Implementation of Finite Deformation Triphasic Modeling in the Finite Element Code FEBio

2012 ◽  
Author(s):  
Gerard A. Ateshian ◽  
Steve Maas ◽  
Jeffrey A. Weiss
Keyword(s):  
Soft Tissues ◽  
Mixture Theory ◽  
Solid Matrix ◽  
Osmotic Swelling ◽  
Solute Content ◽  
Counter Ions ◽  
The Matrix ◽  
Triphasic Theory ◽  
Ion Species ◽  
And Diffusion

Many biological soft tissues exhibit a charged solid matrix, most often due to the presence of proteoglycans enmeshed within the matrix. The predominant solute content of the interstitial fluid of these tissues consists of the monovalent counter-ions Na+ and Cl−. The electrical interactions between the mobile ion species and fixed charge density of the solid matrix produces an array of mechano-electrochemical effects, including Donnan osmotic swelling, and streaming and diffusion potentials and currents. These phenomena have been successfully modeled by the triphasic theory of Lai et al. [1], which is based on the framework of mixture theory [2]. Other similar frameworks have also been proposed [3, 4]. The equations of triphasic theory are nonlinear, even in the range of infinitesimal strains. Therefore, numerical schemes are generally needed to solve all but the simplest problems using this framework.

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.

scholarly journals Ductile saline ice

1994 ◽  
Vol 40 (136) ◽  
pp. 566-568
Author(s):  
G. A. Kuehn ◽  
E. M. Schulson
Keyword(s):  
Fresh Water ◽  
Tensile Ductility ◽  
Compressive Deformation ◽  
Dislocation Creep ◽  
Diffusion Creep ◽  
Water Ice ◽  
The Matrix ◽  
And Diffusion

AbstractExperiments have shown that tensile ductility of about 5% or more can be imparted to columnar, saline ice by pre-compressing the material by about 3.5%. This effect is similar to that observed in granular, fresh-water ice and is attributed to the operation of both dislocation creep and diffusion creep within that part of the matrix which recrystallized during the pre-compressive deformation.

scholarly journals Effect of Aluminum, Iron and Chromium Alloying on the Structure and Mechanical Properties of (Ti-Ni)-(Cu-Zr) Crystalline/Amorphous Composite Materials

Metals ◽  
2020 ◽  
Vol 10 (7) ◽  
pp. 874
Author(s):  
Andrey A. Tsarkov ◽  
Vladislav Yu. Zadorozhnyy ◽  
Alexey N. Solonin ◽  
Dmitri V. Louzguine-Luzgin
Keyword(s):  
Mechanical Properties ◽  
Testing Machine ◽  
High Strength ◽  
Alloying Elements ◽  
Solid Matrix ◽  
Complete Suppression ◽  
Structural Investigations ◽  
X Ray ◽  
Composites Materials ◽  
The Matrix

High-strength crystalline/amorphous composites materials based on (Ti-Ni)-(Cu-Zr) system were developed. The optimal concentrations of additional alloying elements Al, Fe, and Cr were obtained. Structural investigations were carried out using X-ray diffraction equipment (XRD) and scanning electron microscope (SEM) with an energy-dispersive X-ray module (EDX). It was found that additives of aluminum and chromium up to 5 at% dissolve well into the solid matrix solution of the NiTi phase. At a concentration of 5 at%, the precipitation of the unfavorable NiTi2 phase occurs, which, as a result, leads to a dramatic decrease in ductility. Iron dissolves very well in the solid solution of the matrix phase due to chemical affinity with nickel. The addition of iron does not cause the precipitation of the NiTi2 phase in the concentration range of 0–8 at%, but with an increase in concentration, this leads to a decrease in the mechanical properties of the alloy. The mechanical behavior of alloys was studied in compression test conditions on a universal testing machine. The developed alloys have a good combination of strength and ductility due to their dual-phase structure. It was shown that additional alloying elements lead to a complete suppression of the martensitic transformation in the alloys.

