Mechanical Engineering Principles

THE EFFECTS OF FORCES ON MATERIALS 7

proportionality a graph of stress/strain will be as
shown in Figure 1.8, and is a similar shape to the
force/extension graph of Figure 1.7.

Strain ε

Stress

σ

Figure 1.

1.8 Hooke’s law

Hooke’s law states:

Within the limit of proportionality, the extension of a

material is proportional to the applied force

It follows, from Section 1.7, that:

Within the limit of proportionality of a material, the

strain produced is directly proportional to the stress

producing it

Young’s modulus of elasticity

Within the limit of proportionality, stressαstrain,
hence
stress=(a constant)×strain

This constant of proportionality is calledYoung’s
modulus of elasticityand is given the symbolE.
The value ofEmay be determined from the gradient
of the straight line portion of the stress/strain graph.
The dimensions ofEare pascals (the same as for
stress, since strain is dimension-less).

E=

σ

ε

Pa

Some typical values for Young’s modulus of
elasticity, E, include: Aluminium alloy 70 GPa
(i.e. 70× 109 Pa), brass 90 GPa, copper 96 GPa,

titanium alloy 110 GPa, diamond 1200 GPa, mild

steel 210 GPa, lead 18 GPa, tungsten 410 GPa, cast

iron 110 GPa, zinc 85 GPa, glass fibre 72 GPa,

carbon fibre 300 GPa.

Stiffness

A material having a large value of Young’s modulus

is said to have a high value of material stiffness,

where stiffness is defined as:

Stiffness=

forceF

extensionx

For example, mild steel is a much stiffer material

than lead.

SinceE=

σ

ε

,σ=

F

A

andε=

x

L

,

thenE=

F

A

x

L

=

FL

Ax

=

(

F

x

)(

L

A

)

i.e. E=(stiffness)×

(

L

A

)

Stiffness

(

=

F

x

)

is also the gradient of the

force/extension graph, hence

E=(gradient of force/extension graph)

(

L

A

)

SinceLandAfor a particular specimen are con-

stant, the greater Young’s modulus the greater the

material stiffness.

Problem 10. A wire is stretched 2 mm by a

force of 250 N. Determine the force that

would stretch the wire 5 mm, assuming that

the limit of proportionality is not exceeded.

Hooke’s law states that extensionxis proportional

to forceF, provided that the limit of proportionality

is not exceeded, i.e.x∝Forx=kFwherekis a

constant.