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Chapter 3
Constitutive
Models  Relations between Stress and Strain
3.8 Small strain viscoplasticity: creep and high
strain rate deformation of crystalline solids
Viscoplastic
constitutive equations are used to model the behavior of polycrystalline
materials (metals and ceramics) that are subjected to stress at high
temperatures (greater than half the melting point of the solid), and also to
model the behavior of metals that are deformed at high rates of strain
(greater than 100 per second).
Viscoplasticity
theory is a relatively simple extension of the rate independent plasticity
model discussed in Section 3.6. You
may find it helpful to review this material before attempting to read this
section.
3.8.1 Features of creep behavior
 Creep
under uniaxial loading
1. If a tensile specimen of a crystalline solid is
subjected to a time independent stress, it will progressively increase in
length. A typical series of
length-v-time curves is illustrated in the figure.
2. The length-v-time plot has three stages: a transient
period of primary creep, where the
creep rate is high; a longer period of secondary
creep, where the extension rate is constant; and finally a period of tertiary creep, where the creep rate
again increases. Most creep laws
focus on modeling primary and secondary creep. In fact, it is often sufficient to model
only secondary creep.
3. The rate of extension increases with stress. A typical plot of secondary creep rate as a
function of stress is shown in the figure.
There are usually three regimes of behavior: each regime can be fit
(over a range of stress) by a power-law with the form . At low stresses, ;
at intermediate stresses ,
and at high stress m increases
rapidly and can exceed 10-20.
4. The rate of extension increases with
temperature. At a fixed stress, the
temperature dependence of strain rate can be fit by an equation of the form ,
where is an activation energy; is the Boltzmann constant, and T is temperature. Like the stress exponent m,
the activation energy can transition from one value to another as
the temperature and stress level is varied.
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5. The various regimes for m and are associated with different mechanisms of
creep. At low stress, creep occurs mostly
by grain boundary sliding and diffusion.
At higher stresses it occurs as a result of thermally activated
dislocation motion. H.J. Frost and
M.F. Ashby. “Deformation Mechanism Maps,” Pergamon Press,
Elmsford, NY (1982) plot charts that are helpful to get a rough
idea of which mechanism is likely to be active in a particular
application. A schematic of a
deformation mechanism map is illustrated in the picture: the figure shows
various regimes of plastic flow as a function of temperature (normalized by
melting temperature), and stress (normalized by shear modulus)
6. The creep behavior of a material is strongly
sensitive to its microstructure (especially grain size and the size and
distribution of precipitates) and composition.
 Creep
under multi-axial loading
Under
proportional multi-axial loading, creep shows all the same characteristics as
rate independent plasticity: (i) plastic strains are volume preserving; (ii)
creep rates are insensitive to hydrostatic pressure; (iii) the principal
strain rates are parallel to the principal stresses; (iv) plastic flow obeys
the Levy-Mises flow rule. These
features of behavior are discussed in more detail in Sect 3.6.1.
3.8.2 Features of high-strain rate
behavior
Stress-strain
curves for metals have been measured for strain rates as high as /sec.
The general form of the stress-strain curve is essentially identical
to that measured at quasi-static strain rates (see Sect 3.6.1 for an
example), but the flow stress increases with strain rate. As an example, the experimental data of
Klopp, Clifton and Shawki, Mechanics of Materials, 4, p. 375 (1985) for the high strain
rate behavior of iron is reproduced in the figure. The flow stress rises slowly with strain
rate up to a strain rate of about ,
and then begins to rise rapidly.
3.8.3 Small-strain, viscoplastic
constitutive equations
Viscoplastic constitutive
equations are almost identical to the rate independent plastic equations in
Section 3.6. The main concepts in
viscoplasticity are,
1. Strain
rate decomposition into elastic and
plastic components;
2. Elastic
stress-strain law, which specifies
the elastic part of the strain rate in terms of
stress rate;
3. The plastic
flow potential, which determines the magnitude of the plastic strain
rate, given the stresses and the resistance of the material to flow.
4. State Variables which characterize
the resistance of the material to flow (analogous to yield stress)
5. The
plastic flow rule, which specifies
the components of plastic strain rate under multiaxial loading. Recall that
in rate independent plasticity the flow rule was expressed as the derivative
of the yield surface with respect to stress.
In viscoplasticity, the flow rule involves the derivative of the
plastic flow potential
6. Hardening
laws which specify the evolution of
the state variables with plastic strain.
These are discussed in
more detail below.
Strain
rate decomposition
We assume infinitesimal
deformation, so shape changes are characterized by .
The strain rate is
decomposed into elastic and plastic parts as
Elastic
constitutive equations
The elastic strains are
related to the stresses using the standard linear elastic stress-strain
law. The elastic strain rate follows
as
where are the components of the elastic compliance
tensor. For the special case of an
isotropic material with Young’s modulus and Poisson’s ratio
Plastic
flow potential
The
plastic flow potential specifies the magnitude of the plastic strain rate, as
a function of stress and the resistance of the material to plastic flow. It is very similar to the yield surface
for a rate independent material. The
plastic flow potential is constructed as follows.
1. Define the plastic strain rate magnitude as
2. Let denote the stresses acting on the material,
and let denote the principal stresses;
3. Experiments show that the plastic strain rate is
independent of hydrostatic pressure.
The strain rate must be a function of only the deviatoric stress
components, defined as
4. Assume that the material is isotropic. The strain rate can therefore only depend
on the invariants of the
deviatoric stress tensor. The
deviatoric stress has only two nonzero invariants. It is convenient to choose
In practice, only the first of these (the von Mises
effective stress) is used in most flow potentials.
