Analytical Solutions for Contaminant Diffusion in Double-Layered Porous Media

Analytical solutions for conservative solute diffusion in one-dimensional double-layered porous media are presented in this paper. These solutions are applicable to various combinations of fixed solute concentration and zero-flux boundary conditions (BC) applied at each end of a finite one-dimensional domain and can consider arbitrary initial solute concentration distributions throughout the media. Several analytical solutions based on several initial and BCs are presented based on typical contaminant transport problems found in geoenvironmental engineering including (1) leachate diffusion in a compacted clay liner (CCL) and an underlying stratum; (2) contaminant removal from soil layers; and (3) contaminant diffusion in a capping layer and underlying contaminated sediments. The analytical solutions are verified against numerical solutions from a finite-element method based model. Problems related to leachate transport in a CCL and an underlying stratum of a landfill and contaminant transport through a capping layer over contaminated sediments are then investigated, and the suitable definition of the average degree of diffusion is considered.     

Multiple-Porosity Contaminant Transport by Finite-Element Method

An exponential finite-element model for multiple-porosity contaminant transport in soils is proposed. The model combines three compartments for dissolved contaminants: a primary compartment of diffusion–advection transport with nonequilibrium sorption, a secondary compartment with diffusion in rectangular or spherical soil blocks, and a tertiary compartment for immobile solutions within the primary compartment. Hence the finite-element model can be used to solve four types of mass-transfer problems which include: (1) intact soils, (2) intact soils with multiple sources of nonequilibrium partitioning, (3) soils with a network of regularly spaced fissures, and (4) structured soils. Hitherto, mobile/immobile compartments, fissured soils, and nonequilibrium sorption have been treated separately or in pairs. A Laplace transform is applied to the governing equations to remove the time derivative. A Galerkin residual statement is written and a finite-element method is developed. Both polynomial and exponential finite elements are implemented. The solution is inverted to the time domain numerically. The method is validated by comparison to analytical and boundary element predictions. Exponential elements perform particularly well, speeding up convergence significantly. The scope of the method is illustrated by analyzing contamination from a set of four waste repositories buried in fissured clay.     

Computer simulation of diffusion in multiphase systems

A general model to treat multicomponent diffusion in multiphase dispersions is presented. The model is based on multicomponent diffusion data and basic thermodynamic data and contains no adjustable parameters. No restriction is placed on the number of components or phases that take part in the calculations, as long as the necessary thermodynamic and kinetic data are available. The new model is implemented into the DICTRA software, which makes use of THERMO-CALC to handle the thermodynamics. The model is applied to carburization of Ni alloys and heat treatment of welded joints between dissimilar materials. In both cases, the diffusion is accompanied by carbide formation or dissolution. A good agreement between experiments and calculations is found, despite the fact that no adjustable parameters are needed.     

Analytical Solution for Diffusion of VOCs through Composite Landfill Liners

Analytical solutions are presented for analyzing volatile organic compound (VOC) diffusion through intact composite landfill liners for two scenarios with boundary conditions at the base of either a VOC concentration of zero or a VOC mass flux of zero. A time-dependent concentration top boundary condition is included in the presented analytical solutions to model typical variations of VOC concentration in the leachate over time. The presented solutions are verified against alternative numerical solutions and applied to analyze dichloromethane diffusion through a composite liner. The analytical solutions are found to provide useful predictions of VOC concentration and mass flux for the design of composite liners. VOC concentrations and fluxes at the base of the composite liner at 30?years predicted by consideration of representative transient variation in leachate concentration, for an example problem, are nearly half of those when a constant leachate concentration assumed.     

Numerical modeling of multiphase diffusion in the process of homogenizing in binary alloys

A general model to treat multiphase diffusion in an impregnated semi-infinite couple, a finite/semi-infinite couple, and a finite/finite couple, under conditions of zero zurface flux, is presented. The model is based both on the numerical solutions previously presented for the multiphase diffusion in semi-infinite and infinite media to which Boltzmann's transformation can be applied and on the finite difference solutions with a variable grid for the multiphase diffusion to which the transformation cannot be applied. Not only critical layer widths for the occurrence of a new phase and the disappearance of a pre-existing phase but also criteria to calculate or not a change in concentration for a terminal phase are introduced in the model in order to diminish the number of iterations, in the calculation. Typical examples to which the model has been applied are discussed.     

