Continuum Thermodynamics of Constrained Reactive Mixtures

2021 ◽  
Author(s):  
Gerard A. Ateshian ◽  
Brandon Zimmerman
Keyword(s):  
Heat Supply ◽  
Damage Mechanics ◽  
Reactive Power ◽  
Mixture Theory ◽  
Biological Tissues ◽  
Heat Of Reaction ◽  
Chemical Potentials ◽  
Ideal Gases ◽  
Reactive Processes ◽  
Energy Of Formation

Abstract Mixture theory models continua consisting of multiple constituents with independent motions. In constrained mixtures all constituents share the same velocity but they may have different reference configurations. The theory of constrained reactive mixtures was formulated to analyze growth and remodeling in living biological tissues. It can also reproduce and extend classical frameworks of damage mechanics and viscoelasticity under isothermal conditions, when modeling bonds that can break and reform. This study focuses on establishing the thermodynamic foundations of constrained reactive mixtures under more general conditions, for arbitrary reactive processes where temperature varies in time and space. By incorporating general expressions for reaction kinetics, it is shown that the residual dissipation statement of the Clausius-Duhem inequality must include a reactive power density, while the axiom of energy balance must include a reactive heat supply density. Both of these functions are proportional to the molar production rate of a reaction, and they depend on the chemical potentials of the mixture constituents. We present novel formulas for the classical thermodynamic concepts of energy of formation and heat of reaction, making it possible to evaluate the heat supply generated by reactive processes from the knowledge of the specific free energy of mixture constituents as well as the reaction rate. We illustrate these novel concepts with mixtures of ideal gases, and isothermal reactive damage mechanics and viscoelasticity, as well as reactive thermoelasticity. This framework facilitates the analysis of reactive tissue biomechanics and physiological and biomedical engineering processes where temperature variations cannot be neglected.

scholarly journals Three-dimensional numerical simulation of soft-tissue wound healing using constrained-mixture anisotropic hyperelasticity and gradient-enhanced damage mechanics

2020 ◽  
Vol 17 (162) ◽  
pp. 20190708 ◽  
Author(s):  
Di Zuo ◽  
Stéphane Avril ◽  
Haitian Yang ◽  
S. Jamaleddin Mousavi ◽  
Klaus Hackl ◽  
...  
Keyword(s):  
Wound Healing ◽  
Damage Mechanics ◽  
Healing Process ◽  
Three Dimensional ◽  
Biological Tissues ◽  
Collagen Fibres ◽  
Length Scales ◽  
Constrained Mixture ◽  
Proposed Model ◽  
Intrinsic Length

Healing of soft biological tissues is the process of self-recovery or self-repair after injury or damage to the extracellular matrix (ECM). In this work, we assume that healing is a stress-driven process, which works at recovering a homeostatic stress metric in the tissue by replacing the damaged ECM with a new undamaged one. For that, a gradient-enhanced continuum healing model is developed for three-dimensional anisotropic tissues using the modified anisotropic Holzapfel–Gasser–Ogden constitutive model. An adaptive stress-driven approach is proposed for the deposition of new collagen fibres during healing with orientations assigned depending on the principal stress direction. The intrinsic length scales of soft tissues are considered through the gradient-enhanced term, and growth and remodelling are simulated by a constrained-mixture model with temporal homogenization. The proposed model is implemented in the finite-element package Abaqus by means of a user subroutine UEL. Three numerical examples have been achieved to illustrate the performance of the proposed model in simulating the healing process with various damage situations, converging towards stress homeostasis. The orientations of newly deposited collagen fibres and the sensitivity to intrinsic length scales are studied through these examples, showing that both have a significant impact on temporal evolutions of the stress distribution and on the size of the damage region. Applications of the approach to carry out in silico experiments of wound healing are promising and show good agreement with existing experiment results.

Thermodynamics of methanesulfonic acid revisited

2000 ◽  
Vol 78 (10) ◽  
pp. 1295-1298 ◽  
Author(s):  
J Peter Guthrie ◽  
Roger T Gallant
Keyword(s):  
Methanesulfonic Acid ◽  
Heat Of Formation ◽  
Methyl Methanesulfonate ◽  
Thermodynamic Quantities ◽  
Heat Of Reaction ◽  
Reaction Conditions ◽  
Free Energy Of Formation ◽  
Sulfite Ion ◽  
Energy Of Formation

Recently we reported a study of the thermodynamics of methanesulfonic acid and some of its derivatives. The foundation of these results was a measurement of the heat of reaction of S-methyl thioacetate with aq sodium hypochlorite, leading to methanesulfonic acid. We have reinvestigated this reaction and discovered that contrary to the initial stoichiometry experiments, the stoichiometry under the reaction conditions is not as was believed and that the heat of reaction observed was spuriously high. We have found a new reaction, that of sulfite ion with methyl methanesulfonate, which does allow a clean determination of the heat of formation of methanesulfonic acid. Revised thermodynamic quantities for methanesulfonic acid, methanesulfonyl chloride, and methyl methanesulfonate are reported here.Key words: sulfonic acid, heat of reaction, free energy of formation, SN2 reaction.

