Principal directions and stresses

This brings us to a fact that plays a central role in what follows the compressive stress across a plane with normal n is nfo-j + n tr2 + nla.  

For an immediate example, let the principal stresses be 6, 4, and 3 MPa and let n be the direction that makes the same angle with all three axes so 

Then compressive stresses on planes normal to the three directions are 3.6, 

respectively. As we move away fi om the X3 direction where the normal stress component is 3.0, the normal stress increases more rapidly for 26° of rotation toward Xi and less rapidly for 26° of rotation toward Xj. Along the intermediate direction, the increase is interesting in the first 26° of rotation the increase is only 0.4, whereas in rotating on to the direction considered first (nf = n = nf = 1/3) the further increase is 0.93. The extra rotation is 29° so that the increase per degree of rotation is much larger. 

The point just made is general and is used later for directions close to a principal direction, the normal stress changes only slightly for say 5° change in direction, whereas for directions close to the octahedral direction the change in stress for a 5° change in direction is greater. 

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Figure 22.2 Principal directions and stress components for an orthotropic material is the in-plane shear modulus as in plane No. 3 above.) Source Reprinted from University of London, Imperial College of Science, Technology and Medicine, Course entitled Mechanical Testing of Advanced Fibre Composites, 1995.

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