Strain

When forces are applied to a body, deformation, the change in length or shape, occurs. The change in length, divided by the original length, is the strain. We use strain to normalize deformations with respect to the size of the geometry.

Units

Dimensionless [mm/mm, in/in, or %]

Normal Strain

Fig: Normal Strain

Normal strain

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

$$ \varepsilon_n = lim_{B \xrightarrow{} A \rm\ along \rm\ n} \frac{\Delta s - \Delta s'}{\Delta s} \ $$

Average Normal (extensional) Strain

Fig: Avg Normal Strain

Average normal strain

Length change divided by total length.

$$ \epsilon_n = \frac{\delta}{L}\ $$

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

Engineering strain #sta-eng

$$ \varepsilon_{eng} = \frac{\delta}{L_0}\ $$
True (normal) strain: Integrate infinitesimal normal strains.

Fig: Shear Strain

Shear strain

True strain #sta-ts

$$ \varepsilon_{true} = ln(1 + \frac{\delta}{L_0})\ $$

$$ \begin{align} \varepsilon_{true} &= \Sigma\Delta\varepsilon_0 = \int d\varepsilon \\ \varepsilon_{true} &= \int_{L_0}^{L_f}\frac{1}{L}dL \\ \varepsilon_{true} &= ln(L)|_{L_0}^{L_f} = ln(\frac{L_f}{L_0}) \\ \varepsilon_{true} &= ln(\frac{L_0+\delta}{L_0}) = ln(1+\frac{\delta}{L_0}) \\ \varepsilon_{true} &= \varepsilon_{eng}-\frac{1}{2}\varepsilon_{eng}^2 + \frac{1}{3}\varepsilon_{eng}^{3} + ...\end{align} $$
For small strain:

Small strain approximation #sta-ssa

$$ \varepsilon_{true} \approx \varepsilon_{eng} $$

Shear Strain

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

Fig: Avg Shear Strain

Average shear strain

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

When strains are small, the small angle approximation, $ \sin(\theta)\approx \theta $, results in

$$ \gamma = \frac{\pi}{2} - \theta \approx \frac{\delta}{L}\ $$

Average Shear Strain

Fig: extensometer.jpg

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

$$ tan(\gamma) = \frac{\delta}{L} \xrightarrow{} \gamma = \frac{\delta}{L}\ $$

$$ \gamma = \alpha + \beta\ $$

$$ \gamma = \frac{\delta_x}{L_y} + \frac{\delta_y}{L_x}\ $$

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.

$$ E = \begin{bmatrix} \varepsilon_{x} & \gamma_{xy} & \gamma_{xz} \\ \gamma_{yx} & \varepsilon_{y} &\gamma_{yz} \\ \gamma_{zx} & \gamma_{zy} &\varepsilon_{z} \end{bmatrix} \ $$

Three normal strain components: $ \varepsilon_x, \varepsilon_y, \varepsilon_z $
Six shear strain components: $ \gamma_{xy} =\gamma_{yx}, \gamma_{xz}=\gamma_{zx}, \gamma_{yz}=\gamma_{zy} $

The first subscript describes the surface orientation in the normal direction. The second subscript describes the direction of the stress.

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:

Axial strain formula #sta-axs

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

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.

Fig: strainGauge

Rosette strain gauge arrangement and example

Classic 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.

Fig: VibratingWire

Vibrating wire strain gauge attached to the side of a bridge

Vibrating Wire Strain Gauge

A calibrated wire is set into vibration and its frequency is measured. Small changes in the length of the wire as a result of strain produce a measurable change in frequency, allowing for accurate strain measurements over relatively long gauge lengths.

Fig: dic

Experiment set up. The diffuse light source consists of two fluorescent tube lights that produce white light, behind a translucent plastic sheet.

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.

Fig: dic2

$ \varepsilon_{yy} $ strain calculated through DIC of straight-curved specimen with an applied load of 114 N from TAM 456, UIUC.

Fig: True Strain

True strain

TAM 2xx References