Mohrkurver

Mohrkurver, or Mohr’s circles, are graphical tools used in materials science and structural engineering to analyze the state of stress at a point in a material under planar conditions. They visualize how normal and shear stresses transform when the coordinate axes are rotated, and they help identify principal stresses and maximum shear stress. The concept was introduced by the German engineer Christian Otto Mohr in the late 19th century.

Construction: For plane stress with stresses on the x- and y- faces, sigma_x, sigma_y, and the in-plane shear tau_xy, the Mohr circle is drawn with its center at ((sigma_x + sigma_y) / 2, 0) and radius R = sqrt(((sigma_x - sigma_y)/2)^2 + tau_xy^2). The circle passes through the points representing the stresses on the axes: (sigma_x, tau_xy) and (sigma_y, -tau_xy).

Reading results: The principal normal stresses sigma_1 and sigma_2 are located at the ends of the diameter, equal to the center ± R. The orientation of the principal stresses satisfies tan(2 theta_p) = 2 tau_xy / (sigma_x - sigma_y). The maximum shear stress equals R and occurs on planes rotated by 45 degrees from the principal directions. This visualization simplifies the transformation of stress components under rotation.

Extensions and limitations: While Mohr’s circle provides a clear 2D visualization, real materials may involve plane strain or three-dimensional states; for fully 3D states a Mohr’s sphere is used. Mohrkurver are a geometric reference tool for understanding stress transformation, principal stresses, and shear, but they do not by themselves predict yielding or failure, which depend on material properties and failure criteria.