Civil Engineering Reference
In-Depth Information
Figure 3.7
Relationship between strength, stiffness and rigidity.
where
E
ε
f
is the area beneath the stiffness-strain curve. This result, that the strength
of a linear material is equal to the area beneath the stiffness strain curve is an important
one. In fact, with a few restrictions, this result holds for all materials and for states
before failure.
3.7 Constitutive equations
During a general loading in the ground both shear and normal stresses are likely to
change simultaneously so there will be shearing and volumetric straining together.
For soils it turns out that shearing and volumetric effects are coupled so that shearing
stresses cause volumetric strains and normal stresses cause shear strains. This is quite
surprising and we will see later how the particulate nature of soils gives rise to shear
and volumetric coupling.
A simple constitutive equation relating shearing and volumetric stress-strain
behaviour can be written as
δ
S
11
δε
q
S
12
s
=
(3.21)
p
δ
S
21
S
22
δε
v
where [
S
] is a stiffness matrix containing stiffness moduli. The components of [
S
] are
q
∂ε
s
=
=
∂
3
G
S
11
(3.22)
p
∂ε
S
22
=
∂
K
v
=
(3.23)
q
∂ε
v
=
=
∂
J
1
S
12
(3.24)
p
∂ε
S
21
=
∂
J
2
s
=
(3.25)
