Macroscopic Continuum Theory Research Articles

Analytical Model of Hydraulic Fracturing for Low Permeability Hot Dry Rock Reservoirs and DEM Simulation Base on Fluid-Solid Coupling

The formation of a rich underground-seam network is the key problem in the development of low-permeability hot dry rock (HDR) resources. Considering the lack of macroscopic continuum theory to study hydraulic fracturing having preset fracture-interface element, the particle-flow method of micro-mechanical discrete-element theory is introduced to simulate the mechanical behavior of hydraulic fracturing for HDR low permeability reservoirs. The reservoir is simulated as a round particle; the fracturing fluid movement is described by the seepage field equation, and rock movement is described by the displacement field equation. Finally, the particle-flow numerical model of hydraulic fracturing for HDR low permeability reservoirs is established under the condition of fluid-solid coupling: the model contains two parts (rock and fracture). Based on the parallel-bond model, a definition of micro-fractures of hydraulic fracturing is given. The relation between the fracturing effect and influence parameters is discussed. The results show that the fracture-initiation pressure is proportional to the magnitude of minimum horizontal stress, particle normal-contact stiffness, and particle normal- and tangential-connection strengths; the pressure is also independent of maximal horizontal stress and tangential contact stiffness. At the same time, the formation temperature of dry hot rock will reduce the strength of the rock, so particle-flow numerical models of hydraulic fracturing in different temperatures are discussed. Results show that fracture length and width show a trend of increase before decrease with the increase of injection pressure, an inverse relationship with minimums horizontal principal stress, and a positive relationship with HDR reservoir permeability.

Electric-Field-Induced Instabilities in Nematic Liquid Crystals

Systems involving nematic liquid crystals subjected to magnetic fields or electric fields are modeled using the Oseen-Frank macroscopic continuum theory, and general criteria are developed to assess the local stability of equilibrium solutions. The criteria take into account the inhomogeneity of the electric field and the mutual influence of the electric field and the liquid-crystal director field on each other. The criteria show that formulas for the instability thresholds of electric-field Freedericksz transitions cannot in all cases be obtained from those for the analogous magnetic-field transitions by simply replacing the magnetic parameters by the corresponding electric parameters, contrary to claims in standard references. This finding is consistent with [Arakelyan, Karayan, Chilingaryan, Sov. Phys. Dokl., 29 (1984) 202-204]. A simple analytical test is provided to determine when an electric-field-induced instability can differ qualitatively from the analogous magnetic field-induced instability. For the systems we study, it is found that taking into account the full coupling between the electric field and the director field can either elevate or leave unchanged an instability threshold (never lower it), compared to the threshold provided by the magnetic-field analogy. The physical mechanism that underlies the effect of elevating an instability threshold is the added free energy associated with a first-order change in the ground-state electric field caused by a perturbation of the ground-state director field. Examples are given that involve classical Freedericksz transitions and also periodic instabilities. The inclusion of flexoelectric terms in the theory is studied, and it is found that these terms are not capable of altering the instability thresholds of any of the classical Freedericksz transitions, consistent with known results for splay transitions.

Association Rules between the Microstructure and Physical Mechanical Properties of Rock-mass under Coupled Effect of Freeze-thaw Cycles and Large Temperature Difference

The mechanical properties of fractured rock mass are largely dependent on the fracture structure under the coupling of freeze-thaw cycles and large temperature difference. Based on the traditional macroscopic continuum theory, the thermal and mechanical model and the corresponding theories ignore the material internal structure characteristics, which add difficulty in describing the mesoscopic thermal and mechanical behavior of the fractured rock mass among different phases. In order to uncover the inherent relationship and laws among the internal crack development, structural change and the physical and mechanical properties of rock under strong cold and frost weathering in cold area, typical granite and sandstone in cold region were analyzed in laboratory tests. The SEM scanning technology was introduced to record the microstructural change of rock samples subject to freeze-thaw cycles and large temperature difference. Association rules between the microstructure and the physical mechanical properties of rock mass were analyzed. The results indicated that, with the increase of the cyclic number, the macroscopic physical and mechanical indexes and the microscopic fracture index of granite and sandstone continuously and gradually deteriorate. The width of original micro crack continues to expand and extend and new local micro cracks are generated and continue to expand. The fracture area and width of the rock increase and the strength of the rock is continuously damaged. In particular, the strength and elastic modulus of granite decrease by 20.2% and 33.36%, respectively; the strength and elastic modulus of sandstone decrease by 33.4% and 36.43%, respectively.

