Continuum Phase Research Articles

Finite size scaling in 2d causal set quantum gravity

We study the dependence on size, N, of 2d causal set quantum gravity. This theory is known to exhibit a phase transition as the analytic continuation parameter β, akin to an inverse temperature, is varied. Using a scaling analysis we find that the asymptotic regime is reached at relatively small values of N. Focussing on the causal set action S, we find that scales like where the scaling exponent ν takes different values on either side of the phase transition. For we find that which is consistent with our analytic predictions for a non-continuum phase in the large β regime. For we find that , consistent with a continuum phase of constant negative curvature thus suggesting a dynamically generated cosmological constant. Moreover, we find strong evidence that the phase transition is first order. Our results strongly suggest that the asymptotic regime is reached in causal set quantum gravity for .

A modified Gurson-type plasticity model at finite strains: formulation, numerical analysis and phase-field coupling

The modeling of failure in ductile materials must account for complex phenomena at the micro-scale, such as nucleation, growth and coalescence of micro-voids, as well as the final rupture at the macro-scale, as rooted in the work of Gurson (J Eng Mater Technol 99:2–15, 1977). Within a top–down viewpoint, this can be achieved by the combination of a micro-structure-informed elastic–plastic model for a porous medium with a concept for the modeling of macroscopic crack discontinuities. The modeling of macroscopic cracks can be achieved in a convenient way by recently developed continuum phase field approaches to fracture, which are based on the regularization of sharp crack discontinuities, see Miehe et al. (Comput Methods Appl Mech Eng 294:486–522, 2015). This avoids the use of complex discretization methods for crack discontinuities, and can account for complex crack patterns. In this work, we develop a new theoretical and computational framework for the phase field modeling of ductile fracture in conventional elastic–plastic solids under finite strain deformation. It combines modified structures of Gurson–Tvergaard–Needelman GTN-type plasticity model outlined in Tvergaard and Needleman (Acta Metall 32:157–169, 1984) and Nahshon and Hutchinson (Eur J Mech A Solids 27:1–17, 2008) with a new evolution equation for the crack phase field. An important aspect of this work is the development of a robust Explicit–Implicit numerical integration scheme for the highly nonlinear rate equations of the enhanced GTN model, resulting with a low computational cost strategy. The performance of the formulation is underlined by means of some representative examples, including the development of the experimentally observed cup–cone failure mechanism.

Classification of Morphogenetic Types of Mossy Litter in Paludine Spruce Forests Based on Humus Content

We have classified morphogenetic types of mossy litter by a multivariate statistical analysis of a fractional group of organic-matter composition. Three clusters of mossy litters—peats, peaty, and high-ash peaty— are recognized. This results in 94% prediction. Indicators contribute to discriminants in the following descending order: C: N > cellulose > HA-2 > HA-3 > FA-3 > hemicellulose = FA-1 = HA-1. According to canonical analysis, there were two significant roots in cluster determination. The first segregates mainly the peat cluster from two others. The share of canonical function of the root one is 58% of possible discrimination, mainly due to the weight of cellulose and C: N. Canonical function 2, describing 42% of the explained dispersion, discriminates the peaty cluster from the others due to the dominant contribution of FA-1 and FA-3. The classification function for the recognition of new objects was calculated and evaluated. The humus content of various types and clusters of mossy litters was examined. Morphogenetic classification follows the transformation of forest litter in the course of litter formation (continuum phase), quantitative biochemical discrimination is a discrete phase of this process.

Phase Field Modeling of Ductile Fracture in Soil Mechanics

AbstractThis work outlines a rigorous framework for the ductile failure of frictional materials in elastic‐plastic soil mechanics undergoing large strains. Describing soil crack formation can be achieved in a convenient way by recently developed continuum phase field approaches to fracture, which are based on the regularization of sharp crack discontinuities [1]. This avoids the use of complex discretization methods for crack discontinuities, and can account for complex crack patterns. For frictional materials, a non–associative Drucker–Prager‐type elastic‐plastic constitutive model suitable for a wide range of applications in soil mechanics is developed. It is linked to a failure criterion in terms of the elastic‐plastic work density that drives the fracture phase field. We demonstrate the modeling capabilities and algorithmic performance of the proposed formulation by a representative numerical example that describes soil crack formation using elastic‐plastic fracture mechanics. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Simulation of stiffness of randomly-distributed-graphene/epoxy nanocomposites using a combined finite element-micromechanics method

