1
A mathematical representation of the multiaxial Bauschinger effect
C.O. Frederick
a
and P.J. Armstrong
b
a
Rail consultant, Derbyshire, UK
E-mail: cofdalbury@sky.com
b
Emeritus Professor, The Management School. University of Leicester
E-mail: p.armstrong@le.ac.uk
(Reproduction of original 1966 CEGB internal research report in Materials at High
Temperatures Vol. 24, No. 1 pp. 1-26
1. Editorial - R. P. Skelton
Consultant and corresponding author, c/o Science Reviews, PO Box 314, St. Albans,
Herts AL1 4ZG, UK
E-mail: pskelton@scilet.com
The above title and authorship belong to an internal Company Report that was issued
in 1966 [1] and is still often referred to, but it has never been published in the open
literature. The report is reproduced in full below as a paper, but Materials at High
Temperatures has first taken the opportunity to explore the background to what is
regarded by many in the field as a classic work.
1.1 Technical background
The structural integrity of power generating plant components operating at elevated
temperatures depends on the amount of creep and fatigue damage accumulated over
long lifetimes. However, before this material damage can be calculated, the
deformation response of a structure to externally applied loads or constraints must be
assessed. Quite often, the effects of plasticity and creep are allowed for by using
power law equations as an adjustment to initially elastic conditions. This can lead to
pessimistic (i.e., conservative) results and recourse must be made to a fully detailed
inelastic analysis of the structure. For realistic predictions therefore, good
‘constitutive relations’ are required. These should accurately reflect true material
behaviour as when, for example, during start-up and shut-down operations in service,
the location of interest is taken into tension followed by an excursion into
compression. If the material of construction ‘remembers’ its previous history,
subsequent behaviour should be predicted by the equations.
In the study of materials science this history effect had already been given to some
degree by the Bauschinger effect [2]. This may be defined as the lowering of the
absolute value of the elastic limit in compression following a previous tensile loading
and vice versa. Similarly, if deformation in one direction only is considered, loading a
specimen beyond the elastic limit raises the elastic limit for a subsequent load in that
direction. Early interpretations, which are effectively still used today, were couched in
terms of an internal ‘back stress’ which changes sign according to the epoch in the
corresponding stress-strain cycle, which since plasticity is invoked, takes the form of
a hysteresis loop. Masing, a noted investigator, was able to produce a simple model
for the Bauschinger effect during stress reversal [3,4] writing some time later [5] that
“ ... a metal is no mere rigid, dead, body, but is a material that is endowed with
2
something almost like a life of its own, as a result of which many and complex
processes may go on within it.” In other words, a metal may possess a limited
memory of its past history.
For engineering assessment, the mathematical form of the constitutive laws governing
stress-strain behaviour thus depends on (a) the deformation characteristics of the
material, (b) the complexity of the structure and (c) the loading spectrum. There are
many constitutive models, see below, which ideally should take the following into
account:
Non-linear stress-strain response of the material
Material behaviour with repeated cycling (steady-state response, cyclic hardening,
or cyclic softening)
Isotropic hardening (or softening) due to change in yield stress
Kinematic hardening (or softening) due to change in plastic slope
Presence of Bauschinger effect
Relaxation of mean stress under strain control, or strain ratchet under stress
control
Other memory of past deformation
Several constitutive models were reviewed recently [6]. Their properties are very
briefly summarised below. All models [7-13] distinguish between monotonic
(unidirectional) and cyclic behaviour and all take the Bauschinger effect into account.
The list is by no means exhaustive.
Table 1: Capability of various models during cyclic deformation
(after Hales et al., (2002)
Model
Loop
curvature
Cyclic
hardening
or
softening
Ratchet
Memory
effect
Bi-linear kinematic (ORNL)
[7,8]
No
No
No*
No
As above + plastic work term
No
Yes
No
No
As above + memory effect
Yes
Yes
No
Yes
Mroz [9]
Yes
Yes
No
No
Dafalias [10,11]
Yes
Yes
No
No
FRSV [12]
Yes
Yes
No
No
Armstrong-Frederick [1]
Yes
No
Yes
No
Chaboche (non-linear K + I)
[13]
Yes
I or K
§
Partial
No
As above + memory effect [13]
Yes
I or K
§
Partial
Yes
I = Isotropic K = Kinematic
* Allows a change between monotonic and cyclic response
§
Depending on number and form of linear kinematic terms
3
Models such as those listed above would be used when more simple approaches (e.g.,
a power law) fails to predict a stabilised response for example when a component (a)
experiences only a few large cycles, (b) comprises isolated regions undergoing
plasticity at differing strain ranges or (c) is subjected to a complex loading history [6].
