Continuum Mixture Theory Research Articles

Universal phase-field mixture representation of thermodynamics and shock-wave mechanics in porous soft biologic continua.

A continuum mixture theory is formulated for large deformations, thermal effects, phase interactions, and degradation of soft biologic tissues suitable at high pressures and low to very high strain rates. Tissues consist of one or more solid and fluid phases and can demonstrate nonlinear anisotropic elastic, viscoelastic, thermoelastic, and poroelastic physics. Under extreme deformations or shock loading, tissues may fracture, tear, or rupture. Existing models do not account for all physics simultaneously, and most poromechanics and soft-tissue models assume incompressibility of some or all constituents, generally inappropriate for modeling shock waves or extreme compressions. Motivated by these prior limitations, a thermodynamically consistent formulation that combines a continuum theory of mixtures, compressible nonlinear anisotropic thermoelasticity, viscoelasticity, and phase-field mechanics of fracture is constructed to resolve the pertinent physics. A metric tensor of generalized Finsler space supplies geometric insight on effects of rearrangements of microstructure, for example degradation, growth, and remodeling. Shocks are modeled as singular surfaces. Hugoniot states and shock decay are analyzed: Solutions account for concurrent viscoelasticity, fracture, and interphase momentum and energy exchange not all contained in previous analyses. Suitability of the framework for representing blood, skeletal muscle, and liver is demonstrated by agreement with experimental data and observations across a range of loading rates and pressures. Insight into previously unresolved physics is obtained, for example importance of rate sensitivity of damage and quantification of effects of dissipation from viscoelasticity and phase interactions on shock decay.

Thermodynamically consistent diffuse-interface mixture models of incompressible multicomponent fluids

The prototypical diffuse-interface model for incompressible fluid mixtures is the Navier–Stokes Cahn–Hilliard (Allen–Cahn) model. Despite its foundation in continuum mixture theory, it is not fully compatible with this theory due to the diffusive flux approximation. This paper introduces a class of thermodynamically consistent diffuse-interface incompressible fluid mixture models that is fully compatible with the continuum theory of mixtures. The proposed models can be formulated in either constituent or mixture quantities, enabling a direct comparison with the Navier–Stokes Cahn–Hilliard (Allen–Cahn) model with non-matching densities. This comparison reveals the key modelling simplifications employed in the latter.

Multiscale Modeling of Elastic Waves in Carbon-Nanotube-Based Composite Membranes

A multiscale model is developed for vertically aligned carbon nanotube (CNT)-based membranes that are made for water purification or gas separation. As a consequence of driving fluids through the membranes, they carry stress waves along the fiber direction. Hence, a continuum mixture theory is established for a representative volume element to characterize guided waves propagating in a periodically CNT-reinforced matrix material. The obtained coupled governing equations for the CNT-based composite are found to retain the integrity of the wave propagation phenomenon in each constituent, while allowing them to coexist under analytically derived multiscale interaction parameters. The influence of the mesoscale characteristics on the continuum behavior of the composite is demonstrated by dispersion curves of harmonic wave propagation. Analytically established continuum mixture theory for the CNT-based composite is strengthened by numerical simulations conducted in COMSOL for visualizing mode shapes and wave propagation patterns.

Modelling internal erosion using 2D smoothed particle hydrodynamics (SPH)

This paper presents a stabilised multi-phase smoothed particle hydrodynamics (SPH) model applicable to seepage-induced internal erosion and the resulting deformation in soils. Based on the continuum mixture theory, a new single-layer SPH model in the u-w-p formulation is derived for the mathematical description of the erodible porous material. A new modified constitutive model is formulated to account for the influence of erosion on the mechanical behaviour of soils. Extensions of a first-order consistent boundary condition as well as the diffusion algorithm for hydro-mechanical coupling in SPH are also proposed to accommodate the analysis of erosion-transport process. A novel viscous dissipation term is designed to mitigate the numerical instability commonly encountered in coupled problems. In contrast to existing stabilisation terms in the SPH literature, which operate on both the bulk component and shear component, this new stabilisation term offers the possibility to increase the dissipation of unphysical oscillation due to the shear deformation. The proposed method is validated through a series of benchmark tests. The results demonstrate the effectiveness of the proposed approach in addressing the coupled problem in porous materials involving multi-phase flow, phase interaction/transfer, and large deformation with enhanced accuracy, stability, and robustness.