A Hybrid Reactive Multiphasic Mixture with a Compressible Fluid Solvent

2021 ◽  
Author(s):  
Jay J. Shim ◽  
Gerard A. Ateshian
Keyword(s):  
Fluid Dynamics ◽  
Lagrange Multiplier ◽  
Deformation Gradient ◽  
Fluid Pressure ◽  
Volumetric Strain ◽  
Mixture Theory ◽  
Biological Tissues ◽  
Solid Matrix ◽  
State Function ◽  
Wide Range

Abstract Mixture theory is a general framework that has been used to model mixtures of solid, fluid, and solute constituents, leading to significant advances in modeling the mechanics of biological tissues and cells. Though versatile and applicable to a wide range of problems in biomechanics and biophysics, standard multiphasic mixture frameworks incorporate neither dynamics of viscous fluids nor fluid compressibility, both of which facilitate the finite element implementation of computational fluid dynamics solvers. This study formulates governing equations for reactive multiphasic mixtures where the interstitial fluid has a solvent which is viscous and compressible. This hybrid reactive multiphasic framework uses state variables that include the deformation gradient of the porous solid matrix, the volumetric strain and rate of deformation of the solvent, the solute concentrations, and the relative velocities between the various constituents. Unlike standard formulations which employ a Lagrange multiplier to model fluid pressure, this framework requires the formulation of a function of state for the pressure, which depends on solvent volumetric strain and solute concentrations. Under isothermal conditions the formulation shows that the solvent volumetric strain remains continuous across interfaces between hybrid multiphasic domains. Apart from the Lagrange multiplier-state function distinction for the fluid pressure, and the ability to accommodate viscous fluid dynamics, this hybrid multiphasic framework remains fully consistent with standard multiphasic formulations previously employed in biomechanics. With these additional features, the hybrid multiphasic mixture theory makes it possible to address a wider range of problems that are important in biomechanics and mechanobiology.

Modeling Bioheat Transport at Macroscale

2010 ◽  
Vol 133 (1) ◽  
Author(s):  
Liqiu Wang ◽  
Jing Fan
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Porous Media ◽  
Mixture Theory ◽  
Biological Tissues ◽  
Blood Velocity ◽  
Dual Phase ◽  
Bioheat Equation ◽  
Density Vector ◽  
Energy Equations

Macroscale thermal models have been developed for biological tissues either by the mixture theory of continuum mechanics or by the porous-media theory. The former uses scaling-down from the global scale; the latter applies scaling-up from the microscale by the volume averaging. The used constitutive relations for heat flux density vector include the Fourier law, the Cattaneo–Vernotte (Cattaneo, C., 1958, “A Form of Heat Conduction Equation Which Eliminates the Paradox of Instantaneous Propagation,” Compt. Rend., 247, pp. 431–433; Vernotte, P., 1958, “Les Paradoxes de la Théorie Continue de I’equation de la Chaleur,” Compt. Rend., 246, pp. 3154–3155) theory, and the dual-phase-lagging theory. The developed models contain, for example, the Pennes (1948, “Analysis of Tissue and Arterial Blood Temperature in the Resting Human Forearm,” J. Appl. Physiol., 1, pp. 93–122), Wulff (1974, “The Energy Conservation Equation for Living Tissues,” IEEE Trans. Biomed. Eng., BME-21, pp. 494–495), Klinger (1974, “Heat Transfer in Perfused Tissue I: General Theory,” Bull. Math. Biol., 36, pp. 403–415), and Chen and Holmes (1980, “Microvascular Contributions in Tissue Heat Transfer,” Ann. N.Y. Acad. Sci., 335, pp. 137–150), thermal wave bioheat, dual-phase-lagging (DPL) bioheat, two-energy-equations, blood DPL bioheat, and tissue DPL bioheat models. We analyze the methodologies involved in these two approaches, the used constitutive theories for heat flux density vector and the developed models. The analysis shows the simplicity of the mixture theory approach and the powerful capacity of the porous-media approach for effectively developing accurate macroscale thermal models for biological tissues. Future research is in great demand to materialize the promising potential of the porous-media approach by developing a rigorous closure theory. The heterogeneous and nonisotropic nature of biological tissue yields normally a strong noninstantaneous response between heat flux and temperature gradient in nonequilibrium heat transport. Both blood and tissue macroscale temperatures satisfy the DPL-type energy equations with the same values of the phase lags of heat flux and temperature gradient that can be computed in terms of blood and tissue properties, blood-tissue interfacial convective heat transfer coefficient, and blood perfusion rate. The blood-tissue interaction leads to very sophisticated effect of the interfacial convective heat transfer, the blood velocity, the perfusion, and the metabolic reaction on blood and tissue macroscale temperature fields such as the spreading of tissue metabolic heating effect into the blood DPL bioheat equation and the appearance of the convection term in the tissue DPL bioheat equation due to the blood velocity.

