A Strain Energy Function for Lung Parenchyma

1985 ◽  
Vol 107 (1) ◽  
pp. 81-86 ◽  
Author(s):  
D. Stamenovic ◽  
T. A. Wilson
Keyword(s):  
Surface Energy ◽  
Strain Energy ◽  
Large Deformations ◽  
Strain Energy Function ◽  
Lung Parenchyma ◽  
Elastic Behavior ◽  
Recoil Pressure ◽  
Linear Elastic ◽  
Tissue System ◽  
Line Elements

The strain energy for the air-filled lung is calculated from a model of the parenchymal microstructure. The energy is the sum of the surface energy and the elastic energies of two tissue components. The first of these is the peripheral tissue system that provides the recoil pressure of the saline-filled lung, and the second is the system of line elements that form the free edges of the alveolar walls bordering the alveolar ducts. The computed strain energy is consistent with the observed linear elastic behavior of parenchyma and the data on large deformations around blood vessels.

Mechanical properties of pleural membrane

1984 ◽  
Vol 57 (4) ◽  
pp. 1189-1194 ◽  
Author(s):  
D. Stamenovic
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Large Deformations ◽  
Strain Energy Function ◽  
Shear Forces ◽  
Stress Strain ◽  
Oriented Line ◽  
Strain Equation ◽  
Line Elements ◽  
Good Agreement

The pleural membrane is modeled as a planar collection of interconnected randomly oriented line elements. By assuming that the line elements follow the strain field of a continuum, a strain-energy function is formulated. From the strain-energy function, an explicit stress-strain equation for large deformations is derived. In the linear approximation of the stress-strain equation the shear modulus and the area modulus of the membrane are respectively found to be 2.4 and 2.8 times the tension at the reference state. The stress-strain equation for large deformations is used to predict the displacement field around a circular hole in pleura. Good agreement is found between these predictions and measurements made on ablated pleura from dog lungs. From these theoretical and experimental results the conclusion is drawn that the pleura has a significant role in carrying shear forces and maintaining the lung's shape.

The Habit Plate Shift and the Morphology of ∝" Nitride Precipitate in ∝-Fe*

1982 ◽  
Vol 40 ◽  
pp. 728-729
Author(s):  
Y. C. Shih ◽  
J. W. Morris
Keyword(s):  
Surface Energy ◽  
Habit Plane ◽  
Elastic Strain ◽  
Strain Energy ◽  
Lattice Mismatch ◽  
Parent Phase ◽  
Elastic Strain Energy ◽  
Size And Shape ◽  
Linear Elastic ◽  
New Phase

A new phase which precipitates from a parent matrix has a size and shape which reflects its difference from parent phase. If the lattice mismatch is significant, the elastic strain energy is more important than the surface energy in determining the morphology and the preferred habit plane. The preferred habit of a tetragonal inclusion in a cubic matrix has been predicted by minimizing the elastic strain energy as from the Ktachaturyan linear elastic formula.

Nonlinear Finite Element Analysis of Crack Tip of Rubber-Like Material

2007 ◽  
Vol 353-358 ◽  
pp. 1013-1016
Author(s):  
Jian Bing Sang ◽  
Su Fang Xing ◽  
Xiao Lei Li ◽  
Jie Zhang
Keyword(s):  
Finite Element ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Elastic Behavior ◽  
Element Analysis ◽  
Finite Element Program ◽  
User Subroutine ◽  
Element Program ◽  
Notch Tip

It has been well known that rubber-like material can undergo large deformation and exhibit large nonlinear elastic behavior. Because of the geometrically nonlinear of rubber like material, it is more difficult to analyze it with finite element near the notch tip. What is more, because there are varieties of the strain energy functions, implementation of these models in a general finite element program to meet the need of industry applications can be time consuming. In order to make use of the constitutive equation of Y.C. Gao in 1997 and analyze the notch tip of rubber-like material, a framework to implement the rubber-like material model is established within the general-purpose finite element program MSC.Marc. It will be very convenient to implement this isotropic hyperelastic model into the program with a user subroutine. This paper starts with the theoretical analysis based on the strain energy function given by Y.C. Gao in 1997. A user subroutine is programmed to implement this strain energy function into the program of MSC.Marc, which offer a convenient method to analyze the stress and strain of rubber-like material with the strain energy function that is needed. Though analysis with MSC.Marc, it is found that the result with finite element is consistent with the analytical result that given by Y.C. Gao in 1997, which testify that analyzing rubber like material with this method is reasonable and convenient.

