NON-MONOTONIC STRAIN HARDENING AND ITS
CONSTITUTIVE REPRESENTATION
R.K. Boger a, R.H. Wagoner b*, F. Barlat c, W. Gan d,
a
ABAQUS Central, 9075 Centre Pointe Drive, Suite 410, West Chester, OH 45069, USA
b
Ohio State University, Department of Materials Science and Engineering, 477 Watts
Hall, 2041 College Road, Columbus, OH 43210, USA
c
Materials Science Division, Alcoa Technical Center, 100 Technical Drive, Alcoa
Center, PA 15069-0001, USA
c
Edison Welding Institute, 1250 Arthur E. Adams Dr., Columbus, OH 43221, USA
* Corresponding Author
Abstract
Modeling sheet metal forming operations requires understanding of the plastic behavior
of sheet alloys along complex strain paths. In most materials, plastic deformation in one
direction will affect subsequent deformation in another direction. For one-dimensional
deformation, this phenomenon is known as the Bauschinger effect. The Bauschinger
effect in heat-treatable aluminum alloys is heavily dependent on the presence of
hardening precipitates.
In this work, aluminum alloys 2524 and 6013, were studied as a function of artificial
aging. Not only is there a substantial reduction of the reverse yield stress with aging, the
period of the transient behavior following the load reversal is also lengthened. In overaged materials this effect is observed after prestrains as small as 0.4%. Materials with
less aging had shorter transient periods, but showed some prestrain dependent permanent
softening.
A constitutive model was developed, based upon a nonlinear kinematic hardening model,
which is capable of describing the reduction in the reverse yield stress and the hardening
transient observed after the reversal. The ability to model the permanent offset was
introduced by the addition of an additional linear backstress into the formulation, which
can be defined as a function of plastic strain. This model has been implemented in a finite
element program that successfully reproduces the main features of the experimental
results for both uniaxial and more general strain paths. Simulations of draw-bead forces
showed significant reductions, up to 25%, because of the Bauschinger effect.
Simulations of the springback angle using the new model were within 1.5° of the
experimental results for the 2524 over-aged and peak-aged temper, which show the
largest Bauschinger effects studied.
Keywords
strengthening mechanisms, yield condition, constitutive behaviour, cyclic loading,
inhomogeneous material, finite elements, mechanical testing
* Corresponding author.
Tel.: +1 614 292 2079; fax: +1 614 292 6530.
E-mail address: wagoner.2@osu.edu (R.H. Wagoner).
Features of the Bauschinger Effect
The Bauschinger effect is named after Johann Bauschinger, who in 1886 observed that
the compressive yield strength of steel samples is reduced by prior tensile deformation
(Mughrabi, 1987). Figure 1 shows the reverse loading curve for aluminum alloy 6022, in
which, the compressive flow curve has been rotated into the first quadrant by plotting
stress magnitude versus total accumulated true strain. The curve for continued monotonic
tension is shown for comparison. Comparing the reverse curve to that obtained by
monotonic tension illustrates some of the features usually associated with the
Bauschinger effect. Common attributes are a reduced yield stress followed by a period of
increased strain-hardening with permanent softening at high strains where the two curves
achieve parallelism. These features are not observed in all materials, and when they are
present, they may take different forms. For example, in some materials the two hardening
curves recombine and no permanent softening is observed. In yet other cases, such as
mild steel subjected to orthogonal deformation, the reloading curve is higher than the
monotonic curve because of latent hardening (Teodosiu and Hu, 1998).
The body of research on the sources of the Bauschinger effect can be loosely divided into
two main groups. The first focuses on the residual stresses that develop during forward
deformation, and how these stresses aid the external force after load reversal. The second
theory is based on the behavior of dislocations during the deformation. This description
may consider the anisotropy that develops on individual slip planes during deformation,
or may be more phenomenologically based and consider the rearrangement and
redistribution of many dislocation in some averaged way.