Numerical modeling of multiphase diffusion in the process of homogenizing in binary alloys

A general model to treat multiphase diffusion in an impregnated semi-infinite couple, a finite/semi-infinite couple, and a finite/finite couple, under conditions of zero surface flux, is presented. The model is based both on the numerical solutions previously presented for the multiphase diffusion in semi-infinite and infinite media to which Boltzmann’s transformation can be applied and on the finite difference solutions with a variable grid for the multiphase diffusion to which the transformation cannot be applied. Not only critical layer widths for the occurrence of a new phase and the disappearance of a pre-existing phase but also criteria to calculate or not a change in concentration for a terminal phase are introduced in the model in order to diminish the number of iterations in the calculation. Typical examples to which the model has been applied are discussed.     

Theoretical treatment of interaction parameters in multicomponent metallic solutions

A new theoretical equation for interaction parameter in multicomponent metallic solutions is developed using the pseudopotential formalism coupled with the free energy of the hard sphere system. The approximate expression for the pseudopotential term is given in terms of the heat of solution at infinite dilution, to allow easy evaluation of the interaction parameter in various multicomponent systems. This theory has been applied to 23 non-ferrous alloys based on Pb, Sn, Bi and In. Comparison with the results of previous theoretical calculations using only the hard sphere model suggests that the inclusion of the pseudopotential term yields a quantitatively more correct prediction of interaction parameters in multicomponent metallic solutions. Numerical calculations were also made for 320 Fe-base solutions relevant to steelmaking and the agreement between calculation and experimental data appears reasonable, with 90% reliability in predicting the correct sign.     

A generalized formulation of latent heat functions in enthalpy-based mathematical models for multicomponent alloy solidification systems

A systematic and generalized procedure for mathematical formulation of latent heat functions, as applicable for enthalpy-based solidification modeling of multicomponent alloy systems, is developed. The method uses a metallurgically appropriate thermo-solutal coupling strategy, in conjunction with updating of respective phase fractions, in order to obtain solutions of field variables that are consistent with the pertinent phase-change morphology. The present approach can model the solidification of multicomponent alloys for a wide range of local scale diffusion behavior of the constituent species.     

Mechanics of Composite Sinusoidal Honeycomb Cores

Lightweight and heavy-duty fiber-reinforced polymer (FRP) composite honeycomb sandwich structures have been increasingly used in civil infrastructure. Unique cellular core configurations, such as sinusoidal core, have been applied in sandwich construction. Due to specific core geometry, the solutions for core effective stiffness properties are not readily available. This paper presents a mechanics of materials approach to evaluate the effective stiffness properties of sinusoidal cores. In particular, the internal forces of a curved wall in a unit cell are expressed in terms of resultant forces, and based on the energy method and principle of equivalence analysis, the in-plane stiffness properties of sinusoidal cores are derived. Both finite-element modeling and experimental testing are carried out to verify the accuracy of the proposed analytical formulation. To illustrate the present analytical approach as an efficient tool in optimal analysis and size selection of sinusoidal cores, several design plots are provided and discussed. The simplified analysis and formulation presented for sinusoidal cores can be used in design application of FRP honeycomb sandwich and optimization of efficient cellular core structures.     

Wave Propagation in a Pipe Pile for Low-Strain Integrity Testing

This paper presents an analytical solution methodology for a tubular structure subjected to a transient point loading in low-strain integrity testing. The three-dimensional effects on the pile head and the applicability of plane-section assumption are the main problems in low-strain integrity testing on a large-diameter tubular structure, such as a pipe pile. The propagation of stress waves in a tubular structure cannot be expressed by one-dimensional wave theory on the basis of plane-section assumption. This paper establishes the computational model of a large-diameter tubular structure with a variable wave impedance section, where the soil resistance is simulated by the Winkler model, and the exciting force is simulated with semisinusoidal impulse. The defects are classified into the change in the wall thickness and Young’s modulus. Combining the boundary and initial conditions, a frequency-domain analytical solution of a three-dimensional wave equation is deduced from the Fourier transform method and the separation of variables methods. On the basis of the frequency-domain analytic solution, the time-domain response is obtained from the inverse Fourier transform method. The three-dimensional finite-element models are used to verify the validity of analytical solutions for both an intact and a defective pipe pile. The analytical solutions obtained from frequency domain are compared with the finite-element method (FEM) results on both pipe piles in this paper, including the velocity time history, peak value, incident time arrival, and reflected wave crests. A case study is shown and the characteristics of velocity response time history on the top of an intact and a defective pile are investigated. The comparisons show that the analytical solution derived in this paper is reliable for application in the integrity testing on a tubular structure.     