scholarly journals A homeostatic-driven turnover remodelling constitutive model for healing in soft tissues

2016 ◽  
Vol 13 (116) ◽  
pp. 20151081 ◽  
Author(s):  
Ester Comellas ◽  
T. Christian Gasser ◽  
Facundo J. Bellomo ◽  
Sergio Oller
Keyword(s):  
Constitutive Model ◽  
Damage Mechanics ◽  
Healing Rate ◽  
Soft Tissues ◽  
Healing Process ◽  
Mechanical Damage ◽  
Continuum Damage Mechanics ◽  
Biological Tissues ◽  
Mechanical Stimuli ◽  
Open System Thermodynamics

Remodelling of soft biological tissue is characterized by interacting biochemical and biomechanical events, which change the tissue's microstructure, and, consequently, its macroscopic mechanical properties. Remodelling is a well-defined stage of the healing process, and aims at recovering or repairing the injured extracellular matrix. Like other physiological processes, remodelling is thought to be driven by homeostasis, i.e. it tends to re-establish the properties of the uninjured tissue. However, homeostasis may never be reached, such that remodelling may also appear as a continuous pathological transformation of diseased tissues during aneurysm expansion, for example. A simple constitutive model for soft biological tissues that regards remodelling as homeostatic-driven turnover is developed. Specifically, the recoverable effective tissue damage, whose rate is the sum of a mechanical damage rate and a healing rate, serves as a scalar internal thermodynamic variable. In order to integrate the biochemical and biomechanical aspects of remodelling, the healing rate is, on the one hand, driven by mechanical stimuli, but, on the other hand, subjected to simple metabolic constraints. The proposed model is formulated in accordance with continuum damage mechanics within an open-system thermodynamics framework. The numerical implementation in an in-house finite-element code is described, particularized for Ogden hyperelasticity. Numerical examples illustrate the basic constitutive characteristics of the model and demonstrate its potential in representing aspects of remodelling of soft tissues. Simulation results are verified for their plausibility, but also validated against reported experimental data.

Triphasic Theory for Swelling Properties of Hydrated Charged Soft Biological Tissues

1990 ◽  
Vol 43 (5S) ◽  
pp. S134-S141 ◽  
Author(s):  
V. C. Mow ◽  
W. M. Lai ◽  
J. S. Hou
Keyword(s):  
Osmotic Pressure ◽  
Soft Tissues ◽  
Interstitial Water ◽  
Biological Tissues ◽  
Ion Concentration ◽  
Solid Matrix ◽  
Swelling Properties ◽  
Chemical Potentials ◽  
Triphasic Theory ◽  
Repulsive Forces

Swelling phenomenon of biological soft tissues, such as articular cartilage, depends on their fixed charge densities, the stiffness of their collagen-proteoglycan solid matrix and the ion concentration in the interstitium. Based on the thermodynamic continuum mixture theory, a multiphasic mixture model is developed to describe the equilibrium and transient swelling properties. For articular cartilages in a single salt environment (e.g. NaCl), a three phase model (triphasic theory) suffices to describe its swelling behavior. The three phases are: solid matrix, interstitial water and the mobile salt. The equations of motion in this theory shows that the driving forces for interstitial water and salt are the gradients of their chemical potentials. Constitutive equations for the chemical potentials of the phases and for the total stress under infinitesimal strain but large variation of salt concentration are presented based on the physico-chemical theory for polyelectrolytic solutions and continuum theory. Application of this theory to equilibrium problems yields the well known Donnan equilibrium ion distribution and osmotic pressure equations. The theory indicates that at equilibrium the applied load on the tissue is shared by 1) the solid matrix elastic stress due to deformation; 2) the Donnan osmotic pressure; and 3) the chemical expansion stress due to the charge-to-charge repulsive forces between the charged groups in the solid matrix. For the transient isometric swelling problem, the theory is shown to describe the experimentally observed responses very well.

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