are mostly accountable for the solute flow across the faults, whereas in the bottom layers, the horizontal pressure gradient is primarily responsible for the solute transport.

Equivalent Circuit and Continuum Modeling of the Impedance of Electrolyte-Filled Pores

Equivalent Circuit and Continuum Modeling of the Impedance of Electrolyte-Filled Pores

Continuum theory for confluent cell monolayers: Interplay between cell growth, division, and intercalation

Mechanical forces generated by dynamic cellular activities play a crucial role in the morphogenesis and growth of biological tissues. While the influence of mechanics is clear, many questions arise regarding the way by which mechanical forces communicate with biological processes at the level of a confluent cell population. Some answers may be found in the development of mathematical models that are capable of describing the emerging behavior of a large population of active agents based on individualistic rules (single-cell response). In this perspective, the present work presents a continuum-scale model that can capture, in an average sense, the active mechanics and evolution of a confluent tissue with or without external mechanical constraints. For this, we conceptualize a confluent cell population (in a monolayer) as a deformable dynamic network, where a single cell can modify the topology of its neighborhood by swapping neighbors or dividing. With this description, we use concepts from statistical mechanics and the transient network theory to derive an equivalent active visco-elastic continuum model, which can recapitulate some of the salient features of the underlying network at the macroscale. Without loss of generality, the cell network is here assumed to follow well-known rules used in vertex model simulations, which are: (a) cell elasticity based on its bulk and cortical elasticity, (b) cell intercalation (or T1 transition), and (c) cell proliferation (expansion and division). We show, through examples and illustrations, that the model is able to characterize complex cross-talk between mechanical forces and biological processes, which are likely to drive the emergent growth and deformation of cell aggregates.

Modelling of flow through naturally fractured geothermal reservoirs, Taupō Volcanic Zone, New Zealand

BackgroundNumerous fractures are observed in fractured geothermal reservoirs on borehole images in the Taupō Volcanic Zone (TVZ), Aotearoa New Zealand. These fractures are necessary to explain the sustained reservoir permeabilities despite the low matrix porosity. However, conventional continuum models do not adequately represent fluid flow through these fractured rocks.MethodsWe present new Discrete Fracture Network (DFN) codes that model fractures and associated fluid flow in 2-D at reservoir scales to represent typical rock types found in TVZ reservoirs. Input parameters are derived from interpretations of borehole images at the Rotokawa and Wairakei geothermal fields where fractures have high dip magnitudes (> 60–70°). This paper focuses on the effect of fracture density along virtual boreholes (P10), that is in average 0.6 m−1 in sheet-like andesites; 0.8 m−1 in ignimbrites and 1.7 m−1 in rhyolite lavas.ResultsThe number of fractures in the models scale linearly with the input P10 in virtual boreholes. The percolation threshold, where the backbone of fractures is connected across the entire model domain, is reached for P10 > 0.24 m−1. Above this threshold, mean flow measured along the mean fracture direction scales linearly with P10. For P10 > 0.4 m−1 the permeability anisotropy lies in the interval 13 ± 3, with the scatter decreasing as P10 increases. The pressure distributions in individual DFN realisations are highly variable, but averages of 50 realisations converge towards those given by equivalent continuum models. Probability density functions resulting from DFN realisations can therefore be used to constrain continuum models. Tracing of fluid particles through the DFN shows that particles can take numerous pathways to define a swath of paths. The travel time of particles over 1 km follows a distribution similar to real tracer tests, with arrivals peaking at 1–2 days and a long tail stretching to over 200 days.ConclusionsThe new codes, calibrated to real measurements of fracture geometries in borehole images of the TVZ, reproduce patterns of flows in fractured geothermal systems. Mean flows and permeability anisotropies derived from the DFNs can be used to improve modelling of flows through fractured geothermal reservoirs using continuum models at a limited computational cost.

A general equivalent continuum model and elastic wave velocity analysis of jointed rock masses

The presence of joints dramatically affects the mechanical behavior of rock mass. For highly fractured jointed rock mass, it is more convenient to treat the jointed rock mass as equivalent continuum material. Consequently, several researchers have proposed continuum models to describe the mechanical behavior of jointed rock mass. These models are reviewed and their limitations are discussed. From the perspective of energy conservation, a general continuum model for jointed rock mass is proposed in this study, which incorporates a fourth-rank compliance tensor for joints. The stiffness, areas and orientations of joints are included in the compliance tensor. The proposed compliance tensor is in a general form, which is applicable for joints of arbitrary shape, such as penny-shaped joints and multi-sets of persistent/non-persistent joints etc. The directional modulus and Poisson's ratio of jointed rock mass are deduced from the proposed model. Besides, the elastic wave velocities of jointed rock mass are analyzed from the proposed model according to Christoffel equation. It is found that the square of longitudinal wave velocity is approximately proportional to directional modulus of jointed rock mass. Effectiveness of the proposed model is verified by comparisons with both in situ tests and numerical simulations.