Residual Stress
One of the first attempts to describe the origin of the Bauschinger effect was developed
by Heyn around 1918 (Abel, 1987; Heyn, 1918) and supported by experiments by
Masing (1923, 1926; cited in Skelton et al, 1997). Heyn showed that residual stresses
which develop in a material during deformation can give rise to Bauschinger-like
behavior. In materials with hard second-phase particles, the plasticity mismatch between
the two phases is significant this residual stresses effect can be important. The influence
of residual stresses on a deforming matrix have been researched for a variety of materials,
including metal matrix composites (Johannessson et al, 2001), dispersion hardened
aluminum (Reynolds and Lyons, 1997), dual phase steels (Ma et al, 1989), and structural
aluminum alloys. Wilson (1965), Abel and Ham (1966), Moan and Embury (1979), and
Bate et al. (1982) studied the Bauschinger effect as a function of precipitate structure in
Al-4 wt % Cu alloys, and found the reverse yield stress was sensitive to the presence of
non-coherent precipitates that lead to residual stresses in the matrix.
This observation led to the elastic inclusion model (Barlat and Liu, 1998). Inspired by
Eshelby’s solution of a stress field around an elliptical inclusion (1957), this model
assumes the backstress in the matrix, , is
 
α  f σ i  fC I  S p
(1)
where i is the stress caused by the ith precipitate, f is the precipitate volume fraction, C is
the elastic modulus matrix of the precipitate, I is the forth order identity matrix, S is the
Eshelby tensor and p is the plastic strain in the matrix. In this equation, curly brackets
indicate directional averaging over all the possible precipitate orientations.
While these stresses can be calculated, the way in which they influence the Bauschinger
effect and evolve with plastic strain is not obvious. This original model assumes a perfect
interface between the particle and matrix, so the entire plastic strain in the matrix is
accompanied by an elastic strain in the particle. Depending on the interface behavior
between the second-phase particle and the matrix, this is may not be a realistic
assumption if some of these stresses are relieved by plastic relaxation of the matrix
around the particle. For some systems, experimental results have confirmed that the
elastic back-stresses vanish at larger strains (Wilson, 1965; Abel and Ham, 1966; Moan
and Embury, 1979). To account for this relaxation in some simple way, Barlat multiplied
the above equation by a function h(p), which tends to zero as the plastic strain increases.
In the general case, this decay term may be dependent on all aspects of the precipitate
including size shape and spacing as well as dislocation creation and annihilation.
Continuum simulations of elastic precipitates in a plastic matrix assuming a perfectly
coherent interface indicate that although elastic strains can lead to some Bauschinger-like
reduction of the yield point on reversal, but the effect saturates within 0.4% strain after
the reversal for cases where the hard-particle volume fraction is 3%. In cases where there
is a larger concentration of second phase particles (6%) the strain transient after the
reversal is longer, however is still saturates within 1% strain(Gan, 2005). This period is
dramatically shorter than the long terms transients usually observed in real materials,
such as that shown in Figure 1.
Anisotropic Dislocation Behavior
Deformation in metallic materials is accomplished by the movement of crystal
dislocations. For this reason it would seem possible to describe the mechanical behavior
both before and after a load path change based on the nature of the dislocation mobility.
This is admittedly a difficult task given the shear number of dislocations in a material as
well as the complex behavior governing their creation, annihilation and interactions with
the material structure. However, Orowan showed that dislocation effects should produce
a Bauschiner effect even in the most basic situation, where we neglect the more complex
aspects of dislocation behavior consider a single dislocation in a random field of strongobsticles, as shown in schematic in Figure 3. The dislocation will bypass at a stress that is
inversely proportional to the obstacle spacing. This will continue until the dislocation
encounters an array of obstacles that are sufficiently close to inhibit further motion, as
occurs at position 1 in Figure 3. According to Orowan, “If the applied stress is removed,
the dislocation may adjust itself locally, but in the absence of sufficiently strong backstresses it will stay more or less where it was.” (Orowan, 1958) Now, if the load is
reversed, the stress necessary to move the dislocation in the reverse direction is lower
because the row of obstacles now in front of the dislocation is less dense than the row
that arrested its forward motion, and the dislocation may travel a large distance before it
encounters another distribution of obstacles, at position 2, that are as strong as the
original array, and thus leads to anisotropic behavior—a Bauschinger effect.