Mathematical modelling of avascular-tumour growth

A system of nonlinear partial differential equations is proposed as a model for the growth of an avascular-tumour spheroid. The model assumes a continuum of cells in two states, living or dead, and, depending on the concentration of a generic nutrient, the live cells may reproduce (expanding the tumour) or die (causing contraction). These volume changes resulting from cell birth and death generate a velocity field within the spheroid. Numerical solutions of the model reveal that after a period of time the variables settle to a constant profile propagating at a fixed speed. The travelling-wave limit is formulated and analytical solutions are found for a particular case. Numerical results for more general parameters compare well with these analytical solutions. Asymptotic techniques are applied to the physically relevant case of a small death rate, revealing two phases of growth retardation from the initial exponential growth, the first of which is due to nutrient-diffusion limitations and the second to contraction during necrosis. In this limit, maximal and "linear' phase growth speeds can be evaluated in terms of the model parameters.     

Experimental and Computational Studies on High-Strength Concrete Circular Columns Confined by Aramid Fiber-Reinforced Polymer Sheets

The aim of this paper is to study the properties of high-strength concrete (HSC) circular columns confined by aramid fiber-reinforced polymer (AFRP) sheets under axial compression. A total of 60 specimens were tested, considering the following parameters: the compressive strength of concrete, the number of AFRP layers, and the form of AFRP wrapping. In addition, an analytical model for predicting the stress–strain curves is proposed based on the experimental results. Meanwhile, a three-dimensional nonlinear finite-element model with a Drucker–Prager plasticity model for the concrete core and an elastic model for the AFRP is developed by using the finite-element code ANSYS. It is demonstrated that the strength and ductility of the columns with continuous AFRP wrapping are increased greatly; whereas the strength of the columns with discontinuous AFRP wrapping is also increased, but the ductility is not always increased notably. The analytical model and the finite-element model are validated against the experimental results.     

Modeling gas diffusion into metals with a moving-boundary phase transformation

A fixed-grid, finite-difference model is developed to investigate the moving-boundary phase transformation that occurs during gas charging or discharging from finite specimens. The model is developed in a nondimensional form, and solutions are calculated for plane sheets, cylinders, and spheres. Numerical predictions are compared with existing analytical solutions, as well as experimental data for the β/α transformation of titanium induced by diffusion of oxygen from the surface of a plane sheet. Finally, a parametric study is performed to elucidate the effect of several input parameters on phase transformation kinetics, and simple empirical relations for the transformation time are developed based on this study.     

Quantitative Comparison of Models for Barotropic Response of Homogeneous Basins

Numerous three-dimensional models that solve the shallow water equations have been proposed to describe the processes of circulation and mixing in large bodies of water. The utility of those models is often demonstrated by comparison of computed variables with field observations. However, both the hydrodynamic data and the boundary conditions that drive the model have unknown measurement uncertainties and a limited spatial coverage, which limit the validity of this approach. A series of simple benchmark problems with analytical solutions is proposed to evaluate a particular model’s suitability to efficiently and accurately reproduce a wide range of characteristic hydrodynamic phenomena in natural lakes. The test problems focus on the representation of free and forced oscillations in homogeneous water bodies (barotropic response). This is not intended as a substitute for model validation against field data but, rather, as a necessary step in the initial model testing and selection. To illustrate this approach, the proposed test problems are used to compare a finite-element and a finite-difference free-surface hydrostatic model.     

Mathematical model of the thermal processing of steel ingots: Part I. Heat flow model

A two-dimensional mathematical model has been developed to predict stress generation in static-cast steel ingots during thermal processing with the objective of understanding the role of stress generation in the formation of defects such as panel cracks. In the first part of a two-part paper the formulation and application of a heat-flow model, necessary for the prediction of the temperature distribution which governs thermal stress generation in the ingot, are described. A transverse plane through the ingot and mold is considered and the model incorporates geometric features such as rounded corners and mold corrugations by the use of the finite-element method. The time of air gap formation between mold and solidifying ingot skin is input, based on reported measurements, as a function of position over the ingot/mold surface. The model has been verified with analytical solutions and by comparison of predictions to industrial measurements. Finally, the model has been applied to calculate temperature contours in a 760×1520 mm, corrugated, low-carbon steel ingot under processing conditions conducive to panel crack formation. The model predictions are input to an uncoupled stress model which is described in Part II. B. G. THOMAS, formerly a Graduate Student at the University of British Columbia     

Influence of Fiber Undulations on Buckling of Thin Filament-Wound Cylinders in Axial Compression

The filament-winding process introduces inherent geometric defects into thin-shell cylinders in the form of fiber-bundle, or tow, crossovers. This research identified the micromechanical geometry in the fiber crossover regions of filament-wound cylinders. A stiffness model was developed for the crossover regions based on a modified classical lamination theory. The local stiffness-coupling values predicted by this model were incorporated into a global finite-element model of thin-shell, filament-wound cylinders. Eigenvalue buckling analyses performed using this enhanced finite-element model are compared to eigenvalue analyses performed without incorporating the influence of the stiffness couplings at the fiber crossover regions. Experiments were carried out in which 16 filament-wound cylinders with four different surface patterns were fabricated and tested. The results showed that the accuracy of a finite-element analysis improved significantly when the stiffness coupling effects due to fiber undulations were properly accounted for in the analytical model.     

