Civil Engineering Reference
In-Depth Information
s
(stress)
b
a
ε
(strain)
F
IGURE
7.8
Typical stress-strain curve
to a specimen of material of a given length and cross-sectional area and measuring
the corresponding increases in length. The stress produced by each value of load may
be calculated from Eq. (7.1) and the corresponding strain from Eq. (7.4). A stress-
strain curve is then drawn which, for some materials, would have a shape similar to
that shown in Fig. 7.8. Stress-strain curves for other materials differ in detail but,
generally, all have a linear portion such as ab in Fig. 7.8. In this region stress is directly
proportional to strain, a relationship that was discovered in 1678 by Robert Hooke
and which is known as
Hooke's law
. It may be expressed mathematically as
=
E
ε
σ
(7.7)
where
E
is the constant of proportionality.
E
is known as
Young's modulus
or the
elastic
modulus
of the material and has the same units as stress. For mild steel
E
is of the
order of 200 kN/mm
2
. Equation (7.7) may be written in alternative form as
σ
ε
=
E
(7.8)
For many materials
E
has the same value in tension and compression.
SHEAR MODULUS
By comparison with Eq. (7.8) we can define the
shear modulus
or
modulus of rigidity
,
G
, of a material as the ratio of shear stress to shear strain; thus
τ
γ
G
=
(7.9)
VOLUME OR BULK MODULUS
Again, the
volume modulus
or
bulk modulus
,
K
, of a material is defined in a similar
manner as the ratio of volumetric stress to volumetric strain, i.e.
volumetric stress
volumetric strain
K
=
(7.10)
It is not usual to assign separate symbols to volumetric stress and strain since they may,
respectively, be expressed in terms of direct stress and linear strain. Thus in the case
