An Anisotropic Hyperelastic Constitutive Model With Fiber-Matrix Shear Interaction for the Human Annulus Fibrosus

Based on fiber reinforced continuum mechanics theory, an anisotropic hyperelastic constitutive model for the human annulus fibrosus is developed. A strain energy function representing the anisotropic elastic material behavior of the annulus fibrosus is additively decomposed into three parts nominally representing the energy contributions from the matrix, fiber and fiber-matrix shear interaction, respectively. Taking advantage of the laminated structure of the annulus fibrosus with one family of aligned fibers in each lamella, interlamellar fiber-fiber interaction is eliminated, which greatly simplifies the constitutive model. A simple geometric description for the shearing between the fiber and the matrix is developed and this quantity is used in the representation of the fiber-matrix shear interaction energy. Intralamellar fiber-fiber interaction is also encompassed by this interaction term. Experimental data from the literature are used to obtain the material parameters in the constitutive model and to provide model validation. Determination of the material parameters is greatly facilitated by the partition of the strain energy function into matrix, fiber and fiber-matrix shear interaction terms. A straightforward procedure for computation of the material parameters from simple experimental tests is proposed.

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Finite Elastic Deformation for a Class of Soft Biological Tissues

Journal of Biomechanical Engineering ◽

10.1115/1.3426279 ◽

1977 ◽

Vol 99 (2) ◽

pp. 98-103

Author(s):

Han-Chin Wu ◽

R. Reiss

Keyword(s):

Energy Function ◽

Annulus Fibrosus ◽

Strain Energy ◽

Soft Tissues ◽

Broad Class ◽

Strain Energy Function ◽

Biological Tissues ◽

Tension Test ◽

Soft Biological Tissues ◽

Precise Form

The stress response of soft biological tissues is investigated theoretically. The treatment follows the approach of Wu and Yao [1] and is now extended for a broad class of soft tissues. The theory accounts for the anisotropy due to the presence of fibers and also allows for the stretching of fibers under load. As an application of the theory, a precise form for the strain energy function is proposed. This form is then shown to describe the mechanical behavior of annulus fibrosus satisfactorily. The constants in the strain energy function have also been approximately determined from only a uniaxial tension test.

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A passive strain-energy function for elastic and muscular arteries: correlation of material parameters with histological data

Medical & Biological Engineering & Computing ◽

10.1007/s11517-010-0598-x ◽

2010 ◽

Vol 48 (6) ◽

pp. 507-518 ◽

Cited By ~ 25

Author(s):

Dimitrios P. Sokolis

Keyword(s):

Energy Function ◽

Strain Energy ◽

Strain Energy Function ◽

Material Parameters ◽

Histological Data

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Application and Research of Numerical Simulation for Rubber Like Material with MSC.Marc

Materials Science Forum ◽

10.4028/www.scientific.net/msf.575-578.854 ◽

2008 ◽

Vol 575-578 ◽

pp. 854-858

Author(s):

Jian Bing Sang ◽

Bo Liu ◽

Zhi Liang Wang ◽

Su Fang Xing ◽

Jie Chen

Keyword(s):

Numerical Simulation ◽

Energy Function ◽

Strain Energy ◽

Strain Energy Function ◽

Stress And Strain ◽

Perfect Agreement ◽

Energy Functions ◽

Material Parameters ◽

User Subroutine ◽

Strain Energy Functions

This paper starts with a discussion on the theory of finite deformation and various types strain energy functions of rubber like material, the material parameter of elastic law of Gao[3] is estimated by experiment and numerical simulation. Because there are various types of strain energy functions, a user subroutine is programmed to implement the strain energy function of Gao[3] into the program of MSC.Marc, which offers a convenient method to analyze the stress and strain of rubber-like material with the strain energy function that is needed. Two examples will be presented in this paper to demonstrate the use of the framework for rubber like materials. One is to simulate a foam tube in compression. The other one is to simulate a rectangle board with a circular hole. After numerical analysis, it is proved the numerical results based on Gao model are in perfect agreement with the results based on Mooney model and the estimated material parameters are valid.