In this study, a new approach was developed for prediction of the stiffness of polymer nanocomposites with randomly distributed graphene sheets. This approach is a combination of the finite element and micromechanics methods. First, the stiffness of the nanocomposites containing one layer of an aligned nano graphene embedded in a polymer was modeled by using a finite element method. The matrix was considered as a continuum phase and each covalent bond of graphene sheet was simulated by an equivalent structural beam. Nonlinear springs were used as Van der Waals bonds in the interphase region of graphene and polymer. Considering the real size of graphene nano platelets, the numerical simulation of the representative volume element of nanocomposites with a real size aligned graphene sheets embedded in the matrix is not a feasible task. Therefore, in the present research, a new approach was developed to overcome this problem. In this new approach, by using the moduli of different graphene sheets with different sizes embedded in a representative volume element, the moduli of a real size graphene embedded in the matrix were predicted. The results obtained by the finite element method were used by the micromechanics approach in order to consider the effect of the random distribution of graphene sheets in a polymer. By combining these two methods, the stiffness of nanocomposites with randomly distributed graphene sheet was predicted. The result of the model is in an acceptable agreement with the result of conducted experimental program.

Virial stress–based model to simulate the silica glass densification with the discrete element method

SummaryThe discrete element method (DEM) presents an alternative way to model complex mechanical problems of silica glass, such as brittle fracture. Since discontinuities are naturally considered by DEM, no complex transition procedure from continuum phase to discontinuum one is required. However, to ensure that DEM can properly reproduce the silica glass cracking mechanisms, it is necessary to correctly model the different features characterizing its mechanical behavior before fracture. Particularly, it is necessary to correctly model the densification process of this material, which is known to strongly influence the fracture mechanisms. The present paper proposes a new and very promising way to model such process, which is assumed to occur only under hydrostatic pressure. An accurate predictive‐corrective densification model is developed. This model shows a great flexibility to reproduce extremely complex densification features. Furthermore, it involves only one calibration parameter, which makes it very easy to apply. This new model represents a major step towards accurate modeling of materials permanent deformation with the discrete element method, which has long been a huge challenge in applying this method for continuum problems.

Compression and recovery of carbon nanotube forests described as a phase transition

In this paper we describe experiments and a continuum phase transition model for the compression of carbon nanotube (CNT) forests. Our model is inspired by the observation of one or more moving interfaces across which densified and rarefied phases of the CNT forests co-exist. We use a quasi-static version of the Abeyaratne-Knowles theory of phase transitions for continua with a stick-slip type kinetic law and a nucleation criterion based on the critical stress for buckling of CNT forests to describe the formation and motion of these interfaces in uniaxial compression experiments. We investigate micropillars made from bare CNTs, as well as those coated with different thicknesses of alumina using atomic layer deposition (ALD). The coating thickness affects the moduli of individual CNTs as well as the adhesion energy per contact between CNTs. In order to test the applicability of our model to more complex stress states, we carry out nanoindentation experiments on the CNT pillars and interpret the load-indentation data by incorporating a constitutive law allowing for phase transitions into solutions for the indentation of a linearly elastic half-space. Even though the state of stress in a nanoindentation experiment is more complex than that in a uniaxial compression test, we find that the parameters extracted from the nanoindentation experiments are close to those from uniaxial compression. Our models could therefore aid the design of CNT forests to have engineered mechanical properties, and guide further understanding of their behavior under large deformations.

Electrorheological behavior of suspensions of camphorsulfonic acid (CSA) doped polyaniline nanofibers in silicone oil

The electrorheological (ER) effect is known as the enhancement of the apparent viscosity upon application of an external electric field applied perpendicular to the flow direction. Suspensions of polarizable particles in non-conducting solvents are the most studied ER fluids. The increase in viscosity observed in the suspensions is due to the formation of columns that align with the electric field. This work presents the ER behavior of suspensions, in silicone oil, of camphorsulfonic acid (CSA) doped polyaniline (PANI) nanofibers. The ER properties of the suspensions were investigated with a rotational rheometer, to which an ER cell was coupled, in steady shear, and electrical field strengths up to 2 kV mm−1. The effects of the electric field strength, content of nanostructures and viscosity of the continuum phase, in the shear viscosity and yield stress, were investigated at room temperature. As expected, the ER effect increases with the increase of the electric field as well as with the increase of content of nanofibers and it decreases with the increase of the oil viscosity. The suspensions present giant ER effects (higher than 2 orders of magnitude increase in viscosity for low shear rates and high electric fields), showing their potential application as ER smart materials.

