**Force, Stress, and Strain**

If an object receives an external force from the top, it internally generates a repelling

**force**to maintain the original shape. The repelling**force**is called internal**force**and the internal**force**divided by the cross-sectional area of the object (a column in this example) is called**stress,**which is expressed as a unit of Pa (Pascal) or N/m².**Stress**is the force an object generates inside by responding to an applied external**force**. Suppose that the crosssectional area of the column is A (m²) and the external**force**is P (N, Newton). Since external force = internal force,**stress,**σ (sigma), is: σ = P/A (Pa or N/m²) (1)

When a bar is pulled, it elongates by ΔL, and thus it lengthens to L (original length) + ΔL (change in length). The ratio of this elongation (or contraction), ΔL, to the original length, L, is called

**strain**, which is expressed in ε (epsilon): ε

_{1}= ΔL (change in length) / L (original length)**Strain**in the same tensile (or compressive) direction as the external

**force**is called longitudinal

**strain**. Since

**strain**is an elongation (or contraction) ratio, it is an absolute number having no unit. Usually, the ratio is an extremely small value, and thus a

**strain**value is expressed by suffixing “x10

^{-6}(parts per million) strain,” “μm/m” or “με.”

The pulled bar becomes thinner while lengthening. Suppose that the original diameter, d

_{0}, is made thinner by Δd. Then, the strain in the diametrical direction is:

ε

_{2}= –Δd / d_{0}**Strain**in the orthogonal direction to the external

**force**is called

**lateral strain**. Each material has a certain ratio of

**lateral strain**to

**longitudinal strain**, with most materials showing a value around 0.3. This ratio is called Poisson’s ratio, which is expressed in ν (nu):

ν = |ε

_{2}/ ε_{1}| = 0.3With various materials, the relation between

**strain**and**stress**has already been obtained experimentally. Figure below graphs a typical relation between**stress**and**strain**on common steel (mild steel). The region where**stress**and**strain**have a linear relation is called the proportional limit, which satisfies the Hooke’s law. σ = E.ε or σ/ε = E

The proportional constant, E, between

**stress**and**strain**in the equation above is called the modulus of longitudinal elasticity or Young’s modulus, the value of which depends on the materials. As described above,**stress**can be known through measurement of the**strain**initiated by external**force**, even though it cannot be measured directly.

**Polarity of Strain**

There exist **tensile strain**(elongation) and

**compressive strain**(contraction). To distinguish between them, a sign is prefixed as follows:

- Plus (+) to
**tensile strain**(elongation) - Minus (–) to
**compressive strain**(contraction)

**Strain Gage**

A **strain gage**is a sensor whose resistance varies with applied

**force**; It converts

**force,**

**pressure**,

**tension**,

**weight**, etc., into a change in electrical resistance which can then be measured.

**Structure of Strain Gages**

There are many types of

**strain gage**s. Among them, a universal**strain gage**has a structure such that a grid-shaped sensing element of thin metallic resistive foil (3 to 6μm thick) is put on a base of thin plastic film (15 to 16μm thick) and is laminated with a thin film.

**Principle of Strain Gages **

Metal wires can be used as

**strain gage**s. Stretching of the wire changes its geometry in a way that acts to increase the resistance. For a metal wire, we can calculate the gage factor as follow:R = ρL/A = ρL/πr² = 4ρL/πD²

dR = (4L/πD²) dρ + (ρ/πD²) dL− (8ρL/πD

^{3}) dD

dR/R = dρ/ρ + dL/ L − 2dD/D

Then

K = (dR/R)/(dL/L) = (dρ/ρ)/(dL/L) + 1 − (2dD/D)/(dL/L)

Since

−(dD/D)/(dL/L)

is deﬁned as Poisson’s ratio, v, we have the

**Gage Factor**:

K = 1 + 2υ + (dρ/ρ)/(dL/L)

For different metals, this quantity depends on the material properties, and on the details of the conduction mechanism. In general, metals have

**gage factor**s between 2 and 4.Now, since the stress times the area is equal to the force, and the fractional change in resistance is equal to the gage factor times the fractional change in length (the strain), and stress is Young’s modulus times the strain, we have

F = σA = EA(dL/L) = (EA/K)(dR/R)

or

dR/R=FK/EA

So the fractional change in resistance of a

**strain gage**is proportional to the applied**force**and is proportional to the**gage factor**divided by Young’s modulus for the material. Clearly, we would prefer to have a large change in resistance to simplify the design of the rest of a sensing instrument, so we generally try to choose small diameters, small Young’s modulus, and large**gage factor**s when possible. The elastic limits of most materials are below 1%, so we are generally talking about resistance changes in the 1%–0.001% range. Clearly, the measurement of such resistances is not trivial, and we often see resistance bridges designed to produce voltages that can be fed into ampliﬁcation circuits.#### Examples of strain gage applications

There is an uncountable number of different applications for strain gages. Only a few are listed here:

**Experimental stress analysis. Diagnosis on machines and failure analysis.**

- multi axial stress fatigue testing, proof testing
- residual stress
- vibration measurement
- torque measurement
- bending and deflection measurement
- compression and tension measurement
- strain measurement

**Sensors for machines, automotive, research etc.**

- force measurement in machine tools
- aerospace
- impact sensors
- dental sensors
- medical sensors
- automotive, motor sport
- Biometrics
- tension sensors
- web tension
- force on hydraulic or pneumatic press

## 2 comments:

blogging walking here..

February 24, 2011 at 12:29 AMwell done...

February 24, 2011 at 12:43 AM## Post a Comment