MATHEMATICAL MODEL FOR THE RUPTURE OF CEREBRAL SACCULAR ANEURYSMS THROUGH THREE-DIMENSIONAL STRESS DISTRIBUTION IN THE ANEURYSM WALL

2006 ◽  
Vol 06 (03) ◽  
pp. 325-335 ◽  
Author(s):  
HANS R. CHAUDHRY ◽  
DAWN A. LOTT ◽  
CHARLES J. PRESTIGIACOMO ◽  
THOMAS W. FINDLEY
Keyword(s):  
Mathematical Model ◽  
Stress Distribution ◽  
Strain Energy ◽  
Finite Strain ◽  
Three Dimensional ◽  
Strain Energy Function ◽  
Unruptured Aneurysms ◽  
Aneurysm Wall ◽  
Thin Spherical Shell ◽  
Saccular Aneurysms

A mathematical model for the rupture of cerebral saccular aneurysms is developed through the analysis of three-dimensional stress distribution in the aneurysm wall. We assume in this paper that a saccular aneurysm resembles a thin spherical shell (a spherical membrane), and then develop a strain-energy function valid for finite strain to analyze three-dimensional stress distribution in the aneurysm wall. We find that rupture occurs when the ratio of the wall thickness to the radius of the aneurysm is 6.1 × 10-3. We also conclude from our analysis that rupture can occur when the ratio of thickness to radius of the parent aneurysm equals the ratio of thickness to radius of the daughter aneurysm. These findings may be helpful to the neurosurgeon for predicting the rupture potential in patients presenting with unruptured aneurysms.

Three-Dimensional Stress Distribution in Arteries

1983 ◽  
Vol 105 (3) ◽  
pp. 268-274 ◽  
Author(s):  
C. J. Chuong ◽  
Y. C. Fung
Keyword(s):  
Experimental Data ◽  
Stress Distribution ◽  
Vessel Wall ◽  
Strain Energy ◽  
Arterial Wall ◽  
Three Dimensional ◽  
Strain Energy Function ◽  
Exponential Form ◽  
Strain Relationship ◽  
Longitudinal Stretch

A three-dimensional stress-strain relationship derived from a strain energy function of the exponential form is proposed for the arterial wall. The material constants are identified from experimental data on rabbit arteries subjected to inflation and longitudinal stretch in the physiological range. The objectives are: 1) to show that such a procedure is feasible and practical, and 2) to call attention to the very large variations in stresses and strains across the vessel wall under the assumptions that the tissue is incompressible and stress-free when all external load is removed.

ON THE DEVELOPMENT OF COMPRESSIBLE PSEUDO-STRAIN ENERGY DENSITY FUNCTION FOR HYPERELASTIC MATERIAL: EXPERIMENT, THEORY AND FEM

2005 ◽  
Vol 29 (3) ◽  
pp. 459-475
Author(s):  
Hamid Ghaemi ◽  
A. Spence ◽  
K. Behdinan
Keyword(s):  
Mechanical Behavior ◽  
Density Function ◽  
Energy Function ◽  
Strain Energy ◽  
Three Dimensional ◽  
Strain Energy Function ◽  
Material Parameter Identification ◽  
Constitutive Function ◽  
Strain Energy Density Function ◽  
Biaxial Tests

This study was carried out to develop a compressible pseudo-strain energy function that describes the mechanical behavior of rubber-like materials. The motivation for this work was two fold; first was to define a single-term strain energy function derived from constitutive equations that can describe the mechanical behavior of rubber-like materials and taking into account the coupling between principal stretches and the nearly incompressibility characteristic of elastomers. Second was to implement this strain energy function into the Finite Element Method (FEM) to study the suitability of the model in FEM. A one-term three-dimensional strain energy function based on the principal stretch ratios was proposed. The three dimensional constitutive function was then reduced to describe the behavior of rubber-like materials under biaxial and uniaxial loading condition based on the membrane theory. The work presented here was based on the decoupling of the strain density function into a deviatoric and a volumetric part. Using pure gum, GMS-SS-A40, uniaxial and equi-biaxial experiments were conducted employing different strain rate protocols. The material was assumed to be isotropic and homogenous. The experimental data from uniaxial and biaxial tests were used simultaneously to determine the material parameters of the proposed strain energy function. A GA curve fitting technique was utilized in the material parameter identification. The proposed strain energy function was compared to a few well-known strain energy functions as well as the experimental results. It was determined that the proposed strain energy function predicted the mechanical behavior of rubber-like material with greater accuracy as compared to other models both analytical and numerical results.