Numerical investigation of high pressure condensing flows in supersonic nozzles

High-pressure non-equilibrium condensing flows are investigated in this paper through a quasi-1D Euler model coupled to the method of moments for the physical characterization of the dispersed phase. Two different numerical approaches, namely the so-called (a) the mixture and (b) continuum phase model, are compared in terms of computational efficiency and accuracy. The results are verified against experimental data of high-speed condensing steam measured at high pressure (100.7 bar).The analysis demonstrates that Model (b) markedly outperforms the mixture model in terms of computational cost, while retaining comparable accuracy. However, both models, in their original formulation, lead to considerable deviations in the nucleation onset prediction as well as an overestimation of the average droplet radius.A further investigation is then conducted to figure out the main physical parameters affecting the condensation process, i.e. the surface tension, the growth rate and the nucleation rate. It is eventually inferred that applying proper correction to these three quantities allows to obtain best fit with the experimental data. A final calculation is carried out to show the dependence of these three correcting factors from the thermodynamic conditions of the mixture.

Hydraulic fracture in poro-hydro-elastic media

The prediction of fluid-driven crack propagation in deforming porous media has achieved increasing interest in recent years, in particular with regard to the modeling of hydraulic fracturing, the so-called “fracking”. Here, the challenge is to link at least three modeling ingredients for (i) the behavior of the solid skeleton and fluid bulk phases and their interaction, (ii) the crack propagation on not a priori known paths and (iii) the extra fluid flow within developing cracks. To this end, a macroscopic framework is proposed for a continuum phase field modeling of fracture in porous media that provides a rigorous approach to a diffusive crack modeling based on the introduction of a regularized crack surface. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies including branching. It shows that the quasi-static problem of elastically deforming, fluid-saturated porous media at fracture is related to a minimization principle for the evolution problem. The existence of this minimization principle for the coupled problem is advantageous with regard to a new unconstrained stable finite element design, while previous space discretizations of the saddle point principles are constrained by the LBB condition. This proposed formulation includes a generalization of crack driving forces from energetic definitions towards threshold-based criteria in terms of the effective stress related to the solid skeleton of a fluid-saturated porous medium. Furthermore, a Poiseuille-type constitutive continuum modeling of the extra fluid flow in developed cracks is suggested based on a deformation-dependent permeability, that is scaled by a characteristic length.

Analytical solutions for laminar flow in membrane channels with cylindrical symmetry: Single- and dual- membrane systems

Advances in the design of membranes and membrane modules have led to a variety of new configurations with enhanced separation performance. In order to study the performance of membrane systems with novel geometries, the flow field of the continuum phase must be known. In this study an approximate approach is used to obtain the solution for steady state flow fields in several crossflow membrane filtration systems with cylindrical symmetry. The analytical solutions are presented for a single channel tubular membrane, a tubular membrane with a co-axial solid cylinder at its center, a tubular membrane located at the center of a channel with impervious walls, and a configuration with two tubular co-axial membranes. The results are compared to numerical solutions. The analytical solutions for both velocity and pressure distributions are in agreement with the results of numerical modeling.

A coupled SDPH–FVM method for gas‐particle multiphase flow: Methodology

SummaryThis paper describes a novel methodology that combines smoothed discrete particle hydrodynamics (SDPH) and finite volume method (FVM) to enhance the effective performance in solving the problems of gas‐particle multiphase flow. To describe the collision and fluctuation of particles, this method also increases a new parameter, namely, granular temperature, according to the kinetic theory of granular flow. The coupled framework of SDPH–FVM has been established, in which the drag force and pressure gradient act on the SDPH particles and the momentum sources of drag force are added back onto the FVM mesh. The proposed technique is a coupled discrete‐continuum method based on the two‐fluid model. To compute for the discrete phase, its SDPH is developed from smoothed particle hydrodynamics (SPH), in which the properties of SPH are redefined with some new physical quantities added into the traditional SPH parameters, so that it is more beneficial for SDPH in representing the particle characteristics. For the continuum phase, FVM is employed to discretize the continuum flow field on a stationary grid by capturing fluid characteristics. The coupled method exhibits strong efficiency and accuracy in several two‐dimensional numerical simulations. Copyright © 2016 John Wiley & Sons, Ltd.

