Mohr’s Circle for Strain Analysis

The mathematical structure of the transformation equations for plane strain is identical to that for plane stress. This analogy allows for the use of Mohr’s circle for strain analysis. The horizontal axis represents the normal strain, [latex]\epsilon_n[/latex], and the vertical axis represents half the engineering shear strain, [latex]\gamma_{nt}/2[/latex]. The use of [latex]\gamma/2[/latex] (tensorial shear strain) instead of [latex]\gamma[/latex] (engineering shear strain) is necessary to maintain the circular form of the locus.

Given a state of strain defined by [latex]\epsilon_x[/latex], [latex]\epsilon_y[/latex], and the shear strain [latex]\gamma_{xy}[/latex], the circle is constructed with a center at [latex]C = (\epsilon_{avg}, 0)[/latex], where [latex]\epsilon_{avg} = (\epsilon_x + \epsilon_y)/2[/latex], and a radius [latex]R = \sqrt{\left(\frac{\epsilon_x – \epsilon_y}{2}\right)^2 + \left(\frac{\gamma_{xy}}{2}\right)^2}[/latex]. The intersections with the horizontal axis give the principal strains, [latex]\epsilon_1[/latex] and [latex]\epsilon_2[/latex]. The maximum in-plane shear strain is twice the radius of the circle, [latex]\gamma_{max} = 2R[/latex]. This tool is invaluable in experimental 力学, where strains are often measured directly using strain gauge rosettes. The circle provides a quick graphical 方法 to convert these measured strains into principal strains and their orientations.