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A Simplified Strain Energy Function to Represent the Mechanical Behavior of the Annulus Fibrosus

ASME 2009 Summer Bioengineering Conference, Parts A and B ◽

10.1115/sbc2009-206261 ◽

2009 ◽

Author(s):

Jose J. García ◽

Christian Puttlitz

Keyword(s):

Finite Element ◽

Mechanical Behavior ◽

Energy Function ◽

Annulus Fibrosus ◽

Strain Energy ◽

Degrees Of Freedom ◽

Strain Energy Function ◽

Efficient Simulation ◽

Matrix Interactions ◽

Complex Functions

Models to represent the mechanical behavior of the annulus fibrosus are important tools to understand the biomechanics of the spine. Many hyperelastic constitutive equations have been proposed to simulate the mechanical behavior of the annulus that incorporate the anisotropic nature of the tissue. Recent approaches [1,2] have included terms into the energy function which take into account fiber-fiber and fiber-matrix interactions, leading to complex functions that cannot be readily implemented into commercial finite element codes for an efficient simulation of nonlinear realistic models of the spine (which are generally composed of 100,000+ degrees of freedom). An effort is undertaken here to test the capability of a relatively simple strain energy function [3] for the description of the annulus fibrosus. This function has already been shown to successfully represent the mechanical behavior of the arterial tissue and can be readily implemented into existing finite element codes.

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Lamellar and Interlamellar Shear Compared Across Regions of the Annulus Fibrosus

Volume 1B: Extremity; Fluid Mechanics; Gait; Growth, Remodeling, and Repair; Heart Valves; Injury Biomechanics; Mechanotransduction and Sub-Cellular Biophysics; MultiScale Biotransport; Muscle, Tendon and Ligament; Musculoskeletal Devices; Multiscale Mechanics; Thermal Medicine; Ocular Biomechanics; Pediatric Hemodynamics; Pericellular Phenomena; Tissue Mechanics; Biotransport Design and Devices; Spine; Stent Device Hemodynamics; Vascular Solid Mechanics; Student Paper and Design Competitions ◽

10.1115/sbc2013-14309 ◽

2013 ◽

Author(s):

K. M. Labus ◽

A. H. Hsieh ◽

C. M. Puttlitz

Keyword(s):

Intervertebral Disc ◽

Energy Function ◽

Degenerative Disc Disease ◽

Annulus Fibrosus ◽

Strain Energy ◽

Computational Models ◽

Strain Energy Function ◽

Disc Disease ◽

Interlamellar Region ◽

Single Material

Computational models of the intervertebral disc commonly use continuum descriptions that treat the annulus fibrosus as a single material rather than discretely modeling the lamellae and interlamellar interactions [1,2]. However, modeling the mechanics of individual lamellae and the interlamellar region can aid in the understanding of degenerative disc disease and its treatment. Previous work has demonstrated that fibrous connections between lamellae as well as bridges spanning across layers exist, but the mechanical contributions of these structures have largely remained uncharacterized [3]. Studying interlamellar shear mechanics may provide insights into the structure-function relationships of the annulus. The purpose of this study was to compare the mechanical shear in the interlamellar and lamellar regions, model the stress-stretch relationships of these areas utilizing a hyperelastic strain energy function, and compare the shear properties across multiple locations of the intervertebral disc.

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Nonlinear Thermohyperviscoelastic Constitutive Model for Soft Materials with Strain Rate and Temperature Dependency

International Journal of Applied Mechanics ◽

10.1142/s1758825120500593 ◽

2020 ◽

Vol 12 (06) ◽

pp. 2050059

Author(s):

Zahra Matin Ghahfarokhi ◽

Mehdi Salmani-Tehrani ◽

Mahdi Moghimi Zand

Keyword(s):

Constitutive Model ◽

Strain Rate ◽

Energy Function ◽

Strain Energy ◽

Strain Energy Function ◽

Soft Materials ◽

Material Behavior ◽

Temperature Dependent ◽

Dependent Behavior ◽

Temperature Dependent Behavior

Soft materials, such as polymeric materials and biological tissues, often exhibit strain rate and temperature-dependent behavior when subjected to external loads. To characterize the thermomechanical behavior of isotropic soft material, a thermohyperviscoelastic constitutive model has been developed through an additive decomposition of strain energy function into elastic and viscous parts. A three-term generalized Rivlin strain energy function is utilized to formulate the hyperelastic part of the model, while a new viscous potential function is proposed to describe the effect of strain rate and temperature on material behavior. Toward this end, a new procedure has been proposed to determine the viscous mechanical properties as a function of strain-rate and temperature. Comparing with the previously published experimental data for linear low-density polyethylene reveals that the proposed model can sufficiently capture the nonlinearity, rate- and temperature-dependent behavior of the soft materials.