Twisted geometries, twistors, and conformal transformations

The twisted geometries of spin network states are described by simple twistors, isomorphic to null twistors with a time-like direction singled out. The isomorphism depends on the Immirzi parameter, and reduces to the identity when the parameter goes to infinity. Using this twistorial representation we study the action of the conformal group SU(2,2) on the classical phase space of loop quantum gravity, described by twisted geometry. The generators of translations and conformal boosts do not preserve the geometric structure, whereas the dilatation generator does. It corresponds to a 1-parameter family of embeddings of T*SL(2,C) in twistor space, and its action preserves the intrinsic geometry while changing the extrinsic one - that is the boosts among polyhedra. We discuss the implication of this action from a dynamical point of view, and compare it with a discretisation of the dilatation generator of the continuum phase space, given by the Lie derivative of the group character. At leading order in the continuum limit, the latter reproduces the same transformation of the extrinsic geometry, while also rescaling the areas and volumes and preserving the angles associated with the intrinsic geometry. Away from the continuum limit its action has an interesting non-linear structure, but is in general incompatible with the closure constraint needed for the geometric interpretation. As a side result, we compute the precise relation between the extrinsic geometry used in twisted geometries and the one defined in the gauge-invariant parametrization by Dittrich and Ryan, and show that the secondary simplicity constraints they posited coincide with those dynamically derived in the toy model of [1409.0836].

Behaviour of rippled shocks from ablatively-driven Richtmyer-Meshkov in metals accounting for strength

While numerous continuum material strength and phase transformation models have been proposed to capture their complex dependences on intensive properties and deformation history, few experimental methods are available to validate these models particularly in the large pressure and strain rate regime typical of strong shock and ramp dynamic loading. In the experiments and simulations we present, a rippled shock is created by laser-ablation of a periodic surface perturbation on a metal target. The strength of the shock can be tuned to access phase transitions in metals such as iron or simply to study high-pressure strength in isomorphic materials such as copper. Simulations, with models calibrated and validated to the experiments, show that the evolution of the amplitude of imprinted perturbations on the back surface by the rippled shock is strongly affected by strength and phase transformation kinetics. Increased strength has a smoothing effect on the perturbed shock front profile resulting in smaller perturbations on the free surface. In iron, faster phase transformations kinetics had a similar effect as increased strength, leading to smoother pressure contours inside the samples and smaller amplitudes of free surface perturbations in our simulations.

Hypervelocity launching of flyers at the SG-III prototype laser facility

Experiments of laser-driven hypervelocity flyers have been conducted at the SG-III prototype laser facility. Using the continuum phase plate technique, four laser beams each with a 3-ns quadratic profile are configured to produce relatively uniform irradiated spots of diameter size either 500 μm or 2000 μm. With the former, specifically designed multi-layered flyers (polyimide/copper) were accelerated by shock impedance and reverberation techniques via direct laser ablation to a super-high averaged velocity of 55 km/s, much faster than recently reported results. Light-emission signals of shock breakout and flyer impact on flat or stepped windows were obtained that indicated good planarity and integrity for the flyer. In the latter, single-layered aluminum flyers were gradually accelerated to a terminal velocity of 11 km/s, as measured by optical velocimetry, without melting and vaporization. The results suggest that the SG-III prototype laser facility has the capability to launch high-speed flyers to create extreme conditions for investigating the science of shock compression and its equation of state.

Particle-laden flows forced by the disperse phase: Comparison between Lagrangian and Eulerian simulations

The goal of the present work is to assess the ability of Eulerian moment methods to reproduce the physics of two-way coupled particle-laden turbulent flow systems. Previous investigations have been focused on effects such as preferential concentration, and turbulence modulation, but in regimes in which turbulence is sustained by an imposed external forcing. We show that in such regimes, Eulerian methods need resolutions finer than nominal Kolmogorov scale in order to capture statistics of particle segregation, but gas and disperse phase velocity variances can be captured with resolutions comparable to the Kolmogorov length. The work is then extended to address the question whether Eulerian methods are suitable in scenarios in which the continuum field of interest (temperature or momentum) is itself primarily driven by particles. To this end we have extended our analysis to the problem of turbulence driven by heated particles (Zamansky et al. PoF 2014) and have assessed capabilities of Eulerian methods in capturing particle segregation, as well as statistics of the temperature and velocity fields. Separate investigations are developed for cases with and without buoyancy driven turbulence. For each case corresponding Lagrangian calculations are developed and convergence of statistics with respect to the number of particles is established. Then the statistically-converged Lagrangian and Eulerian results are compared. Results show that accurate capture of segregation by the Eulerian methods always requires resolutions much higher than the nominal Kolmogorov scale. In scenarios for which a continuum phase is forced by particles, results from Eulerian methods show some sensitivity of predicted continuum statistics to the mesh resolution. This sensitivity was found to be largest for the case of a temperature field forced by hot particles, but without presence of buoyancy. In this case a Eulerian method with nominal Kolmogorov resolution was found to be insufficient for capture of temperature statistics. When additional coupling between particles and continuum phase is introduced by including the buoyancy effects, this sensitivity is suppressed in the temperature field, but some sensitivity to the Eulerian mesh resolution were detected in the momentum fields.

