Lecture 1Stress, strain and elasticity

Linear deformation of a solid

Cross-sectional area

A

Cross-sectional area

A

FF FF

L0 + ∆LL0 + ∆L

0

1 FL L

Y A

Proportionality factorProportionality factor

0

FAY

LL

Young’s modulusYoung’s modulus

stressstress

strain (deformation)strain (deformation)

Elastic moduli

Hooke’s law: Hooke’s law: stressstress

strainstrain= constant (elastic

modulus)= constant (elastic

modulus)

Fk

x

(We absorbed A and L0 into k)(We absorbed A and L0 into k)Remember springs?Remember springs?

Unit of stress:

SI: Pascal 1 Pa = 1 N/m2

US: psi (pounds per square inch)

Unit of stress:

SI: Pascal 1 Pa = 1 N/m2

US: psi (pounds per square inch)

This works for small strains.This works for small strains.

Young’s modulus

0

FAY

LL

Measure of stiffnessMeasure of stiffness

MaterialYoung’s modulus

(GPa)

Rubber (small strain) 0.01-0.1

wood 1-10

bone 9-16

concrete 20

steel 200

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:

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.

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.

610

4

stress 5 10 Pa10 Pa

strain 5 10Y

Pressure in a fluid

Fluid = liquid or gasFluid = liquid or gas

Particles are always moving

Particles are always moving

i.e., hitting surfacesi.e., hitting surfaces

i.e., exerting (perpendicular) forces on surfaces i.e., exerting (perpendicular) forces on surfaces

FF

Surface of area ASurface of area A

PressurePressure

Fp

A (units: Pa)(units: Pa)

• does not depend on orientation of surface

• increases with depth

• does not depend on orientation of surface

• increases with depth

DEMO: Plastic glass

with cover

Bulk stress and strain

pressure p0

volume V0

pressure p0

volume V0

0

pB

VV

Bulk

modulus Bulk

modulus stressstress

strainstrain

Put it at the bottom of

Michigan lake

Put it at the bottom of

Michigan lake

pressure p0 + ∆p

volume V0 + ∆V (∆V

< 0)

pressure p0 + ∆p

volume V0 + ∆V (∆V

< 0)

FF

1k

B

Compressibility

Compressibility

Bulk modulus

Bulk modulus (Pa)

H2O SteelGas (STP)Pb

GAS: compressible, small B (density depends a lot on pressure)GAS: compressible, small B (density depends a lot on pressure)

LIQUID and SOLIDS: (nearly) incompressible, large B (density almost “constant”)LIQUID and SOLIDS: (nearly) incompressible, large B (density almost “constant”)

Shear modulus

Top (or bottom) area

AF//

F//

h

x

FAS

xh

Shear modulus

Shear modulus

stressstress

strainstrainstrain tan

xh

Shear angleShear angle

θθ

Example: Jello cube

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?

φ = 10°

fS

N

mg

fSmg sinφ

θ = 12°

3water

2water

2 3 3 2

sinsin

tan tan4 10 m 10 kg/ m 10 m/ s sin10

~300 Patan12

d gd gdS

sin SF mg f

2A d

strain tan

3 3J ello waterm d d

Beyond Hooke’s law

(Reversible deformations)(Reversible

deformations)

(Permanent deformations)(Permanent

deformations)

A little more realistic

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

Arches

compressioncompression

tensiontension

MaterialTensile strength

(MPa)Compressive strength

(MPa)

concrete 2 20

marble - 80

wood (parallel to

grain)40 35

Ok with wood, not stoneOk with wood, not stone

Mostly compression

Mostly compression

Good design for stoneGood design for stone