Ion-Induced Swelling Behavior of Articular Cartilage due to Non-Newtonian Flow and Its Effects on Fluid Pressure and Solid Displacement.

In this study, we examine the behavior of articular cartilage equilibrated in a salt (NaCl) solution during non-Newtonian fluid flow that follows an Ostwald-de Waele model. A linearly elastic and isotropic rectangular strip of cartilage is considered for analysis. A continuum theory of mixtures has been employed to develop a coupled system of partial differential equations for the solid displacement and the fluid pressure by considering the important factor of the ion concentration by assuming the cartilage as a deformable porous media. The coupled system of partial differential equations is solved using the numerical method named method of lines. In most cases, shear-thinning fluid is compared to the shear-thickening fluid to magnify the difference. Graphical results show that shear-thickening fluids bring more solid deformation and shows less fluid pressure in comparison to the shear-thinning fluids.

A novel two-dimensional continuum mixture theory model for paraffin wax melting in a cavity heating from side considering volume change

Paraffin wax presents a considerable potential for thermal storage technologies, given its high latent heat. Therefore, the accurate and systematic modeling of its melting process bears significant importance. Yet, existing mathematical models fail to adequately address the velocity driven by the density change during the melting process. In this research, we introduce a novel model for paraffin wax melting within a square enclosure subject to side heating. Unlike current models, our approach posits that flow within the mushy zone is dictated by the mass equation, not Darcy’s equation, and thus excludes the usage of mushy zone constant. This assumption, premised on the idea that the density-deviation-driven flow precludes the bulk flow from penetrating the mushy zone, is confirmed through a carefully designed visual experiment. Utilizing our melting model, we construct a mathematical model anchored in the continuum mixture theory. Through a comparative analysis of results acquired by numerical methods and experimental data, we find that our innovative mathematical model effectively predicts real-world scenarios with remarkable precision. Furthermore, we explore the influence of flows within the mushy zone on the melting process. Our findings indicate a significant deviation in results when assuming zero velocity in the mushy zone, suggesting that the density-deviation-driven flow is numerically negligible. However, it is crucial to note the physical relevance of the density-deviation-driven flow in accurately predicting the melting process. The absence of this consideration allows the bulk flow in the liquid region to permeate the mushy zone, thereby introducing substantial errors.

A nonlocal theory of heat transfer and micro-phase separation of nanostructured copolymers

Directed self-assembly of block copolymer (BCP) has shown promise as a nano-patterning technology for various applications such as microelectronics, semiconductor manufacturing and biotechnology. In this paper, we present a thermodynamically consistent nonlocal model of phase transformation and heat transfer for BCP via the continuum theory of mixtures. Our model is developed based on the mass and energy balances of each phase and a set of microforce balances to describe the dynamics of microphase separation during self-assembly and the size and memory effects of heat conduction in BCP thin films. We derive constitutive relations for coupling the phase field and temperature through upscaling the self-consistent field theory of polymers, and present the connections between the derived nonlocal heat conduction model and the phonon Boltzmann transport equation. We provide a finite element solution of the proposed model that includes unconditionally stable time-stepping schemes, and we perform various numerical experiments that showcase different features of the model. Furthermore, we conduct three-dimensional numerical experiments of laser zone annealing process to qualitatively investigate the effect of laser velocity on the self-assembly behavior of BCP thin films. Our results show that the proposed model can simulate essential features of thermally induced phase responses of BCPs, consistent with experimental observations and high-fidelity simulations.

A unified framework for Navier–Stokes Cahn–Hilliard models with non-matching densities

Over the last decades, many diffuse-interface Navier–Stokes Cahn–Hilliard (NSCH) models with non-matching densities have appeared in the literature. These models claim to describe the same physical phenomena, yet they are distinct from one another. The overarching objective of this work is to bring all of these models together by laying down a unified framework of NSCH models with non-zero mass fluxes. Our development is based on three unifying principles: (1) there is only one system of balance laws based on continuum mixture theory that describes the physical model, (2) there is only one natural energy-dissipation law that leads to quasi-incompressible NSCH models, (3) variations between the models only appear in the constitutive choices. The framework presented in this work now completes the fundamental exploration of alternate non-matching density NSCH models that utilize a single momentum equation for the mixture velocity, but leaves open room for further sophistication in the energy functional and constitutive dependence.

