rheology polymer

1. 1: elastic solid
even the mountains flowed
before the lord
from the song of Deborah afte her victory
over the Philistines,
Judges 5:5
(translated by M. Reiner, Physics today,
Jan. 1964
Coast of Bretagne, France
Hooke, 1678
(??)
f ~ΔL
force versus extension
By Rita Greer, from
written descriptions,
2009
stress & strain

2. the stress tensor
the stress tensor the stress tensor (notation)

3. the stress tensor (notation) the stress tensor (symmetry)
the stress tensor (pressure)
fluid at rest fluid in motion; extra stress
the stress tensor (normal stress differences)
eliminate pressure
Uniaxial extension by compression
special case: simple shear
N1: first normal stress difference
N2: second normal stress difference

4. the stress tensor (principal stresses & invariants)
principal plane, principal stress
the stress tensor (principal stresses & invariants)
determine the values of σ: eigenvalue problem
first, ,second & third invariant
also eigenvectors (principal stress
directions)
finite deformation tensors
(examples of deformations)
finite deformation tensors (components of F)

5. finite deformation tensors (examples of deformations)
uniaxial extension
simple shear
solid body rotation
finite deformation tensors (examples of deformations)
uniaxial extension (incompressible)
simple shear
solid body rotation
finite deformation tensors
Finger deformation tensor B
Cauchy-Green tensor C
finite deformation tensors
simple shear

6. finite deformation tensors
inverse deformation tensors B-1, C-1
strain tensor E = B - I
principal strains (eigenvalue problem)
Incompressible: IIIB = detF = 0
the other two invariants IB IIB bind
the possible deformations
neo-Hookean solid
Hooke: 1678, Cauchy: 1820 (small deformations. Metals ceramics)
G: elastic shear modulus
T = 0 in rest state: p = G
p = 0 in rest state:
general elastic solid
- Cayley Hamilton theorem:
any tensor satisfies its own characteristic equation
- gi functions of the invariants of B
- incompressible: IIIB = 0
strain energy function
for an elastic solid the stress is a function of the internal energy U only
with W = ρ0 U and for an incompressible, isotropic material:
Examples
Neo-Hookean:
Mooney-Rivlin: