Stress and Strain

1
Stress and Strain

• Deformation and Failure

2
Stress and Strain

• Stress is a directed force distributed over an
  area.
• Strain is the deformation resulting from the
  stress

3
Compressive and tensile strength

• Direction of applied stress
• Compressive stress
• Tensile stress

Resulting Strain
4
How things break or deform

• Brittle and Ductile behavior
• Brittle Fracture results
• Ductile permanent strain
• Behavior depends on
• Magnitude of Stress
• Rate of Stress build up (dropping vs vise effect)
• Temperature of material under stress

5
Specific behavior depends on

• Differential stress (s1 s2)
• Stress and time
• Hydrostatic pressure
• Temperature
• Strain rate

s1
Weaker
s1
Stronger
s2
6
Compressive and tensile strength

• For building cement,
• compressive strength gtgt tensile strength
• 960 kg/cm2 vs 4.5 kg/cm2
• Instantaneous Strength gtgt creep Strength
• Creep strength is 20-60 of instantaneous
  strength

Directed Stresses
7
Interrelationship of stress, time and strain

• The classis Creep Curve
• Primary Creep visco-elastic
• Secondary Creep viscous flow
• Tertiary Creep accelerated viscous strain and
  rupture

Primary
Strain
Tertiary
Secondary
Time
8
Creep Curve
9
Stress Strain Curves
10
Classic Fracture

• Under directed stress, most objects fail along a
  fracture surface oriented 45o to the maximum
  stress axis.

s1
11
Force on a Plane
F
A
AP A/cos(q) Note area of P increases with
increasing q
Fn
q
P
Fs
Fs F sin (q) Fn F cos (q)

• s s/2 sin 2 q
• n s cos2q

12
Resolution of Stress on a plane
13
Stresses in the Earths crust

• Compute the stress due to overburden
• Start with F Mass x Acceleration
• Where Mass r . Volume
• And Acceleration due to gravity (g) 980 cm/sec2
• The gravitational force of 1 cm3 of rock with
  density
• is F r v g r g h a, where height (h) is 1cm
  and area (a) 1 cm2

14
F r g h a

• So,
• Note cgs force unit is a dyne

15
Stress (s)

16
To compute stress at 1 kilometer in depth

• Compressive Stress at Depth (1 km)

17
Stresses in the Earth

• In General, we can say that 1 kilobar vertical
  overburden pressure of 4 kilometers of crustal
  rock. There is a stress gradient

sz (bars)
z
sz vs depth z
18
Horizontal stresses due to body forces

• Poissons Ratio (n) the ratio of transverse
  strain to longitudinal strain.

sz
eL
eT
Strain (e) is defined as
19
Magnitude of stresses in a horizontal plane at
shallow depth is

Typical values of n are listed below
20
Horizontal s for granite at 1km depth

• Note the smaller the poisson ratio, the stiffer
  the material and the less horizontal stress will
  result.

21
Water Pressure
Using the previous formula where the density of
water at 0oC is 0.999 gm/ml and 1 ml 1 cm3.
s rgh 1 gm/cc 980 cm/s2 1 cm
s 980 dynes/cm2 98 pascals
At a depth (or head) of 200 meters
s 1.96 x 106 pascals 284 pounds/in2 (PSI)
22
Issues with water pressure

• Water reservoirs need to be at a sufficient
  height to provide water pressure between 50 and
  80 PSI (fire department requirement).
• Flat areas of the country depend on Water Towers

23
Water Pressure

• Areas with topography, use natural reservoirs
  (lakes), or man made reservoirs (dammed streams).
• Syracuse makes use of the finger lakes which are
  200 above Onondaga County.

24
Water Pressure Dangers

• Back to the underground drift
• Often, exploration is first attempted by drilling
  bore holes and reviewing the cuttings.
• This leaves cylindrical reservoirs which fill
  with water over time.
• What happens then..