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.

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.

On the Ability of Structural and Phenomenological Hyperelastic Models to Predict the Mechanical Behavior of Biological Tissues Submitted to Multiaxial Loadings

2012 ◽  
Author(s):  
Mathieu Nierenberger ◽  
Yves Rémond ◽  
Saïd Ahzi
Keyword(s):  
Experimental Data ◽  
Mechanical Behavior ◽  
Strain Energy ◽  
Soft Tissues ◽  
Least Squares Method ◽  
Biological Tissues ◽  
Physical Parameters ◽  
Cauchy Stress ◽  
Specific Loading ◽  
Hyperelastic Models

Medical surgery is currently rapidly improving and requires modeling faithfully the mechanical behavior of soft tissues. Various models exist in literature; some of them created for the study of biological materials, and others coming from the field of rubber mechanics. Indeed biological tissues show a mechanical behavior close to the one of rubbers. But while building a model, one has to keep in mind that its parameters should be loading independent and that the model should be able to predict the behavior under complex loading conditions. In addition, keeping physical parameters seems interesting since it allows a bottom up approach taking into account the microstructure of the material. In this study, the authors consider different existing hyperelastic models based on strain energy functions and identify their coefficients successively on single loading stress-stretch curves. The experimental data used, come from a paper by Zemanek dated 2009 and concerning uniaxial, equibiaxial and plane tension tests on porcine arterial walls taken in identical experimental conditions. To achieve identification, the strain energy function of each model is derived differently to provide an expression of the Cauchy stress associated to each loading case. Firstly the parameters of each model are identified on the uniaxial tension curve using a least squares method. Then, keeping the obtained parameters, predictions are made for the two other loading cases (equibiaxial and plane tension) using the associated expressions of stresses. A comparison of these predictions with experimental data is done and allows evaluating the predictive capabilities of each model for the different loading cases. A similar approach is used after swapping the loading types. Since the predictive capabilities of the models are really dependent on the loading chosen to determine their parameters, another type of identification procedure is set up. It consists in adding the residues over the three loading cases during identification. This alternative identification method allows a better agreement between each model and the various types of experiments. This study evaluated the ability of some classical hyperelastic models to be used for a predictive scope after being identified on a specific loading type. Besides it brought to light some existing models which can describe at best the mechanical behavior of biological tissues submitted to various loadings.

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.

scholarly journals Constitutive modelling of arteries

2010 ◽  
Vol 466 (2118) ◽  
pp. 1551-1597 ◽  
Author(s):  
Gerhard A. Holzapfel ◽  
Ray W. Ogden
Keyword(s):  
Mechanical Behaviour ◽  
Soft Tissues ◽  
Elastic Strain Energy ◽  
Cylindrical Tube ◽  
Strain Energy Function ◽  
Biological Tissues ◽  
Constitutive Modelling ◽  
Highly Nonlinear ◽  
Wide Range ◽  
Modelling Effort

This review article is concerned with the mathematical modelling of the mechanical properties of the soft biological tissues that constitute the walls of arteries. Many important aspects of the mechanical behaviour of arterial tissue can be treated on the basis of elasticity theory, and the focus of the article is therefore on the constitutive modelling of the anisotropic and highly nonlinear elastic properties of the artery wall. The discussion focuses primarily on developments over the last decade based on the theory of deformation invariants, in particular invariants that in part capture structural aspects of the tissue, specifically the orientation of collagen fibres, the dispersion in the orientation, and the associated anisotropy of the material properties. The main features of the relevant theory are summarized briefly and particular forms of the elastic strain-energy function are discussed and then applied to an artery considered as a thick-walled circular cylindrical tube in order to illustrate its extension–inflation behaviour. The wide range of applications of the constitutive modelling framework to artery walls in both health and disease and to the other fibrous soft tissues is discussed in detail. Since the main modelling effort in the literature has been on the passive response of arteries, this is also the concern of the major part of this article. A section is nevertheless devoted to reviewing the limited literature within the continuum mechanics framework on the active response of artery walls, i.e. the mechanical behaviour associated with the activation of smooth muscle, a very important but also very challenging topic that requires substantial further development. A final section provides a brief summary of the current state of arterial wall mechanical modelling and points to key areas that need further modelling effort in order to improve understanding of the biomechanics and mechanobiology of arteries and other soft tissues, from the molecular, to the cellular, tissue and organ levels.

On the Development of Volumetric Strain Energy Functions

1999 ◽  
Vol 67 (1) ◽  
pp. 17-21 ◽  
Author(s):  
S. Doll ◽  
K. Schweizerhof
Keyword(s):  
Strain Energy ◽  
Volumetric Strain ◽  
Strain Energy Function ◽  
Material Behavior ◽  
Additional Material ◽  
Energy Functions ◽  
Material Parameters ◽  
Physical Conditions ◽  
Starting Point ◽  
Strain Energy Functions

To describe elastic material behavior the starting point is the isochoric-volumetric decoupling of the strain energy function. The volumetric part is the central subject of this contribution. First, some volumetric functions given in the literature are discussed with respect to physical conditions, then three new volumetric functions are developed which fulfill all imposed conditions. One proposed function which contains two material parameters in addition to the compressibility parameter is treated in detail. Some parameter fits are carried out on the basis of well-known volumetric strain energy functions and experimental data. A generalization of the proposed function permits an unlimited number of additional material parameters.  Dedicated to Professor Franz Ziegler on the occasion of his 60th birthday. [S0021-8936(00)00901-6]