In real materials the behavior is more complicated that this because the number of
dislocations as well as their mobility can change as a function of deformation. The
dislocation structure morphology can be a function of several factors—such as the
amount of cross slip or climb, lattice friction, stacking fault energy, grain size, and
precipitate concentrations. Materials with these different observed structures show a
corresponding difference in mechanical behavior. The problem becomes even more
complex when one considers behavior during reversed loadings because of the inherent
polarity of dislocation structure (Pedersen et al, 1981; Stout and Rollett, 1990; Rao and
Laukonis, 1983).
Fortunately, several experimental observations suggest that in many cases it is not critical
to explicitly model all the dislocation interactions to predict the material behavior. Recent
work by Rauch (2005) suggests the mechanical behavior is not sensitive to the particular
morphology of the dislocation network. To illustrate this, samples of low carbon steel
were prestrained at different temperatures to produce either cellular or homogeneous
dislocation structures. Despite the difference in the initial dislocation structure, there was
no difference in the strength or the work hardening rate of these materials when they
were reloaded under common conditions. This result suggests that the material behavior
is determined only by the dislocation mechanics and the dislocation morphology seems to
be derivative of these mechanics rather than a source. Because it may not be necessary
for a model to include all aspects of the morphology to accurately predict the mechanical
behavior. Many models use the dislocation density as a state variable to describe the
strength of the material. One description of the strengthening due to a particular
dislocation density is (Kocks and Mecking, 2003).
  Gb
where
1
2

Gb
L
(2)
is a constant on the order of unity based on the obstacle strength. Because
dislocation density has the units of inverse area, this equation is exactly the Orowan
equation.
More sophisticated models taking into account the effects of dislocation polarity on the
density evolution were proposed by Hasegawa et al (1986)—who made predictions of the
magnitude of the Bauschinger effect caused by untangling dislocation structures—and
Hu, Rauch and Teodosiu (1992) —who incorporated the polarity of the dislocation
structures into a continuum plasticity formulation.
Using dislocation density as a state variable is can be a useful method because one can
obtain or make physical assumptions about the dislocation generation, accumulation and
recovery processes. This provides some predictive capacity to any constitutive model
because it would have the potential to forecast the material behavior assuming the
dislocation mechanics were known. One disadvantage of this particular method is that the
theory by its nature, since it focuses on dislocation-obstacle interactions, may be difficult
to interpret when there are obstacles of varying strength and spacing, such as precipitates,
forest dislocation and grain boundaries. It also neglects non-length scale strengthening
mechanisms such as solute hardening. In addition because the individual dislocation
mechanics can be affected by a variety of chemical and microstructural features, and any
descriptions for a particular material may not be applicable in other systems. Despite
these limitations, this approach seems to have the best opportunity to describe the
Bauschinger effect in a semi-physical way.
The shear strain attainable from this type of deformation can be approximated. The total
dislocation density can be decoupled into the mobile and immobile parts. Immobile
dislocation act as obstacles, and as such as their density increases, so does the hardness of
the material.
This immobile density can be divided into dislocations that are locked and those that are
pinned up against obstacles and can reverse their direction. The locked dislocations lead
to what we will refer to later as isotropic hardening and the pinned dislocation can be
identified with the kinematic hardening. In the most severe cases of the Bauschinger
effect that will be shown later in this document the maximum kinematic hardening
observed is about 108 MPa. Using the relationship between the stress increase and the
dislocation density in the previous equation, this corresponds to a dislocation density of
about 2.4 x 109 cm-2.
The shear strain caused by the motion dislocations of density ρ, a distance,  is
  b
(3)
The Bergers vector for Al is 2.36 x 10-10. Assuming the obsticles are non-shearable, the
effective spacing, , can be estimated as 50.8 nm from Equation 9 and the yield stress of
the OA material, 143 MPa. Using these values the amount of strain expected from these
dislocations is 0.034%, over two orders of magnitude smaller than the transients observed
in this material. Therefore, it appears the Orowan description of the Bauschinger effect
does not tell the entire story.
