Introduction, Elasticity
Elasticity
Aggregates of matter ("stuff") can stretch, shrink, or bend if you push or pull on it
Think of individual particles (atoms, molecules) connected by springs
Tighter springs = stiffer
Looser springs = floppier
Rigid objects like steel rods behave a lot like springs
Stretch them a little and release them, they'll return to their original shape
Mostly they're very stiff, so you don't notice it
If you stretch them a lot, part or all of the stretch can become permanent, with potentially
catastrophic results
Quantifying elasticity
We want a way to describe the elasticity of a material that:
Doesn't depend on the particular force applied - though it may depend on the type of force
(squeezing, bending)
Doesn't depend on the particular object being squeezed
Best way to do it is to relate
Stress: the deforming force per unit area (N/m^2)
Strain: "unit" deformation (cm/cm) ("unit" means "dimensionless")
Applied force and deformation
Stress is proportional to strain in the linear region
In the elastic region, the material will return to its original shape
Beyond the elastic limit, you got permanent deformation
Eventually the object will break completely apart
F
Elastic region
elastic
limit
Three types of stress
¢L
Tensile stress.
Shear stress.
F
¢L
=Y
A
L
F
¢x
=G
A
L
Y- Young's
modulus
S- Shear
modulus
Hydraulic stress
F
¢V
=B
A
V
B- Bulk
modulus
Example: Young's modulus
Car suspended by a steel rod: how much does it stretch?
Car has mass m = 1225 kg (compact car)
Rod has diameter d = 1 cm, length l = 1 m
Young's modulus for steel is 200 x 10^9 N/m^2
F
¢L
=Y
A
L
F = mg = 1200N
2
A = ¼ (0:5cm) = 7:9 £ 10¡5 m2
Y = 200 £ 109 N =m2
FL
= ¢L
AY
¢L = 0:8mm
breaking
point
Linear region