As expected, implementation of many of these models requires a large computational
resource. For the more advanced examples it becomes necessary to describe the
evolution of the stress range (under total strain control) in terms of an appropriate
variable such as the accumulated plastic work (total area of closed hysteresis loops) or
the plastic path length. The latter is defined as the accumulated plastic strain,
independent of sign. Some commercially available finite element codes for structural
analysis make use of some of the above models. Further background descriptions of
constitutive models and their broad classification into ‘Standard’ (decomposition of
total strain into elastic, plastic, creep and anelastic contributions) and ‘Unified’ (creep
and plasticity regarded as arising from the same dislocation source) types can be
found elsewhere [6]. ‘Standard models require a definite yield stress and ‘yield
surface’ and an incremental formulation since FE can distinguish between elastic and
plastic regions. This yield surface can be allowed to expand or contract (isotropic
behaviour) or translate (kinematic behaviour) In principle, distortion of the surface
can also be allowed. Isotropic hardening cannot predict the Bauschinger effect. In
kinematic hardening the degree of translation is a measure of the ‘back stress’. These
are not truly representative of material behaviour but can lead to a high degree of
accuracy.
Amongst the references listed in Table 1 occurs that of Chaboche [13]. This model is
able to capture evolutionary or history-dependent aspects of material response. For
practical use it requires the appropriate hysteresis loop to be described in terms of
several coefficients which are in turn deduced from experimental data. In its simplest
form it employs a kinematic hardening rule to describe a closed hysteresis loop, but
without functions for cumulative plastic strain. The work is a development of the
Armstrong-Frederick model [1] which in turn (as will be seen below) is a
development of the Prager linear kinematic model [7]. Armstrong and Frederick
introduced a ‘recall’ term which influences plastic flow differently for tensile or
compressive loading, depending on the accumulated plastic strain. The model has
become popular and has been incorporated into finite element codes [e.g. 14]. A
feature of the model is a tendency to predict ratchetting under an asymmetrical stress
cycle. This is a result of the operation of the recall term which contains the back
stress.
1.2 Historical setting – structures, specimens, and back stresses
The account given above is a very brief summary of how plasticity theory is currently
conceived and applied in high temperature applications. Thirty to forty years ago
matters were not so clear-cut, and the division between the disciplines of ‘materials
science’ and ‘engineering’ was far sharper than perhaps it is today. The account in
this Section starts in 1977 (strictly, 1975, see below) when Chaboche published a
paper [15] on ‘viscoplastic constitutive equations for the description of cyclic and
anisotropic behaviour of metals’. Computer techniques and capacity had advanced so
that non-linear problems, in this case the deformation response of superalloy IN100
(used in aero-engine turbine blades and discs) at 1000C, could be tackled. A set of
4
constitutive equations, ‘including a hidden internal state parameter’, was employed.
The extra term in the equations accounted for an ‘evanescent strain memory effect’ or
‘delay trace hypothesis’. This was in fact the Bauschinger effect, and the paper
attributed this development to an internal company report by Armstrong and
Frederick [1], issued some eleven years earlier. The analysis was extended to the 3-
dimensional (multiaxial) state. According to the complexity of the model
assumptions, stabilisation of the hysteresis loop could be predicted within a few
cycles, and the results were compared with experiments in load control and strain
control, including periods of stress relaxation.
Further details on the development of the equations are provided below by the authors
themselves. The key point about Chaboche’s (1977) paper [15] is that this was almost
certainly the first time that Armstrong and Frederick’s [1] work had been referred to
in the open literature. The report was produced under the aegis of the erstwhile
Central Electricity Generating Board (CEGB), where publication of unclassified
material was openly encouraged. In the present case this never happened (although an
attempt was made) for reasons given by the authors below. Armstrong and Frederick
were given 10 copies each for distribution amongst their colleagues and the report
was freely available from relevant CEGB libraries. In subsequent publications and
personally, Chaboche himself has always acknowledged his indebtedness to the work.
The report has been cited many times in the open literature (see below) by
investigators who surely cannot have seen the original at first hand. This state of
affairs has arisen because Chaboche’s equations in turn have been incorporated into
finite element and other analytical computer programs [14], see later.
