Direct Eulerian formulation of anisotropic hyperelasticity

2020 ◽  
pp. 1-6
Author(s):  
Konstantin Volokh
Keyword(s):  
Constitutive Equations ◽  
Transverse Isotropy ◽  
Strain Tensor ◽  
Soft Materials ◽  
Biological Tissues ◽  
Current Configuration ◽  
Eulerian Strain ◽  
Eulerian Formulation ◽  
Anisotropic Hyperelasticity ◽  
Structure Tensors

Abstract Abstract Many soft materials and biological tissues comprise isotropic matrix reinforced by fibers in the characteristic directions. Hyperelastic constitutive equations for such materials are usually formulated in terms of a Lagrangean strain tensor referred to the initial configuration and Lagrangean structure tensors defining characteristic directions of anisotropy. Such equations are “pushed forward” to the current configuration. Obtained in this way, Eulerian constitutive equations are often favorable from both theoretical and computational standpoints. Abstract In the present note, we show that the described two-step procedure is not necessary and anisotropic hyperelasticity can be introduced directly in terms of an Eulerian strain tensor and Eulerian structure tensors referring to the current configuration. The newly developed constitutive equation is further applied to the particular case of the transverse isotropy for the sake of illustration.

Topics in Finite Elasticity: Hyperelasticity of Rubber, Elastomers, and Biological Tissues—With Examples

1987 ◽  
Vol 40 (12) ◽  
pp. 1699-1734 ◽  
Author(s):  
Millard F. Beatty
Keyword(s):  
Constitutive Equations ◽  
Mechanical Energy ◽  
Finite Elasticity ◽  
Physical Nature ◽  
Decomposition Theorem ◽  
Elastic Stability ◽  
Biological Tissues ◽  
Field Equations ◽  
Energy Principle ◽  
Hyperelastic Materials

This is an introductory survey of some selected topics in finite elasticity. Virtually no previous experience with the subject is assumed. The kinematics of finite deformation is characterized by the polar decomposition theorem. Euler’s laws of balance and the local field equations of continuum mechanics are described. The general constitutive equation of hyperelasticity theory is deduced from a mechanical energy principle; and the implications of frame invariance and of material symmetry are presented. This leads to constitutive equations for compressible and incompressible, isotropic hyperelastic materials. Constitutive equations studied in experiments by Rivlin and Saunders (1951) for incompressible rubber materials and by Blatz and Ko (1962) for certain compressible elastomers are derived; and an equation characteristic of a class of biological tissues studied in primary experiments by Fung (1967) is discussed. Sample applications are presented for these materials. A balloon inflation experiment is described, and the physical nature of the inflation phenomenon is examined analytically in detail. Results for the different materials are compared. Two major problems of finite elasticity theory are discussed. Some results concerning Ericksen’s problem on controllable deformations possible in every isotropic hyperelastic material are outlined; and examples are presented in illustration of Truesdell’s problem concerning analytical restrictions imposed on constitutive equations. Universal relations valid for all compressible and incompressible, isotropic materials are discussed. Some examples of non-uniqueness, including that of a neo-Hookean cube subject to uniform loads over its faces, are described. Elastic stability criteria and their connection with uniqueness in the theory of small deformations superimposed on large deformations are introduced, and a few applications are mentioned. Some previously unpublished results are presented throughout.

scholarly journals Mesoscale bicontinuous networks in self-healing hydrogels delay fatigue fracture

2020 ◽  
Vol 117 (14) ◽  
pp. 7606-7612 ◽  
Author(s):  
Xueyu Li ◽  
Kunpeng Cui ◽  
Tao Lin Sun ◽  
Lingpu Meng ◽  
Chengtao Yu ◽  
...  
Keyword(s):  
Fatigue Fracture ◽  
Polymer Network ◽  
Hierarchical Structures ◽  
Soft Materials ◽  
Biological Tissues ◽  
Crack Advance ◽  
Self Healing ◽  
Hard Phase ◽  
Soft Phase ◽  
Simple Model System