Estimation of REV size of 2-D DFN models in nonlinear flow: Considering the fracture length-aperture correlation

Determining the representative elementary volume (REV) of fractured rock masses is the foundation for studying the properties of such rock masses using the equivalent continuum model. This paper focuses on estimating the REV sizes of a fractured rock mass, considering the linked trace length and fracture aperture in both linear and nonlinear flow regimes. To achieve this, Monte Carlo simulations were employed to create discrete fracture network (DFN) models based on the geometric characteristics of fractures. The linear element method, modified by the Forchheimer law, was used to calculate the equivalent permeability coefficient tensor K and the equivalent non-Darcy coefficient tensor β. These calculations were performed to determine the REV size of DFNs under nonlinear flow conditions. The findings of this study demonstrate a general consistency in the computed K values for the two flow regimes. It was observed that the REV size, determined by β value, exceeded that obtained using K for the identical fracture network. Furthermore, when considering the same flow regime, the REV size was found to be smallest when the fracture aperture remained constant, followed by cases where it was independent of the trace length and finally when it was correlated with the trace length.

Sensitivity of a homogeneous and isotropic second‐gradient continuum model for particle‐based materials with respect to uncertainties

AbstractThis work concerns the probabilistic analysis of particle‐based materials. More precisely, this work is devoted to the stochastic modeling of the geometric and constitutive microscale parameters associated with particle‐pair interactions of an existing model for particle‐based materials. Such an issue is addressed with a probabilistic methodology that relies on the maximum entropy principle from information theory. After defining and improving the chosen second‐gradient continuum model for particle‐based materials, it is shown that for micro‐homogeneous and micro‐isotropic materials, the involved microscale parameters turn out to be statistically independent. More precisely, the particle‐pair distance between two consecutive particles is a uniformly distributed random variable and the specific micro‐scale stiffness parameters are Gamma‐distributed random variables. A homogeneous and isotropic 2D concrete plate subjected to an axial load is considered for illustration purposes. A stochastic solver based on Monte Carlo numerical simulations and mixed finite element (FE) method are chosen. The FE discretization is applied to the weak formulation of the equivalent continuum model with random mechanical properties. On the contrary, the particle‐pair distance and the micro‐scale stiffness parameters, which are the parameters of the boundary value problem formulated for the equivalent continuum model, are modeled as random variables. Finally, uncertainties propagation is discussed and statistical fluctuations of the macro mechanical response are found to be significant.

An equivalent line element approach for free surface flow through two-dimensional rock mass including fracture networks

In this paper, an equivalent line element method is established to analyze seepage problems with free surface through two-dimensional rock mass, including fracture networks. Low-permeablity rock matrix is characterized by the line elements as well as the high-permeablity fracture networks. Unlike the conventional equivalent continuum model, water flow in a rock mass is restricted to the line elements, and the equivalent hydraulic conductivity is determined according to Darcy's law and flux equivalence principle. As a result, uniform expressions of governing equations and boundary conditions are formulated, and the corresponding numerical procedure is developed by introducing the local coordinate and continuous Heaviside function. Satisfactory agreements with the existing test and numerical data verify this proposed line element method. The numerical solutions on the fractured slope indicate that the geological features of fractures have a substantial effect on locating the free surface. In addition, the proposed line element method can efficiently achieve accurate results with less computing time, iteration, and storage.

Equivalent continuum modeling method for transient response analysis of large space truss structures with nonlinear elastic joints

Equivalent continuum modeling method for transient response analysis of large space truss structures with nonlinear elastic joints

A New Method for Inversion of Dam Foundation Hydraulic Conductivity Using an Improved Genetic Algorithm Coupled with an Unsaturated Equivalent Continuum Model and Its Application.

Seepage is a main cause of dam failure, and its stability analysis is the focus of a dam's design, construction, and management. Because a geological survey can only determine the range of a dam foundation's hydraulic conductivity, hydraulic conductivity inversion is crucial in engineering. However, current inversion methods of dam hydraulic conductivity are either not accurate enough or too complex to be directly used in engineering. Therefore, this paper proposes a new method for the inversion of hydraulic conductivity with high application value in hydraulic engineering using an improved genetic algorithm coupled with an unsaturated equivalent continuum model (IGA-UECM). This method is implemented by a new code that fully considers engineering applicability. In addition to overcoming the premature convergence shortcomings of traditional genetic algorithms, it converges faster than Bayesian optimization and tree-structured Parzen estimator inversion algorithms. This method is verified by comparing the water head from drilling exploration and inversion. The results of the inversion are used to study the influence of a cement grouting curtain layout scheme on the seepage field of the Hami concrete-face rockfill dam in China, which is used as an engineering application case of the IGA-UECM. The law of the seepage field is reasonable, which verifies the validity of the IGA-UECM. The new inversion method of hydraulic conductivity and the proposed cement grouting curtain layout in this study offer possible strategies for the design, construction, and management of concrete-face rockfill dams.

