# Youngs modulus and shear relationship quotes

Poisson ratio, Poisson's ratio tutorial, elasticity, transverse deformation, relation among moduli, bulge in compression, elastic moduli, elastic constants. Figure 1: (a) Deformation of a cube of side length a by shear stress. (b) The prove the relation between the modulus of rigidity, Young's modulus and Poisson's. The work done by nonaffine displacements represents a quote of internal . The excluded-volume correlation term contributing to the elastic.

Poisson's ratio can have positive or negative values of arbitrarily large magnitude in anisotropic materials.

For orthotropic materials, Poisson's ratio is bounded by the ratio of Young's moduli E as follows. Poisson's ratio in viscoelastic materials The Poisson's ratio in a viscoelastic material is time dependent in the context of transient tests such as creep and stress relaxation.

If the deformation is sinusoidal in time, Poisson's ratio may depend on frequency, and may have an associated phase angle. Specifically, the transverse strain may be out of phase with the longitudinal strain in a viscoelastic solid.

Get pdf of a research article on this. Poisson's ratio and phase transformations Poisson's ratio can vary substantially in the vicinity of a phase transformation. Typically the bulk modulus softens near a phase transformation but the shear modulus does not change much. The Poisson's ratio then decreases in the vicinity of a phase transformation and can attain negative values.

Phase transformations are discussed further on the linked page. Poisson's ratio, waves and deformation The Poisson's ratio of a material influences the speed of propagation and reflection of stress waves. In geological applications, the ratio of compressional to shear wave speed is important in inferring the nature of the rock deep in the Earth.

This wave speed ratio depends on Poisson's ratio. Poisson's ratio also affects the decay of stress with distance according to Saint Venant's principle, and the distribution of stress around holes and cracks.

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Analysis of effect of Poisson's ratio on compression of a layer. What about the effect of Poisson's ratio on constrained compression in the 1 or x direction?

Constrained compression means that the Poisson effect is restrained from occurring. This could be done by side walls in an experiment. Also, compression of a thin layer by stiff surfaces is effectively constrained. Moreover, in ultrasonic testing, the wavelength of the ultrasound is usually much less than the specimen dimensions. The Poisson effect is restrained from occurring in this case as well.

In Hooke's law with the elastic modulus tensor as Cijkl we sum over k and l, but, due to the constraint, the only strain component which is non-zero is e Let us find the physical significance of that tensor element in terms of engineering constants.

One may also work with the elementary isotropic form for Hooke's law. There is stress in only one direction but there can be strain in three directions. The ratio of stress to strain is the elastic modulus—stiffness, but the stress, and only the stress, defines the strength of the material. Remember, strength measures how much stress the material can handle before permanent deformation or fracture occurs, whereas the stiffness measures the resistance to elastic deformation.

Simple stress-strain curves illustrating stiff vs non-stiff behavior. Curves A and B correspond to stiffer materials, whereas curve C represents a non-stiff material. Image source Understanding the Concepts of Stress and Strain Stress is an internal force resulting from an applied load; it acts on the cross-section of a mechanical or structural component. Strain is the change in shape or size of a body that occurs whenever a force is applied.

Yield strength is used in materials that exhibit an elastic behavior. Ultimate strength refers to the maximum stress before failure occurs. Fracture strength is the value corresponding to the stress at which total failure occurs.

Stiffness is how a component resists elastic deformation when a load is applied. Hardness is resistance to localized surface deformation.

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Ultimate tensile strength, yield strength, and fracture stress Failure Point. The strength of a material can refer to yield strength, ultimate strength, or fracture strength. Tensile strength can be calculated from hardness and is convenient because hardness tests—such as Rockwell—are usually simple to do, inexpensive, and nondestructive. Only a small penetration is performed on the specimen.

For many metals, tensile strength increases as hardness increases. A reliable online source is www. Stiffness Beam deflection The image below is a finite element analysis FEA of a beam subjected to a loading.

Steady deformations do not occur instantaneously when a steady stress is applied, but the immediate initial deformation is followed by a slower continuing deformation creep to an equilibrium value. Similarly the stress required to maintain a given deformation falls gradually from its initial value to a new equilibrium level stress relaxation.

It is thought to be primarily the smooth muscle which is responsible for the visco-elastic properties of artery walls, although collagen, despite its great stiffness, also exhibits visco-elasticity in experiments in vitro.