Mechanical Modeling of the Bauschinger Effect
In materials that possess non-random grain orientations, the distribution of active slip
systems introduces anisotropy in the mechanical response. If the texture is known,
polycrystal models can construct the yield surface by calculating the resolved shear stress
on each slip system. These numerical surfaces can be complicated to implement into
associated plasticity formulations because the surface normal is difficult to compute.
Because of this situation, differentiable, analytical representations of the yield surface are
desired for numerical modeling.
The most common of these, the Von Mises yield criteria, (1928) is an isotropic yield
surface based on the assumption that plastic deformation occurs when the distortional
energy reaches a certain critical value. For sheet metal forming, other yield potentials,
such as those proposed by Hosford (1972) and Hill (1948), have been found to accurately
represent the yield surface when the anisotropy axis of the material align with the
principal straining directions. A more general anisotropic form is Barlat’s YLD2000-2D
yield function (Barlat, Brem, et al, 2003). This yield surface is created from a linear
combination of two convex potentials, which assures the convexity of the resultant
surface. This yield surface can be computed from the yield stresses and r-values of tensile
tests in the directions 0º, 45º and 90º from the rolling direction and the balanced biaxial
test. YLD2000 assures convexity and improves the yield surface prediction over the prior
Barlat yield functions, in addition to simplifying the material parameter calculation
(Yoon, Barlat et al, 2004).
The above yield equations were proposed to account for anisotropy caused by material
texture, and do a reasonable job of describing the initial yield surface of the material. As
equally important as the initial shape of the yield surface, is how it evolves and changes
with strain. Since, by definition, all stress states lie on or within the yield surface, if the
flow stress is to increase the surface must necessarily change its size, shape or position to
accommodate this new stress state.
During deformation, there is a gradual rotation of the crystal orientations, producing a
change in the material texture, which can be reasonably represented by a strain dependent
change of the yield surface exponent in a non-quadratic yield function (Suh et al, 1996;
Wagoner, 1980). However, in contrast to this gradual texture evolution, most materials
show comparatively rapid changes to the yield stress with deformation as evidenced by
work hardening and the development of the Bauschinger effect. If these effects can not be
explained by texture evolution, they must be caused by other mechanisms. If all slip
systems and directions are hardened equivalently, the material is said to experience
isotropic hardening. In this situation, the plastic potential retains the same form and the
yield surface uniformly expands with a constant shape.
One major deficiency of isotropic hardening is its inability to account for the Bauschinger
effect, where the yield stress is lowered by prior deformation in the opposite direction.
Experimentally, the yield surface has been measured after prestraining using small offset
strains on the order of 10-4 - 10-6 (Cheng and Krempl, 1991; Helling et al, 1986), which
show a dramatic flattening of the yield surface in the direction opposite the original
straining direction. Models that account for this type of distortion have been proposed for
both viscoplastic (Helling and Miller, 1987) and irreversible thermodynamic (Francois,
2001) methods as well as elasto-plastic formulations. The elasto-plastic methods involve
either defining the yield surface as some strain dependent parametric equation (Vincent et
al, 2002, 2004) or by multiplying the original yield surface by some strain dependent
anisotropy tensor (Vouiadjis et al, 1990, 1995).
Typically, for metal forming analysis, these microplastic effects are neglected, and the
Bauschinger effect is modeled by kinematic hardening. Simple linear kinematic
hardening methods, proposed by Prager (1956) and Ziegler(1959), introduced a term,
typically called the backstress, α, which acts to translate the center of the yield surface.
f   eq σ  α   k  0
(4)
The term “backstress” has its origins in the residual stress explanations of the
Bauschinger effect, but is often used broadly without regard to the mechanisms of the
mechanical behavior. In Prager’s model the translation of the yield surface occurs in the
direction of the plastic strain increment
dα  c dε p
(5)
whereas Ziegler’s model assumes the translation is along the radial direction along the
vector connecting the current stress to the center of the yield surface, (σ - α)
dα 
c
e
σ  α dp
(6)
where dp is the modulus of the plastic strain rate, |dεp| for the uniaxial case. Ziegler’s
assumption prevents any inconsistencies that may exist between the yield surface normal
in the 1D, 2D and 3D cases (Shield and Zeigler, 1958; Crisfield, 1991; Hodge, 1957).
