Finite Elastic Deformation for a Class of Soft Biological Tissues

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.

Proposal and validation of polyconvex strain-energy function for biological soft tissues

2021 ◽  
pp. 1-14
Author(s):  
Takashi Funai ◽  
Hiroyuki Kataoka ◽  
Hideo Yokota ◽  
Taka-aki Suzuki
Keyword(s):  
Curve Fitting ◽  
Energy Function ◽  
Strain Energy ◽  
Soft Tissues ◽  
Strain Energy Function ◽  
Biological Tissues ◽  
Stress Strain ◽  
Increasing Trend ◽  
Strain Energy Functions ◽  
Derived Function

BACKGROUND: Mechanical simulations for biological tissues are effective technology for development of medical equipment, because it can be used to evaluate mechanical influences on the tissues. For such simulations, mechanical properties of biological tissues are required. For most biological soft tissues, stress tends to increase monotonically as strain increases. OBJECTIVE: Proposal of a strain-energy function that can guarantee monotonically increasing trend of biological soft tissue stress-strain relationships and applicability confirmation of the proposed function for biological soft tissues. METHOD: Based on convexity of invariants, a polyconvex strain-energy function that can reproduce monotonically increasing trend was derived. In addition, to confirm its applicability, curve-fitting of the function to stress-strain relationships of several biological soft tissues was performed. RESULTS: A function depending on the first invariant alone was derived. The derived function does not provide such inappropriate negative stress in the tensile region provided by several conventional strain-energy functions. CONCLUSIONS: The derived function can reproduce the monotonically increasing trend and is proposed as an appropriate function for biological soft tissues. In addition, as is well-known for functions depending the first invariant alone, uniaxial-compression and equibiaxial-tension of several biological soft tissues can be approximated by curve-fitting to uniaxial-tension alone using the proposed function.

On the Choice of Strain Energy Function for Mechanical Characterisation of Soft Biological Tissues

1984 ◽  
Vol 13 (1) ◽  
pp. 11-14 ◽  
Author(s):  
K B Sahay
Keyword(s):  
Energy Function ◽  
Constitutive Equations ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Biological Tissues ◽  
Energy Functions ◽  
Soft Biological Tissues ◽  
Mechanical Characterisation ◽  
Strain Energy Functions ◽  
Made In

Constitutive equations that describe stress–strain relations of soft biological tissues require parameters such as the strain energy functions, or certain of their derivatives. An attempt has been made in this paper to examine the suitability of the various strain energy functions reported in the literature. Certain criteria are proposed for the same.

A Novel Approach Towards Modeling the Collagen Fibril Network for Use in Nonlinear Anisotropic Polyconvex Mixture Models of Articular Cartilage

2010 ◽  
Author(s):  
Reza Shirazi ◽  
Pasquale Vena ◽  
Robert L. Sah ◽  
Stephen M. Klisch
Keyword(s):  
Articular Cartilage ◽  
Energy Function ◽  
Strain Energy ◽  
Soft Tissues ◽  
Continuous Distribution ◽  
Strain Energy Function ◽  
Material Point ◽  
Distribution Functions ◽  
Anatomical Location ◽  
Novel Approach

Despite distinct mechanical functions, biological soft tissues have a common microstructure in which a ground matrix is reinforced by a collagen (COL) fibril network. The highly anisotropic, heterogeneous, and asymmetric material properties caused by the microstructural nature of the COL fibril network suggest the importance, as well as the challenges, of accurately modeling soft tissue biomechanics. For soft fibrous tissues with multiple constituents, mathematical distribution functions have represented dispersed and continuous (i.e. non-discrete) fibrils oriented in all directions depending on the type of (and anatomical location in) the tissue under investigation [1–2]. These types of continuous fibril models have been used recently for articular cartilage [3–5]. The strain energy of the COL fibril network is calculated based on the response of individual fibrils in tension in different directions and integrated over a unit sphere at a material point. The specific aims of the current study were to: 1. introduce a novel approach to modeling a continuous distribution of COL fibrils in soft tissues; 2. develop a strain energy function for the COL network based on the proposed distribution function of COL fibrils; 3. derive the stress and material elasticity tensors for the COL network that may be “pre-stressed” in a stress-free natural configuration of the tissue; 4. propose a special model that may be appropriate for immature tissue and establish its suitability for use in a polyconvex tissue strain energy function.