Finding the Constitutive Relation for a Specific Elastomer

1993 ◽  
Vol 115 (3) ◽  
pp. 329-336 ◽  
Author(s):  
Yun Ling ◽  
Peter A. Engel ◽  
Wm. L. Brodskey ◽  
Yifan Guo
Keyword(s):  
Experimental Data ◽  
Energy Function ◽  
Strain Energy ◽  
Pure Shear ◽  
Strain Energy Function ◽  
Invariant Polynomial ◽  
Energy Functions ◽  
Finite Element Program ◽  
Biaxial Extension ◽  
Strain Energy Functions

The main purpose of this study was to determine a suitable strain energy function for a specific elastomer. A survey of various strain energy functions proposed in the past was made. For natural rubber, there were some specific strain energy functions which could accurately fit the experimental data for various types of deformations. The process of determining a strain energy function for the specific elastomer was then described. The second-order invariant polynomial strain energy function (James et al., 1975) was found to give a good fit to the experimental data of uniaxial tension, uniaxial compression, equi-biaxial extension, and pure shear. A new form of strain energy function was proposed; it yielded improved results. The equi-biaxial extension experiment was done in a novel way in which the moire techniques (Pendleton, 1989) were used. The obtained strain energy functions were then utilized in a finite element program to calculate the load-deflection relation of an electrometric spring used in an electrical connector.

Neural Network Based Constitutive Model for Rubber Material

2004 ◽  
Vol 77 (2) ◽  
pp. 257-277 ◽  
Author(s):  
Y. Shen ◽  
K. Chandrashekhara ◽  
W. F. Breig ◽  
L. R. Oliver
Keyword(s):  
Neural Network ◽  
Curve Fitting ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Phenomenological Approach ◽  
Neural Network Approach ◽  
Energy Functions ◽  
The Neural Network ◽  
Strain Energy Functions

Abstract Rubber hyperelasticity is characterized by a strain energy function. The strain energy functions fall primarily into two categories: one based on statistical thermodynamics, the other based on the phenomenological approach of treating the material as a continuum. This work is focused on the phenomenological approach. To determine the constants in the strain energy function by this method, curve fitting of rubber test data is required. A review of the available strain energy functions based on the phenomenological approach shows that it requires much effort to obtain a curve fitting with good accuracy. To overcome this problem, a novel method of defining rubber strain energy function by Feedforward Backpropagation Neural Network is presented. The calculation of strain energy and its derivatives by neural network is explained in detail. The preparation of the neural network training data from rubber test data is described. Curve fitting results are given to show the effectiveness and accuracy of the neural network approach. A material model based on the neural network approach is implemented and applied to the simulation of V-ribbed belt tracking using the commercial finite element code ABAQUS.

Finite Bending and Straightening of Hyperelastic Materials: Analytical Solution and FEM

2019 ◽  
Vol 11 (09) ◽  
pp. 1950084 ◽  
Author(s):  
Sara Sheikhi ◽  
Mohammad Shojaeifard ◽  
Mostafa Baghani
Keyword(s):  
Finite Element ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Optimization Procedure ◽  
Element Analysis ◽  
Energy Functions ◽  
Hyperelastic Materials ◽  
Finite Bending ◽  
Strain Energy Functions ◽  
Analytical Approaches

In this research, an incompressible, isotropic, nonlinear elastic rectangular block and a circular cylindrical sector are studied under bending and straightening moments, respectively. Analytical approaches are presented on implementing of the left Cauchy–Green tensor and Cauchy stresses. In addition, finite element analysis of both problems is carried out using UHYPER user-defined subroutine in ABAQUS to verify the analytical methods. Four different invariant-based strain energy functions, including neo-Hookean, Mooney–Rivlin, Arruda–Boyce, and recently proposed polynomial Exp-Exp models, are examined, and the results are compared. Material parameters of silicon rubber for the strain energy functions are identified by applying an optimization procedure. Finite element method results confirmed the analytical approach with great compatibility. Results showed that the length of the unbent beam does not affect the stress. Likewise, the initial angle of curved structure does not affect the unbending moment and stresses. Moreover, the Exp-Exp model had a slightly different result rather than other strain energies, which means that this model is more conservative than its counterparts. Furthermore, the Exp-Exp strain energy function is calibrated for tissue-like phantom and is compared with experimental data.

scholarly journals A generalized strain approach to anisotropic elasticity

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
M. H. B. M. Shariff
Keyword(s):  
Experimental Data ◽  
Energy Function ◽  
Constitutive Equations ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Single Variable ◽  
Energy Functions ◽  
Anisotropic Strain ◽  
Strain Energy Functions ◽  
Strain Function

AbstractThis work proposes a generalized Lagrangian strain function $$f_\alpha$$ f α (that depends on modified stretches) and a volumetric strain function $$g_\alpha$$ g α (that depends on the determinant of the deformation tensor) to characterize isotropic/anisotropic strain energy functions. With the aid of a spectral approach, the single-variable strain functions enable the development of strain energy functions that are consistent with their infinitesimal counterparts, including the development of a strain energy function for the general anisotropic material that contains the general 4th order classical stiffness tensor. The generality of the single-variable strain functions sets a platform for future development of adequate specific forms of the isotropic/anisotropic strain energy function; future modellers only require to construct specific forms of the functions $$f_\alpha$$ f α and $$g_\alpha$$ g α to model their strain energy functions. The spectral invariants used in the constitutive equation have a clear physical interpretation, which is attractive, in aiding experiment design and the construction of specific forms of the strain energy. Some previous strain energy functions that appeared in the literature can be considered as special cases of the proposed generalized strain energy function. The resulting constitutive equations can be easily converted, to allow the mechanical influence of compressed fibres to be excluded or partial excluded and to model fibre dispersion in collagenous soft tissues. Implementation of the constitutive equations in Finite Element software is discussed. The suggested crude specific strain function forms are able to fit the theory well with experimental data and managed to predict several sets of experimental data.

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