Postcritical Behavior of Nonlocal Strain Gradient Arches: Formulation and Differential Quadrature Solution

Arches are important components in nanostructures and systems. Because of the remarkable strength of nanomaterials, nanoarches are slender, thus enhancing their vulnerability to geometric instability, such as buckling. Although molecular simulations are often employed to analyze nanostructures, their use in routine analysis/design is formidable due to prohibitively exhaustive computation. Equivalent continuum models were developed as alternatives. The buckling and postcritical phenomena of the classical arch also form an important benchmark problem in nonlinear mechanics. This study investigates the buckling and the postcritical behavior of nanoarch subjected to inward pressure using nonlocal (NL), strain gradient (SG) continuum theory. The governing equations are derived in the form of a sixth-order nonlinear integrodifferential equation, unlike the fourth-order for a classical arch. The equation is then solved numerically using a differential quadrature (DQ) with appropriate boundary conditions. An incremental-iterative arc-length continuation is employed for the solution of the resulting algebraic system of equations. The equilibrium paths are traced for the possible instability modes, such as symmetric/antisymmetric bifurcations, snap-through and limit point instability, ascertained with the internal-external force diagrams. A particular instability mode is triggered at a certain threshold/range of the slenderness ratio of the arch, which is significantly influenced by the NL and SG interactions. These interactions not only cause quantitative changes in the instability behavior but also lead to qualitative changes, such as cessation, shift, and conversion of modes, more prominently for SG arches. Similar to classical arches, the prebuckling nonlinearity is shown to be significant.

Numerical modelling of spatially and temporally distributed on-fault induced seismicity: implication for seismic hazards

Induced seismicity is strongly related to various engineering projects that cause anthropogenic in-situ stress change at a great depth. Hence, there is a need to estimate and mitigate the associated risks. In the past, various simulation methods have been developed and applied to induced seismicity analysis, but there is still a fundamental difference between simulation results and field observations in terms of the spatial distribution of seismic events and its frequency. The present study aims to develop a method to simulate spatially distributed on-fault seismicity whilst reproducing a complex stress state in the fault zone. Hence, an equivalent continuum model is constructed, based on a discrete fracture network within a fault damage zone, by employing the crack tensor theory. A fault core is simulated at the center of the model as a discontinuous plane. Using the model, a heterogeneous stress state with stress anomalies in the fault zone is first simulated by applying tractions on the model outer boundaries. Subsequently, the effective normal stress on the fault plane is decreased in a stepwise manner to induce slip. The simulation result is validated in terms of the b-value and other seismic source parameters, hence demonstrating that the model can reproduce spatially and temporally distributed on-fault seismicity. Further analysis on the parameters shows the variation of frequency-magnitude distribution before the occurrence of large seismic events. This variation is found to be consistent with field observations, thus suggesting the potential use of this simulation method in evaluating the risk for seismic hazards in various engineering projects.

Three-dimensional cleat scale modelling of gas transport processes in deformable fractured coal reservoirs

To understand the flow processes in naturally fractured coal reservoirs, a 3D numerical model for coupled gas flow, adsorption and deformation at the scale of coal cleat and matrix blocks is presented in this study. A discrete fracture matrix (DFM) modelling approach has been adopted where flow patterns in fractures and matrices are described separately and explicitly. Different from previous studies in which constant diffusion coefficient, equilibrium adsorption and lumped deformation of matrix and fracture are assumed, in this study, adsorbed gases are treated as an independent phase and the mass exchange process between free phase and adsorbed phase is described using the Langmuir kinetic model. Different gas transport mechanisms in a porous coal matrix are considered for both phase gas transport. Particularly, an equivalent poroelastic continuum model is applied to represent deformation of fracture-matrix system, in which impacts of fracture deformation on the bulk matrix-fracture deformation is accounted for. The hybrid dimensional elements have been employed to discretize the governing equations where fractures are discretized using lower-dimensional interface elements. The accuracy of developed model is validated against experimental results collected from literatures. The simulation results indicate that the gas diffusion process in coal matrices is pressure dependent, surface diffusion of adsorbed gas can contribute to the bulk gas diffusion in coal matrices. Individual cleat initially exhibits a slight opening, followed by significant closure due to adsorption-induced swelling. Ignoring the effect of fracture on bulk deformation, the aperture change is overestimated.

