Y. C. Fung

On residual stresses in arteries.

In the study of vascular elasticity the unloaded state (one with zero transmural pressure and zero axial load) is commonly used as the reference state in which stresses and strains are considered as zero everywhere. Strains at loaded states are defined with respect to this state. Stress-strain relationships are identified under the assumption that the vessel wall is stress-free at this unloaded state. Evidence of the existence of residual stresses in the arterial wall at the unloaded state is given in Fung [4]. With a longitudinal cut along the vessel wall the unloaded specimen springs open and its cross section becomes a sector. The opening angle of the vessel wall is time-dependent after the sudden relief of the initial residual stress. It shows that the artery is not stress-free at the unloaded state. It is important to identify the stress-free state. When we use pseudoelasticity [3] to characterize the arterial wall, we need a stress-free state as the reference state for strain measurements. Correspondingly, we also want to define stress with respect to this same reference state so that we can relate stresses to strains easily. Presence of the residual stress at the unloaded tube state will certainly affect the evaluation of stress distribution in the arterial wall due to actual loadings in the physiological range. In this note we present a method to describe the geometry of the opened-up stress-free state of the artery, which is taken to be the reference state. An algorithm is outlined for the identification of the stress-strain relationship of the arterial wall. Residual stresses, and strains in the unloaded tube are evaluated. With the consideration of residual stresses the stress distributions due to loadings in the physiological range are also evaluated.

Injury and Repair of the Musculoskeletal Soft Tissues

This reference work summarises current knowledge in all of the musculoskeletal soft-tissue areas and focuses on the structural aspects of soft-tissue injury.

Three-Dimensional Stress Distribution in Arteries

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.

Classical and Computational Solid Mechanics

Tensor analysis stress tensor analysis of strain conservation laws elastic and plastic behaviour of materials linearized theory of elasticity solutions of problems in linearized theory of elasticity by potentials two-dimensional problems in linearized theory of elasticity variational calculus, energy theorems, Saint-Venants principle Hamiltons principle, wave propagation, applications of generalized co-ordinates elasticity and thermodynamics irreversible thermodynamics and viscoelasticity thermoelasticity viscoelasticity large deformation incremental approach to solving some nonlinear problems finite element methods mixed and hybrid formulations finite element methods for plates and shells finite element modelling of nonlinear elasticity, viscoelasticity, plasticity, viscoplasticity and creep.

Residual Stress in Arteries

Knowledge of stress distributions in the arterial wall is important for many reasons. In the study of the propagation of pulse waves, one must know the incremental modulus of the elasticity that changes with the stress level. In the study of circulation control, the action of the vascular smooth muscle, which depends on its local stress level (see review in Fung, 1984), must be evaluated. In the study of atherogenesis, one must know the stress distribution in the vessel wall because the tensile and shear stress can alter the local wall permea-bility and pressure gradient which is the force that drives the fluid in or out of the vessel wall (Chuong and Fung, 1983). Accurate evaluation of stress distributions in the arterial wall is therefore an important step toward a better understanding of various physiological functions and pathological mechanisms associated with the circulatory system.

Relationship Between Hypertension, Hypertrophy, and Opening Angle of Zero-Stress State of Arteries Following Aortic Constriction

Examination of changes occurring in the zero-stress state of an organ provides a way to study cellular growth in the organ due to change of physical stresses. The zero-stress state of the aorta is not a tube. It is a sector with an opening angle that varies with the location on the aorta and changes with cellular remodeling. Blood vessel remodeling can be induced by imposing a constriction on the abdominal aorta by a metal clip (aortic banding), which causes an increase of blood pressure, hypertrophy of the aortic wall, and large change of opening angle. The correlation of the opening angle with the blood vessel wall thickness and blood pressure changes in rats aorta due to aortic banding is presented in this report. The opening angle changes daily following the aortic banding. Blood pressure rises in vessels of the upper body, but that in the lower body decreases at first and then rises to an asymptotic value. Blood vessel wall thickness increases in rough proportion to blood pressure. Vessel diameter changes also. But the most dramatic is the course of change of the zero-stress state. Typically, the time to reach 50 percent of asymptotic hypertrophy of blood vessel wall thickness is about 3-5 days. The corresponding time for blood pressure is about 7 days. The opening angle of the zero-stress state, however, increases rapidly at first, reaches a peak in about 2 to 4 days, then decreases gradually to a reduced asymptote. The exact values of the time constants depend on the location along the aortic tree. In general, the course of change of residual strain is very different from those of the blood pressure and the blood vessel wall thickness.

Compressibility and constitutive equation of arterial wall in radial compression experiments

A large number of papers treat the tensile properties of arterial wall, but few treat compressive properties. Almost everybody assumes incompressibility; few have measured the vessel wall fluid extrusion due to compressive loading. In this work, uniaxial compressive force is applied directly on rabbit thoracic artery in the radial direction to study its constitutive equation under compressive stresses. The resulting stress-strain curves show that the wall material becomes increasingly stiffer at larger compressive strain, quite similar to the behavior in tension. A pseudo-strain energy function of the exponential type which has been applied successfully on the tension side is used to identify the material constants on the compression side. The material constants are identified in two ways: with and without the assumption of incompressibility. To determine the compressibility of the wall, the fluid extrusion accompanying this type of loading is measured, and is found to be in the range of 0.50-1.26% of the undeformed tissue volume per 10 kPa compressive stress loading in the radial direction. At compressive stresses higher than 30 kPa, the percentage of fluid extrusion per unit compressive stress decreases. At this degree of fluid extrusion the tissue is only slightly compressible (or nearly incompressible). However, the use of incompressibility assumption in the stress-strain relationship results in a set of material constants which is very different from that derived without that assumption.

Mechanical properties of the heart muscle in the passive state

Abstract The viscoelastic behaviour of the heart muscle (papillary muscle) in the passive unstimulated) state is studied by such methods as stress relaxation, creep, vibration and stress-strain testing. The tests are conducted on a newly developed electromechanical muscle testing device which is suitable for conducting active and passive tests on biological materials.

Elastic and inelastic properties of the canine aorta and their variation along the aortic tree

Abstract Data on the nonlinear and inelastic mechanical properties of the canine aorta are presented. Major differences in the properties of the aorta from those of the mesentery, the skin, the muscle, etc, are pointed out. Mathematical representations of the experimental results are given. The relaxation spectrum is analyzed.

Longitudinal strain of canine and porcine aortas

The in situ longitudinal strain of canine and porcine aortas was investigated. Marks of black water-resistant ink were placed on the aortas and the axial lengths between the marks were measured in situ and in vitro. When the aortas were cut, the retraction was measured and described by the stretch ratio, which is defined as the length of a segment in situ divided by the length at no-load state. Results show that the stretch ratios of both porcine and canine aortas increase monotonically from 1.2 in the descending region to about 1.5 in abdominal region. Species differences are seen in the middle region. In both animals, the stretch ratio is correlated to the cross-sectional area of the vessel wall.