Stress & Strain

Ductile materials

Fig 2. A stress–strain curve typical of structural steel 1. Ultimate Strength 2. Yield Strength 3. Rupture 4. Strain hardening region 5. Necking region. A: Apparent stress (F/A0) B: Actual stress (F/A)

Steel generally exhibits a very linear stress–strain relationship up to a well defined yield point (Fig.2). The linear portion of the curve is the elastic region and the slope is the modulus of elasticity or Young's Modulus. After the yield point, the curve typically decreases slightly because of dislocations escaping from Cottrell atmospheres. As deformation continues, the stress increases on account of strain hardening until it reaches the ultimate strength. Until this point, the cross-sectional area decreases uniformly because

of Poisson contractions. The actual rupture point is in the same vertical line as the visual rupture point.

However, beyond this point a neck forms where the local cross-sectional area decreases more quickly than the rest of the sample resulting in an increase in the true stress. As shown in Fig.2, On an engineering stress–strain curve this is seen as a decrease in the apparent stress. However if the curve is plotted in terms of true stress and true strain the stress will continue to rise until failure. Eventually the neck becomes unstable and the specimen ruptures (fractures).

Less ductile materials such as aluminum and medium to high carbon steels do not have a well-defined yield point. For these materials the yield strength is typically determined by the "offset yield method", by which a line is drawn parallel to the linear elastic portion of the curve and intersecting the abscissa at some arbitrary value (most commonly 0.2%). The intersection of this line and the stress–strain curve is reported as the yield point. Plastic region is the point where the material will stay deformed, The elastic region is the point where the material can stretch no further. Failure point is when the object breaks.

Brittle materials

Fig.3 Stress Strain Curve for Brittle materials

Brittle materials such as concrete and carbon fiber do

not have a yield point, and do not strain-harden.

Therefore the ultimate strength and breaking strength

are the same. A most unusual stress-strain curve is

shown in Fig.3. Typical brittle materials like glass do

not show any plastic deformation but fail while the

deformation is elastic. One of the characteristics of a

brittle failure is that the two broken parts can be

reassembled to produce the same shape as the original

component as there will not be a neck formation like

in the case of ductile materials. A typical stress strain

curve for a brittle material will be linear. Testing of

several identical specimen, cast iron, or soil, tensile

strength is negligible compared to the compressive

strength and it is assumed zero for many engineering

applications. Glass fibers have a tensile strength

stronger than steel, but bulk glass usually does not.

This is because of the stress intensity factor associated

with defects in the material. As the size of the sample

gets larger, the size of defects also grows. In general,

the tensile strength of a rope is always less than sum

of the tensile strength of its individual fibers.(less than

2%)

A universal testing machine

1.tensile machine:

Tensile testing, also known as tension testing,[1] is a fundamental materials science test in which a sample is subjected to uniaxial tension until failure. The results from the test are commonly used to select a material for an application, for quality control, and to predict how a material will react under other types of forces. Properties that are directly measured via a tensile test are ultimate tensile strength, maximum elongation and reduction in area.[2] From these measurements the following properties can also be determined: Young's modulus, Poisson's ratio, yield strength, and strain-hardening characteristics.[3]

2.three point pending machine :

Poisson's ratio

Figure 1: A cube with sides of length L of an

isotropic linearly elastic material subject to tension along the x axis, with a Poisson's ratio of 0.5. The

green cube is unstressed, the red is expanded in the

x direction by ΔL due to tension, and contracted in

the y and z directions by ΔL'.

Poisson's ratio (ν), named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain (perpendicular to the applied load), to the extension or axial strain (in the direction of the applied load).

When a material is compressed in one direction, it usually tends to expand in the other two directions perpendicular to the direction of compression. This phenomenon is called the Poisson effect. Poisson's ratio ν (nu) is a measure of the Poisson effect. The Poisson ratio is the ratio of the fraction (or percent) of expansion divided by the fraction (or percent) of compression, for small values of these changes.

Conversely, if the material is stretched rather than compressed, it usually tends to contract in the directions transverse to the direction of stretching. Again, the Poisson ratio will be the ratio of relative contraction to relative stretching, and will have the same value as above. In certain rare cases, a material will actually shrink in the transverse direction when compressed (or expand when stretched) which will yield a negative value of the Poisson ratio.

