CONTINUUM MECHANICS for ENGINEERS

FIGURE 4.5
Typical Mohr’s circles for strain.
for the Mohr’s circles, the shear strain axis (ordinate) has units of as
shown by the typical diagram of Figure 4.5. The infinitesimal spherical strain
tensor is represented by a diagonal matrix having equal elements denoted
1 1
by εM = εii
= e , known as the mean normal strain. The infinitesimal deviator
3 3
strain tensor is defined by
and in matrix form
ij ij
1
3 ij kk ij ij M
η = ε − δ ε = ε −δ
ε
⎡η11
η12 η13⎤
⎡ε11
− ε ε ε ⎤
M
12 13
⎢
⎥ ⎢
⎥
⎢η12
η22 η23⎥
= ⎢ ε12 ε22 − ε ε23
⎥
M
⎣
⎢η13
η23 η33⎦
⎥ ⎢ ε13 ε23 ε33 − ε ⎥
⎣
M
⎦
(4.7-20)
(4.7-21)
Note that as with its stress counterpart, the first invariant of the deviator
strain is zero, or
and the principal deviator strains are given by
η ii = 0 (4.7-22)
η (q) = ε (q) – ε M, (q = 1,2,3) (4.7-23)
where ε (q) is a principal value of the infinitesimal strain tensor.
A state of plane strain parallel to the X 1X 2 plane exists at P if
1
2 γ
ε 33 = γ 13 = γ 31 = γ 23 = γ 32 = 0 (4.7-24)