5. The plastic flow potential can be represented
graphically by plotting it as a function of the three principal stresses,
exactly as the yield surface is shown graphically for a rate independent
material. An example is shown in the
picture. The lines show contours of
constant plastic strain rate.
6. For Drucker stability, the contours of constant
strain rate must be convex, and the plastic strain rate must increase with
strain rate in the direction shown in the figure.
7. Just as the yield stress of a rate independent
material can increase with plastic strain, the resistance of a viscoplastic
material to plastic straining can also increase with strain. The resistance to flow is characterized by
one or more material state variables,
which may evolve with plastic straining.
The
most general form for the flow potential of an isotropic material is thus
where
must satisfy for all (with state variables held fixed), and must
also be a convex function of
Examples of
flow potentials: Von-Mises flow potential with power-law rate sensitivity
Creep is often modeled using the a flow potential of the
form
where
,
and ,
i=1..N are material properties ( are the activation energies for the various
mechanisms that contribute to creep); k
is the Boltzmann constant and T is
temperature. The model is most often
used with N=1, but more terms are
required to fit material behavior over a wide range of temperatures and
strain rates. The potential has several state variables, . To model steady state creep, you can take to be constant; to model transient creep, must increase with strain. An example of an evolution law for is given below.
High strain rate deformation is also modeled using a power law Mises flow
potential: the following form is sometimes used:
where
and are material properties, while is a strain, strain rate and temperature
dependent state variable, which represents the quasi-static yield stress of
the material and evolves with deformation as described below. In this equation, so as to model the transition in strain rate
sensitivity at high strain rates; while controls the point at which the transition
occurs.
Plastic
flow rule
The
plastic flow rule specifies the components of plastic strain rate resulting
from a multiaxial state of stress. It
is constructed so that:
1. The plastic strain rate satisfies the Levy-Mises
plastic flow rule
2. The viscoplastic stress-strain law satisfies the
Drucker stability criterion
3. The flow rule predicts a plastic strain magnitude
consistent with the flow potential.
Both (1) and (2) are
satisfied by
If g depends on stress only through the Von-Mises effective
stress, this expression can be simplified to
For the particular case
of the Power-law von Mises flow
potential this gives
Hardening
rule
The
hardening rule specifies the evolution of state variables with plastic
straining. Many different forms of
hardening rule are used (including kinematic hardening laws such as those
discussed in 3.6.5). A simple example
of an isotropic hardening law which
is often used to model transient creep is
where is the accumulated strain associated with
each mechanism of creep, and and are material constants. The law is usually used only with N=1.
Similar
hardening laws are used in constitutive equations for high strain rate
deformation; but in this case the flow strength is made temperature
dependent. The following formula is
sometimes used
where
is the total accumulated strain, T is temperature, and are material properties. More sophisticated hardening laws make the
flow stress a function of strain rate see Clifton “High strain rate behavior of metals.” Applied
Mechanics Reviews 43 (5) (1990)
S9-S22 for more details.
3.8.4 Representative values of
parameters for viscoplastic models of creeping solids
Fitting
material parameters to test data is conceptually straightforward: the flow
potential has been constructed so that for a uniaxial tension test with ,
all other stress components zero, the uniaxial plastic strain rate is
so
the properties can be fit directly to the results of a series of uniaxial
tensile tests conducted at different temperatures and applied stresses. To
model steady-state creep can be taken to be constant. One, or two terms in the series is usually
sufficient to fit material behavior over a reasonable range of temperature
and stress.
Creep
rates are very sensitive to the microstructure and composition of a material,
so for accurate predictions you will need to find data for the particular material
you plan to use. Frost and Ashby
“Deformation Mechanism Maps,” Pergamon Press, 1982 provide approximate
expressions for creep rates of a wide range of materials, as well as
references to experimental data. As a
rough guide, approximate values for a 1-term fit to creep data for
polycrystalline Al alloys subjected to stresses in the range 5-60MPa are
listed in the table.
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Approximate
creep parameters for polycrystalline Al alloys
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(sec-1)
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(MPa)
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(J)
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20
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4
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3.8.5 Representative values of
parameters for viscoplastic models of high strain rate deformation
The
material parameters in constitutive models for high strain rate deformation
can also be fit to the results of a uniaxial tension or compression
test. For the model described in
Section 3.7.3, the steady-state uniaxial strain rate as a function of stress
is
The
material constants ,
and the flow stress can be determined from a series of uniaxial
tension tests conducted at different temperatures and levels of applied
stress; while can be found from .
If strain hardening can be neglected, is a temperature dependent constant, which
could be approximated crudely as ,
where are constants and T is temperature. Viscoplastic
properties of materials are very strongly dependent on their composition and
microstructure, so for accurate predictions you will need to find data for
the actual material you intend to use.
Clifton, R.J. “High strain rate behavior of metals.”
Applied Mechanics Reviews 43 (5)
(1990) S9-S22 describes several experimental techniques for testing material
at high strain rates, and contains references to experimental data. As a rough guide, parameter values for
1100-0 Al alloy (fit to data in Clifton’s paper cited earlier) are listed in the table
below. The value of was estimated by assuming that the solid
loses all strength at the melting point of Al (approximately 650C).
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Approximate
constitutive parameters for high strain rate behavior of Al alloy
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100
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1.6
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15
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0.1
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50
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0.00157
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298
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