Mech Nonlin Mats 15.0 L02 Creep

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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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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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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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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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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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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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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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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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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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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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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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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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

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Mech Nonlin Mats 15.0 L02 Creep