Introductory Solid Mechanics

Strain

Units

Dimensionless [mm/mm etc, or %]

 

Normal Strain

Relative change in length of a line element oriented in arbitrary direction $n$

\[ \epsilon_n = lim_{B->A \rm\ along \rm\ n} \frac{\Delta s = \Delta s'}{\Delta s} \]

Three normal strain component: $\epsilon_x, \epsilon_y, \epsilon_z$

Average Normal (extensional) Strain: Length change divided by total length:

\[ \epsilon_n = \frac{\delta}{L}\]

Engineering or nominal (normal) strain: Average normal strain using original (undeformed) total length.

True (normal) strain: Integrate infinitesimal normal strains.

\[ \varepsilon_{eng} = \frac{\delta}{L_0}\] \[ \varepsilon_{true} = ln(1 + \frac{\delta}{L_0})\]

 

Shear Strain

Change in angle between line segments oriented in perpendicular directions $n$ and $t$ :

\[ \gamma_{nt} = lim_{\begin{matrix} B->A \rm\ along \rm\ n\\ C->A \rm\ along \rm\ t \end{matrix}} (\frac{\pi}{2} - \theta ') \]

Six shear strain components, but symmetric: $\gamma_{xy} =\gamma_{yx}, \gamma_{xz}=\gamma_{zx}, \gamma_{yz}=\gamma_{zy}$

Average shear strain:

\[\gamma = \frac{\pi}{2} - \theta \approx \frac{\delta}{H}\]

Engineering (shear) strain: Compute angle from length changes and original (undeformed) total length.

True (shear) strain: Integrate infinitesimal angle changes.

 

Strain Tensor

The components of normal and shear strain can be combined into the strain tensor. This is a symmetric matrix.

\[ E = \begin{bmatrix} \varepsilon_{x} & \gamma_{xy} & \gamma_{xz} \\ '' & \varepsilon_{y} & \gamma_{yz} \\ '' & '' & \varepsilon_{z} \end{bmatrix} \]

 

Measurement of Strain

Direct measurement

Initial and final lengths of some section of the specimen are measured, perhaps by some handheld device such as a ruler. Axial strain computed directly by following formula:

\[\varepsilon = \frac{\delta}{L} = \frac{L_{final} - L_{initial}}{L_{initial}} \]

Accurate measurements of strain in this way may require a fairly large initial length

Contact extensometer

A clip-on device that can measure very small deformations. Two clips attach to a specimen before testing. The clips are attached to a transducer body. The transducer outputs a voltage. Changes in voltage output are converted to strain

A tensile test in the Materials Testing Instructional Laboratory, Talbot Lab, UIUC

Strain gauges

Small electrical resistors whose resistance charges with strain. Change in resistance can be converted to strain measurement. Often sold as "rosettes", which can measure normal strain in two or more directions. Can be bonded to test specimen.

Digital image correlation (DIC)

Image placed on surface of test specimen. Image may consist of speckles or some regular pattern. Deformation of image tracked by digital camera. Image analysis used to determine multiple strain component

Experiment set up. The diffuse light source consists of two fluorescent tube lights that produce white light, behind a translucent plastic sheet.
Eyy strain calculated through DIC of straight-curved specimen with an applied load of 114 N from TAM 456, UIUC.