Equivalent Continuum Model Research Articles

Selecting Generalized Continuum Theories for Nonlinear Periodic Solids Based on the Instabilities of the Underlying Microstructure

In the context of architected materials, it has been observed that both long-wavelength instabilities leading possibly to localization and short-wavelength instabilities leading to the apparition of a deformation pattern could occur. This work proposes for the first time a comparison of the ability of two families of higher order equivalent media, namely strain-gradient and micromorphic media, to capture both patterning and long-wavelength macroscopic instabilities in those materials. The studied architected material consists in a simple one-dimensional arrangement of non-linear springs, thus allowing for analytical or nearly analytical treatment of the problem, avoiding any uncertainties or imprecisions coming from a numerical method. A numerical solution of the problem is then used to compare the post-buckling prediction of both models. The study concludes that, micromorphic media are the appropriate choice of equivalent continuum models to emply when dealing with the possibility of patterning inside a structured medium, but if long-wavelength global instability is of interest, a strain-gradient type equivalent medium is well suited.

Equivalent Dynamic Model for Large Parabolic Cylindrical Deployable Mechanism

With the development of the space industry and the demand of future complex space missions, large space deployable antennas require high deployable ratio, high precision, and lightweight. A large parabolic cylindrical deployable (PCD) mechanism with a multilayer tape-spring (MLTS) hinge was proposed. The energy method was adopted to establish the equivalent dynamic model of the large PCD mechanism with MLTS hinges. The new problems encountered in the dynamic analysis of the new mechanism are solved. First, the nonlinear effects of the tape-spring hinges and cable in the PCD mechanism were considered. The relationship equations of strain and kinetic energy were derived. Then, the stiffness and mass matrices of the PCD mechanism were obtained. Based on Timoshenko beam theory, the modal analysis of the PCD mechanism in cantilever state was conducted. The finite element method was used to verify the correctness of the equivalent continuum model of the large PCD mechanism with MLTS hinges. The equivalent dynamic analytical method can be applied to the spatial large deployable mechanism.

System reliability analysis of foundation stability of gravity dams considering anisotropic seepage and multiple sliding surfaces

PurposeFor gravity dams built on foundations with directional joint sets, the seepage in the foundation possesses anisotropic characteristics and may have adverse effects on the foundation stability. A methodology for system reliability analysis of gravity dam foundations considering anisotropic seepage and multiple sliding surfaces is proposed in this paper.Design/methodology/approachAnisotropic seepages in dam foundations are simulated using finite element method (FEM) with the equivalent continuum model (ECM), and their effect on dam foundation stability is involved by uplift pressures acting on the potential sliding surfaces. The system failure probability of the dam foundation is efficiently estimated using Monte Carlo method (MCM) combined with response surface method (RSM).FindingsThe case study shows that it is necessary to consider the possibly adverse effect of anisotropic seepage on foundation stability of gravity dams and the deterministic analysis of the foundation stability may be misleading. The system reliability analysis of the dam foundation is justified, as the uncertainties in shear strength parameters of the foundation rocks and joint sets as well as aperture, connectivity and spacing of the joint sets are quantified and the system effect of the multiple potential sliding surfaces on the foundation reliability is reasonably considered.Originality/value(1) A methodology is proposed for efficient system reliability analysis of foundation stability of gravity dams considering anisotropic seepage and multiple sliding surfaces (2) The influence of anisotropic seepage on the stability of gravity dam foundation is revealed (3) The influence of estimation errors of RSMs on the system reliability assessment of dam foundation is investigated.

Numerical simulation of cold and hot water injection into naturally fractured porous media using the extended–FEM and an equivalent continuum model

AbstractIn this paper, a computational technique is presented for the isothermal and non‐isothermal water injection into naturally fractured oil reservoirs. A remarkable number of naturally fractured reservoirs contain relatively heavy oils that could not be extracted economically; hence, the thermal recovery methods are extensively used for such reservoirs. In this study, the effectiveness of hot water injection over cold (isothermal) water injection in oil production is quantified. The influence of long and short fractures and their alignments on oil recovery are discussed. To this end, a 2D model for two‐phase fluid flow and heat transfer is presented. The medium is assumed to be partially saturated with water as the wetting phase and oil as the non‐wetting phase. The governing equations consist of the energy balance equation together with the continuity equations of both fluid phases. To take the long (macro) fractures into account, the extended finite element method is employed by considering permeable fracture faces; accordingly, the mass and heat transfer between the fracture and the matrix are captured accurately. In addition, the network of short (micro) fractures is incorporated by an equivalent continuum model. Several numerical examples are presented to demonstrate the influence of injection temperature, initial reservoir temperature, and the long and short fractures on oil production. Furthermore, the simultaneous effect of the macro‐ and micro‐fractures along with the injection temperature are examined. The results show that the proposed computational model is a promising tool that can be used in the cold or hot water injection, and can be employed in the economic feasibility studies of recovery methods in the future.