Progressive damage of laminated cylindrical/conical panels under meridional compression

Abstract The objective of this paper is to investigate the progressive failure behaviour of laminated cylindrical/conical panels under meridional compression considering geometric nonlinearity and evolving material damage. The evolving microscopic damage such as fiber breakage, matrix cracking, fiber matrix debonding etc. is modeled through a generalized macroscopic continuum theory within the framework of irreversible thermodynamics. The analysis is carried out using field consistent finite element approach based on first-order shear deformation theory. The nonlinear governing equations are solved using the Newton–Raphson iterative technique coupled with the adaptive displacement control method to trace the equilibrium path. The damage evolution equations are solved at every Gauss point using Newton–Raphson iterative technique within each iteration of a loading/displacement increment. To accurately model the transverse shear strain energy, shear correction factors are calculated using layers' properties and lamination scheme. The detailed study is carried out to highlight the influences of evolving damage, span-to-thickness ratio, lamination scheme, radius-to-span ratio, boundary conditions and semi-cone angle on the postbuckling response and failure load of laminated panels.

Melt flow effect on interface stability during directional solidification

In the framework of the phenomenological macroscopic continuum theory using the approximation of a flat frontier layer the stability of solid-liquid interface at the directional solidification under melt motion along the interface is studied. The stability conditions are reduced to determination of eigenvalues of boundary value problem for infinitesimal disturbances of stationary process. In case of stagnant melt it is shown that in the plane “wave number-pulling rate” there are two areas of instability for low and large pulling rates divided by the area of steady-steady growth. Neutral stability curve calculated for rather large pulling rates for succinonitrile-acetone (SCN-Ac) system is close to the relevant values received by Mullins and Sekerka, while the absolute values of critical growth rates are of the same order of magnitude as the experimental ones. Melt flow along the interface leads to emergence of the third area of instability which is characterized by small values of wave numbers. When increasing the melt flow rate the area of instability extends towards great values of wave numbers.

Calculation of solvation free energies with DCOSMO-RS.

The concept of dielectric continuum models has turned out to be very fruitful for the qualitative description of solvation effects in quantum chemical calculations, although from a theoretical perspective its basis is questionable, at least if applied to polar solvents, because the electrostatic nearest neighbor interactions in polar solvents are much too strong to be described by macroscopic dielectric continuum theory. On the basis of this insight, the Conductorlike Screening Model for Realistic Solvation (COSMO-RS) had been developed, which gives a thermodynamically consistent, quantitative description of solvation effects in polar and nonpolar solvents, even in mixtures and at variable temperature, starting from quantum chemical calculations of solute and solvent molecules embedded in a virtual conductor (COSMO). Though COSMO-RS usually only requires quantum chemical calculations in the conductor and thus does not allow for studying of the concrete solvent influence on the solute electron density, the direct COSMO-RS (DCOSMO-RS) has been introduced, which uses the σ-potential, i.e., a solvent specific response function provided by COSMO-RS, as a replacement of the conductor or dielectric response employed in continuum solvation models. In this article we describe the current status of DCOSMO-RS and demonstrate the performance of the DCOSMO-RS approach for the prediction of free energies of solvation.

Postbuckling response of composite laminated plates with evolving damage

In this paper, postbuckling behavior and progressive failure of laminated plates considering geometric nonlinearity and evolving material damage under uniaxial, biaxial compressive and in-plane shear loadings are studied. The damage model is based on a generalized macroscopic continuum theory within the framework of irreversible thermodynamics and enables to predict the progressive damage and failure load. Damage variables are introduced for the phenomenological treatment of degradation of the stiffness properties of laminated plates due to microscopic defects. The analysis is carried out using field-consistent finite element approach based on first-order shear deformation theory. The nonlinear governing equations are solved using the Newton–Raphson iterative technique coupled with the adaptive displacement control method to trace the pre- and postbuckling equilibrium path. The damage evolution equations are solved at every Gauss point using Newton–Raphson iterative technique within each loading/displacement increment iteration. The detailed parametric study is carried out to investigate the influences of evolving damage, span-to-thickness ratio, lamination scheme, boundary conditions and aspect ratio on the postbuckling response and failure load of laminated plates under in-plane loading. The ratio of failure load to buckling load increases with increase in thickness ratio due to the dominating restoring action of in-plane stretching forces in postbuckling region of thin plates whereas the thick plates depict material failure prior to buckling point. Thin plates with SSFF boundary conditions with evolving damage depict softening postbuckling response whereas the plates with other boundary conditions considered depict hardening response.

