Download - Mech Nonlin Mats 15.0 L02 Creep
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2013 ANSYS, Inc. July 31, 2014 1
ANSYS Mechanical Advanced Nonlinear Materials
Lecture 2 Rate Dependent Creep
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Chapter Overview
This chapter will address the wide range of implicit creep laws available in Mechanical.
We will cover the following topics:
A. Background on Creep
B. Definition of Terms
C. General Creep Equation
D. Available Creep Models
E. Material Input
F. Analysis Settings for Creep
G. Review Creep Results
H. Workshop
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A. Background on Creep
In crystalline materials, such as metals, creep mechanism is linked to diffusional flow of vacancies and dislocation movement.
Vacancies are point defects, and they tend to favor grain boundaries that are normal, rather than parallel, to the applied stress. Vacancies tend to move from regions of high to low concentrations. Diffusional flow can occur at low stresses but usually require high temperatures.
Dislocations in grains are line defects. The movement of dislocations (climb, glide, deviation) tend to be activated by high stresses, although it may also occur at intermediate temperatures.
Grain boundary sliding is sometimes considered as a separate mechanism which also contributes to creep deformation.
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... Background on Creep
Although a detailed discussion of material science is beyond the scope of this seminar, it may suffice to say that the aforementioned physical mechanics contribute to creep.
The dependency of creep deformation on stress, strain, time, and temperature are generally modeled with a form similar to the following:
The functions f1-4 are dependent on the creep law selected.
Associated creep constants are usually obtained through various tensile tests at different rates and temperatures.
Assuming isotropic behavior, the von Mises equation is used to compute the effective stress, and the equivalent strain is used in the creep strain rate equation (similar to rate-independent plasticity).
Tftfffcr 4321
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... Background on Creep
WB Mechanical uses the additive strain decomposition when calculating elastic, plastic, and creep strain:
Plastic strains (flow rule) are calculated in a similar fashion as described in the lecture on plasticity. Creep strains are evaluated based on the creep strain rate equations, specific forms of which will be discussed later.
The elastic, creep, and plastic strains are all evaluated on the (current) stress state, but they are calculated independently (not based on each other).
crplel Additive decomposition
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... Background on Creep
Creep, like plasticity, is an irreversible (inelastic) strain which is based on deviatoric behavior. The material is assumed to be incompressible under creep flow.
On the other hand, creep, unlike rate-independent plasticity, has no yield surface at which inelastic strains occur.
Hence, creep does not require a higher stress value for more creep strain to occur. Creep strains are assumed to develop at all non-zero stress values.
In engineering usage, creep is generally used to describe a thermally-activated process with a low strain rate. Rate-independent plastic and implicit creep strains
are treated in a weakly coupled manner.
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B. Definition of Terms
Three stages of creep:
Under constant load, the uniaxial strain vs. time behavior of creep is shown below.
In the primary stage, the strain rate decreases with time. This tends to occur over a short period. The secondary stage has a constant strain rate associated with it. In
the tertiary stage, the strain rate increases rapidly until failure (rupture).
t
Secondary
Tertiary
Primary
Rupture
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... Definition of Terms
Three stages of creep (contd):
The creep strain rate may be a function of stress, strain, temperature, and/or time.
For engineering analysis, the primary and secondary stages of creep are usually of greatest interest.
Tertiary creep is usually associated with the onset of failure (necking, damage) and is short-lived. Hence, tertiary creep is not modeled in Mechanical.
The strain rate associated with primary creep is usually much greater than those associated with secondary creep.
However, the strain rate is decreasing in the primary stage whereas it is usually nearly constant in the secondary stage (for the aforementioned uniaxial test case at constant stress and temperature).
Also, primary creep tends to be of a shorter period than secondary creep.
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... Definition of Terms
Creep
Under constant applied stress, strain increases.
Stress Relaxation
Under constant applied strain, stress decreases.
t
t
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... Definition of Terms Time-hardening
Assumes that the creep strain rate depends only upon the time from
the beginning of the creep process.
In other words, the curve shifts
up/down. As stress changes from
1 to 2, the different creep rates
are calculated at points A to B.
