Force, stress, strain and elastic deformation

1
Force, stress, strain and elastic deformation
2
Force, stress and strain Mass in kilograms, kg
often called weight Produces, in gravity, Force
in Newtons. 1kg produces a force of 9.8
Newtons A force pulling (or pushing) on a rod of
a cross-sectional area A Produces a stress,
measured in Pascals (Newtons per square
meter) Stress Force/(Area of cross
section) For a weight of 10 kg hanging from a
rod that is 1cm (0.01meters) in diameter Force
98 N Area pr2 78.5x10-6 m2 Stress 1.2x106
Pa, 1.2 MegaPascals This will cause a strain
(change in length/original length) in rod.
3
Mechanical Properties Response to Stress
Elastic reversible strain (e.g a
spring) Plastic permanent strain (e.g. a bent
paperclip) Fracture propagation of a crack
(e.g. breaking glass) Viscous flow of a
liquid Visco-elastic slow elastic strain and
recovery (e.g. toffee) Fatigue slowly growing
crack in cyclic loading Creep slow permanent
strain due to vacancy motion (e.g. old lead pipe)
4
Mechanical testing needs a machine much stiffer
than the sample. A load cell in the crosshead
measures force as the sample is stretched at a
constant rate by the screw drive (or compressed
or bent)
5
For now we are concerned with the initial
reversible, usually linear, part of the curve
that precedes yield and permanent deformation.
6
Elastic modulus in tension Linear
elasticity Hookes Law, 1660 Deformation ?
Load Youngs modulus 1807 Stress Modulus x
Strain, ? E . ? Engineering Stress
Load/Original Area Engineering Strain
Extension/Length At large deformations the
cross-sectional area reduces significantly from
the original and we need to redefine stress. True
Stress Load/ Actual Area True strain log
(L/Lo)
7
Stiffness is good for Stiff low elastic
deflection under load Aircraft stiff light
aluminum alloy, carbon fiber composite Columns
prone to elastic compressive buckling
pillars, legs, bicycle frames Radio
masts Stiffness is bad for Trees Buildings
in earthquakes Bungee cords Structures subject to
impacts bicycle frames, auto suspensions, bodies
8
Other moduli Shear tg.e g1/3E Bulk
PK.DV/V Poissons ratio, n Dw/DL
1/2 And many more.
9
Elastic Bending In bending, one surface is in
tension, one is in compression and the center is
neutral. Shear stress is highest at the center.
For a load P, length L, width b, thickness d, the
stress at the top surface is
For our weight lifter P 3000 Newtons, L 1
meter (2x bone length) b 2 cm, d2cm Stress
562 MPa, enough to break the bone easily.
10
In tension or bending elasticity may lead to
deformation that is inconvenient. Buckling is
also an important failure mode. A column under a
centric axial load can exhibit elastic buckling,
where it suddenly bends out and then kinks.
K column effective length factor, whose value
depends upon the conditions of end support of the
column, as follows. For both ends pinned
(hinged, free to rotate), K 1.0. For both ends
fixed, K 0.50. For one end fixed and the other
end pinned, K 0.70. For one end fixed and the
other end free to move laterally, K 2.0. l
unsupported length of column, I Moment of inertia