Most attempts to improve the description of strain hardening seek to generalize these two
types of hardening to better match the behavior seen in materials. One of the first
attempts was Hodge’s (1957) mixed hardening, which assumes the hardening rate is a
linear decomposition of isotropic and kinematic hardening. Other more advanced
techniques have been proposed, such as the multi-surface models proposed by Mroz
(1967) and the family of nonlinear kinematic hardening models, first introduced by
Armstrong and Frederick (1966) and then expanded by many researchers including
Chaboche (1986), Teodoisu (1998) and Geng and Wagoner (2002a; 2002b).
The Armstrong Frederick (1966) model, forms the basis for the heavily used and updated
models by Chaboche (1986, 1989, 1994) and assumes the backstress evolves by the
following differential equation.
dα  c dε p  γ α dp
(7)
where c and γ are material constants. The second term on right-hand side of the equation
is a recall term, whose direction depends on the current value of the backstress and
provides the strain path dependence. The evolution of the backstress before and after a
strain reversal produces different behavior because the sign of the first term reverses,
while the second term does not. For a long proportional path, the backstress saturates to a
value of c/ γ at a rate determined by the magnitude of γ. This saturation behavior is
equivalent to an “implied” two-surface model where the radius of the bounding surface is
larger than the yield surface by the constant c/ γ. For monotonic deformation the
backstress can be integrated explicitly as
  p  
1  e 

c
  p
(8)
Chaboche (1986) showed that this model can be generalized by introducing several
backstresses, i.e.,
m
α   αi
(9)
i 1
dαi  ci dε   i αi dp
p
This approach significantly improves the degrees of freedom available, and makes it
much easier to match the model to the material data because the total backstress is not
determined by a single exponential, but by a sum of, i, independent exponential
functions. The addition of multiple backstresses has one other useful advantage. Adding
an additional linear backstress, with the recall term omitted (γ=0), changes the “implied”
two-surface model to allow kinematic hardening of the bounding surface. In this case, the
backstress for the monotonic situation would be.
 ε p  
c1
1
1  e  c ε
  1
p
2
p
(10)
This calculation for the backstress does not saturate. If some saturation of the linear term
is desired, it can be introduced through a strain dependence of c2 resulting in a flexible
model which can reproduce a wide variety of material responses.
The Bauschinger Effect in Aluminum Alloys 2524 and 6013
Material Alloying and Heat Treatment
Aluminum alloys 6013 and 2524 were obtained in sheet form to study the effects
of precipitation on non-proportional loading. The nominal composition of both alloys is
shown in Table 1 (Alloy Digest, 1988; 2004). Both 6013 and 2524 were solution heat
treated at 530°C and water quenched. Some material was stored in dry ice for testing in
the solution heat treated (SHT) condition, the remaining material was artificially aged to
three tempers: peak-hardness (PA) and similar hardness in the under-aged (UA) and overaged conditions (OA). The aging curves for each of the alloys at 220°C are shown in
Figure 4. The 6013 samples were heat treated for 20 minutes to produce the UA temper,
4 hours for PA and at 250°C for 96 hours to obtain OA. The 2524 was aged at 220°C for
20 minutes for UA, 90 minutes for PA and 72 hours for OA.