Load-bearing biological tissues, such as muscles, are highly fatigue-resistant, but how the exquisite hierarchical structures of biological tissues contribute to their excellent fatigue resistance is not well understood. In this work, we study antifatigue properties of soft materials with hierarchical structures using polyampholyte hydrogels (PA gels) as a simple model system. PA gels are tough and self-healing, consisting of reversible ionic bonds at the 1-nm scale, a cross-linked polymer network at the 10-nm scale, and bicontinuous hard/soft phase networks at the 100-nm scale. We find that the polymer network at the 10-nm scale determines the threshold of energy release rateG0above which the crack grows, while the bicontinuous phase networks at the 100-nm scale significantly decelerate the crack advance until a transitionGtranfar aboveG0. In situ small-angle X-ray scattering analysis reveals that the hard phase network suppresses the crack advance to show decelerated fatigue fracture, andGtrancorresponds to the rupture of the hard phase network.

scholarly journals Image-based analysis of uniaxial ring test for mechanical characterization of soft materials and biological tissues

Soft Matter ◽  
2019 ◽  
Vol 15 (16) ◽  
pp. 3353-3361 ◽  
Author(s):  
Eline E. van Haaften ◽  
Mark C. van Turnhout ◽  
Nicholas A. Kurniawan
Keyword(s):  
Mechanical Properties ◽  
Mechanical Characterization ◽  
Soft Materials ◽  
Biological Tissues ◽  
Ring Test ◽  
Analysis Approach

We propose a simple image-based analysis approach to accurately estimate the mechanical properties of ring-shaped materials.

Strain-Rate Sensitive Constitutive Modeling of Anisotropic Visco-Hyperelastic Materials

2012 ◽  
Author(s):  
Sahand Ahsanizadeh ◽  
LePing Li
Keyword(s):  
Strain Rate ◽  
Evolution Equations ◽  
Mechanical Response ◽  
Viscoelastic Behavior ◽  
Biological Tissues ◽  
Internal Variables ◽  
Current Configuration ◽  
Hyperelastic Materials ◽  
History Of ◽  
Viscoelastic Constitutive Model

Integral-based formulations of viscoelasticity have been widely used to describe the mechanical behavior of soft biological tissues and polymers. However, it is suggested that they are not suitable to be used under high strain rates. On the other hand, strain-rate sensitive models with an explicit dependence on the strain-rate have been developed for a certain class of materials. They predict the viscoelastic behavior during ramp loading more accurately while fail to account for the relaxation response. In order to overcome these drawbacks, a viscoelastic constitutive model has been proposed in this study based on the concept of internal variables. While the behavior of elastic materials is uniquely determined by the current state of deformation or external variables, the mechanical response of inelastic materials are regulated also by internal variables. The internal variables are associated with the dissipative mechanisms in the material and along with the evolution equations introduce the effect of history of the deformation to the current configuration. The current study employs short-term and long-term internal variables to account for the viscoelastic response during loading and relaxation respectively.

scholarly journals A novel apparatus and methodology for the high frequency mechanical characterisation of ultrasoft materials

2021 ◽  
Vol 250 ◽  
pp. 01033
Author(s):  
Aaron Graham ◽  
Clive R Siviour
Keyword(s):  
Mechanical Response ◽  
Soft Materials ◽  
Biological Tissues ◽  
Image Correlation ◽  
Full Field ◽  
Mechanical Characterisation ◽  
Wide Range ◽  
Experimental Apparatus ◽  
Low Modulus ◽  
Elastic Shear

Characterising the mechanical response of ultra-soft materials is challenging, particularly at high strain rates and frequencies [1]. Time Temperature Superposition (TTS) can sometimes be used to mitigate these limitations [2], however not all materials are suitable for TTS. Biological tissues are particularly difficult to test: in addition to the extreme softness, challenges arise due to specimen inhomogeneity, sensitivity to boundary conditions, natural biological variability, and complex post-mortem changes. In the current study, a novel experimental apparatus and methodology was developed and validated using low modulus silicone elastomers as model materials. The full field visco-elastic shear response was characterised over a wide range of deformation frequencies (100-1000+ Hz) and amplitudes using Digital Image Correlation (DIC) and the Virtual Fields Method (VFM). This methodology allows for the extraction of fullfield material properties that would be difficult or impossible to obtain using traditional engineering techniques.

