Stretch ratio
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 Failed to parse (Missing texvc executable;
please see math/README to configure.): L_0
to a current, deformed length Failed to parse (Missing texvc executable;
please see math/README to configure.): L
. Stretch ratio, also known as relative elongation[1] is a measure of the deformation defined as:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \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:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \mathbf {\epsilon = \lambda - 1}
.
From this we can derive:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \epsilon = {e \over L_0} = \int {dL \over L_0}
where Failed to parse (Missing texvc executable;
please see math/README to configure.): 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:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \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 Failed to parse (Missing texvc executable;
please see math/README to configure.): A_0
and current area Failed to parse (Missing texvc executable;
please see math/README to configure.): A
, the work conjugate of nominal strain is nominal stress Failed to parse (Missing texvc executable;
please see math/README to configure.): \sigma = F / A_0
and that of logarithmic strain is true stress Failed to parse (Missing texvc executable;
please see math/README to configure.): \sigma = F / A
. This condition is necessary to ensure that the product of stress and strain is energy per unit initial volume Failed to parse (Missing texvc executable;
please see math/README to configure.): V_0
or current volume Failed to parse (Missing texvc executable;
please see math/README to configure.): V
.
In more-than-one-dimension, the stretch ratio and both strain measures generalise to become second order tensors.
[edit] See also
[edit] References
|