STRESS-RELAXATION BEHAVIOR OF DEFORMABLE POROUS SHELLS DURING PASSAGE OF POWER-LAW FLUIDS IN COMPRESSION

Multiphasic deformation of the porous solids can be described using the low-viscosity fluid flow through the viscously deformable and permeable solid matrix. The purpose of the present paper is to develop a better understanding of the governing equations that have been derived for this purpose while considering two-phase flows. An insight into the stress-relaxation behavior of a deformable porous shell has been formulated during passage of power-law fluids, in compression. The fluid flow was considered outwardly directed during the act of loading at the deformable inner radius of the shell. The outer boundary was taken as a rigid mesh that offers negligible resistance during the passage of fluids. The governing dynamics were derived using the continuum mixture theory approach, whereas non-Newtonian flow behavior was incorporated in the governing equations using the power-law model. A coupled system of partial differential equations was derived for the porosity and solid deformation to consider the nonlinear interaction between the fluid and solid. In the case of transient problem, a numerical solution is computed along with an exact solution of a steady-state problem. The propagation of porosity of the shell is bounded by the viscosity; otherwise, fluid flow resists in the solid matrix. The viscous stresses cause more deformation in the radial geometry, when compared to the planer geometry. In this setting, an additional pressure gradient is required for the fluid flow around the obstacles. When viscous resistance is increased in the nonlinear governing models, porosity controls the solid matrix deformation.

Generalized continuum mixture theory for multi-material shock physics

A long-standing area of research for computational shock physics has been the treatment of mixed materials. A mixed material volume is defined as a control volume space where two or more materials coexist as immiscible materials having separate densities and energies that have non-equilibrium stress and thermal states. In traditional computational shock physics framework, a mixture of materials can occupy a control volume having a common velocity corresponding to the mixture center of mass. In this work, this assumption is relaxed and separate densities, energies and velocities are used to characterize non-equilibrium material behavior. To accomplish this task, this work builds upon the original two-phase Baer and Nunziato model by extending the two-phase theory to N-phases. Specifically, this work is based on the development by Drumheller to derive a new N-phase mixture theory that incorporates deformation mechanics using the deformation gradient with a full stress tensor.

Formation and growth of multiple, distinct ice lenses in frost heave

AbstractThis paper presents a fully coupled thermo‐hydro‐mechanical (THM) model which simulates frost heave in fully saturated soils. The model is able to simulate the formation and growth of multiple distinct ice lenses. The basic equations of the system were derived using the continuum theory of mixtures, nonequilibrium thermodynamics, and fracture mechanics, considering skeleton deformation, water flow and heat transport. Central to this model is the coupled transport of mass due to the temperature gradient across the frozen fringe, which acts as the main driving force of the phenomenon. The model is formulated in terms of measurable physical properties and thus no ad hoc parametrization is required. In an ice‐lens‐free state, the system is solved as a continuum using the finite element method (FEM). It is then locally treated as a discontinuous system upon the formation of ice lens, by enriching the elements carrying the embedded ice lens(es) using the extended finite element method (X‐FEM). The accuracy and efficiency of the proposed model has been verified using several laboratory tests on Devon silt samples at different overburden pressures and thermal boundary conditions. Shut‐off pressures have been also estimated and compared with the experimental results.

A five-phase approach, SPH framework and applications for predictions of seepage-induced internal erosion and failure in unsaturated/saturated porous media

Seepage-induced internal erosion and failure in unsaturated/saturated porous media is challenging for computational simulations as they involve the behaviour, interactions (solid, air, water) and transformation (fluidisation and deposition of fines grains) of different phases. Tackling this challenging problem requires correct mathematical descriptions of phase interactions and transformation together with a robust computational framework, both of which are addressed in this paper. The new mathematical model and coupled governing equations based on the continuum mixture theory enable the use of a single set of SPH particles for the descriptions of behaviour, interactions and phase transformation of all five phases of the porous media (soil skeleton, erodible fines particles, fluidised particles, water, and air), including the effect of both saturation and erosion on the shear strength of porous media. A fully explicit and stabilised SPH framework that allows accurate SPH approximations of spatial gradients is proposed for the numerical solutions of coupled governing equations. The proposed computational framework performs well in benchmark tests against available analytical and numerical solutions and achieved reasonable agreements with experiments. Numerical results obtained from the predictions of seepage-induced erosion and failure demonstrate that the proposed computational framework is efficient for addressing challenging problems involving coupled flow-deformation, seepage-induced internal erosions, and large deformation failures of unsaturated/saturated porous media.