Some Forms of the Strain Energy Function for Rubber

1993 ◽  
Vol 66 (5) ◽  
pp. 754-771 ◽  
Author(s):  
O. H. Yeoh
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Large Strain ◽  
Strain Energy Function ◽  
Phenomenological Theory ◽  
Material Model ◽  
Elastic Behavior ◽  
Strain Behavior ◽  
Strain Invariants ◽  
Infinite Power

Abstract According to Rivlin's Phenomenological Theory of Rubber Elasticity, the elastic properties of a rubber may be described in terms of a strain energy function which is an infinite power series in the strain invariants I1, I2 and I3. The simplest forms of Rivlin's strain energy function are the neo-Hookean, which is obtained by truncating the infinite series to just the first term in I1, and the Mooney-Rivlin, which retains the first terms in I1 and I2. Recently, we proposed a strain energy function which is a cubic in I1. Conceptually, the proposed function is a material model with a shear modulus that varies with deformation. In this paper, we compare the large strain behavior of rubber as predicted by these forms of the strain energy function. The elastic behavior of swollen rubber is also discussed.

scholarly journals The Exponentiated Hencky Strain Energy in Modeling Tire Derived Material for Moderately Large Deformations

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
Giuseppe Montella ◽  
Sanjay Govindjee ◽  
Patrizio Neff
Keyword(s):  
Constitutive Model ◽  
Strain Energy ◽  
Large Deformations ◽  
Large Strain ◽  
Strain Tensor ◽  
Strain Energy Function ◽  
Cold Forging ◽  
Central Feature ◽  
Hencky Strain ◽  
Highly Nonlinear

This work presents a hyperviscoelastic model, based on the Hencky-logarithmic strain tensor, to model the response of a tire derived material (TDM) undergoing moderately large deformations. The TDM is a composite made by cold forging a mix of rubber fibers and grains, obtained by grinding scrap tires, and polyurethane binder. The mechanical properties are highly influenced by the presence of voids associated with the granular composition and low tensile strength due to the weak connection at the grain–matrix interface. For these reasons, TDM use is restricted to applications involving a limited range of deformations. Experimental tests show that a central feature of the response is connected to highly nonlinear behavior of the material under volumetric deformation which conventional hyperelastic models fail in predicting. The strain energy function presented here is a variant of the exponentiated Hencky strain energy, which for moderate strains is as good as the quadratic Hencky model and in the large strain region improves several important features from a mathematical point of view. The proposed form of the exponentiated Hencky energy possesses a set of parameters uniquely determined in the infinitesimal strain regime and an orthogonal set of parameters to determine the nonlinear response. The hyperelastic model is additionally incorporated in a finite deformation viscoelasticity framework that accounts for the two main dissipation mechanisms in TDMs, one at the microscale level and one at the macroscale level. The new model is capable of predicting different deformation modes in a certain range of frequency and amplitude with a unique set of parameters with most of them having a clear physical meaning. This translates into an important advantage with respect to overcoming the difficulties related to finding a unique set of optimal material parameters as are usually encountered fitting the polynomial forms of strain energies. Moreover, by comparing the predictions from the proposed constitutive model with experimental data we conclude that the new constitutive model gives accurate prediction.

A Mathematical Model of Lung Parenchyma

1980 ◽  
Vol 102 (2) ◽  
pp. 124-136 ◽  
Author(s):  
A. D. Karakaplan ◽  
M. P. Bieniek ◽  
R. Skalak
Keyword(s):  
Strain Energy ◽  
Strain Energy Function ◽  
Strain Curve ◽  
Lung Parenchyma ◽  
The Finite Element Method ◽  
Stress Strain Curve ◽  
Proposed Model ◽  
Elastic Membranes ◽  
Mammalian Lung ◽  
Nonlinear Elastic

The geometry of the proposed model of the parenchyma of a mammalian lung reproduces a cluster of alveoli arranged around a lowest-level air duct. The alveolar walls are assumed to be nonlinear elastic membranes, whose properties are described in terms of a strain energy function which reflects the hardening character of the stress-strain curve. The effect of the surfactant is included in terms of a variable (area-dependent) surface tension. Analyses of various mechanical processes in the parenchyma are performed with the aid of the finite element method, with the geometric and physical nonlinearities of the problem taken into account.