Incorporating defects into model predictions of metal lattice-structured materials

Metal lattice structures are optimized for high specific strength properties and are now customizable using additive manufacturing methods. However, many of these methods, like selective laser melting, can introduce defects such as porosity, parasitic material, and disconnected struts into the structure, which can negatively affect mechanical behavior. While there are many computational models used to predict the mechanical response of lattice-structured materials, most use an idealized structure and are often not robust enough to account for defects. In this study, metal lattice structures are characterized for defects and mechanically tested in compression. The defect types and distributions are characterized using x-ray micro-tomography and the tomography analyses is used in two different model predictions. First, in an equivalent continuum model, where the defects are used to predict the variability in mechanical properties, and second, in a finite element analysis, where the predicted stress-strain response for both realistic and idealized structures are compared. Investigating this further, a finite element analysis of an octet lattice quantifies the reduction in strength associated with disconnected struts and captures a dependence on the strut's orientation relative to the loading direction. Overall, incorporating defect information gleaned from tomography data improves predictions of mechanical properties by capturing a more realistic deformation response for lattice-structured material.

Survey on Equivalent Continuum Modeling for Truss Structures and Their Nonlinear Dynamics and Vibration Control

Survey on Equivalent Continuum Modeling for Truss Structures and Their Nonlinear Dynamics and Vibration Control

Investigating the effects of intersection flow localization in equivalent-continuum-based upscaling of flow in discrete fracture networks

Abstract. Predicting effective permeabilities of fractured rock masses is a crucial component of reservoir modeling. Its often realized with the discrete fracture network (DFN) method, whereby single-phase incompressible fluid flow is modeled in discrete representations of individual fractures in a network. Depending on the overall number of fractures, this can result in high computational costs. Equivalent continuum models (ECMs) provide an alternative approach by subdividing the fracture network into a grid of continuous medium cells, over which hydraulic properties are averaged for fluid flow simulations. While continuum methods have the advantage of lower computational costs and the possibility of including matrix properties, choosing the right cell size to discretize the fracture network into an ECM is crucial to provide accurate flow results and conserve anisotropic flow properties. Whereas several techniques exist to map a fracture network onto a grid of continuum cells, the complexity related to flow in fracture intersections is often ignored. Here, numerical simulations of Stokes flow in simple fracture intersections are utilized to analyze their effect on permeability. It is demonstrated that intersection lineaments oriented parallel to the principal direction of flow increase permeability in a process we term intersection flow localization (IFL). We propose a new method to generate ECMs that includes this effect with a directional pipe flow parameterization: the fracture-and-pipe model. Our approach is compared against an ECM method that does not take IFL into account by performing ECM-based upscaling with a massively parallelized Darcy flow solver capable of representing permeability anisotropy for individual grid cells. While IFL results in an increase in permeability at the local scale of the ECM cell (fracture scale), its effects on network-scale flow are minor. We investigated the effects of IFL for test cases with orthogonal fracture formations for various scales, fracture lengths, hydraulic apertures, and fracture densities. Only for global fracture porosities above 30 % does IFL start to increase the systems permeability. For lower fracture densities, the effects of IFL are smeared out in the upscaling process. However, we noticed a strong dependency of ECM-based upscaling on its grid resolution. Resolution tests suggests that, as long as the cell size is smaller than the minimal fracture length and larger than the maximal hydraulic aperture of the considered fracture network, the resulting effective permeabilities and anisotropies are resolution-independent. Within that range, ECMs are applicable to upscale flow in fracture networks.