Elastin, on the other hand, is known to be purely elastic; it is this elastic component in blood vessel walls which prevents creep or stress relaxation from continuing indefinitely. Visco-elasticity is important only when the stress or strain varies with time; for example, if a segment of artery is subjected to a cyclically varying strain of given frequency and amplitude, the stress required for a given strain is greater during extension than during recoil Fig.

This phenomenon is an example of hysteresis. Three consecutive pressure-volume curves of a femoral artery. The sequence of inflation and deflation is shown by the arrows the last two deflations followed the same course. The vessel appears to become rather less distensible during the series of inflations. The static elastic properties of the arterial wall, p.

The properties of visco-elastic materials cannot usually be represented by a small number of constants in the way that linearly elastic materials can be represented by the Young's modulus and Poisson's ratio. Indeed, it is not usually possible to describe in exact mathematical terms how a visco-elastic material will respond to unsteady applied stresses of different kinds. And when it is possible, the mathematics is often so complicated as to be of little help.

Sometimes, however, a simple approximate description of the behaviour of the material can be made, and is very useful. This is true of blood vessels when the deformations are small, and vary cyclically with time under the action of cyclically varying applied stresses i. The response of the strain to changes in stress is delayed slightly, so that as the applied stress varies cyclically, the variations in strain lag behind the stress, and have a smaller amplitude than they would have if the material were purely elastic Fig.

When a cyclically varying stress is applied to a material, the resulting strain stays in phase if the material is purely elastic curve I.

However, if it has visco-elastic properties, the strain lags behind the stress, and varies with a smaller amplitude than if it were purely elastic curve II. It is possible to represent this behaviour by means of two constants, a 'dynamic Young's modulus' Edyn, and an 'effective viscosity' h. Using these formulae, the values of Edyn and h can be inferred when a known, cyclically varying stress is imposed on a piece of artery, and the resulting time-varying strain measured. Some results of such measurements are shown in Fig.

Both Edyn and h w are approximately constant over a wide range of frequencies. It is found that Edyn is usually greater than the value of E measured in static experiments.

For frequencies below about 2 Hz i. It is approximately constant for frequencies between 2 Hz and 20 Hz, and exceeds the static value by a ratio which varies from about 1. The viscous component h w also varies with frequency, but for frequencies above 5 Hz, the quantity h w is approximately constant and takes a value between 0.

In other words, we shall examine the dynamics of blood and of vessel walls. First, however, it is necessary to look at the balance of forces exerted on the walls of a tube which is completely at rest and contains a fluid which is also stationary; i. Consider, for example, an artery, which is stretched longitudinally by the tethering forces, and distended by an internal pressure which exceeds the pressure outside atmospheric pressure in many cases by approximately l.

We shall ignore gravity in the force balance; this is because when a blood vessel is distended by a large transmural pressure, the stress forces set up are very large compared with the gravitational forces.

## Shear modulus

When the transmural pressure is not large, gravity can be important for example, it can cause a thin-walled flexible tube in air to collapseand in that case our conclusions would be strictly accurate only if the vessel were both filled and surrounded by a fluid which had the same density as itself; then the weight of the wall would be balanced by the hydrostatic pressure in the fluid.

In the circulation there is an additional reason why gravity cannot always be neglected, since the pressure inside blood vessels varies hydrostatically quite apart from the pressure variations associated with blood flow. For example, vessels in the foot will have an internal pressure which, in a man standing upright, can be greater than that at the level of the heart by about 1. In the head, the blood pressure is correspondingly smaller.

The pressure outside blood vessels is approximately uniform, and close to atmospheric pressure except in the thorax where it can vary from about 3 x Nm-2 23 mm Hg or 30 cm H2O below atmospheric pressure to about the same amount above it during heavy breathing.

Together with the fact that the transmural pressure varies between blood vessels of different types at the same level being greater in arteries than in veinsthis means that there is a wide variation of transmural pressure in the cardiovascular system. As we shall see, this proves to be important in determining whether or not certain blood vessels may be close to collapse. Let us now examine the static balance of forces on a small element of the wall of a tube, neglecting gravity except in so far as it determines the pressure within the tube.

We call the internal and external pressures pi and pe respectively, and suppose that the tube has a circular cross-section with internal and external radii ri and re see Fig. The directions of the coordinate axes are parallel to the tube axis 1radially outwards 2and tangential to a cross section of the tube 3.

The forces on the element are marked with heavy arrows; those on the curved faces come from the internal and external pressures; those on the flat faces parallel to the axis of the tube come from the 'hoop' stress Sq.