Boundary Finite-Element Method Coupling Finite-Element Method for Steady-State Analyses of Dam-Reservoir Systems

The boundary finite-element method (BFEM) is extended for steady-state analyses of dam-reservoir system problems. In this study, the dam is assumed to be rigid and subjected to horizontal ground motions, and the liquid in the reservoir is assumed to be semiinfinite. The semiinfinite reservoir domain is partitioned into two subdomains: a near-field domain and a far-field domain. In it, the near-field domain is modeled by using the finite-element method (FEM), whereas the far-field domain is modeled by BFEM and is treated as a layered semiinfinite fluid domain. A BFEM/FEM coupling procedure is employed to solve the steady-state response of the reservoir. The coupling procedure is easy to implement and suitable for all frequencies, be it real or complex. The BFEM/FEM coupling procedure is validated in the frequency domain. Numerical results that are based on the present procedure are in good agreement with analytical and other available numerical solutions.     

Equilibrium Model of Precipitation in Microalloyed Steels

The formation of precipitates during thermal processing of microalloyed steels greatly influences their mechanical properties. Precipitation behavior varies with steel composition and temperature history and can lead to beneficial grain refinement or detrimental transverse surface cracks. This work presents an efficient computational model of equilibrium precipitation of oxides, sulfides, nitrides, and carbides in steels, based on satisfying solubility limits including Wagner interaction between elements, mutual solubility between precipitates, and mass conservation of alloying elements. The model predicts the compositions and amounts of stable precipitates for multicomponent microalloyed steels in liquid, ferrite, and austenite phases at any temperature. The model is first validated by comparing with analytical solutions of simple cases, predictions using the commercial package JMat-PRO, and previous experimental observations. Then it is applied to track the evolution of precipitate amounts during continuous casting of two commercial steels (1004 LCAK and 1006Nb HSLA) at two different casting speeds. This model is easy to modify to incorporate other precipitates, or new thermodynamic data, and is a useful tool for equilibrium precipitation analysis.     

Dynamic Testing for Structural Identification of a Bridge

Structural identification via modal analysis in structural mechanics is gaining popularity in recent years, despite conceptual difficulties connected with its use. This paper is devoted to illustrating the advantages and also the indeterminacy characterizing structural identification problems for bridge structures, even in rather simple instances. In particular, an identification procedure based on modal analysis and finite-element model updating is presented for the characterization of a concrete bridge whose constructive typology is quite diffuse in the area of Friuli Venezia Giulia, Italy. Experience has suggested (so as to restrict the indeterminacy frequently affecting identification issues) resorting to all the a priori information on the system and mindfully selecting the parameters to be identified. The analysis pointed out some particularities in the modeling of bridge typology under study, otherwise not a priori detectable from an analytical point of view.     

Shrinkage and splitting of microcracks under pressure simulated by the finite-element method

The two-dimensional finite-element method is applied to analyze the shrinkage and splitting of microcracks regularly arranged on or perpendicular to a grain boundary under pressure. Grain-boundary and surface diffusions are coupled by the boundary conditions at the triple point of the microcrack surface and the grain boundary. The shrinkage and splitting processes for the two kinds of microcracks are revealed by detailed finite-element analyses. For the microcrack lying on a grain boundary, it first shrinks to a small void shape, then the void is split by the grain boundary and the two split voids assume a cylindrical shape under the capillary force of the surface. For the microcrack perpendicular to the grain boundary, it is split into two segments by the grain boundary during the early stage of shrinkage. Then, the split microcracks stop shrinking and evolve into two cylindrical channels with a circular section by the capillary force of the surface. These evolution processes are controlled by the applied pressure, microcrack spacing, ratio of grain-boundary diffusion to surface diffusion, and equilibrium dihedral angle, defined by surface and grain-boundary tensions. The influences of these controlled parameters on the evolution processes are numerically clarified based on a great number of finite-element analyses.