Matrix Analysis of Shell Structures with Flexible Frames

1958 ◽  
Vol 9 (4) ◽  
pp. 361-394 ◽  
Author(s):  
J. S. Przemieniecki
Keyword(s):  
Thermal Loading ◽  
Arbitrary Cross Section ◽  
Shell Structures ◽  
Matrix Analysis ◽  
Internal Load ◽  
Flexibility Matrix ◽  
The Matrix ◽  
Simple Matrix ◽  
Curved Panels ◽  
And Diffusion

SummaryA simple matrix method is presented for the deflection and stress analysis of cylindrical shell structures of arbitrary cross section stiffened by flexible frames. The method is an extension to fuselage structures of the Matrix Force Method developed by Argyris, in which the internal load system in the structure consists of two parts:—(a) synthetic load distribution, represented by the matrix b0, satisfying the external and internal equations of equilibrium, and(b)self-equilibrating load systems, represented by the matrix b1, which are introduced to satisfy compatibility conditions. The magnitudes of these self-equilibrating load systems are determined from the generalised compatibility equations formulated using the flexibility matrix f for the un-assembled elements of the structure. The self-equilibrating systems are non-orthogonal, but are arranged in such a way that the mixing between one system and another is kept to a minimum and, consequently, the resulting compatibility equations are well-conditioned. The three basic matrices, b0, b1;and f, are compiled using only very simple formulae. The matrices b0and b1depend on the geometry of the structure, while the flexibility matrix f is a function of geometry and elastic properties. The present analysis is applied to cut-out problems in fuselage structures. It can also be used for problems involving thermal loading and diffusion of loads in curved panels stiffened by flexible frames.

Relationship Between Meniscal Sample Swelling and the Compressive Properties of Bovine Menisci

2013 ◽  
Author(s):  
Stephen H. J. Andrews ◽  
Nigel G. Shrive ◽  
Janet L. Ronsky
Keyword(s):  
Vertical Direction ◽  
Axial Direction ◽  
Solid Matrix ◽  
Compressive Properties ◽  
Connective Tissues ◽  
Drag Forces ◽  
The Matrix ◽  
Two Phases ◽  
And Storage ◽  
Phosphate Buffered Saline

The menisci are anisotropic hydrated connective tissues, situated in the tibiofemoral joint. The menisci transmit approximately 50% of the load across this joint [1, 2]. In this tissue, compression would only be experienced in the axial (vertical) direction, and as such, many studies have tested samples in the axial direction to determine the compressive properties [3–5]. The material behaviour of the menisci has been described as biphasic, meaning the response of the tissue to applied load is time dependent and determined by both the solid constituents and their interaction with the fluid component [3]. Due to the low permeability of the tissue, deformation results in relative movement of the solid matrix and the fluid it contains, resulting in the creation of drag forces between the two phases. Fluid exudation from the matrix governs the viscoelastic behaviour of the tissue, including stress relaxation and creep [6]. The swelling behaviour of meniscal samples in varying osmotic environments was evaluated in our lab (unpublished data), where they swelled significantly, approximately 30% volumetrically in iso-osmotic phosphate buffered saline (PBS). It was hypothesized that the material properties of the tissue would be affected by this significant swelling. To date, no study has evaluated the effect of sample swelling, due to sample preparation and storage, on the behaviour of the menisci in compression. Therefore, the purpose of this study was to evaluate this relationship. We hypothesized that meniscal samples would be less stiff and more permeable in a swollen state than when they are compressed to the ‘fresh’, non-swollen, thickness prior to initiation of the protocol.

Mechanisms of Residual Stress in Soft Tissues

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
Yoram Lanir
Keyword(s):  
Residual Stress ◽  
Soft Tissues ◽  
Local Level ◽  
Mixture Theory ◽  
Meso Level ◽  
Physical Mechanisms ◽  
Organ Specific ◽  
Tissue Space ◽  
Micro Level