The tensile forces on the flat ends balance out and are omitted. The wall thickness h is equal to re - ri. We consider the equilibrium of an element of axial length Dl, subtending a small angle q at the axis Fig. The net forces on this element result from perpendicular normal stresses on its six faces; no tangential stresses can be present because of the symmetry of the arrangement. The net components of force acting on the element in three mutually perpendicular directions must be zero; we choose these directions to be parallel to the axis 1radially out from the axis 2and tangential to the tube in a plane perpendicular to the axis 3.

The force balance in the axial direction 1 shows that the longitudinal tension T on each end must be the same equal to the area of an end times the average normal stress over that end. Similarly the force balance in the tangential direction 3 shows that the normal forces on the two flat faces of the element parallel to the axis must also be the same equal to the area of a face, hDl, times the average normal stress Sq over that face.

The other contributions to the radial component of force come from the internal and external pressures; the outwards force from the internal pressure is pi times the area of the inner curved face Dl ri qand the inwards force from the external pressure is pe times the area of the outer curved face Dl re q. Thus the radial force balance gives 7. We can therefore cancel out Dl q on either side of Equation 7. From our everyday experience with balloons, we intuitively expect a distension, caused by an excess of internal over external pressure, to be associated with tension in the wall.

That is, when pi exceeds pe, we expect the tension Sqh to be positive.

However, we can see from Equation 7. In other words there is a compressive stress in the wall. First, though, let us calculate whether the hoop stress in blood vessels is a tension or a compression.

Consider, for example, an artery within which the pressure exceeds atmospheric by l.

### What is Poisson's ratio?

If it is like the dog's femoral artery, for example, whose properties are given in Table I at the front of the book, it will have an internal radius of 0.

Then, from Equation 7. If the internal pressure were greater, by only 50 mmHg as in the foot, perhaps the wall stress would be a tension; if it were smaller, the compression would be more marked. In fact, if one calculates the wall stress in a great variety of blood vessels by this means, one concludes that the vast majority of them have a compressive hoop stress in their walls.

The exceptions are the biggest vessels aorta and vena cavaand other large vessels below the level of the heart, where the internal pressure is relatively high. The largest compressive stresses occur in small arteries and arterioles, whose walls are very thick, so that re exceeds ri by a large amount.

The fact that the underlying hoop stress may be compressive is not necessarily very important for the mechanical behaviour of blood vessels in vivo. This is because in the circulation we normally wish to analyse small departures from some equilibrium state. As we have explained earlier, it does not matter if there is some initial stress present in that equilibrium state or not, as long as the elastic properties of the material are represented by the effective Young's modulus, measured for small departures from the same equilibrium state.

For this reason, it is common to quote values of tension and pressure relative to a given equilibrium state, and to ignore the absolute values. In physiology, we usually quote pressures relative to atmospheric. The corresponding tensions are quoted relative to their value when the pressures both inside and outside the tube are atmospheric: When that is the case, the hoop stress, say S0, is given by Equation 7. If, during a departure from this situation, pe remains atmospheric, if pi exceeds patm by an amount p', and if S0 exceeds -patm by an amount S', then Equation 7.

It shows that the pressure required to distend a tube against a given tension in the wall is inversely proportional to the radius of the tube. A balloon, for example, is more difficult to blow up when its radius is very small than when it is somewhat larger.

In the circulation, the law indicates that, relative to atmospheric pressure, the tension required to balance a certain distending pressure decreases as the radius of the tube decreases. This is sometimes used to explain why the walls of capillaries can be so thin: However, that argument ignores the dominant tethering effect of the surrounding tissue, and may not be relevant. The law of Laplace, then, can be used to analyse small departures from the equilibrium state in which all pressures are atmospheric.

It cannot be used to give an accurate description of the absolute stress level in a vessel wall, unless the distending pressure is so large that the stress S0 Equation 7. This is usually not the case. If we come to analyse when vessels are likely to collapse, it is important to know the absolute stress level. By collapse of a vessel, we mean a drastic decrease in its cross-sectional area, which occurs spontaneously in response to a small change in applied stresses.

It would also usually involve a change in cross-sectional shape, which may go from the circular to the elliptical or to more complicated shapes Fig. A tube will clearly collapse if the pressure outside it is very much greater than the pressure inside.

The reason for the collapse is associated with the large compressive circumferential stress which, from Equation 7.