MULTIAXIAL LOADING:
GENERALIZED HOOKE's LAW
Relation Between Relative Volume Change and Strain
For simplicity, consider a rectangular block of material with dimensions a0, b0,
and c0.
c0 Its volume V0 is given by
by,
When the block is loaded by stress, its volume will change since each
dimension no
now incl
includes
des a direct strain meas
measure.
re To calc
calculate
late the volume
ol me
when loaded Vf, we multiply the new dimensions of the block,
Products of strain measures will be much smaller than individual strain measures
when the overall strain in the block is small (i.e. linear strain theory). Therefore,
we were able to drop the strain products in the equation above
above.
The relative change in volume is found by dividing the volume difference by the
initial volume,
Hence, the relative volume change (for small strains) is equal to the sum of the 3
direct strains.
Problem 3: A 56 mm long piece of square stock (structural steel) 3 mm on a side is
clamped in a press in one lateral direction but not in the other, using 5.5 kN, while a
tension of 3 kN is applied
pp
axially.
y The press
p
has rectangular
g
clamps
p 5 mm wide and 7
mm long. What are the values of the three strain components? What are the dimensional
changes in each of the three dimensions?
Solution: From Appendix B, structural steel has an elastic modulus of 220 GPa, and a
modulus of rigidity
g y of 77.2 GPa. The cross-sectional area is ((0.003 m))2 = 9 x 10-6 m2.
Taking the axial direction as the z-direction, and the two lateral directions as x and y,
with the 5.5 kN applied in the x-direction, the normal stress is:
The stresses in the other two directions are (since the area to which the force is applied is
3 mm x 7 mm):
Poisson's ratio is given by equation 2.43:
original length in the zz-direction
direction is 56 mm but the third equation applies to only 7
mm of this
However the remaining 49 mm of length is affected by an axial strain only:
However,
. Multiplying this by 49 mm, I get 7.42 x 10-2 mm. So the total change in length of the bar is the sum of these: 8.84 x 10-2 mm.
Problem 2: Determine the fractional change in volume of cold-worked
cupronickel under an overpressure of 50 atm, assuming the sample to be
isotropic.
Solution: E = 140 GPa and G = 52 GPa.
As a percentage, that's approximately - 0.003 %. Incidentally, 50 atm of overpressure is
equivalent to approximately 517 m of depth in fresh water. As small as they may seem to be,
volume changes are extremely important for submersibles.
Problem 1: The entire isotropic homogeneous rod assembly stretches , and
the diameter of rod BC decreases by , under the load shown. Determine the
elastic modulus and Poisson's ratio.
The cross-sectional areas are:
The stresses are:
Equating these, since both rods are of the same material, I have the two equations:
Substituting back into the second equation for E, I get E = 81.5 GPa.
Poisson's ratio is the negative of the ratio of the lateral to the axial strain, where
Strain tensor ( Stress matrix)
Stress tensor ( Stress matrix)
where u, v, and w are the displacements in
the x, y, and z directions respectively
.
Generalized Hooke's Law (Anisotropic Form)
Cauchy generalized Hooke's law to three dimensional elastic bodies
and stated that the 6 components of stress are linearly related to the
6 components of strain.
The stress-strain relationship written in matrix form
form, where the 6
components of stress and strain are organized into column vectors, is,
ε = S·σ
σ = C·ε
where C is the stiffness matrix, S is the compliance matrix, and S = C-1.
Isotropic Form
Hooke's Law for Plane Stress
Hooke's Law for Plane Strain