Nonlinear static analysis of composite laminated plates with evolving damage

Considering geometric nonlinearity and damage evolution, the static response characteristics of laminated composite plates subjected to uniformly distributed loading are investigated using finite element approach based on the first-order shear deformation theory. The damage evolution is modeled employing generalized macroscopic continuum theory within the framework of irreversible thermodynamics. The governing nonlinear equations are solved using Newton–Raphson iterative technique. The resulting finite element-based continuum damage model enables to predict the progressive damage and failure load. A detailed parametric study is carried out to investigate the influences of damage evolution, boundary conditions, span-to-thickness ratio, and lamination scheme on the static response of laminated plates undergoing moderately large deformation. It is revealed that the in-plane stretching forces owing to geometric nonlinearity significantly influence the failure load, damage, and stress distribution for immovable thin laminates.

Continuum damage modeling of composite laminated plates using higher order theory

The aim of the paper is to check the accuracy of global plate models in predicting progressive damage and failure load of cross-ply laminated composite plates. The damage evolution model is based on a generalized macroscopic continuum theory within the framework of irreversible thermodynamics. The progressive damage analysis is carried out using global higher order shear deformation theory with/without thickness stretch and zig–zag terms, and first order shear deformation theory with shear correction factor of 5/6 and the one calculated accounting for layers’ properties and lamination scheme. For comparison, progressive damage modeling employing 3-D finite element is also done. The parametric study is carried out to assess the accuracy of higher-order model based on continuum damage mechanics for different span-to-thickness ratio, lamination scheme and boundary conditions. It is concluded that higher-order model with zig–zag terms in in-plane displacement and thickness stretch terms in transverse displacement predicts failure load and stress distributions quite accurately. The higher order model is computationally efficient since the number of unknowns is independent of number of layers. For two-layered thick laminates, first order model with shear correction factor calculated based on lamination scheme predicts significantly better results compared to that with shear correction factor of 5/6.

Continuum damage mechanics approach to composite laminated shallow cylindrical/conical panels under static loading

Static response characteristics and failure load of laminated composite shallow cylindrical and conical panels subjected to internal/external lateral pressure are investigated using continuum damage mechanics approach considering geometric nonlinearity and damage evolution. The damage model is based on a generalized macroscopic continuum theory within the framework of irreversible thermodynamics and enables to predict the progressive damage and failure load. Damage variables are introduced for the phenomenological treatment of the state of defects and its implications on the degradation of the stiffness properties. The analysis is carried out using finite element method based on the first order shear deformation theory. The nonlinear governing equations are solved using Newton–Raphson iterative technique coupled with the adaptive displacement control method to efficiently trace the equilibrium path. The detailed parametric study is carried out to investigate the influences of geometric nonlinearity, evolving damage, span-to-thickness ratio, lamination scheme and semi-cone angle on the static response and failure load of laminated cylindrical/conical panels. It is revealed that the membrane forces due to geometric nonlinearity significantly influence the damage distribution and failure load.

Theory of dipole-exchange spin waves in metallic ferromagnetic nanotubes of large aspect ratio

A macroscopic continuum theory is presented for the dipole-exchange spin waves in nanometer-sized cylindrical tubes with a large length-to-diameter aspect ratio. The magnetization and applied magnetic field are taken parallel to the cylinder axis, and the properties of the hybridized surface and bulk magnetic excitations are studied in relation to the inner and outer interfaces of the nanotubes. The calculations describe the radial and angular quantization of the different modes for both unpinned and effective pinned cases. The results for tube geometries are found to be in contrast with the limiting (single-interface) special cases of wires and antiwires. Numerical examples are presented mainly for Ni materials.