Goaf and Slope Stability: A Case Study on the Yangla Copper Mine

Considering the wide distribution range of deposits, the complicated geological conditions, and the potential detriment to slope stability during the caving period, underground goaf and slope stability is a critical concern in the Yangla copper mine. In this paper, goaf and slope stability is studied by implementing empirical analysis and numerical simulations with an equivalent continuum model. An overall understanding of underground goaf stability is obtained by the Laubscher graph method, and an approach composed of the geological modeling technology GoCAD and the specialized meshing tool Griddle is proposed to manage the geological database and prepare the numerical model for stability analysis with FLAC 3D . The Hoek–Brown constitutive model is introduced to simulate the complicated mechanical behavior of rock mass in the process of mining, i.e., the brittleness and confinement effect of rock mass. The key orebodies KT2 and KT5 develop in reverse inclination from slope surface to the inner slope with a maximum depth of 900 m, which means the rock mass bears a high stress induced by mining. The mechanical parameters are calibrated carefully by performing mining simulations that are consistent with the field instability phenomenon and measurement results. In particular, rock-mass behavior is predicted by sensitivity modeling in consideration of the goaf span, mining sequence, and material filling, the aim of which is to optimize the mining design to maintain slope stability during the third mining phase. The calculation results indicate that large deformations at the roof of the goaf and pillar are a critical problem in the Yangla copper mine as they are detrimental to slope stability. Moreover, many goafs collapse during the second and third phases of underground mining, which is also detrimental to slope stability, and the span of the goaf with a deep buried depth needs to be adjusted from 36 to 24 m in the third mining phase. The slope is stable in the second mining phase, but, in the third phase, it enters the large-scale instability state due to the destruction of the rock-mass structure near the slope surface by underground caving. Accordingly, goafs with the potential for collapse should be filled with high-strength materials to maintain slope stability.

A Novel Approach for Design and Simulation of Equivalent Continuum Model of Lattice Structures Used in Tunnels and Caverns

Developing a continuum model of a lattice structure means expressing a complicated structure into a simpler one, and this is done by determining equivalent stiffness of the original structure, which can further be used to transform into desired simpler geometry for multiple benefits. Objective of this paper is to find a reliable method for determination of equivalent stiffness of lattice girder, comparing all the techniques suggested by various researchers. In present work the actual lattice girder structures and their equivalents were analysed using Finite Element Method tool Strand7. The primary focus is on lattice girder, which is a primary rock support system employed in tunnels and caverns. The equivalent stiffness so obtained is used as input parameter in numerical analysis tool STRAND7 for analysis of equivalent lattice beam. The maximum deflection and bending stress at centre of equivalent lattice beam were calculated and compared. In an attempt to develop a more accurate equivalent beam model a novel approach which is modification of the Method-B has been developed by authors. This Noval approach is the most convenient approach to develop an equivalent section with an extremely uncompromised representation of stress development and deformation characteristic. In this approach, the focus was made on the physical resemblance of a proposed equivalent beam with the actual lattice girder, leading us to adopt a beam of I-cross-section. This approach yielded similar stress and deflections in the equivalent beam as that of the actual lattice girder.

Computational assessment of Stone-Wales defects on the elastic modulus and vibration response of graphene sheets

In this work we assess the influence of Stone-Wales defects on the elastic modulus and vibration responses of single-layered graphene sheets. To this end, the atomic position of the C-atoms within the defective layers are inserted as initial conditions into finite element calculations used to estimate the mechanical properties and natural frequencies of carbon monolayers. Moreover, we consider two sets of initial atomic positions: the harmonic solution predicted by discrete dislocation theory and its fully-relaxed configuration, which is a novelty in this type of study. Thus, the mechanical properties of the beam elements are depicted based on the equivalence between the interatomic potential energy of the atomic model and the strain energy of the equivalent continuum model. Furthermore, based on the results, we propose an extension of the classical beam model that accurately defines the mechanical properties of the elements in terms of the actual length of each atomic bond. The finite element model is implemented in the commercial software Abaqus/Standard. The results obtained are in reasonable agreement with previous data presented by other authors. In this work we have specifically ascertained the importance of using a non-linear model to predict the Young’s modulus and vibration responses of defective graphene sheets.

Equivalent Continuum Coupling-Based Slope Stability Analysis of Zhouning Pumped Storage Power Station

A 3D equivalent continuum coupling analysis of the slope of the Zhouning pumped storage power station was proposed in this study. Firstly, the hydraulic properties of rock mass, groundwater, and deformation mechanism under high-frequency fluctuations of the water level are analyzed. Secondly, the three-dimensional Digital Terrain Model (DTM) with surface image information was established by digital photogrammetry technology for cataloging field fracture information. Third, stress-seepage coupling simulation using a hybrid grid approach and the FLAC3D equivalent continuum model was implemented. The results show that the permeability of rock mass is stress-dependent and anisotropic, and the direction of maximum principal permeability of both bank slopes is approximately parallel to the river. Slope deformations induced by impoundment during the operation period are the superposition results of two mechanisms, referred to as “mattress effect” and “swelling effect.” Attention should be paid to the tensile strength of the rock mass during operation.

