Ratcheting and Cyclic Plasticity

Considerations
for Code Analysis
Presented to:
Working Group Design Methodologies (Sc-III)

February 12, 2013

By: Dave Dewees P.E.

The Equity Engineering Group, Inc.
 Ratcheting Definitions
• “Ratchetting: Unsymmetric cycles of stress between prescribed
limits will cause progressive ‘creep’ or ‘ratchetting’ in the direction of
the mean stress [1]”

Figure
from [1]

• From Paragraph 5.5.1.5, Combinations of steady state and cyclic

loadings may result in ratcheting or incremental cyclic growth of a
component.
• Appears coke drums were first component observed to do this,
sodium-cooled fast (nuclear) reactors were then suspected of having
the potential to do it
• The failure mechanism is instability from growth and burst from
thinning

2
 Bree Diagram
• The Code ratcheting rules (for elastic
analysis results) are based on the Bree
diagram
– Bree published his famous paper in
1967 [2] – it re-derived the results of
Miller [3] from a decade earlier
– Original Code criteria document [4]
sites the Miller paper as the basis for
the Code rules

Figure
from [3]

Figure from [2]

3
 Bree Diagram
• Bree made the following
analysis assumptions to obtain
a problem that had a close-
formed solution:
– Cylinder wall approximated by
1D slab (hoop direction stress
kept)
– Elastic-perfectly plastic material
with constant (temperature-
independent) yield stress
– Constant membrane (pressure)
stress, P∙R/t
– Thermal stress equal to
E∙a∙DT/(2(1-n))
– DT assumed to alternate
between zero and max. value
(i.e. not fully reversed like
might happen in a start-
up/shut-down cycle)

Figure from [2]

4
 Bree Diagram
• To summarize:
– Primary stress: membrane only, constant through time
– Secondary stress: alternates between zero and pure bending (i.e. linear
temperature gradient, though parabolic gradient has also been solved and
included in Code)
– Fictitious elastic stresses are always used to enter into the diagram (see
Slide 12.8)
• The Bree Diagram is an amazing bit of practical engineering and a
great screening tool
– but it only applies to a pretty specific problem
– Different loading conditions can give a very different result!

Figure
from [2]

5
 Extensions of the Bree Diagram
• Moreton verified the Bree/Miller ratchet boundaries experimentally,
and showed that the ratchet strains predicted by their model were
extremely conservative (which they predicted originally)
• He also numerically generated Bree-type diagrams for alternate
loading conditions

Figure
from [5]

6
 Extensions of the Bree Diagram
• The Elastic Code ratcheting rules
do not account for any of these
cases, but are generally
conservative (sometimes
extremely so) for them according
to the Moreton analysis
• The most extreme illustration of
this is the constant thermal,
alternating pressure case, which
is predicted to never show
ratcheting, only shakedown

Figure
from [5]

7
 Cyclic Stress-Strain Behavior
• Life is simple if you use an elastic-perfectly plastic model like the Bree
diagram is based on
• For this case, the common incremental plasticity models in FEA (isotropic and
kinematic hardening) are the same
• For real metals, things are much more complex, and models that try to deal
with this reality are correspondingly much more complex
ratcheting

Figure from [7]

8
 Cyclic Stress-Strain Behavior
• “Cyclic” is a bit deceptive – any time the load is not held indefinitely,
there is the potential for cyclic behavior because there is unloading
• Cyclic behavior is complex and shows increased (or decreased)
stresses at fixed strain range cycling before eventual stabilization
• Stabilized means that the stress amplitude has stopped changing
with cycling at a fixed strain range

(from Fundamentals of Metal Fatigue Analysis by Bannantine et. al., Prentice Hall 1989)

9
Cyclic Material Data
60000
40000

Stress (psi)
20000
0
-20000
-40000
-60000
0 100 200 300 400
Cycle

60000
Initial monotonic 40000
response (curve)

Stress (psi)
20000
0
-20000
-40000
-60000
0 5 10
Cycle

10
 Cyclic Plasticity
• The cyclic stress-strain curve (and its equations) simply represents
the results of a fully reversed strain controlled 1D test – nothing
about these equations will reproduce ratcheting
• Ratcheting is simulated by taking the cyclic stress-strain results and
using them in an appropriate kinematic hardening incremental
plasticity model

1 1
a  r  ncss a  a  ncss
 tr   2   ta   
Ey  css 
2 K E y  K css 

Figure
from [8]

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 Isotropic vs. Kinematic Hardening
• Essential difference between the two
models is treatment of elastic range
for unloading/reloading
• In the isotropic model, elastic range
increases with plastic straining (total
elastic range is twice the value of the
largest equivalent stress obtained)
• In the kinematic model, the elastic
strain range is fixed at twice the initial
yield stress value and never
increases:
– Any stress increase beyond yield
(termed the backstress or drag stress)
will be removed from the yield strength
upon reverse loading
– Termed the Bauschinger Effect

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 Kinematic Hardening
• There are MANY kinematic hardening plasticity models
• The first and simplest is still widely used:
– Linear kinematic hardening
– Single plastic stress-strain slope
• The linear model is very useful, BUT:
– It cannot predict several cyclic plasticity phenomena
– Most importantly, it cannot predict ratcheting
• There are many nonlinear kinematic hardening (NLKH)
models as well:
– Several of these also fundamentally cannot predict
ratcheting
– The most widely known nonlinear model of this class is
the Mroz or multilinear model
– The most widely known NLKH model that CAN predict
ratcheting is the Frederick-Armstrong model

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 Nonlinear Kinematic Hardening
• Basic Frederick-Armstrong model predicts the back stress or drag
stress (stress past yield, denoted here as o):
C
dα   σ  α  dp    α  dp
o
• Chaboche generalized the model as a sum of multiple backstress so
that realistic curve shapes could be predicted
n
Ci
dα    σ  α  dp   i  α i  dp
i 1 o

Example (2.25Cr-1Mo) showing

total curve (red) and 4 separate
backstress curves that sum to it.

