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Wednesday, November 3, 2010

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):

        ε1 = 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, d0, is made thinner by Δd. Then, the strain in the diametrical direction is:

       ε2 = –Δd / d0

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 = |ε2 / ε1| = 0.3

With 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 gages. 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 gages. 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/πD3) dD
       dR/R = dρ/ρ + dL/ L − 2dD/D 


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


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 factors 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

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