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3 =
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P
P
= 3 2 .
L 1 2
L
2
The stress-stretch relations take the form
P
2
1 3 = 12 3 ,
L2
) (
(
)
P
2
2
2 3 = 2 3 .
2
L
These two equations, togeth
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fields on the applied stress. The distribution of the fields still need be determined
by solving the boundary-value problem.
The proof of this theorem
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Also marked on the W ( ) curve are the two inflection points, where
d 2W ( )
= 0.
d2
The part of the W ( ) curve between the inflection points is
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The equivalent stress is yet another way to write the second invariant. Under
uniaxial tension, the equivalent stress coincides with the applied stres
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Thixotropy
Dynamic microstructure. The flow of a material requires that its
constituting particles change neighbors. A liquid of small molecules, such
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For an incompressible fluid, Dkk = 0 . The second invariant is nonnegative for all
rates of deformation, Dij Dij 0 .
The third invariants, however, ca
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The second-invariant model uses the flow curve = g measured under shear
()
to predict the relation between stress and rate of deformation for all type
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large. This shear-thinning behavior is often modeled by a power law. Bingham
(1922) proposed an alternative model:
" 0,
for < Y
$
$
= #
Y
, for > Y
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(
) ( )
Write the change in the value of the function as dQ = Q v + dv Q v .
We translate the above result in analysis into a statement using words in
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11 22 = N 1 ,
22 33
()
= N () ,
2
Reversing the direction of the shear will not affect the normal stresses, so that
N 1 = N 1 and N 2 = N 2 ; they a
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Invariants of stress tensor. A state of stress is a physical fact,
independent of how we choose a basis. Once we choose a basis e1 ,e2 ,e 3 , we
pictu
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Extensional flow curve. Often it is more convenient to test a material
under uniaxial tension. We represent the experimentally measured curve of the
t
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becomes possible. Observe that the condition of vanishing tangent modulus is
the same as the Considre condition.
In the previous treatment of neck
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has been analyzed recently by E. Hohlfeld and L. Mahadevan (Phys. Rev. Letts.
106, 105702, 2011) In creasing, the amplitude of the field deviates
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prescribed stress is below the peak, two states of equilibrium exist. Of the two
states of equilibrium, the one with smaller stretch is stable, bu
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s(X , t )
2 x (X , t )
.
=
X
t 2
Material model. The material is taken to be nonlinearly elastic with the
stress-stretch relation
s = g( ).
This
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NONLINEAR VISCOSITY
Nonlinear viscosity. A purely viscous fluid has no memory. When the
state of stress changes, the rate of deformation changes insta
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small, so that formulas in the differential geometry of curves take simplified
forms. The slope of the column is
=
( ).
y X ,t
X
The curvature of
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Rheology of material. The model of linear viscosity relates the state of
stress and the rate of deformation:
Dkk = 0 ,
sij = 2 Dij ,
where the deviato
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n
When the fluid obeys the power law, = / A , we write the scaling
(
)
relation as
n
Q " Ga %
$
' .
a3 # A &
This scaling relation can be justified us
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d r 2
= f Dr .
dr r
The boundary conditions are r a = 0 and r = appl . Integration gives that
( )
()
( )
( ) drr .
appl = 2 f Dr
a
Recall the express
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Forces as independent variables. Rayleighs model of viscosity uses
the n velocities as independent variables. Alternatively, we can use the n forces
a
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dQ = fdv .
()
Define the Legendre transform of the function Q v by
R = fv Q .
Note that
dR = vdf .
()
function R ( f ) . In this case, the relation dR
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( ) ( ) (
proving that the function Q ( v ) is convex.
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) ( )
Q w < Q u + 1 Q v ,
()
Q u
Q
( u v)
()
dQ v
dv
()
Q v
v
u
( )
# Q u Q v &
% ( ) ( )( > 0
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STRAIN HARDENING
Rheology. Rheology is the science of deformation. This science poses a
()
question for every material: given a history of stress, t ,
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Slip lines around cracks (Rice 1968; Hutchinson 1979).
Small-scale yielding. The plastic zone is small compared to the sample
size. The slip-line fiel
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Flow under General State of Stress
Viscous flow. We have studied the viscous flow under a general state of
stress (Suo 2014). Here we list the key res
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Rigid-Plastic Flow under the Plane Strain Conditions
Compatibility of geometry. When a body flows under the plane strain
conditions, the in-plane velo
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equation comes from the yield criterion, and two partial differential equations
come from the balance of forces. Consequently, The problem is statical
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VISCOSITY
These notes develop the mechanics of viscous deformation. We begin with
viscosity in shear and in dilation. We then consider a fluid in a ho