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A Microstructurally Based Orthotropic Hyperelastic Constitutive Law

Journal of Applied Mechanics ◽

10.1115/1.1485754 ◽

2002 ◽

Vol 69 (5) ◽

pp. 570-579 ◽

Cited By ~ 126

Author(s):

J. E. Bischoff ◽

E. A. Arruda ◽

K. Grosh

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Constitutive Model ◽

Energy Function ◽

Strain Energy ◽

Strain Energy Function ◽

Entropy Change ◽

Constitutive Law ◽

Elastic Behavior ◽

The Finite Element Method

A constitutive model is developed to characterize a general class of polymer and polymer-like materials that displays hyperelastic orthotropic mechanical behavior. The strain energy function is derived from the entropy change associated with the deformation of constituent macromolecules and the strain energy change associated with the deformation of a representative orthotropic unit cell. The ability of this model to predict nonlinear, orthotropic elastic behavior is examined by comparing the theory to experimental results in the literature. Simulations of more complicated boundary value problems are performed using the finite element method.

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Extending Marlow’s general first-invariant constitutive model to compressible, isotropic hyperelastic materials

Engineering Computations ◽

10.1108/ec-05-2020-0251 ◽

2020 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Florian Hüter ◽

Frank Rieg

Keyword(s):

Finite Element ◽

Constitutive Model ◽

Test Data ◽

Curve Fitting ◽

Energy Function ◽

Strain Energy ◽

Strain Energy Function ◽

Content Type ◽

Hyperelastic Materials ◽

Fitting Procedures

Purpose A general first-invariant constitutive model has been derived in literature for incompressible, isotropic hyperelastic materials, known as Marlow model, which reproduces test data exactly without the need of curve-fitting procedures. This paper aims to describe how to extend Marlow’s constitutive model to the more general case of compressible hyperelastic materials. Design/methodology/approach The isotropic constitutive model is based on a strain energy function, whose isochoric part is solely dependent on the first modified strain invariant. Based on Marlow’s idea, a principle of energetically equivalent deformation states is derived for the compressible case, which is used to determine the underlying strain energy function directly from measured test data. No particular functional of the strain energy function is assumed. It is shown how to calibrate the volumetric and isochoric strain energy functions uniquely with uniaxial or biaxial test data only. The constitutive model is implemented into a finite element program to demonstrate its applicability. Findings The model is well suited for use in finite element analysis. Only one set of test data is required for calibration without any need for curve-fitting procedures. These test data are reproduced exactly, and the model prediction is reasonable for other deformation modes. Originality/value Marlow’s basic concept is extended to the compressible case and applied to both the volumetric and isochoric part of the compressible strain energy function. Moreover, a novel approach is described on how both compressive and tensile test data can be used simultaneously to calibrate the model.

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A Strain Energy Function for the Annulus Fibrosus Specified Using Biaxial Experimental Data

10.1115/imece1999-0457 ◽

1999 ◽

Author(s):

Elisa C. Bass ◽

Jeffrey C. Lotz

Keyword(s):

Energy Function ◽

Annulus Fibrosus ◽

Strain Energy ◽

Mechanical Response ◽

Biaxial Tension ◽

Elastic Strain Energy ◽

Strain Energy Function ◽

Constitutive Formulation

Abstract The mechanical behavior of the annulus fibrosus has typically been characterized through the use of uniaxial tests. In contrast, its in vivo constraints are multiaxial and likely result in a mechanical response very different from that observed to date in vitro. The goal of this study was to test the annulus in biaxial tension and use these data to determine an elastic strain energy function for the annulus. Our results demonstrate that the mechanical response of the annulus is dramatically influenced by a biaxial constraint, and that these experiments provide important data for the determination of the constitutive formulation for this strongly anisotropic and nonlinear tissue.

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