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Z. Suo
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, together with the condition for incompressibility 123 = 1 ,
)

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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 is simple: the above form satisfies all the
governing

Solid Mechanics
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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 concave, where a
single phase is unstable. Parts of the

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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 stress.
We can write the second-invariant model by
sij
Dij =

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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 as
water, does not form any microstructure. The flow o

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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, can be either positive or negative. If we
choose c as a c

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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 types of flow.
The model achieves unusual economics: buy on

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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
$
$
%
stress,
Y
rate of deformation,
The model chara

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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
geometry and algebra: the change in the scalar, dQ, is

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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 are even functions. Experiments show
( )
()
( )
()
that

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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
picture a unit cube in the fluid, with the faces of the cube

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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
tensile stress and the rate of extension as a function:

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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 necking, we restricted ourselves to
homogeneous and time-in

Solid Mechanics
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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 greatly from
the homogenous state, even though the spat

Solid Mechanics
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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, but the one with
larger stretch is unstable. This stateme

Solid Mechanics
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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 function is measured by applying to a short segment of

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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 instantly, with no delay.
A model of viscosity specifies a r

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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 the column is
=
2 y X ,t
( ).
X 2
Buckling. We first l

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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 deviatoric stress is sij = ij mij , and the mean stress is m =

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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 using the Ilyushin theorem, which we will
study later. Th

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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 expression for the rate of deformation in the radial direction

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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
as independent variables.
Start with a scalar function o

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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 = vdf
dR ( f )
v=
.
Provided f v is a one-to-one funct

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( ) ( ) (
proving that the function Q ( v ) is convex.
Z. Suo
) ( )
Q w < Q u + 1 Q v ,
()
Q u
Q
( u v)
()
dQ v
dv
()
Q v
v
u
( )
# Q u Q v &
% ( ) ( )( > 0 .
Theorem B. A differentiable function Q v is convex if

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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 , how do we find the
()
history of strain, t ?
We can ce

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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 field around the crack tip is similar to that of
indentatio

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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 results. Let ij be a state of stress. The
mean stress is m

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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 velocities are functions of two coordinates of the plane,
v

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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 statically
determinate provided the boundary conditions are tra

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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 homogeneous
state, define stress and velocity gradient, d