scholarly journals On some method of the heredity kernel parameters determination of nonlinear viscoelastic materials under the complex stress state

2019 ◽  
pp. 178-181
Author(s):  
V. S. Reznik
Keyword(s):  
Stress State ◽  
Constitutive Equations ◽  
Strain Tensor ◽  
Complex Stress ◽  
Complex Stress State ◽  
Nonlinear Viscoelastic ◽  
Nonlinear Viscoelastic Materials ◽  
The Relationship ◽  
Kernel Parameters

The deformation of viscoelastic medium given by means of constitutive equations of the hereditary type. These equations establish the relationship between the components of strain tensor, the components of stress tensor and the integral time operator, and contain the set of function and coefficients that are determined from the basic experiments. А method of the heredity kernel parameters determination of nonlinear viscoelastic materials is developed. As the visco-elastic model, the constitutive equations of the hereditary type are chosen in which the relationship between the components of the strain tensor and the stress tensor is given based on the hypothesis of the deviators proportionality. The nonlinearity of the viscoelastic properties is given by the equations of Ratotnov’s type. The method is based on the relations between the creep kernels under complex stress state and the creep kernels under one-dimensional stress state. The method verified experimentally for the problems of determination of creep deformations under combined loading applied to the thin-walled tubular elements made of polyethylene of high density.

scholarly journals An Eulerian formulation of inelasticity: from metal plasticity to growth of biological tissues

2019 ◽  
Vol 377 (2144) ◽  
pp. 20180071 ◽  
Author(s):  
M. B. Rubin
Keyword(s):  
Applied Mathematics ◽  
Biological Tissues ◽  
Total Deformation ◽  
Theme Issue ◽  
Medical Treatments ◽  
Metal Plasticity ◽  
Related Disease ◽  
Eulerian Formulation ◽  
Control Growth ◽  
Intermediate Configuration

The purpose of this paper is to review and contrast the Lagrangian and Eulerian formulations of inelasticity as they apply to metal plasticity and growth of biological tissues. In contrast with the Lagrangian formulation of inelasticity, the Eulerian formulation is unaffected by arbitrary choices of the reference configuration, an intermediate configuration, a total deformation measure and an inelastic deformation measure. Although the Eulerian formulation for growth of biological tissues includes a rate of mass supply and can be used to understand the mechanics of growth, it does not yet model essential mechanobiological processes that control growth. Much research is needed before this theory can help design medical treatments for growth related disease. This article is part of the theme issue ‘Rivlin's legacy in continuum mechanics and applied mathematics’.

Residual strain effects in needle-induced cavitation

Soft Matter ◽  
2019 ◽  
Vol 15 (37) ◽  
pp. 7390-7397 ◽  
Author(s):  
Christopher W. Barney ◽  
Yue Zheng ◽  
Shuai Wu ◽  
Shengqiang Cai ◽  
Alfred J. Crosby
Keyword(s):  
Residual Strain ◽  
Soft Materials ◽  
Biological Tissues ◽  
Strain Effects ◽  
Fracture Properties

Needle-induced cavitation (NIC) locally probes the elastic and fracture properties of soft materials, such as gels and biological tissues.

Brain Tissue Modelling: Calibration and Validation of a Fractional Calculus Based Viscoelastic Constitutive Equation

2011 ◽  
Author(s):  
Vincent Libertiaux ◽  
Serge Cescotto
Keyword(s):  
White Matter ◽  
Constitutive Equation ◽  
Fractional Calculus ◽  
Constitutive Equations ◽  
Neural Tissue ◽  
Biological Tissues ◽  
Macroscopic Description ◽  
Calibration And Validation ◽  
The Brain ◽  
Tissue Modelling

Constitutive equations for biological tissues can be sorted in two categories: structural and phenomenological. The first approach is based on the microstructure of the tissue while the second relies on a macroscopic description. Due to the nature of the neural tissue, structural approach is rarely used. Meaney [1] proposed a model for the white matter of the brain.

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.

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