The Cosserat Point Element as an Accurate and Robust Finite Element Formulation for Implicit Dynamic Simulations

2019 ◽  
Vol 17 (01) ◽  
pp. 1844006
Author(s):  
Mahmood Jabareen ◽  
Yehonatan Pestes
Keyword(s):  
Finite Element ◽  
Energy Function ◽  
Strain Energy ◽  
Three Dimensional ◽  
Nonlinear Dynamic Analysis ◽  
Strain Energy Function ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Dynamic Simulations ◽  
Cosserat Point

The reliability of numerical simulations manifested the need for an accurate and robust finite element formulation. Therefore, in the present study, an eight node brick Cosserat point element ( CPE ) for the nonlinear dynamic analysis of three-dimensional (3D) solids including both thick and thin structures is developed. Within the present finite element formulation, a strain energy function is proposed and additively decoupled into two parts. One part is characterized by any 3D strain energy function, while the other part controls the response to inhomogeneous deformations. Several example problems are presented, which demonstrate the accuracy and the robustness of the developed CPE in modeling the dynamic response of elastic structures.

scholarly journals Residually Stressed Fiber Reinforced Solids: A Spectral Approach

Materials ◽  
2020 ◽  
Vol 13 (18) ◽  
pp. 4076
Author(s):  
Mohd Halim Bin Mohd Shariff ◽  
Jose Merodio
Keyword(s):  
Constitutive Equation ◽  
Energy Function ◽  
Strain Energy ◽  
Finite Strain ◽  
Strain Energy Function ◽  
Spectral Approach ◽  
Fiber Reinforced ◽  
Elastic Solids ◽  
Preferred Direction ◽  
The Right

We use a spectral approach to model residually stressed elastic solids that can be applied to carbon fiber reinforced solids with a preferred direction; since the spectral formulation is more general than the classical-invariant formulation, it facilitates the search for an adequate constitutive equation for these solids. The constitutive equation is governed by spectral invariants, where each of them has a direct meaning, and are functions of the preferred direction, the residual stress tensor and the right stretch tensor. Invariants that have a transparent interpretation are useful in assisting the construction of a stringent experiment to seek a specific form of strain energy function. A separable nonlinear (finite strain) strain energy function containing single-variable functions is postulated and the associated infinitesimal strain energy function is straightforwardly obtained from its finite strain counterpart. We prove that only 11 invariants are independent. Some illustrative boundary value calculations are given. The proposed strain energy function can be simply transformed to admit the mechanical influence of compressed fibers to be partially or fully excluded.

Nonlinear Modeling of 3D Open-Cell Foams

1999 ◽  
Author(s):  
Yu Wang ◽  
Alberto M. Cuitiño
Keyword(s):  
Strain Energy ◽  
Nonlinear Modeling ◽  
Three Dimensional ◽  
Strain Energy Function ◽  
Nonlinear Material ◽  
Open Cell ◽  
Wide Range ◽  
Open Cell Foams ◽  
Explicit Correlation ◽  
Strain Energy Functions

Abstract In this article, we present a hyperelastic model for light and compliant open cell foams with an explicit correlation between microstructure and macroscopic behavior. The model describes a large number of three dimensional structures with regular and irregular cells. The theory is based on the formulation of strain-energy function accounting for stretching which is the main deformation mechanism in this type of materials. Within the same framework, however, bending, shear and twisting energies can also be incorporated. The formulation incorporates nonlinear kinematics which traces the evolution of the structure during loading process and its effects on the constitutive behavior, including the cases where configurational transformations are present leading to non-convex strain-energy functions. Also nonlinear material effects at local or beam level are introduced to accommodate a wide range of different material behaviors. Since the micromechanical formulation presented here has explicit correlation with the foam structure, it preserves in the constitutive relation the symmetries or directional properties of the corresponding structures, including the cases of re-entrant foams which exhibit negative Poisson’s ratio effects. The model captures the central features exhibit by these materials. Predictions of the model for macroscopic uniaxial strain are presented in this article.

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.