Transmission electron microscopy (TEM) was performed on these heat treated
samples to compare the precipitate arrangement to that reported in the literature
(Tiryakioglu and Staley, 2003; Katgerman and Eskin, 2003; Wilson and Partridge, 1965;
Shih et al, 1996; Anderson, 1959; Thanaboonsombut and Sanders, 1994; Chakrabarti and
Laughlin, 2003; Barbosa et al, 2002; Dutkiewicz and Litynska, 2002). For 2534, the
large, dark phases on the order of hundreds of nanometers shown in Figure 5 are
constituent or dispersoid phases. The hardening phases are the finer, lower contrast,
features. As aging proceeds, the density of these phases increases from the relatively
sparse situation in the under-aged sample to the more dense distribution in the longeraged samples. 6013, shown in Figure 6, also shows larger constituent and dispersoid
phases, with the important hardening precipiates appearing as finer structure.
Mechanical Testing Methods
Sheet metal exhibits anisotropic texture different from bulk specimens.
Unfortunately, techniques for mechanical testing sheet material is often limited by elastic
instabilities and buckling in the thickness direction. The majority of the testing
performed in this work is in-plane compression followed by tension, using a specialized
method device developed as part of this research effort (Boger, 2005). This testing
method gave significant insight to the long term plastic effects as it enabled compressive
strains on the order of 0.20 without elastic buckling.
Because classical plasticity holds that the flow stress is independent of the mean
pressure stress, in the absence of testing artifacts, one would expect the monotonic
tension and compression flow curves to be identical. For measurements of very large
strains, a compressive test might even be superior due to the absence of necking. This
would be especially advantageous for materials which display limited tensile ductility.
Figures 7 and 8 show both the monotonic tension and compression curves for aged 6013
and 2524. As low strains the tension and compression data are very similar, but as the
strains increase, the curves do diverge. This is most likely due to locking of the sample
between the supporting plates as the sample thickness increases during the test. The
onset of this locking is nearly impossible to detect or account for, which produces some
difficulty when trying to quantitatively assess large strain features. For this reason, the
compression-tension test is more reliable than the tension-compression tests for
Bauschinger work because it allows the reverse flow curve to be conducted in tension
where this locking is not present.
Quantification of the Bauschinger Effect
When comparing and discussing the Bauschinger behavior, it is often useful to
have some method of quantifying the response using a scalar measurement. Various
scalar measures of the Bauschinger effect have been used in the literature (Abel, 1987).
The most useful measurement will highlight the interesting Bauschinger features, while
mitigating the effects of experimental error or evaluator bias.
The stress based Bauschinger measures are, generally, indicators of the difference
between the forward and reverse flow stresses. A commonly used parameter is, β


 forward

 forward   reverse
 forward
(11).
β describes the fractional reduction in the yield stress magnitude after reversal, and is
zero if there is no Bauschinger effect, one if the material yields at zero applied stress, and
two if the material yields immediately upon reloading. The normalization inherent in the
parameter β eases the comparison of the Bauschinger effect between materials with
different flow stresses. In this calculation, the forward flow stress is simply the final flow
stress before the load reversal. Several choices can be made for the reverse stress,
depending on the offset used to determine the reverse yield stress. Figure 11 shows β
values calculated as a function of aging time for 2524, computed using various plastic
strain offsets to determine the reverse yield stress. If the offset is too small, the results
can be significantly clouded by very small continuum elastic effects or experimental
anomalies (Gan, 2005). These continuum effects can be eliminated by using a yield
offset definition of 0.4%, which is double the commonly used 0.2% offset. Using this
larger offset removes the scatter while preserving the trends in the data. Larger offsets
decrease the value of Δσ, effectively compressing the β axis, which makes differentiating
between samples with similar Bauschinger effects more difficult. In this particular
material, the magnitude of the Bauschinger change is large enough that there is virtually
no disadvantage to using offsets as large as 2%.
In addition to the flow stress difference, it is also helpful to quantify the transient
period observed after the load reversal. A very basic transient measure is the strain
needed on reversal for the material to regain the original forward flow stress. Although
simple, this measure is very robust, in that it does not require any definition of yield and
can be measured even in the absence of the monotonic flow curve. For materials with a
large permanent offset, this parameter can be unbounded, however, if this becomes a
problem, a smaller percentage of the flow stress can be used as a reference value with
little loss in accuracy, as shown in Figure 13.
