## 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 4-15: 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

ei

e2

e3

= det

0.8

0.6

0

0.6e1

0

0

1

n(D ]

0.8

0.6

0

n(2)

=

0

0

1

n(3)

1

0.6

-0.8

0

0.8

0.6

0

1

V3

0

" 0.8

0

0.6

" 2.3

0

0

0

0

1

V3

0

0

0.6

0

-0.8

=

0

1

0

0.6

-0.8

0

0

0

1

0

1

0

0

0

Construct the Mohr's circle for the following plane strain case:

 0 0 0 0 5 V3 0 V3 We note that since E(i) = 0 is a principal value for plane strain, ttwo of the circles are drawn as shown. 4.9 Initial or Thermal Strains -yian
0 0