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 Chaboche NLKH
• This model is widely used and has been shown to represent the
behaviors of many engineering alloys quite well
– It can be integrated to give the monotonic stress-strain response (curve)
as a simple function
– If the integration is continued for reversed loading(s), an analytical
function can be obtained for the cyclic stress-strain curve
– Regression can be used to determine the model constants from curve
fitting to the monotonic curve or the cyclic curve (or ideally, both)
• Monotonic curve form resulting from integration:

1  e 
n
Ci
  p  o    i p

i i
• Cyclic curve form resulting from continued integration:

D Dp
 Dp    o   i tanh   i 
n
C
2 i i  2 

n
Ci
 max   o  
i 1 i

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 Verification
• How do you know for sure if your chosen hardening model can
predict ratcheting?
• Test it in a simple 1 element test that enforces an applied STRESS
range that is NOT fully-reversed
• In general, it’s essential to check your model in this fashion; for
example, does application of fully reversed STRAIN ranges give back
the intended cyclic stress strain curve that the C’s and ’s were
calibrated to in the first place?

(From FEA single element

N Cj
Dσ  Dα total    σ  αtotal  Dp   j  α j  Dp test results)

j 1  o

*Plastic, hardening=combined, data type=parameters, number backstresses=4

42351.4, 4.586E+05,24.341, 1.292E+06,135.2, 5.027E+06,593.4, 2.921E+07,
3156

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 Code Implementation
• The only generally available form of cyclic stress-strain data
is the cyclic stress-strain curve
– These curves come from tests on “typical” (not minimum)
strength material
– They further include (isotropic) hardening for many materials
(like carbon and stainless steel) and so are even further
strengthened relative to a Code minimum yield strength
• Use of any cyclic data for a ratcheting assessment
REQUIRES a nonlinear kinematic model as a minimum
– Even then, not all nonlinear models can fundamentally predict
ratcheting
– Calibration of these models is not trivial either
+ For example, in the F-A or Chaboche models, the ’s should not vary
with temperature
+ Additionally, the C’s need to vary smoothly with temperature, or when
the FEA interpolates for temperatures other than those specified,
nonsense will likely be returned
+ Not a trivial task

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 Code Implementation
• And even after all of that work:
– Only very advanced models with very specialized (and
overwhelmingly non-existent) material test data can
accurately predict actual ratcheting strain magnitudes
per cycle
– Or deal with known complexities like non-proportional
loading
• As a Code working group, we also need to ensure that
we’re not biased towards a particular commercial
software package when trying to make sense (and
rules) with respect to all of the preceding
• If guidance is developed, it should probably focus on
required verification and demonstration activities,
what strength level is intended, etc.

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 Code Implementation
• Perhaps the simplest rule we could make as a start
would recommend perfect-plasticity (as required for
any FEA ratcheting analysis in the current Section VIII
Division 2)
– Code minimum strength is used unambiguously
– Margin is consistent with Bree simplified rules
– You literally cannot screw up the hardening model
implementation – both isotropic and kinematic give the
same answer in this case, and ratcheting IS predicted
– Substantial benefit is often gained relative to the
simplified Bree rules:
+ FEA even in this simple case will capture the Bree diagram
extensions discussed earlier (pressure/thermal loading
pattern)
+ Biggest benefit is that if any elastic core exists, shakedown
will always be predicted

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 References
1. Abaqus Analysis User’s Manual, Version 6.9, Section 19.2.2 “Models for metals subjected to cyclic
loading,” Dassault Systèmes, Simulia.
2. Bree, J., “Elastic-Plastic Behaviour of Thin Tubes Subjected to Internal Pressure and Intermittent
High-Heat Fluxes with Application to Fast-Nuclear-Reactor Fuel Elements,” Journal of Strain
Analysis, V2 N3, 1967.
3. Miller, D.R., “Thermal-Stress Ratchet Mechanism in Pressure Vessels,” Journal of Basic
Engineering, Transactions of the ASME, June 1959.
4. “Criteria of the ASME Boiler and Pressure Vessel Code for Design by Analysis in Sections III and
VIII, Division 2”, 1969, The American Society of Mechanical Engineers, New York.
5. Moreton, D.N. and Ng, H.W., “The Extension and Verification of the Bree Diagram,” Trans. of the
6th International Structural Mechanics in Reactor Technology (SMiRT) Conference, Paris, 1981.
6. Reinhardt, Wolf, “On the Interaction of Thermal Membrane and Thermal Bending Stress in
Shakedown Analysis,” Proceedings of PVP2008, 2008 ASME Pressure Vessels and Piping Division
Conference, Chicago, 2008.
7. Khan, A.S. and Huang, S., 1995, Continuum Theory of Plasticity, Wiley-Interscience, New York.
8. Bannantine, J., Comer, J. and Handrock, J., 1989, Fundamentals of Metal Fatigue Analysis,
Prentice Hall, New York.
9. Lemaitre, J. and Chaboche, J.-L., 1994, Mechanics of Solid Materials, Cambridge University Press.

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