Although the importance of knowing the magnitude of residual stress (RS) and its functional significance are widely recognized, there is still disagreement and confusion regarding the nature of physical mechanisms giving rise to RS in tissues and organs. Here an attempt is made to examine the various mechanisms which may be involved in producing RS, and to estimate their roles and significance based on previously published experimental observations. Two concepts are introduced. The first establishes a hierarchy of different possible RS producing mechanisms from the micro (local) level of the tissue space, through the meso-level of the whole tissue, to the macro (organ) one. Whereas micro-level RS seem to be present in all soft tissues, the existence of macro- and meso-level mechanisms are tissue and organ specific. The second concept introduced highlights the significance of tissue swelling as an RS producing mechanism in the local micro-level. The implications of RS mechanism hierarchy are discussed regarding the interpretations of commonly used experimental methods aimed to study RS or to estimate its magnitude. Of the three categories of RS mechanisms, the local micro-RS is the least understood. It is analyzed here in terms of the tissue’s multiconstituent structure, in the framework of mixture theory. It is shown that the micro-RS can stem either from interactions between the solid tissue constituents or between its solids and its fluidlike matrix. The latter mode is associated with osmotic-driven tissue swelling. The feasibility of these two mechanisms is analyzed based on published observations and measured data. The analysis suggests that under conditions not too remote from the in vivo homeostatic one, osmotic-driven tissue swelling is a predominant RS producing mechanism. The analysis also suggests that a true stress-free configuration can be obtained only if all RS producing mechanisms are relieved, and outlines a manner by which this may be achieved.

A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage

1991 ◽  
Vol 113 (3) ◽  
pp. 245-258 ◽  
Author(s):  
W. M. Lai ◽  
J. S. Hou ◽  
V. C. Mow
Keyword(s):  
Articular Cartilage ◽  
Charge Density ◽  
Fixed Charge ◽  
Solid Matrix ◽  
Confined Compression ◽  
Fixed Charge Density ◽  
Chemical Expansion ◽  
Chemical Potentials ◽  
Free Swelling ◽  
Triphasic Theory

Swelling of articular cartilage depends on its fixed charge density and distribution, the stiffness of its collagen-proteoglycan matrix, and the ion concentrations in the interstitium. A theory for a tertiary mixture has been developed, including the two fluid-solid phases (biphasic), and an ion phase, representing cation and anion of a single salt, to describe the deformation and stress fields for cartilage under chemical and/or mechanical loads. This triphasic theory combines the physico-chemical theory for ionic and polyionic (proteoglycan) solutions with the biphasic theory for cartilage. The present model assumes the fixed charge groups to remain unchanged, and that the counter-ions are the cations of a single salt of the bathing solution. The momentum equation for the neutral salt and for the intersitial water are expressed in terms of their chemical potentials whose gradients are the driving forces for their movements. These chemical potentials depend on fluid pressure p, salt concentration c, solid matrix dilatation e and fixed charge density cF. For a uni-uni valent salt such as NaCl, they are given by μi = μoi + (RT/Mi)ln[γ±2c (c + c F)] and μW = μow + [p − RTφ(2c + cF) + Bwe]/ρTw, where R, T, Mi, γ±, φ, ρTw and Bw are universal gas constant, absolute temperature, molecular weight, mean activity coefficient of salt, osmotic coefficient, true density of water, and a coupling material coefficient, respectively. For infinitesimal strains and material isotropy, the stress-strain relationship for the total mixture stress is σ = − pI − TcI + λs(trE)I + 2μsE, where E is the strain tensor and (λs,μs) are the Lame´ constants of the elastic solid matrix. The chemical-expansion stress (− Tc) derives from the charge-to-charge repulsive forces within the solid matrix. This theory can be applied to both equilibrium and non-equilibrium problems. For equilibrium free swelling problems, the theory yields the well known Donnan equilibrium ion distribution and osmotic pressure equations, along with an analytical expression for the “pre-stress” in the solid matrix. For the confined-compression swelling problem, it predicts that the applied compressive stress is shared by three load support mechanisms: 1) the Donnan osmotic pressure; 2) the chemical-expansion stress; and 3) the solid matrix elastic stress. Numerical calculations have been made, based on a set of equilibrium free-swelling and confined-compression data, to assess the relative contribution of each mechanism to load support. Our results show that all three mechanisms are important in determining the overall compressive stiffness of cartilage.

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