Principal stresses rotation angle

With both the two principal stresses and the rotation angle known the stress field is completely determined. At 45 degrees to the principal stress field orientation, one finds the directions of maximum shear stress ... 

The Mohr circle representation (Fig. 9.6c) is a graphical method of relating stress components in different sets of axes. When the axes in the material rotate by an angle B, the diameter of the circle rotates by an angle 2 B. If the material yields, the circle has radius k, the constant in the Tresca yield criterion. The axes of the Mohr diagram are the tensile and shear stress components. Thus, in the left-hand circle, representing the stresses at A in Fig. 9.6b, the ends of the horizontal diameter are the principal stresses. The principal axes are parallel and perpendicular to the notch-free surface. There is a tensile principal stress Ik parallel to the surface, and a zero stress perpendicular to the surface. The points at the ends of the vertical diameter represent the stress components in the a)3 axes, rotated by 45° from the principal axes. In the a/3 axes, the shear stresses have a maximum value k, and there are equal biaxial tensile stresses of magnitude = k (the coordinate of the centre of the circle). 

A short exercise can illustrate how to use a Mohr s circle to get the principal stresses. A priori the angles are not needed for this. Rgure 1.22 indicates the method when no rotation of the coordinate system has occurred. The magnitudes of ffx [Pg.34]

An exception is flow birefringence where differences in the principal stresses and their angle of rotation are measured directly see Section 9.4. 

The rotation angle x ofthe principal stress axes is used in analyzing flow birefringence data as discussed in Section 9,4.1. In this example x = 4S C and 2 - xj is the plane of shearing. Then the results above satisfy eqs. 9.4.2 and 9,4.3,... 

These equations show the relationship between the normal and shear t3q>es of stress on the principal types of stress and the angle a that represents a force balance for the volume element. The equations can be represented the Mohr stress circle (Figure 12.35) with a radius of (o-j — o-2)/2 and its center at (ci + 0 2)/ on the abscissa. The shear stress T y is plotted on the ordinate and the normal forms of stress r and Tyy are plotted on the abscissa. Tbe points of the circle intersection with the abscissa give the principal t3q>es of stress ai and a2- The normal stress and the associated shear stress, t, are fixed by the radius arm which is rotated through an angle 2a from the abscissa. The normal stress r, now appears as the projection of the radius arm onto the abscissa, and the shear stress appears as the projection onto the ordinate. It can be seen from the Mohr circle that the shear stress has its maximum values for a = 45° and 135°. 

The stresses and the refractive indices are expressed here in the principal axes (noted I and II), in which the tensors are diagonal. These axes generally do not coincide with the laboratory coordinate system. They are rotated with an angle X, which corresponds to the extinction angle determined by the isoclinic fringes. 

Figure 1.39 is reproduced from Fig. 1.21 in terms of strain in the two-dimensional case to show the principal strains. A transformation to the principal direction is performed by rotating the x, y axes to x, y, the principal directions of those axes. The principal strains are sj and Sn- Due to the similarity between the plane-stress and plane-strain transformation equations, the orientation of the principal axes and the principal strains are given below. First, there is an angle, 0p, at which the shear strain, xy, vanishes. In analogy to Eq. (1.35a), this is now given as ... 

The principal scheme of the torque pendulum is shown in Figure 5.21. A disk of radius R is placed at the surface of the liquid or at the interface between the two liquids (a polar aqueous surfactant solution phase and a nonpolar phase, that is, hydrocarbon or fluorocarbon). The disk is suspended on a thin wire that serves as a dynamometer. Turning this wire by an angle f produces a torque M, while the turn of a cuvette by an angle ( ) with respect to the disk leads to the total shear deformation of the adsorption film. The principle of operation of this instrument is similar to that of a rotation viscometer, with one principal difference in the torque pendulum it is not possible to utilize a thin gap between a disk and a cuvette. Because of this the stresses and the deformations in the film are not uniform. 

Consider the case where the principal material axes, L and T, are rotated counterclockwise through an angle 0 relative to the body reference axes (x, y) as shown in Figure 8.15. The stress transformation equations relating known stresses in the x, y coordinate system to stresses in the L, T coordinate system are... 

---