On H. Hencky’s Approximate Strain-Energy Function for Moderate Deformations

1979 ◽  
Vol 46 (1) ◽  
pp. 78-82 ◽  
Author(s):  
L. Anand
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Finite Strain ◽  
Large Deformations ◽  
Strain Energy Function ◽  
Strain Measure ◽  
Logarithmic Measure ◽  
Infinitesimal Strain ◽  
Isotropic Elasticity ◽  
Good Agreement

It is shown that the classical strain-energy function of infinitesimal isotropic elasticity is in good agreement with experiment for a wide class of materials for moderately large deformations, provided the infinitesimal strain measure occurring in the strain-energy function is replaced by the Hencky or logarithmic measure of finite strain.

Collagen and elastin fibers in human pulmonary alveolar mouths and ducts

1987 ◽  
Vol 63 (3) ◽  
pp. 1185-1194 ◽  
Author(s):  
M. Matsuda ◽  
Y. C. Fung ◽  
S. S. Sobin
Keyword(s):  
Strain Energy ◽  
Unit Length ◽  
Strain Energy Function ◽  
Lung Parenchyma ◽  
Fiber Bundles ◽  
Elastin Fiber ◽  
Analytic Expressions ◽  
Quantitative Basis ◽  
The Mean ◽  
The Right

To provide a quantitative basis for the understanding of the mechanical properties of the lung tissue, the morphology of the pulmonary alveolar ducts was studied and the dimensions of the collagen and elastin fiber bundles in the alveolar mouths were measured. Statistical data are presented in this report. It was found that the probability frequency functions of the widths of both the collagen and elastin fibers are skewed to the right, but the fourth roots of the widths are normally distributed. Hence, knowing the mean and standard deviation of (width)1/4, the probability of finding fiber bundles of width D is known. On the other hand, we already know the analytic expressions of the strain energy per unit length of fibers of given width. With the probability distribution of width and an estimation of the length of alveolar mouths, the strain energy of the alveolar mouths can be computed. Then the contribution of the alveolar mouths to the stress in the lung parenchyma due to any strain can be obtained by a differentiation of the strain energy function with respect to that strain.

scholarly journals Measuring and Modelling Nonlinear Elasticity of Ex Vivo Mouse Muscles

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
E. Rizzuto ◽  
R. De Luca ◽  
A. Musarò ◽  
Z. Del Prete
Keyword(s):  
Elastic Modulus ◽  
Young’S Modulus ◽  
Linear Model ◽  
Soft Tissues ◽  
Large Deformations ◽  
Ex Vivo ◽  
Young's Modulus ◽  
Elastic Behavior ◽  
Linear Region ◽  
Linear Elastic

Elastography is a noninvasive imaging technique that provides information on soft tissue stiffness. Young’s modulus is typically used to characterize soft tissues’ response to the applied force, as soft tissues are often considered linear elastic, isotropic, and quasi-incompressible materials. This approximation is reasonable for small strains, but soft tissues undergo large deformations also for small values of force and exhibit nonlinear elastic behavior. Outside the linear regime, the elastic modulus is dependent on the strain level and is different for any kind of tissue. The aim of this study was to characterize, ex vivo, the mechanical response of two different mice muscles to an external force. A system for transverse force-controlled uniaxial compression enabled obtaining the stress-strain (σ-ε) curve of the samples. The strain-dependent Young’s modulus (SYM) model was adopted to reproduce muscle compression behavior and to predict the elastic modulus for large deformations. After that, a recursive linear model was employed to identify the initial linear region of the σ-ε curve. Results showed that both muscle types exhibited a strain hardening effect and that the SYM model provided good fitting of the entire σ-ε curves. The application of the recursive linear model allowed capturing the initial linear region in which the approximation of these tissues as linear elastic materials is reasonable. The residual analysis displayed that even if the SYM model better summarizes the muscle behavior on the entire region, the linear model is more precise when considering only the initial part of the σ-ε curve.

A Microstructurally Based Orthotropic Hyperelastic Constitutive Law

2002 ◽  
Vol 69 (5) ◽  
pp. 570-579 ◽  
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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