Final answer:
The student's question involves determining the principal stresses and the angle to the principal stress plane for a given stress state using Mohr's circle, which is a fundamental concept in the field of Engineering Mechanics of Materials.
Step-by-step explanation:
The student's question involves calculating the principal normal and shear stresses, the angle from the x-axis to σ1, and drawing the Mohr's circle for a given plane stress state, which relates to the Mechanics of Materials in Engineering. In this scenario, the given stresses are σx = −4 MPa (compression), σy = 12 MPa (tension), and τxy = 7 MPa (shear stress acting counter-clockwise).
The calculation of principal stresses and the angle to the principal plane involves finding the center of Mohr's circle, its radius, and using trigonometric relationships. According to Mohr's theory, the radius of the circle (R) is the shear stress, and the average of σx and σy determines its center (σavg). The principal stresses (σ1 and σ2) are found at the intersections of the circle with the horizontal axis, and the angle to the principal plane can be found by 2θ, where θ is the angle made by a radius line with the horizontal axis starting from σavg to the point of tangency by the line representing the given shear stress.
To draw Mohr's circle, plot a point for the state of stress (σx, τxy) and construct a circle centered at (σavg, 0) with a radius equal to R. The principal stresses are found at the horizontal axis intercepts, and the maximum shear stress is equal to the radius of the circle. Regarding the stress elements, these are depicted by a square that has been rotated by θ, indicating the direction of the principal stresses.