Effects of Gouge Fill on Elastic Wave Propagation in Equivalent Continuum Jointed Rock Mass

The presence of gouge in rock joints significantly affects the physical and mechanical properties of the host rock mass. Wave-based exploration techniques have been widely used to investigate the effects of gouge fill on rock mass properties. Previous research on wave propagation in gouge-filled joints focused on analytical and theoretical methods. The lack of experimental methods for multiple rock joint systems, however, has limited the verification potential of the proposed models. In this study, the effects of gouge material and thickness on wave propagation in equivalent continuum jointed rocks are investigated using a quasi-static resonant column test. Gouge-filled rock specimens are simulated using stacked granite rock discs. Sand and clay gouge fills of 2 and 5 mm thicknesses are tested to investigate the effects of gouge material and thickness. Comprehensive analyses of the effects of gouge thickness are conducted using homogeneous isotropic acetal gouge fills of known thickness. The results show that gouge fill leads to changes in wave velocity, which depend on the characteristics of the gouge fill. The results also show that particulate soil gouge is susceptible to preloading effects that cause permanent changes in the soil fabric and contact geometry and that increased gouge thickness causes a more significant stiffness contribution of the gouge material properties to the overall stiffness of the equivalent continuum specimen. The normal and shear joint stiffnesses for different gouge fill conditions are calculated from the experimental results using the equivalent continuum model and suggested as input parameters for numerical analysis.

Dynamics of pull and release of graphene nanoribbons

The pull and release molecular dynamics simulations of graphene nanoribbons (GNR) lead to their length dependent several folded conformations in the ground states at finite temperatures. The folding occurs when the potential energy of the stretched GNR is sufficient to overcome the energy barriers between the flat GNR and its folded conformations. We explore the dynamics of GNRs during their pull and release as the flat GNR develops intrinsic ripples as soon as thermal fluctuations are introduced and rich mechanical phenomena are found even before the folding process starts. Nudged elastic band calculations were performed to understand the nature of energy barriers and the minimum energy paths between the flat and the rippled nanoribbons. The potential and the kinetic energies of nanoribbons show strong oscillatory characters with attenuation in amplitudes at the start and the end of the pulling process, but as soon as the release process starts the oscillatory nature becomes weak giving rise to small fluctuations. Both thermostatted and unthermostatted nanoribbons were studied with some important observations. Different equivalent continuum models were analyzed which can mimic these experiments. It has been found that the pull and release of the GNRs without any intrinsic rippling can be successfully simulated with a thin plate continuum model using Hooke’s law which does not account for geometric nonlinearities and therefore is aptly able to capture small deformations of the flat GNR. This relation fails when the GNR develops ripples and then undergoes through the pull and release experiments. The rippled GNR was modeled by introducing ripples in the flat continuum GNR by nonlinear curve fitting of the atomistic z-coordinates with a sinusoidal function. Then three different constitutive relations were tried and it was found that modeling GNRs as St. Venant–Kirchhoff materials, which assume a linear relation between the second Piola–Kirchhoff stress tensor and the Green-St. Venant strain tensor correctly reproduce the atomistic behavior before the folding starts. The temperature results were also reproduced by solving the energy balance equation where a thermoelastic constitutive relation between the second Piola–Kirchhoff stress tensor and the Green-St. Venant strain tensor subtracted by the thermal strain tensor was assumed.

Thermodynamically consistent non-local continualization for masonry-like systems

Masonry-like systems composed by modular stiff units bonded by soft connections represent an efficient, versatile and ultimately successful strategy for natural and artificial macro-scale architectures. The static and dynamic behavior of masonry-like materials characterized by a running bond periodic pattern of rigid rectangular blocks and elastic interfaces is described by formulating equivalent nonlocal continuum models. The paper discusses the thermodynamic restrictions limiting the consistency of the standard continualization strategies. The inherent pathologies recognized in the macroscopic quasi-static response and/or in the dynamic dispersion properties of different continuum models motivate the original proposal of an enhanced continualization strategy. Based on the series expansion of the pseudo-differential functions accounting for shift operators and proper downscaling laws, the enhanced continualization scheme allows formulating homogeneous non-local continuum models of increasing orders, analytically featured by characteristic non-local constitutive and inertial terms. The enhanced continualization shows thermodynamic consistency in the definition of the overall elastic moduli, as well as qualitative agreement and convergent matching of the frequency dispersion functions. The theoretical findings are successfully verified though the solution of representative static and dynamic benchmark problems.