Biomechanical response of soft tissues during passage of synovial fluid in compression

The present study provides a novelty approach for the computational biological model based on continuum mixture theory in combination with power-law model for incorporating an accurate governing model for the synovial fluids. We investigated the biomechanical response of a soft tissue while passage of non-Newtonian fluid during act of loading at the rigid bony interface. A special kind of multiphasic deformation has been reported in these types of problems that justify nonlinear coupling between the fluid and solid. In modeling these types of problems, general assumption of mixture constituents incompressibility is often provoked. The mixture components are considered intrinsically incompressible; however, in the derivation of governing equations, viscoelastic behavior of the solid along with interstitial fluid was developed. The nonlinear interaction between the fluid–solid is modeled using strain-dependent permeability and is experimentally determined. This manipulation of linear model with nonlinear permeability required attention for the computational point of view. A system of nonlinear coupled partial differential equations is derived for the local fluid pressure along with an equation for solid deformation. The governing system of equations is solved numerically for the case of permeability dependent flow, whereas an exact solution is given for constant permeability case. Various interesting features, such as, pressure changes within the tissue, swelling behavior of the solid matrix, and effects of power law index on the tissue deformation have been presented graphically. A good qualitative agreement has been noticed between the exact and numerical solutions for constant permeability case.

Dynamic water potential waves in unsaturated soils

Quantitative descriptions of the temporal and spatial variations of volumetric water content and total water potential in unsaturated soils have for more than nine decades focused on diffusive mechanisms as represented by the Richards equation (Richards, 1931). However, recent laboratory studies (Lo et al., 2017) have revealed short-time transient behavior of water in non-deforming unsaturated soils that is distinctly oscillatory, not diffusive. In this paper, we establish a theoretical framework to generalize the Richards equation in a systematic way to account for non-diffusive mechanisms giving rise to oscillatory behavior in the water content and total water potential. A partial differential equation is developed, with the total water potential as the dependent variable, based on the coupling of mass and linear momentum balance within the continuum theory of mixtures. This new differential equation becomes equivalent to the Richards equation when non-diffusive mechanisms can be neglected, but when this condition is not met, the governing equation describes damped propagating wave behavior of the total water potential. This dynamic wave differs from a poroelastic wave in respect to solid framework motions and wave speed, as well as time and length scales. For a rigid, homogeneous unsaturated soil, the governing equation reduces to the telegraph equation, which can be solved analytically. An inherent feature of our approach is the appearance of natural time and length scales characteristic of total water potential waves. Consideration of these natural scales for three representative soils of differing texture helps to prescribe more precisely the domain of applicability of the Richards equation.

Analysis of shock waves in a mixture theory of a thermoelastic solid and fluid with distinct temperatures

The continuum theory of mixtures is used to describe a medium consisting of a solid phase and fluid phase, both compressible and attributed distinct deformation, temperature, entropy, and internal energy fields. Continuum balance laws are presented, and constitutive equations consistent with thermodynamic restrictions are postulated. Jump conditions for shock waves are established. Evolution equations for transient, i.e., unsteady, shock wave propagation are newly derived for an ideal mixture consisting of an arbitrary number of nonlinear thermoelastic solids, whereby inviscid thermoelastic fluid behavior is attained for a given constituent under more restrictive constitutive assumptions. Detailed calculations of Hugoniot response and shock decay are specialized to the case of a two-phase mixture of a single solid and a single fluid, where physical properties pertinent to lung parenchymal tissue containing air are assigned. Local constitutive behaviors for this lung mixture model correspond to a compressible neo-Hookean solid phase and an ideal gas fluid phase. Predictions of the model offer new information on origins of soft tissue trauma, in the context of prior experimental observations, for dynamic impact or shock loading of the lung.

Time-fractional Cahn–Hilliard equation: Well-posedness, degeneracy, and numerical solutions

In this paper, we derive the time-fractional Cahn–Hilliard equation from continuum mixture theory with a modification of Fick's law of diffusion. This model describes the process of phase separation with nonlocal memory effects. We analyze the existence, uniqueness, and regularity of weak solutions of the time-fractional Cahn–Hilliard equation. In this regard, we consider degenerating mobility functions and free energies of Landau, Flory–Huggins and double-obstacle type. We apply the Faedo–Galerkin method to the system, derive energy estimates, and use compactness theorems to pass to the limit in the discrete form. In order to compensate for the missing chain rule of fractional derivatives, we prove a fractional chain inequality for semiconvex functions. The work concludes with numerical simulations and a sensitivity analysis showing the influence of the fractional power. Here, we consider a convolution quadrature scheme for the time-fractional component, and use a mixed finite element method for the space discretization.

