Principal Strains Strain Invariants Mohr Circle
so Determination of the principal strains (E(3) < E(2) < E(i), strain invariants and the Mohr circle for strain parallel the one for stresses (Sect. 2.4) and will not be repeated here.
where the symbols IE, IIE and IIIE denote the following scalar expressions in the strain components:
IIe = — (E11E22 + E22E33 + E33E11 ) + E223 + E31 + E222 (4.165)
87 In terms of the principal strains, those invariants can be simplified into
IIe = —(E(1) E(2) + E(2)E(3) + E(3)EW) (4.170) IIIe = E(1)E(2)E(3) (4.171)
ss The Mohr circle uses the Engineering shear strain definition of Eq. 4.91, Fig. 4.8 ■ Example 415: Strain Invariants & Principal Strains
Determine the planes of principal strains for the following strain tensor
Solution:
The strain invariants are given by
The principal strains by
E(1) 
— A(1)  
E(2) 
— A(2)  
E(3) 
The eigenvectors for Em = give the principal directions n(1): V3 0 which gives n31) n31) V 12 For the second eigenvector A(2) = 1: which gives (with the requirement that n(2)n(2) = 1) Finally, the third eigenvector can be obrained by the same manner, but more easily from nv nv ; xn Therefore and this results can be checked via

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