mt4101=05-1

Prévia do material em texto

1
ISSUES TO ADDRESS...
• Stress and strain: What are they and why are
they used instead of load and deformation?
• Elastic behavior: When loads are small, how much 
deformation occurs? What materials deform least?
• Plastic behavior: At what point does permanent 
deformation occur? What materials are most 
resistant to permanent deformation?
Mechanical Properties Concepts of Stress and Strain
• Various types of stress
–Normal (Tensile / Compressive)
–Shear
–Hydrostatic
Strength Definitions
Force = M x a (mass multiplied by acceleration, units: N (Newton))
e.g. Weight = mass x g (Acceln due to gravity)
An apple has mass ~ 0.1 kg. g ~ 10 m s-1
∴ An apple weighs ~ 1N
Stress = force / unit area Units N.m-2 = Pa (Pascal)
A Pascal is a very small unit. e.g. if an apple stalk has area 4 mm2, 
tensile stress = 1/(4 x 10-6) = 0.25 MPa.
Strain = the change in length / the original length (unitless)
Tensile Stress
F
• Normal or direct stress acts perpendicular to the 
plane
• Acting outwards - tensile stress (+ve)
• Acting inwards - compressive stress (-ve)
F
Shear Stress
• A shear force is one that is applied parallel or tangentially 
to the surface
• Shear stress is defined as the shear force per unit area
• Denoted by τ
F
F
φ
y
x
F
F
Hydrostatic Stress
• Special state of direct 
stress
• Occurs deep in the 
earth's crust
• Deep in the ocean
• For a body in a fluid
σx = σy = σz = P
• Change in volume
F
F
F
F
2
∴ Stress has units:
N/m2 or lbf/in2
Engineering Stress
• Shear stress, τ:
Area, A
Ft
Ft
Fs
F
F
Fs
τ = Fs
Ao
• Tensile stress, σ:
original area 
before loading
Area, A
Ft
Ft
σ = Ft
Ao
2
f
2m
Nor
in
lb=
• Simple tension: cable
τ
Note: τ = M/AcR here.
Common States of Stress
Ao = cross sectional 
area (when unloaded)
FF
o
σ = F
A
o
τ = Fs
A
σσ
M
M Ao
2R
FsAc
• Torsion (a form of shear): drive shaft Ski lift
Canyon Bridge, Los Alamos, NM
o
σ = F
A
• Simple compression:
Note: compressive
structure member
(σ < 0 here).
Other Common Stress States (1)
Ao
Balanced Rock, Arches 
National Park
• Bi-axial tension: • Hydrostatic compression:
Pressurized tank
σ < 0h
Other Common Stress States (2)
Fish under water
σz > 0
σθ > 0
Deformation
• Deformation can occur in a material due to
– External loads applied
– Temperature changes
– Irradiation
• Bending, torsion, compression and shear are all 
common modes of deformation
• In some materials deformation is highly visible even 
when small loads are applied
– Rubber, plastic
• Metals and Ceramics
– Deformation tends to be very small when a small 
load is applied
Elastic means reversible!
Elastic Deformation
1. Initial 2. Small load 3. Unload
F
δ
bonds 
stretch
return to 
initial
F
δ
Linear-
elastic
Non-Linear-
elastic
3
Plastic means permanent!
Plastic Deformation 
F
δ
linear 
elastic
linear 
elastic
δplastic
1. Initial 2. Small load 3. Unload
planes 
still 
sheared
F
δelastic + plastic
bonds 
stretch 
& planes 
shear
δplastic
Engineering Strain
• Strain is defined as the 
change in length over 
original length
• Original length = l0
• Final length = lf
• Extension = Δl
• Δl = lf-l0
• Tensile Strain = ε
FF
lo
lf
Δl
• Tensile strain: • Lateral strain:
• Shear strain:
Strain is always
dimensionless.
Engineering Strain
θ
90º
90º - θy
Δx θγ = Δx/y = tan 
ε = δ
Lo
−δεL = L
wo
δ/2
δL/2
Lowo
Volumetric Strain
• Hydrostatic stress will give rise to a change in 
volume which is expressed per unit of the original 
volume
εv V
V
= Δ