Page 75 - Physical Principles of Sedimentary Basin Analysis

3.11 Linear stress–strain relations 57

We have already defined the shear modulus as the ratio of shear stress over strain with
equation (3.6). The definition of the shear modulus for plane strain in (3.6) becomes
generalized as

shear stress σ xy σ xy
G = = = . (3.91)
deformation angle ∂u x /∂y + ∂u y /∂x 2ε xy
Equation (3.88) shows that the shear strain and the shear stress (deformation angle) are
proportional. The relationship between the shear stress and shear strain for the xz- and
yz-directions are similar, and all three relationships are

1 + ν
ε xy = σ xy (3.92)
E
1 + ν
ε xz = σ xz (3.93)
E
1 + ν
ε yz = σ yz (3.94)
E
where it is used that
E
G = . (3.95)
2(1 + ν)
Exercise 3.18 derives equation (3.95) for the shear modulus G in terms of E and ν.The
all-together six stress–strain relationships (3.88)–(3.90) and (3.92)–(3.94) can be collected
into one equation
1 ν
− ε ij = (1 + ν) σ ij − σ kk δ ij (3.96)
E E
where the summation convention applies to the term σ kk (which is σ xx + σ yy + σ zz ), and
where δ ij is the Kronecker delta defined by

1 i = j
δ ij = . (3.97)
0 i = j
Notice that a minus sign has been added to the left-hand side of equation (3.96). The reason
for this sign reversal is that compressive stress states are more common in the crust than
stress states of tension, and it is therefore conventional in rock mechanics to associate a
positive stress with compression (negative strain).
Equation (3.96) for the strain as a linear function of stress can be inverted to an
expression for stress as a linear equation of strain (see Exercise 3.14):

(3.98)
− σ ij = 2Gε ij + λε kk δ ij
where
νE
λ = . (3.99)
(1 + ν)(1 − 2ν)