Mathematical Modeling: Cisplatin Binding to Deoxyribonucleic Acid

The discovery of the cisplatin drug attracted considerable research attention as scientists strove to understand the drug’s mechanism in the human body that is responsible for destroying cancer cells, particularly the coordination between the cisplatin drug and deoxyribonucleic acid. Here, the binding energies of a cisplatin molecule relative to double-stranded deoxyribonucleic acid are obtained. The interactions of the system are determined by performing double integrals, and the analytical expressions are derived from the Lennard–Jones function and the continuum approximation; here, it is assumed that a discrete atomic structure might be replaced by surfaces with a constant average atomic density. The results observed that the cisplatin molecule is binding to the double-stranded deoxyribonucleic acid at either the minor or major grooves. By minimizing the interaction energies between the cisplatin molecule and the minor and major grooves, for arbitrary distances λ and arbitrary tilt angles φ from the axis of the helix of the double-stranded deoxyribonucleic acid, the binding energies are determined, and their values are ≈−6 and ≈−12.5 (kcal/mol), respectively. Thus, we may deduce that the major groove in double-stranded deoxyribonucleic acid is the most preferred groove for linking with the cisplatin molecule. The current analysis might help in the equivalent continuum modeling of deoxyribonucleic acids and nanocomposites.

Equivalent discontinuum analysis of stress and displacement in underground rock masses

The complexity and high frequency of fractures in rock masses make discrete fracture network (DFN) models suffer from burdensome computational consumption when every fracture is explicitly simulated. Meanwhile, the uncertainty in the spatial distribution of fractures also leads to the unnecessary explicit simulation of small fractures to achieve more accurate results. In this study, an equivalent discontinuum analysis (EDA) method is proposed to deal with complexly fractured rock masses in the deep underground. An equivalent discrete fracture network (E-DFN) with simple geometric configurations is extracted based on a defined significance index of fractures. The influence of fractures that are neglected in E-DFNs is superposed on the remaining fractures by reassigning their material properties. Using the proposed EDA method, the anisotropy of the deformation modulus of the original DFN model is well represented by the E-DFN models. Compared with the equivalent continuum and discontinuum models, the E-DFN models with reassigned fracture properties have a better potential for maintaining stress variability and displacement discontinuity in rock masses.

Scattering in quantum wires and junctions of quantum wires with edge states of quantum spin Hall insulators

An integral part of scattering theory calculations in continuum quantum systems involves identifying appropriate boundary conditions in addition to writing down the correct Hamiltonian. In the simplest problem of scattering in one dimensional lattice, scattering due to an on-site potential and scattering due to an unequal bond (in otherwise translationally invariant lattice) give different results for scattering amplitudes. While the scattering problems in the continuum and the lattice models can be mapped to one another for scattering due to on-site potential, the equivalent continuum model for scattering due to an unequal bond is missing. We introduce a new parameter c in the boundary condition of the continuum model that is equivalent to scattering due to an unequal bond on the lattice. Further, we study a junction between a normal metal quantum wire and a one dimensional edge of quantum spin Hall insulator (QSHI) in continuum using the parameter c. In the case of a junction between a normal metal quantum wire and the edge of QSHI, we identify the boundary condition that permits maximum transmission. Further, we solve the scattering problem between the junction of quantum wire and QSHI using a lattice model and map it to continuum model results. The problem of transport between four channels of spinful normal metal quantum wire and two channels of QSHI edge is not well-defined. We rectify this situation by formulating the scattering problem in terms of a junction of a semi-infinite normal metal quantum wire with an infinite edge of QSHI, gapping out one semi-infinite section of the QSHI edge by a Zeeman field and applying an appropriate boundary condition at the junction. We calculate the scattering amplitudes analytically.

Evaluation of groundwater flow through a high rockfill dam foundation in karst area in response to reservoir impoundment

Constructing high dams in karst valleys remains a challenging issue as karst conduits provide preferential channels for flow through the foundations. The uncertainty of the karst system drastically increases the risk of leakage and hence the difficulty in seepage control at the foundations. This study examined the transient flow behaviors through a karst foundation on which a rockfill dam with a height of 219 m had been constructed. During reservoir filling, an abnormal pressure rise occurred in the foundation downstream the grout curtains, typically with a magnitude of 53∼69 m higher than the pore pressure head at the neighboring positions. This phenomenon was comprehensively investigated by site characterization, inverse modelling, numerical simulation and cross-correlation analysis, demonstrating that the abnormal pressure rise is locally induced by eroded fault (left bank) or contact zones between karst formations (right bank). These conduits have an effective permeability value one order of magnitude higher than that of the karst formations, which quickly transmit the reservoir pressure pulse across the grout curtains, as evidenced by a higher cross-correlation and a smaller lag between pool level and piezometer measurements. The grout curtains have sufficiently low permeability in limiting the amount of leakage through the foundation, and hence ensure the safety of the reservoir operation in spite of the pressure rise. This study confirms the applicability of the equivalent continuum model in addressing groundwater flow problems in karst formations, as long as the site conditions are properly characterized.