Solid Mechanics
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must be positive-definite for any
infinitesimal change in the stretches, 1
Z. Suo
2 < < +
and 2 . This condition, according to a
theorem in linear algebra, requires that
the following three conditions be sati

Plasticity
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Z. Suo
Denote the rates of extension along the three coordinates by D11 , D22 and D33 .
()
State at time t, l t
(
State at time t + dt, l t + dt
)
Rate of dilation. A piece of fluid, consisting of a fixed number

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f
.
a
The fluid is in a homogeneous state. The traction is independent of the shape and
the area of the region, but depends on the direction n of the region. For instance,
for the chewing gum in tension, a

Plasticity
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Isotropy. We further assume that the fluid is isotropic. For an isotropic
fluid, the linear relation between two symmetric tensors takes the general form:
!
$
1
ij + pij = 2 # Dij Dkkij & + Dkkij ,
3
"
%

Plasticity
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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

Plasticity
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Z. Suo
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

Plasticity
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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

Plasticity
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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

Plasticity
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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

Plasticity
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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

Plasticity
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RIGID-PLASTIC FLOW
Toothpaste retains its shape when stress is small, but flows when stress is
sufficiently large. Thus, the toothpaste does not drip or spread under its own
weight, but extrudes from a tub

Plasticity
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Elastic, isotropic hardening flow in multiaxial state. Once again
we adopt the assumption that the rate of deformation is the sum of the elastic and
plastic parts:
D = De + D p .
We adopt the Mises yield c

Plasticity
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D = s
forbidden
rigid
yield surface
1
s s = k2
2 ij ij
stress space
The material obeys the Levy-Mises flow rule, D = s , where is a scalar
measure of the flow, and depends on the state of stress as follows

Plasticity
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many viscoelastic models (Tanner 2000, Irgens 2014). Sorting what models
predict what phenomena is not easy, and is still an ongoing topic of research.
Inhomogeneous Deformation
When a body of the material

Plasticity
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Flow rule. The second-invariant model evolves the state of deformation
according to the Levy-Mises flow rule, D = s . We calculate the scalar using
the experimentally measured stress-strain curve. The flow

Plasticity
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RHEOLOGICAL MODELS
Rheology is the science of deformation. This science poses a question for
every material: Given a history of stress, how do we predict the history of strain,
or the other way around?
Rhe

Plasticity
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ij n j = ti ,
on St . Consequently, the field ij ij* satisfies the equations of force balance,
with vanishing body force and vanishing traction on St .
Applying the principle of virtual power, we obtain t

Plasticity
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Let ij satisfy
ij
x j
+ bi = 0 ,
in the body, and
ij n j = ti
on St . The field ij need not satisfy any other equations of the boundary-value
problem. The field is known as the virtual stress.
Define
=

Plasticity
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n+1
( )
F e
A !e $
# &
=
n +1 # A &
" %
.
This relation expresses the flow potential as a function of the equivalent stress,
i.e., a function of the second invariant.
Dissipation function. We can also use

Plasticity
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( )
so does the function k I2 . The second invariant is a convex function, I2 = Dij Dij .
( )
Consequently, W D is a convex function.
Uniqueness of Solution
Boundary-value problems. The equations for the b

Plasticity
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A nonlinear model that satisfies the thermodynamic inequality.
We construct a model by assuming that the dissipation function depends on the
velocities through a single variable, a positive-definite quadra

Plasticity
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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

Plasticity
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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

Plasticity
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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

Plasticity
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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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(
) ( )
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

Plasticity
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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

Plasticity
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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

Plasticity
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Z. Suo
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

Plasticity
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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