Stretch ratio

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In the uniaxial tensile test commonly carried out to determine some properties of engineering materials, a small testpiece is stretched from an initial, undeformed length $L 0$ to a current, deformed length $L$ . Stretch ratio, also known as relative elongation^[1] is a measure of the deformation defined as:

$\lambda = {L \over L_0}$ .

Undeformed material then has a stretch ratio of 1.

Stretch ratio is a good measure of deformation for materials such as elastomers, which can sustain stretch ratios of 3 or 4 before they break. However, traditional engineering materials break at much lower stretch ratios, perhaps of the order of 1.001. The whole of the deformation information is then contained in the fourth significant figure. This can lead to large error in calculations. What is required is a measure of deformation in which the information is contained in the first significant figure. This type of measure is called a strain.

The undeformed material should have a strain of 0 and one way to ensure this is simply to define the strain ε as:

$\mathbf {\epsilon = \lambda - 1}$ .

From this we can derive:

$\epsilon = {e \over L_0} = \int {dL \over L_0}$

where $e = L - L 0$ is the extension of the testpiece. This is called engineering strain or nominal strain but it is not the only possible strain measure. Another common definition is the logarithmic or so-called true strain:

$\epsilon = \ln \lambda = \int {dL \over L}$ .

Any of these deformation measures is perfectly acceptable, the only requirement being that it is used in constitutive equations alongside its work conjugate stress measure. For a testpiece with initial area $A 0$ and current area $A$ , the work conjugate of nominal strain is nominal stress $σ = F / A 0$ and that of logarithmic strain is true stress $σ = F / A$ . This condition is necessary to ensure that the product of stress and strain is energy per unit initial volume $V 0$ or current volume $V$ .