Strain-hardening
Assumes that the creep rate depends only on the existing strain of the
material. In other words, the curve
shifts left/right. As stress changes
from 1 to 2, the different creep strain
rates are calculated at points A to B.
t
c 1
2 A
B
n
cr
t
c 1
2 A
B
n
cr t
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... Definition of Terms
Implicit creep
Implicit creep refers to the use of backward Euler integration for creep strains. This method is numerically unconditionally stable. This means that it does not require as
small a time-step as the explicit creep method, so it is much faster overall.
For implicit creep plus rate-independent plasticity, the plasticity correction and creep correction done at the same time, not independently. Consequently, implicit creep is
generally more accurate than explicit creep, but it is still dependent on the time-step
size. A small enough time-step must be used to capture the path-dependent behavior
accurately.
,,, ttttttcr Tf
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C. General Creep Equation
As noted earlier, the creep equations are usually of a rate form similar to the one below:
However, the type of material being analyzed determines the choice of a specific creep equation. Some general characteristics will be discussed presently. Specific models will be covered in the implicit creep sections.
The implicit creep equations are also covered in the Elements Manual, Ch. 2.5.
Tftfffcr 4321
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... General Creep Equation
Primary creep usually exhibits either time- or strain-hardening.
Time-hardening is the inclusion of a time-dependent term:
Strain-hardening is the inclusion of a strain-dependent term:
Determination of which to use (strain- or time-hardening) is based upon material data available. Strain-hardening tends to approximate primary creep of metals more
accurately although time-hardening tends to be more popular.
Secondary creep does not exhibit time- or strain-hardening. Creep strain rate is usually constant for secondary stage.
m
cr t
n
crcr
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... General Creep Equation
Temperature-dependency
Creep effects are thermally activated, and its temperature dependence is usually expressed through the Arrhenius law:
Where Q is the activation energy, R is the universal gas constant, and T is absolute temperature.
RT
Q
cr e
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... General Creep Equation
Stress dependency
Creep strain is also usually stress-dependent, especially with dislocation creep. The steady-state creep behavior (secondary creep) is expressed in various ways.
Nortons law (a.k.a. power law):
A common modification to the above is the exponential law:
The hyperbolic sine law is yet another common function used to describe secondary (constant) creep rate:
n
cr
Ccr e
Acr sinh
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... General Creep Equation
Below is a summary of implicit creep laws available in Mechanical which will be reviewed in the following section:
Creep Equation Description Type
Strain Hardening Primary
Time Hardening Primary
Generalized Exponential Primary
Generalized Graham Primary
Generalized Blackburn Primary
Modified Time Hardening Primary
Modified Strain Hardening Primary
Generalized Garofalo (Hyperbolic sine) Secondary
Exponential Form Secondary
Norton Secondary
Time Hardening Both
Rational Polynomial Both
Generalized Time Hardening Primary
User Creep
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D. Available Creep Models
1) Strain Hardening
Primary creep
2) Modified Strain Hardening
Primary creep
These two creep laws contain Nortons law as well as a strain-hardening
term. Since the constant C3 is usually negative, these laws are able to
model primary creep where the creep strain rate decreases as e increases.
They can also capture some secondary creep effects since, as e increases,
the creep strain rate can become nearly constant. Note that these laws
also contain the Arrhenius equation.
/TCCC
cr eC432
1
/TCCCCcr eCC 4332 11
31 1
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... Available Creep Models 3) Time Hardening
Primary creep
4) Generalized Time Hardening
Primary Creep
5) Modified Time Hardening
Primary creep
The above three creep laws include the Arrhenius equation and Nortons law, as well as
a time-hardening term. The exponential term for t is usually between -0.5 and -1.0 to
model the decreasing creep strain rate for primary creep. Hence, this model may also
approximate a significant part of secondary creep where creep strain rate is constant.
/TCCC
cr etC432
1
54
3
3
2
21
6
CCr
CCCf
eft TC
r
cr
13
1
1432
C
etC
/TCCC
cr
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... Available Creep Models
6) Generalized Blackburn
Primary creep
7) Generalized Graham
Primary creep
8) Generalized Exponential
Primary creep
These are some variants of time-hardening creep laws (see previous slide for
discussion on time-hardening creep). Note that Generalized Blackburn uses
exponential law instead of Nortons law and, like Generalized Exponential, it
includes an exponential form for the time-hardening term.