On the Constitutive Equations of Biological Materials

1975 ◽  
Vol 42 (1) ◽  
pp. 242-243 ◽  
Author(s):  
H. Demiray
Keyword(s):  
Energy Function ◽  
Constitutive Equations ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Transversely Isotropic ◽  
Biological Materials ◽  
Biological Tissues

This paper deals with a simple possible form of the strain-energy function for biological tissues which are assumed to be transversely isotropic. Also the solution of a problem is studied and the result is compared with experiments.

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

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.

Lamellar and Interlamellar Shear Compared Across Regions of the Annulus Fibrosus

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.

Isotropic incompressible hyperelastic models for modelling the mechanical behaviour of biological tissues: a review

2015 ◽  
Vol 60 (6) ◽  
Author(s):  
Cora Wex ◽  
Susann Arndt ◽  
Anke Stoll ◽  
Christiane Bruns ◽  
Yuliya Kupriyanova
Keyword(s):  
Mechanical Behaviour ◽  
Strain Energy ◽  
Soft Tissues ◽  
Strain Energy Function ◽  
Biological Tissues ◽  
Tissue Response ◽  
Large Strains ◽  
Energy Functions ◽  
Strain Energy Functions ◽  
Hyperelastic Models

AbstractModelling the mechanical behaviour of biological tissues is of vital importance for clinical applications. It is necessary for surgery simulation, tissue engineering, finite element modelling of soft tissues, etc. The theory of linear elasticity is frequently used to characterise biological tissues; however, the theory of nonlinear elasticity using hyperelastic models, describes accurately the nonlinear tissue response under large strains. The aim of this study is to provide a review of constitutive equations based on the continuum mechanics approach for modelling the rate-independent mechanical behaviour of homogeneous, isotropic and incompressible biological materials. The hyperelastic approach postulates an existence of the strain energy function – a scalar function per unit reference volume, which relates the displacement of the tissue to their corresponding stress values. The most popular form of the strain energy functions as Neo-Hookean, Mooney-Rivlin, Ogden, Yeoh, Fung-Demiray, Veronda-Westmann, Arruda-Boyce, Gent and their modifications are described and discussed considering their ability to analytically characterise the mechanical behaviour of biological tissues. The review provides a complete and detailed analysis of the strain energy functions used for modelling the rate-independent mechanical behaviour of soft biological tissues such as liver, kidney, spleen, brain, breast, etc.

Inversion of a Class of Nonlinear Stress-Strain Relationships of Biological Soft Tissues

1979 ◽  
Vol 101 (1) ◽  
pp. 23-27 ◽  
Author(s):  
Y. C. Fung
Keyword(s):  
Stress Tensor ◽  
Energy Function ◽  
Strain Energy ◽  
Soft Tissues ◽  
Strain Tensor ◽  
Nonlinear Function ◽  
Strain Energy Function ◽  
Stress Strain ◽  
Highly Nonlinear ◽  
Nonlinear Stress

The mechanical properly of soft tissues is highly nonlinear. Normally, the stress tensor is a nonlinear function of the strain tensor. Correspondingly, the strain energy function is not a quadratic function of the strain. The problem resolved in the present paper is to invert the stress-strain relationship so that the strain tensor can be expressed as a nonlinear function of the stress tensor. Correspondingly, the strain energy function is inverted into the complementary energy function which is a function of stresses. It is shown that these inversions can be done quite simply if the strain energy function is an analytic function of a polynomial of the strain components of the second degree. We have shown previously that experimental results on the skin, the blood vessels, the mesentery, and the lung tissue can be best described by strain energy functions of this type. Therefore, the inversion presented here is applicable to these tissues. On the other hand, a popular strain energy function, a polynomial of third degree or higher, cannot be so inverted.

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

2005 ◽  
Vol 73 (5) ◽  
pp. 815-824 ◽  
Author(s):  
X. Q. Peng ◽  
Z. Y. Guo ◽  
B. Moran
Keyword(s):  
Constitutive Model ◽  
Energy Function ◽  
Annulus Fibrosus ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Material Parameters ◽  
The Matrix ◽  
Fiber Matrix ◽  
Matrix Shear ◽  
Fiber Interaction

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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