Multiscale Modeling of Polymer Gels—Chemo-Electric Model versus Discrete Element Model

Polyelectrolyte gels are a very attractive class of actuation materials with remarkable electronic and mechanical properties with a great similarity to biological contractile tissues. They consist of a polymer network with ionizable groups and a liquid phase with mobile ions. Absorption and delivery of solvent lead to a large change of volume. This mechanism can be triggered by chemical (change of salt concentration or pH of solution surrounding the gel), electrical, thermal or optical stimuli. Due to this capability, these gels can be used as actuators for technical applications, where large swelling and shrinkage is desired. In the present work chemically stimulated polymer gels in a solution bath are investigated. To adequately describe the different complicated phenomena occurring in these gels, they can be modeled on different scales. Therefore, models based on the statistical theory and porous media theory, as well as a coupled multi-field model and a discrete element formulation are derived and employed. A refinement of the different theories from global macroscopic to microscopic are presented in this paper: The statistical theory is a macroscopic theory capable of describing the global swelling or bending, e.g., of a gel film, while the general theory of porous media (TPM) is a macroscopic continuum theory which is based on the theory of mixtures extended by the concept of volume fractions. The TPM is a homogenized model, i.e., all geometrical and physical quantities can be seen as statistical averages of the real quantities. The presented chemo-electro-mechanical multi-field formulation is a mesoscopic theory. It is capable of giving the concentrations and the electric potential in the whole domain. Finally the (micromechanical) discrete element (DE) theory is employed. In this case, the continuum is represented by distributed particles with local interaction relations combined with balance equations for the chemical field. This method is predestined for problems involving large displacements and strains as well as discontinuities. In this paper, the coupled multi-field model and the discrete element model for chemical stimulation of a polymer gel film with and without domain deformation are employed. Based on these results, the presented formulations are compared and conclusions on their applicability in engineering practice are finally drawn.

MAGNETOSTATIC MODES IN NANOMETER-SIZED FERROMAGNETIC AND ANTIFERROMAGNETIC TUBES

A theory is presented for the magnetostatic modes in ferromagnetic and antiferromagnetic nanotubes, which have a large length-to-radius aspect ratio and an external magnetic field parallel to the cylindrical axis. The surface and bulk magnetic excitations (or magnetostatic spin waves) are studied for cases where the dipole–dipole interactions are dominant in the spin dynamics. This situation can be realized at sufficiently small wavevectors by inelastic light scattering or magnetic resonance techniques. A macroscopic continuum theory is developed, using the magnetostatic form of Maxwell's equations and the electromagnetic boundary conditions, and the characteristic equations (or dispersion relations) are deduced for the magnetostatic modes. Numerical calculations are presented for ferromagnetic and antiferromagnetic nanostructures, taking Ni 80 Fe 20 and MnF 2, respectively. The spatial variations of the mode amplitudes are also investigated.

Optical piezoelectric transducer for nano-ultrasonics

Piezoelectric semiconductor strained layers can be treated as piezoelectric transducers to generate nanometer-wavelength and THz-frequency acoustic waves. The mechanism of nano-acoustic wave (NAW) generation in strained piezoelectric layers, induced by femtosecond optical pulses, can be modeled by a macroscopic elastic continuum theory. The optical absorption change of the strained layers modulated by NAW through quantum-confined Franz-Keldysh (QCFK) effects allows optical detection of the propagating NAW. Based on these piezoelectric-based optical principles, we have designed an optical piezoelectric transducer (OPT) to generate NAW. The optically generated NAW is then applied to one-dimensional (1-D) ultrasonic scan for thickness measurement, which is the first step toward multidimensional nano-ultrasonic imaging. By launching a NAW pulse and resolving the returned acoustic echo signal with femtosecond optical pulses, the thickness of the studied layer can be measured with <1 nm resolution. This nano-structured OPT technique will provide the key toward the realization of nano-ultrasonics, which is analogous to the typical ultrasonic techniques but in a nanometer scale.

Interaction of Electrons with Polar Optical Phonons in Semiconductor Superlattices

On the basis of the microscopic theory of lattice dynamics, simulation of the electric potentials created by optical phonons in semiconductor superlattices is performed. It is shown that the spatial distribution of the amplitudes of electric potentials differs from that in a dielectric continuum predicted by the conventional macroscopic model without dispersion. A modified macroscopic continuum theory is proposed that takes into account the dispersion of short-range interatomic forces and allows one to obtain analytical expressions for the potentials of electron-phonon interaction.