Equivalent Beam Model for Spatial Repetitive Lattice Structures with Hysteretic Nonlinear Joints

This study presented an equivalent modeling method for analysis of the fundamental harmonic response of the spatial repetitive lattice structures with hysteretic nonlinear joints. Firstly, the joint was modelled as a spatial nonlinear spring-damper system with bilinear hysteresis, the equivalent linear stiffness and viscous damping coefficients of the hysteretic joint were derived using the describing function method. Subsequently, high-precision displacement and rotation fields that consider the warping and distortion of the cross section of the lattice structure were presented for the repeating element and the equivalent continuum modeling method based on the principle of energy equivalence was adopted to obtain an 8-DOFs spatial beam element for the repeating element of the lattice structure. Afterwards, the equations of motion of the equivalent beam model for the lattice structure were established in the frequency domain and solved by using the Newton-Raphson method. In the numerical studies, the joints with axial nonlinearity and rotational nonlinearity were considered, and the influences of joint parameters and excitation amplitude on the dynamic response of the lattice structure were investigated. The correctness of the presented modeling method was verified by comparison with the nonlinear finite element model of the original structure.

Understanding the failure mechanism of a large underground cavern in steeply dipping layered rock mass using an enhanced ubiquitous-joint model

To understand the mechanical behavior and failure mechanism of the surrounding layered rock masses during the excavation of underground caverns for a hydropower station, an enhanced equivalent continuum model based on the ubiquitous-joint concept is developed and compiled as a plugin DLL file in FLAC3D. This model is then applied to analyze two engineering geological issues arisen during the excavations of a large underground powerhouse. Both cases are the typical responses controlled primarily by the internal structure of layered rock mass. The first case is mainly concerned with the continuous increase in the displacement of the upstream sidewall after removal of an auxiliary tunnel crown. Numerical simulation reveals the mixed shear-tensile fractures developed along bedding planes. The preserved crown thickness can affect the maximum displacement of sidewall. Failure region will progressively deepen into the interior if no effective reinforcements are adopted. The second case exhibits as the gradual cracking or slabbing of the shotcrete at downstream roof. Stress concentration is the main cause of shotcrete cracking as shown by numerical simulation. The degree of stress concentration can be influenced largely by the angle between rock strata and cavern axis. This degree also varies with the initial stress level and with excavation process, which is confirmed by field monitoring data. The validity and capability of the newly developed model are thus verified by the actual engineering issues.

A novel hybrid model for multiphase flow in complex multi-scale fractured systems

We present a multi-level discrete fracture model (MLDFM) to guarantee a robust and efficient solution for naturally fractured reservoir simulation. In MLDFM, we apply a triple continuum model using structured grid for forward simulation where large-scale fractures are represented with numerical embedded discrete fracture model (EDFM) and the secondary fractures are upscaled as third continuum. What makes the triple continuum model different from the previous work is that both the numerical EDFM and the third continuum are treated in a dynamic approach by considering the effect of flow direction on the complex local-scale flow response. For that purpose, we construct a finer unstructured discrete fracture matrix (DFM) grid which represents all fractures explicitly and is conformal to the boundary of coarse structured grid. During a simulation run, we apply a basis function to generate the local boundary conditions at fine scale using the global solution. Benefit from that, we can use a more accurate flow-based approach in the extended local upscaling to re-compute the transmissibility in triple continuum model. Moreover, we apply a local-global upscaling formalism to guarantee dynamically updated local boundary conditions for upscaling. Besides, we present several cases using synthetic and realistic fractured networks to demonstrate the performance of MLDFM. The results prove that the proposed MLDFM approach more accurately captures the flow in complex fractured systems than EDFM solutions by comparing against fine-scale DFM. At the same time, MLDFM is more computationally efficient in comparison with fine scale DFM. • An advanced hybrid model for multiphase flow fractured systems is presented. • The model compared with state-of-the-art Discrete Fracture Model (DFM) and Embedded DFM. • The model provides a better accuracy than EDFM with Equivalent Continuum Model used for complex multi-scale fractures. • A set of benchmarks includes complex synthetic and realistic fractured networks.