The Poisson's ratio of a stable, isotropic, linear elastic material cannot be less than −1.0 nor greater than 0.5 due to the requirement that Young's modulus, the shear modulus and bulk modulus have positive values [1]. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. A perfectly incompressible material deformed elastically at small strains would have a Poisson's ratio of exactly 0.5. Most steels and rigid polymers when used within their design limits (before yield) exhibit values of about 0.3, increasing to 0.5 for post-yield deformation (which occurs largely at constant volume.) Rubber has a Poisson ratio of nearly 0.5. Cork's Poisson ratio is close to 0: showing very little lateral expansion when compressed. Some materials, mostly polymer foams, have a negative Poisson's ratio; if these auxetic materials are stretched in one direction, they become thicker in perpendicular directions. Anisotropic materials can have Poisson ratios above 0.5 in some directions.

Assuming that the material is stretched or compressed along the axial direction (the x axis in the diagram):

where

ν is the resulting Poisson's ratio,

is transverse strain (negative for axial

tension (stretching), positive for axial

compression)

is axial strain (positive for axial tension,

negative for axial compression

Isotropic materials

For a linear isotropic material subjected only to compressive (i.e. normal) forces, the deformation of a material in the direction of one axis will produce a deformation of the material along the other axes in three dimensions. Thus it is possible to generalize Hooke's Law (for compressive forces) into three dimensions:

or

where

, and are strain in the direction of x, y and z axis σx , σy and are stress in the direction of x, y and z axis E is Young's modulus (the same in all directions: x, y and z for isotropic materials) ν is Poisson's ratio (the same in all directions: x, y and z for isotropic materials)

These equations will hold in the general case which includes shear forces as well as compressive forces, and the full generalization of Hooke's law is given by:

where δij is the Kronecker delta and

Negative Poisson's ratio materials(important)

Some materials known as auxetic materials display a negative Poisson’s ratio. When subjected to positive strain in a longitudinal axis, the transverse strain in the material will actually be positive (i.e. it would increase the cross sectional area). For these materials, it is usually due to uniquely oriented, hinged molecular bonds. In order for these bonds to stretch in the longitudinal

direction, the hinges must ‘open’ in the transverse direction, effectively exhibiting a positive strain.[6]

Applications of Poisson's effect

One area in which Poisson's effect has a considerable influence is in pressurized pipe flow. When the air or liquid inside a pipe is highly pressurized it exerts a uniform force on the inside of the pipe, resulting in a radial stress within the pipe material. Due to Poisson's effect, this radial stress will cause the pipe to slightly increase in diameter and decrease in length. The decrease in length, in particular, can have a noticeable effect upon the pipe joints, as the effect will accumulate for each section of pipe joined in series. A restrained joint may be pulled apart or otherwise prone to failure.[7]

Another area of application for Poisson's effect is in the realm of structural geology. Rocks, like most materials, are subject to Poisson's effect while under stress. In a geological timescale, excessive erosion or sedimentation of Earth's crust can either create or remove large vertical stresses upon the underlying rock. This rock will expand or contract in the vertical direction as a direct result of the applied stress, and it will also deform in the horizontal direction as a result of Poisson's effect. This change in strain in the horizontal direction can affect or form joints and dormant stresses in the rock.[8].

The use of cork as a stopper for wine bottles is the result of the fact that cork has a Poisson ratio of practically zero. This means that, as the cork is inserted into the bottle, the upper part which is not yet inserted will not expand as the lower part is compressed. The force needed to insert a cork into a bottle arises only from the

compression of the cork and the friction between the cork and the bottle. If the stopper were made of rubber, for example, (with a Poisson ratio of about 1/2), there would be a relatively large additional force required to overcome the expansion of the upper part of the rubber stopper.

Temperature and Its Effects

* 2 basic effects:

• expansion / contraction • change of material properties

Look, first, at the former:

Concept of Thermal Stresses and Strains

Materials and structures expand

and contract as the temperature

changes. Thus: (the thermal

strain will be discussed in the pdf

that had been attached with this

chapter) "Thermal strain.pdf"

Kites team