Solvation Free Energies Estimated from Macroscopic Continuum Theory: An Accuracy Assessment

The solvation thermodynamics of a set of small organic molecules is studied using a simple continuum model. Transfer from vapor or cyclohexane to water is decomposed into three steps: first the atomic partial charges of the solute are removed, then it is transferred into aqueous solution, and then its partial charges are restored. The electrostatic steps are treated using continuum electrostatics; the hydrophobic transfer step is treated using atomic «solvation parameters» or «surface tensions». Vapor-to-water transfer free energies of 35 neutral compounds were estimated, including analogues of 17 amino acid side chains. Variants of the model were tested, using from one to three atom types, each with its own surface tension

Dynamic susceptibilities for magnetic layered structures.

Previous theories of dynamic response for many magnetic layered structures appear to be in conflict with macroscopic electromagnetic theory. We present a method for constructing dynamic magnetic susceptibilities that resolves these difficulties. Our results are valid for a large class of easy-plane magnetic systems and are particularly interesting for systems with antiferromagnetic coupling. The macroscopic approach is justified by microscopic calculations which use exact dipolar and exchange fields and thus avoid the assumptions of a macroscopic continuum theory.

Effective lagrangian for the two-dimensional [formula omitted]P 2 model at low temperatures

Recent Monte Carlo simulations for the two-dimensional R P N−1 models have shown that the matrix non-linear sigma models (NL σM) have a long-distance behavior which is different from that for the corresponding vector models (S N−1 ). This discrepancy cannot be explained by the renormalization group (RG) applied to the naive continuum lagrangians (O( N)-(vector NL σ) model). Motivated by this fact, we study the low-temperature long-distance effective lagrangian for the two-dimensional R P 2 model by means of the saddle-point (SP) plus renormalization methods. Introducing a matrix order parameter Q via the Hubbard-Stratonovich transformation we analyze the system of a macroscopic effective continuum theory for the Q-field. After taking the O(1)×O(2)-invariant SP for the potential we renormalize the massive fluctuations suppressed by the cut-off. This procedure allows us to find, in addition to the O(3)-model, a new term for the massless modes. Thanks to the new term, it is expected that the infrared RG behavior for the R P 2 model could be distinguished from the S 2 model. The effective lagrangian itself turns out to be perturbatively non-renormalizable. The RG analysis is formulated in the space of general O(2)-invariant non-linear lagrangians with an infinite number of coupling constants. The first-order RG analysis using the same field-renormalization constant as of the O(3)-model suggests that at long distances the effective lagrangian in the vicinity of zero temperature deviates from the O(3)-model.

Optical phonons and electron-phonon interaction in quantum wires.

A unified macroscopic continuum theory for the treatment of optical-phonon modes in quantum-wire structures is established. The theory is based on a Lagrangian formalism from which the equations of motion are rigorously derived. They consist of four coupled second-order differential equations for the vibrational amplitude and electrostatic potential. The matching boundary conditions are obtained from the fundamental equations. It is shown that no incompatibility exists between mechanical and electrostatic matching boundary conditions when a correct mathematical treatment of the problem is given. The particular case of a GaAs quantum wire buried in AlAs, where the phonons can be considered completely confined, is analyzed and the vector displacement and electron-phonon interaction potential are illustrated for several modes.

The electron-optical phonon interaction in semiconductor microstructures

The description of the interaction of electrons and optical phonons in semiconductor superlattices and quantum well structures requires analytical expressions for atomic displacements and macroscopic electrostatic potentials accompanying the excitation of polar phonons. Recent discussion of this problem has revealed that the conventional macroscopic dielectric continuum theory contains uncertainties giving rise to controversial results even in the long-wavelength limit. The authors demonstrate that the uncertainties can be lifted if the complete branch dispersion of the underlying bulk phonons and boundary conditions derived from microscopic equations of motion are taken into the continuum theory. In the limit of vanishing dispersion the resulting envelopes confirm the predictions of Huang and Zhu as well as of Bechstedt and Gerecke within the unified model.