This document discusses stress, strain, elasticity, and material properties related to mechanical deformation. It defines stress as force over area, strain as the ratio of deformation to original length, and introduces Hooke's law which states that stress is proportional to strain for small deformations. Young's modulus is defined as the ratio of stress to strain, providing a measure of material stiffness. Examples are given of the Young's modulus for various common materials like rubber, wood, bone, and steel. Shear stress and modulus are also defined. The document compares tensile and compressive strengths of materials and discusses how arches effectively use compression strengths.

1. Lecture 1
Stress, strain and elasticity

2. Linear deformation of a solid
L0 + ∆L
F
F
∆L =
Cross-sectional
area A
1 F
L0
Y A
Proportionality factor
Young’s
modulus
F
Y = A
∆L
L0
stress
strain
(deformation)

3. Elastic moduli
Hooke’s law:
stress
strain
= constant (elastic modulus)
This works for small strains.
Remember springs?
F
=k
∆x
(We absorbed A and L0 into k)
Unit of stress:
SI: Pascal
1 Pa = 1 N/m2
US: psi (pounds per square inch)

4. Young’s modulus
Y =
F
A
∆L
L0
Measure of stiffness
Material
Young’s modulus (GPa)
Rubber (small strain)
0.01-0.1
wood
1-10
bone
9-16
concrete
20
steel
200

5. In-class example: Young’s modulus
When a tensile stress of 5 × 106 Pa is applied to the ends of a bar, it is
desired that the strain be about 5 × 10−4. The most appropriate material to
use for this bar would be:
A. Rubber ( Y ~ 3 × 107 Pa)
B. Wood ( Y ~ 1 × 1010 Pa)
C.
Brass ( Y ~ 1 × 1011 Pa)
D. New material ( Y ~ 1012 Pa)
E.
None of these come close.
stress 5 × 106 Pa
Y =
=
= 1010 Pa
strain
5 × 10 −4

6. Pressure in a fluid
DEMO:
Plastic glass
with cover
Fluid = liquid or gas
Particles are
always moving
i.e., hitting
surfaces
i.e., exerting (perpendicular)
forces on surfaces
Pressure
F⊥
p=
A
Surface of
area A
F⊥
(units: Pa)
• does not depend on
orientation of surface
• increases with depth

7. Bulk stress and strain
F⊥
Put it at the
bottom of
Michigan lake
pressure p0
pressure p0 + ∆p
volume V0
∆p
Bulk
B =−
modulus
∆V
V0
volume V0 + ∆V
stress
strain
(∆V < 0)
Compressibility
1
k =
B

8. Bulk modulus
GAS: compressible, small B (density depends a lot on pressure)
LIQUID and SOLIDS: (nearly) incompressible, large B
(density almost “constant”)
H2O
Gas (STP)
104
105
Pb
106
107
108
109
Bulk modulus (Pa)
1010
Steel
1011

9. Shear modulus
Top (or bottom)
area A
F//
h
F//
Shear
modulus
FP
S =
x
A
h
stress
strain
x
Shear angle
θ
strain = tan θ =
x
h

10. Example: Jello cube
N
A Jello cube of side d = 4 cm
is placed on a 10° incline. It
tilts to an angle of 12°. What is
the shear modulus of Jello?
θ = 12°
fS
fS
mg
φ = 10°
FP = mg sin ϕ = fS
m = d 3 ρJello ≈ d 3 ρwater
A =d 2
strain = tan θ
d 3 ρwater g sin ϕ
d 2 = d ρwater g sin ϕ
S =
tan θ
tan θ
4 × 10 −2 m 103 kg/m3 10 m/s2 sin10°
=
~ 300 Pa
tan12°
(
)(
)(
)
mg sinφ

11. Beyond Hooke’s law
(Permanent
deformations)
(Reversible
deformations)

12. A little more realistic

13. Compressive vs. tensile strength
Material
Tensile strength (MPa)
Compressive strength (MPa)
steel
500
500
cast iron
170
550
concrete
2
20
marble
-
80
wood (parallel to
grain)
40
35
bone
130
170

14. Arches
Material
Tensile strength
(MPa)
Compressive strength
(MPa)
concrete
2
20
marble
-
80
wood (parallel
to grain)
40
35
compression
Mostly
compression
tension
Ok with wood, not stone
Good design for stone