Solid Mechanics Reference

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    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

    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

    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.
      True 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 \( t \):

    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

    Average shear strain
    $$ tan(\gamma) = \frac{\delta}{L} \xrightarrow{} \gamma = \frac{\delta}{L}\ $$
    $$ \gamma = \alpha + \beta\ $$
    $$ \gamma = \frac{\delta_x}{L_y} + \frac{\delta_y}{L_x}\ $$
    • 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.

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

    Electrical Resistance 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.

    Rosette strain gauge arrangement and example

    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.

    Vibrating wire strain gauge attached to the side of a bridge

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