Three-Dimensional Fractional Derivative Models for Finite Deformation

2011 ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Fractional Derivative ◽  
Stress Tensor ◽  
Memory Effect ◽  
Strain Energy ◽  
Three Dimensional ◽  
Strain Energy Function ◽  
Stress Strain ◽  
Different Types ◽  
Strain Tensors ◽  
The Continuum

Fractional derivative stress-strain relations are derived for compressible viscoelastic materials based on the continuum mechanics. Several types of stress tensor and strain tensors are specified to describe the dynamics of continuous media. Consequently there are many equivalent expressions of stress-strain relations. If memory effect is not taken into account, these relations are equivalently transformed from one to another by suitable tensor operations. However, if memory effect is included in the mechanics of the materials, different types of stress-strain relations can be derived depending on the choice of the type of stress tensor, or equivalently the choice of the strain energy function. In this paper, several types of fractional derivative stress-strain relations are proposed.

3D Mechanical Properties of the Layered Esophagus: Experiment and Constitutive Model

2006 ◽  
Vol 128 (6) ◽  
pp. 899-908 ◽  
Author(s):  
W. Yang ◽  
T. C. Fung ◽  
K. S. Chian ◽  
C. K. Chong
Keyword(s):  
Constitutive Model ◽  
Fiber Orientation ◽  
Strain Energy ◽  
Physiological State ◽  
Three Dimensional ◽  
Strain Energy Function ◽  
Esophageal Wall ◽  
Uniaxial Tensile ◽  
Uniaxial Tensile Test ◽  
Esophageal Diseases

The identification of a three dimensional constitutive model is useful for describing the complex mechanical behavior of a nonlinear and anisotropic biological tissue such as the esophagus. The inflation tests at the fixed axial extension of 1, 1.125, and 1.25 were conducted on the muscle and mucosa layer of a porcine esophagus separately and the pressure-radius-axial force was recorded. The experimental data were fitted with the constitutive model to obtain the structure-related parameters, including the collagen amount and fiber orientation. Results showed that a bilinear strain energy function (SEF) with four parameters could fit the inflation data at an individual extension very well while a six-parameter model had to be used to capture the inflation behaviors at all three extensions simultaneously. It was found that the collagen distribution was axial preferred in both layers and the mucosa contained more collagen, which were in agreement with the findings through a pair of uniaxial tensile test in our previous study. The model was expected to be used for the prediction of stress distribution within the esophageal wall under the physiological state and provide some useful information in the clinical studies of the esophageal diseases.

An integrated formulation of anisotropic force–calcium relations driving spatio-temporal contractions of cardiac myocytes

2009 ◽  
Vol 367 (1908) ◽  
pp. 4887-4905 ◽  
Author(s):  
Philippe Tracqui ◽  
Jacques Ohayon
Keyword(s):  
Intracellular Calcium ◽  
Cardiac Myocytes ◽  
Energy Function ◽  
Strain Energy ◽  
Sarcomere Length ◽  
Cardiac Myocyte ◽  
Three Dimensional ◽  
Strain Energy Function ◽  
Calcium Dynamics ◽  
Spatio Temporal

Isolated cardiac myocytes exhibit spontaneous patterns of rhythmic contraction, driven by intracellular calcium waves. In order to study the coupling between spatio-temporal calcium dynamics and cell contraction in large deformation regimes, a new strain-energy function, describing the influence of sarcomere length on the calcium-dependent generation of active intracellular stresses, is proposed. This strain-energy function includes anisotropic passive and active contributions that were first validated separately from experimental stress–strain curves and stress–sarcomere length curves, respectively. An extended validation of this formulation was then conducted by considering this strain-energy function as the core of an integrated mechano-chemical three-dimensional model of cardiac myocyte contraction, where autocatalytic intracellular calcium dynamics were described by a representative two-variable model able to generate realistic intracellular calcium waves similar to those observed experimentally. Finite-element simulations of the three-dimensional cell model, conducted for different intracellular locations of triggering calcium sparks, explained very satisfactorily, both qualitatively and quantitatively, the contraction patterns of cardiac myocytes observed by time-lapse videomicroscopy. This integrative approach of the mechano-chemical couplings driving cardiac myocyte contraction provides a comprehensive framework for analysing active stress regulation and associated mechano-transduction processes that contribute to the efficiency of cardiac cell contractility in both physiological and pathological contexts.

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