There had in fact been a previous external publication by Frederick and Armstrong
(16). The authors showed that ‘…..if two structures differing only in their initial
internal stresses are subject to the same loading history, the distributions of internal
stress will approach one another in regions of creep and plasticity as the loading
proceeds’. The paper was concerned with engineering structures, for it is not
immediately obvious why such a structure should not ratchet (increase its dimensions
in a given direction) under a repeated load. The paper thus concerned shakedown to a
stable condition, and the authors were able to show that a few cycles of computation
should result in the steady cyclic state, regardless of starting conditions. In fact the
same effect can been demonstrated experimentally in a specimen of stainless steel at
elevated temperatures [17]. Starting from the same (compressive) load in each case,
strain rates were varied randomly over a 2 h period over a fixed total strain range. As
expected, the final stress value was path-dependent. But if any one of those random
strain blocks were regularly repeated at 2 h intervals, it was found that the resulting
hysteresis loop closed within a very few cycles. In terms of what was happening in the
microstructure of the material, it was proposed that hardening (dislocation
accumulation) was opposed by recovery (dislocation annihilation) so that at
corresponding stresses in the random cycle, the to-and-fro motion of dislocations
within the internal (cell) structure were imagined to be identical.
At the time the internal Armstrong-Frederick report [1] was conceived, the editor was
a friend and colleague of Peter Armstrong, although we worked in different
departments at the same CEGB establishment. Regular meetings were held between
‘materials’ (mechanical properties) and ‘engineering’ staff so that we might
5
appreciate the different approaches to plasticity, creep and fatigue problems
experienced in power plant. Armstrong recalls that, whereas the published paper [16]
on convergence dealt specifically with an engineering structure, he was struggling to
account for the Bauschinger/internal stress effect as would occur on the
microstructural scale in a specimen in terms of (a) an overall isotropic hardening, (b) a
directional hardening and (c) a scalar softening of the previous hardening. This leads
to the exponential expressions in the report [1].
From a ‘materials’ point of view we would now argue that hardening cannot go on for
ever with every reversed cycle. This has been demonstrated by Halford [18] who,
coincidentally working at the same time as Armstrong and Frederick, showed by
calorimetric experiments that the stored energy of deformation was dissipated as heat
energy twice every complete (tension-compression) cycle. Halford called this the
‘storage/release’ mechanism and was interpreted by Feltner and Laird [19] as due to a
‘flip-flop’ motion of dislocations. (Halford had already shown [20] that the strain
hardening exponent of a material could be identified with the energy stored every half
cycle and went on [21] to calculate the total energy required for fatigue failure.) This
shuttling of dislocations produces pile-ups leading to an opposing back stress, say at
the peak of a compression cycle. This in turn would be destroyed during the
subsequent reversal into tension, and shuttle into a pile-up in the other direction,
giving a back stress of opposite sign [22]. Such destruction requires the onset of
plasticity, and so in the elastic unloading from compression, the residing back stress
acts in the same direction, leading effectively to a reduction in the tensile yield stress
viz., the Bauschinger effect.
These invisible back stresses have an important part to play in explaining material
response [13]. Tanaka and Mura [23] developed the irreversible action of dislocations
into a quantitative model to predict fatigue endurance. Spindler [24] has shown that
the amount of creep damage induced during stress relaxation depends on the starting
position in the hysteresis loop, which can be traced ultimately to the sign of the initial
back stress.
1.3 Citations
The editor has always been aware that the Armstrong-Frederick model [1] has been
widely quoted, despite its never having appeared in the literature. The following
information has been provided by Peter Armstrong. The ISI (Institute of Scientific
Information) index goes back to 1970. Overall there are now 324 citations of the
report. It was not cited at all until 1977 by Chaboche, some 11 years after it was
written. It is possible, though unlikely, that there were citations before 1970. The
citations remained at 3 per year until 1988, i.e., 22 years after the report first
appeared. Thereafter the rate increased rapidly and now runs at around 30 per annum.
Sometimes the rule goes by other names, such as Chaboche, but also as the ‘Ohno-
Wang constitutive equations’. These last authors have based their own work [25] on
that of Chaboche, and hence, by association, on that of Armstrong and Frederick.
It is instructive to compare this citation rate with that for an earlier paper in a
different, but related, discipline published in 1959 by Hull and Rimmer [25] on the
growth of grain boundary cavities at elevated temperature. This was the first
numerical attempt at predicting a quantitative rate of void growth and the work has
