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The principles of Mohr’s circle can also be directly applied to analyze the two-dimensional state of strain at a point. By replacing normal ストレス ([latex]sigma[/latex]) with normal strain ([latex]epsilon[/latex]) and shear stress ([latex]tau[/latex]) with half the shear strain ([latex]gamma/2[/latex]), an analogous circle can be constructed. This graphical tool helps determine principal strains and the maximum shear strain.
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 mechanics, where strains are often measured directly using strain gauge rosettes. The circle provides a quick graphical method to convert these measured strains into principal strains and their orientations.
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