Modeling of the effects of weakness planes in rock masses on the stability of tunnels using an equivalent continuum and damage model

This paper proposes an equivalent continuum model to describe the mechanical behavior of transversely isotropic rocks. In this model, the transverse isotropy of deformation and strength was achieved based on the Mohr–Coulomb and maximum tensile-stress criteria, and the damage was captured by adopting a statistical damage evolution rule. The application of the model is verified through numerical simulation of conventional triaxial tests. The model is then used to reveal the non-uniform mechanical response of the surrounding rock and the secondary lining for a tunnel situated in a weak layered rock mass. The results show that: (1) The proposed model can capture the transverse isotropy in deformation and strength of rocks, and the proposed damage formulation can represent the deterioration and degree of failure of rocks; (2) The fracturing pattern, failure strength and stress–strain curves obtained from the proposed model agree well with test results for three typical rocks with different directional variations in strength; (3) The damage distribution based on the proposed model can identify the failure of layered rock mass; and (4) The damage zones of the surrounding rock and the loads on the secondary lining after tunnel excavation show distinctly asymmetric behavior, that is, the damaged zones are concentrated in the tunnel direction normal to the weak planes, and the positive bending moment and larger axial force are parallel and vertical to the weak planes, respectively.

Advances in statistical mechanics of rock masses and its engineering applications

To efficiently link the continuum mechanics for rocks with the structural statistics of rock masses, a theoretical and methodological system called the statistical mechanics of rock masses (SMRM) was developed in the past three decades. In SMRM, equivalent continuum models of stress–strain relationship, strength and failure probability for jointed rock masses were established, which were based on the geometric probability models characterising the rock mass structure. This follows the statistical physics, the continuum mechanics, the fracture mechanics and the weakest link hypothesis. A general constitutive model and complete stress–strain models under compressive and shear conditions were also developed as the derivatives of the SMRM theory. An SMRM calculation system was then developed to provide fast and precise solutions for parameter estimations of rock masses, such as full-direction rock quality designation (RQD), elastic modulus, Coulomb compressive strength, rock mass quality rating, and Poisson’s ratio and shear strength. The constitutive equations involved in SMRM were integrated into a FLAC3D based numerical module to apply for engineering rock masses. It is also capable of analysing the complete deformation of rock masses and active reinforcement of engineering rock masses. Examples of engineering applications of SMRM were presented, including a rock mass at QBT hydropower station in northwestern China, a dam slope of Zongo II hydropower station in D.R. Congo, an open-pit mine in Dexing, China, an underground powerhouse of Jinping I hydropower station in southwestern China, and a typical circular tunnel in Lanzhou-Chongqing railway, China. These applications verified the reliability of the SMRM and demonstrated its applicability to broad engineering issues associated with jointed rock masses.

Identification of non-local continua for lattice-like materials

The paper is focused on the dynamic homogenization of lattice-like materials with lumped mass at the nodes to obtain energetically consistent models providing accurate descriptions of the acoustic behavior of the discrete system. The equation of motion of the Lagrangian one-dimensional lattice is transformed according to a unitary approach aimed to identify equivalent non-local continuum models of integral-differential and gradient type, the latter obtained through standard or enhanced continualization. The bilateral Z-transform of the difference equation of motion of the lattice, mapped on the unit circle, is matched to the governing integral-differential equation of the equivalent continuum in the transformed Fourier space, which has the same frequency band structure as the Lagrangian one. Firstly, it is shown that the approximation of the kernels via Taylor polynomials leads to the differential field equations of higher order continua endowed with non-local constitutive terms. The field equations derived from such approach corresponds to the ones obtained through the so called standard continualization. However, the differential problem turns out to be ill-posed because the non-positive definiteness of the potential energy density of the higher order continuum. Energetically consistent equivalent continua have been identified through a proper mapping correlating the transformed macro-displacements in the Fourier space with a new auxiliary regularizing continuum macro-displacement field in the same space. Specifically, the mapping here introduced has zeros at the edge of the first Brillouin zone. The integral-differential governing equation and the corresponding differential one has been reformulated through an enhanced continualization, that is characterized by energetically consistent differential equations with inertial and constitutive non-localities. Here, the constitutive and inertial kernels of the integral-differential equation exhibit polar singularities at the edge of the first Brillouin zone. The proposed approach is generalized in a consistent way to two-dimensional lattices by using multidimensional Z- and Fourier transforms, a procedure that may be easily extended to three-dimensional lattices. Finally, two examples of lattice-like systems consisting of periodic pre-stressed cable-nets of point mass at the nodes are analyzed. The resulting gradient continuum models provide dispersion curves who are in excellent agreement with those of the Lagrangian systems.