Rheological effects on the acidizing process in carbonate reservoirs

In this work, we present a new formulation to study the flow of a reactive non-Newtonian fluid in porous media. The mathematical model consists of a two-scale continuum model that couples information from the pore scale to Darcy one. In the place of the traditional Darcy Law, we consider a complete linear momentum equation, based on the Continuum Theory of Mixtures. The upscaling is made by an interaction source term that corresponds to the flow resistance caused by the matrix inside the pores, which considers the non-newtonian nature of the fluid. Motivated by the often applied self diverting acids we analyze the rheological dependence on the local level of pH. Simulations are performed using the finite volume method with the OpenFOAM software. We investigate the influence of the Damköhler number and the power-law index over dissolution patterns and Pore Volume to Breakthrough (PVbt) curves. The results are compared with the corresponding Newtonian cases. The effect of both shear rate and pH on viscosity are also studied. We verify a significant dependency of the dissolution patterns and also in the optimum injection rate on the rheology of the fluid. More shear-thinning fluids lead to lower values of PVbt. Although the divergent acids were really able to find different paths for the breakthrough, higher times were needed for this achievement.

Generalized continuum mixture theory for multi-material shock physics

A long-standing area of research for computational shock physics has been the treatment of mixed materials. A mixed material volume is defined as a control volume space where two or more materials coexist as immiscible materials having separate densities and energies that have non-equilibrium stress and thermal states. In traditional computational shock physics framework, a mixture of materials can occupy a control volume having a common velocity corresponding to the mixture center of mass. In this work, this assumption is relaxed and separate densities, energies and velocities are used to characterize non-equilibrium material behavior. To accomplish this task, this work builds upon the original two-phase Baer and Nunziato model by extending the two-phase theory to N-phases. Specifically, this work is based on the development by Drumheller to derive a new N-phase mixture theory that incorporates deformation mechanics using the deformation gradient with a full stress tensor.

Modeling and simulation of vascular tumors embedded in evolving capillary networks

In this work, we present a coupled 3D–1D model of solid tumor growth within a dynamically changing vascular network to facilitate realistic simulations of angiogenesis. Additionally, the model includes erosion of the extracellular matrix, interstitial flow, and coupled flow in blood vessels and tissue. We employ continuum mixture theory with stochastic Cahn–Hilliard type phase-field models of tumor growth. The interstitial flow is governed by a mesoscale version of Darcy’s law. The flow in the blood vessels is controlled by Poiseuille flow, and Starling’s law is applied to model the mass transfer in and out of blood vessels. The evolution of the network of blood vessels is orchestrated by the concentration of the tumor angiogenesis factors (TAFs); blood vessels grow towards the increasing TAFs concentrations. This process is not deterministic, allowing random growth of blood vessels and, therefore, due to the coupling of nutrients in tissue and vessels, makes the growth of tumors stochastic. We demonstrate the performance of the model by applying it to a variety of scenarios. Numerical experiments illustrate the flexibility of the model and its ability to generate satellite tumors. Simulations of the effects of angiogenesis on tumor growth are presented as well as sample-independent features of cancer.

Dam Breach Simulation with the Material Point Method

Dam embankment breaches caused by overtopping or internal erosion can impact both life and property downstream. It is important to accurately predict the amount of erosion, peak discharge, and the resulting downstream flow. This paper presents a new model based on the material point method to simulate soil and water interaction and predict failure rate parameters. The model assumes that the dam consists of a homogeneous embankment constructed with cohesive soil, and water inflow is defined by a hydrograph using other readily available reach routing software. The model uses continuum mixture theory to describe each phase where each species individually obeys the conservation of mass and momentum. A two-grid material point method is used to discretize the governing equations. The Drucker–Prager plastic flow model, combined with a Hencky strain-based hyperelasticity model, is used to compute soil stress. Water is modeled as a weakly compressible fluid. Analysis of the model demonstrates the efficacy of our approach for existing examples of overtopping dam breach, dam failures, and collisions. Simulation results from our model are compared with a physical-based breach model, WinDAM C. The new model can capture water and soil interaction at a finer granularity than WinDAM C. The new model gradually removes the granular material during the breach process. The impact of material properties on the dam breach process is also analyzed.