CFD investigation on the flow and combustion in a 300 MWe tangentially fired pulverized-coal furnace

The characteristics of the flow, combustion and temperature in a 300 MWe tangentially fired pulverized-coal furnace are numerically studied using computational fluid dynamics. The mathematical model is based on a Eulerian description for the continuum phase and a Lagrangian description for coal particles. The combustion reaction scheme was modeled using eddy dissipation concept. The application of a proper turbulence model is mandatory to generate accurate predictions of flow and heat transfer during combustion. The current work presents a comparative study to identify the suitable turbulence model for tangentially fired furnace problem. Three turbulence models including the standard k-e model, the RNG k–e model and the Reynolds Stress model, RSM are examined. The predictions are compared with the published experimental data of Zheng et al. (Proc Combust Inst 29: 811–818, 2002). The RNG k–e model proves to be the most suitable turbulence model, offering a satisfactory prediction of the velocity, temperature and species fields. The detailed results presented in this paper may enhance the understanding of complex flow patterns and combustion processes in tangentially fired pulverized-coal furnaces.

Development of soy lecithin based novel self-assembled emulsion hydrogels

The current study reports the development and characterization of soy lecithin based novel self-assembled emulsion hydrogels. Sesame oil was used as the representative oil phase. Emulsion gels were formed when the concentration of soy lecithin was >40% w/w. Metronidazole was used as the model drug for the drug release and the antimicrobial tests. Microscopic study showed the apolar dispersed phase in an aqueous continuum phase, suggesting the formation of emulsion hydrogels. FTIR study indicated the formation of intermolecular hydrogen bonding, whereas, the XRD study indicated predominantly amorphous nature of the emulsion gels. Composition dependent mechanical and drug release properties of the emulsion gels were observed. In-depth analyses of the mechanical studies were done using Ostwald-de Waele power-law, Kohlrausch and Weichert models, whereas, the drug release profiles were modeled using Korsmeyer–Peppas and Peppas–Sahlin models. The mechanical analyses indicated viscoelastic nature of the emulsion gels. The release of the drug from the emulsion gels was diffusion mediated. The drug loaded emulsion gels showed good antimicrobial activity. The biocompatibility test using HaCaT cells (human keratinocytes) suggested biocompatibility of the emulsion gels.

Phase field modeling of fracture in multi-physics problems. Part III. Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media

The prediction of fluid- and moisture-driven crack propagation in deforming porous media has achieved increasing interest in recent years, in particular with regard to the modeling of hydraulic fracturing, the so-called “fracking”. Here, the challenge is to link at least three modeling ingredients for (i) the behavior of the solid skeleton and fluid bulk phases and their interaction, (ii) the crack propagation on not a priori known paths and (iii) the extra fluid flow within developed cracks. To this end, a macroscopic framework is proposed for a continuum phase field modeling of fracture in porous media. It provides a rigorous geometric approach to a diffusive crack modeling based on the introduction of a constitutive balance equation for a regularized crack surface and its modular linkage to a Darcy–Biot-type bulk response of hydro-poro-elasticity. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies including branching. A modular concept is outlined for linking of the diffusive crack modeling with the hydro-poro-elastic response of the porous bulk material. This includes a generalization of crack driving forces from energetic definitions towards threshold-based criteria in terms of the effective stress related to the solid skeleton of a fluid-saturated porous medium. Furthermore, a Poiseuille-type constitutive continuum modeling of the extra fluid flow in developed cracks is suggested based on a deformation-dependent permeability, that is scaled by a characteristic length. This proposed modular model structure is exploited in the numerical implementation by constructing a robust finite element method, based on an algorithmic decoupling of updates for the crack phase field and the state variables of the hydro-poro-elastic bulk response. We demonstrate the performance of the phase field formulation of fracture for a spectrum of model problems of hydraulic fracture. A slight modification of the framework allows the simulation of drying-caused crack patterns in partially saturated capillar-porous media.

Martensitic transformations and damage: A combined phase field approach

AbstractA combined continuum phase field model for martensitic transformations and damage is introduced. The present approach considers the eigenstrain within the martensitic phase which leads in the surrounding material to both tensile and compressive stresses. The damage model needs to account for an appropriate differentiation thereof, since compressive stresses should not promote fracture. Interactions between micro crack propagation and the formation of the martensitic phases are studied in two dimensions. In agreement with experimental observations, martensite forms at the crack tip and influences the crack formation. For the numerical implementation finite elements are used while for the transient terms an implicit time integration scheme is employed. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)