5
72
43
61 1
C
CrtC
cr
CCr
teCeeC
/TCCCCCcr etCtCtC 87532 641
/TCC
rtC
cr
eCr
reC
43
2
5
1
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... Available Creep Models
9) Time Hardening
Primary + Secondary
10) Rational Polynomial
Primary + Secondary
Both of these time-hardening laws can be used to model primary and secondary
creep effects directly. If one takes the time derivative of Time Hardening one may
notice that it is both time-hardening and Nortons law. The Rational Polynomial
form is commonly used for steels in the nuclear industry.
121198
43
107
2
1
10 1
CC
m
CC
m
CC
mmc
ccr
CpCc
Ctpt
cpt
t
C
/TCC
/TCCC
cr teCC
etC 76
432
5
3
1
1
1
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... Available Creep Models
11) Generalized Garofalo
Secondary creep
12) Exponential Form
Secondary creep
13) Norton
Secondary creep
These last three creep laws were previously discussed. Because they do not
include any time or strain dependence on creep strain rate, these are suitable to
model secondary creep range (i.e., constant creep strain rate).
/TC Ccr eCC 4321 sinh
/TCC
cr eC32
1
/TC/C
cr eeC32
1
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E. Material Data Input From the Engineering Data Toolbox, open the Creep folder:
Highlight the creep model of interests (in the example below, modified time hardening RMB on the material model and click on Include Property
The creep model will then appear in the Properties Dialogue box.
The yellow blank boxes are now available for user to define the necessary coefficients.
As with all material properties, be sure to use consistent units
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... Material Data Input
Creep models also support temperature dependent properties via Tabular input.
Notes:
The fourth constant (C4) in the modified time hardening model above is also related to temperature via Arrhenius equation .
User has the option to define creep temperature dependency by way of Arrhenius equation with nonzero value for C4 or by multiple sets of temperature dependent data (as in this example) or both.
The Arrhenius function relies on an absolute temperature scale. Temperature units are automatically offset to absolute temperature by solver (See TOFFST command documentation).
13
1
1432
C
etC
/TCCC
cr
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F. Analysis Settings for Creep The Analysis Settings will be similar for most nonlinear problems
Although time has importance in a creep problem, the solution can be static or transient. This would
exclude or include inertial effects.
Ensure that the time step size is small enough to capture the path dependent response adequately.
Large Deflection = ON is recommend
For large models with long run times and potential convergence trouble, consider setting up a Restart
Control strategy in the event that adjustment to time
step range or convergence criteria is necessary
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Analysis Settings for Creep
The creep strain calculation can be turned on or off during an analysis.
This is useful to establish initial conditions. In this situation, a very small ending TIME value (e.g., 1e-8) should be set, and creep effects turned off. Solve initial stress state
as 1st load step. Then, to turn creep effects ON and specify the real end time for load
step 2.
Load Step 1 Load Step 2
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Analysis Settings for Creep
Because creep is a path-dependent phenomenon, it is important to ensure that the response is adequately captured.
One measure of this which the solver uses is the Creep Strain Ratio defined as:
Where cr is the equivalent creep strain increment and et is the modified equivalent elastic strain (see Ch. 4.2/4.3 of the Theory
Manual for details).
Creep Limit Ratio is the maximum allowable limit to the Creep Strain Ratio
et
cr
sC
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Analysis Settings for Creep
If, during a timestep, the solver calculates a Creep Strain Ratio larger than the Creep Limit Ratio (default =1.0), then the solution is automatically bisected until the creep limit is satisfied or the minimum time step is reached.
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Analysis Settings for Creep
Implicit creep is unconditionally stable. However, this does not mean that implicit creep is unconditionally accurate.
Although WB-Mechanical sets this limit to 1.0 by default, a creep limit ratio of 0.1 to 10 (10-1000%) is generally recommended, depending on the magnitude of the equivalent elastic strain developed and the level of accuracy required.
Also, one should be sure to specify a small enough initial, min, and max time step as well.
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G. Reviewing Creep Results In addition to reviewing elastic, thermal, and plastic strains, one can also review creep strains.
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H. Workshop Exercise Please refer to your Workshop Supplement:
Workshop 2A: Stress Relaxation