A very useful method to display Bauschinger data is by plotting Δσ as a function
of the plastic strain after reversal. Using this method clearly shows the length of the
transient, and because it is plotted against plastic strain, the yield stress at any offset can
be identified by a vertical line at that strain. Materials exhibiting a permanent stress
offset can also be easily identified because, Δσ saturates to a non-zero value.
If one assumes the isotropic hardening is a function of the equivalent plastic
strain, and the material is well represented by a Chaboche backstress evolution equation,
then Δσ is the difference between the backstress evolution for continued forward
deformation and the evolution for reverse loading from that point. For uniaxial
deformation, these evolution equations can be solved analytically as,
 

c   p  po 
   o  e
 

c
(12)
where ν = ±1 is the direction of flow, and αo and εpo are the values of the backstress and
plastic strain at the point of the reversal. If the curves in Figure 14 are fit using this
equation, γ is the measure of the transient length. Figure 15 shows γ, which is roughly a
linear function of the inverse plastic strain. This plot shows the best-fit gamma is a
strong function of prestrain, which invalidates the constant γ assumption made in the
curve fitting; however, if this discrepancy is ignored, γ can still be a very effective scalar
parameter to quantify the data.
Results of Bauschinger Testing
Figure 16 and Figure 17 plot the tensile flow curves up to the necking point for
2524 and 6013, respectively. These results show the proposed heat treatments are
effective in producing a peak-aged temper which is significantly higher than both overaged and under-aged conditions. Figure 18 shows the relative increase of the yield stress
and Bauschinger effect for 2024 as it is artificially aged at 230°C for a 4% prestrain. The
values at each time are normalized by the initial values in the SHT condition to give a
relative increase. The first quantity plotted is the yield stress, showing the evolution of
the aging process. The other two parameters are stress and strain based Bauschinger
parameters, β and , introduced in the previous section. As the material is artificially
aged, the yield stress reaches a maximum after several hours of aging, whereas the
magnitude of the Bauschinger effect continues to increase and reaches a maximum value
in the over-aged condition. For the times observed, the strain value appears to saturate at
the same values, whereas the stress parameter decreases. This decrease is partially
attributable to the falling yield stress, but the magnitude of the reverse yield stress is also
increasing in this region. These results are consistent with results of Hidayetoglu, et al.
(1985) who found the Bauschinger effect of 2024 increases during aging at 190°C until
about 100 hours at which the effect stabilizes. They also observed that for extreme
overaged conditions, (2 hours at 300°C) the Bauschinger effect was intermediate between
the initial SHT value and the maximum observed during the continuous aging at 190°C.
The companion curve for 6013 is found in Figure 19. The behavior of this alloy
is similar to 2524, but the trends are not as well defined. Again, the Bauschinger
parameters reach a maximum value at aging times after the peak hardness is attained.
The Bauschinger stress reaches a very sharp peak value at overaged times followed by
decreasing β caused by both a decrease in the forward flow stress as well as an increase
in the magnitude of the reverse flow stress. For this alloy, the strain parameter is
somewhat difficult to interpret because it displays significant variation in the over-aged
condition.
Because these plots show the Bauschinger behavior develops more slowly than
the yield stress, the more kinetically stable UA temper is used for comparison rather than
the SHT, which shows large variability between samples due to rapid natural aging.
When these two tempers are compared with the other tempers as a function of prestrain,
Figure 20, they are similar, suggesting separate analysis of UA and SHT is unnecessary
to capture the general precipitation effects.
To uncover the effect of prestrain on the Bauschinger effect, compression-tension
tests were performed on both alloys in the UA, PA and OA conditions. The
compression-tension curves for 2524 and 6013 are plotted to the UTS (ultimate tensile
strength) in Figure 21 and Figure 22. As was mentioned earlier, a useful way of looking
at the Bauschinger response is to plot Δσ as a function of εp after the reversal. The data is
plotted in this way in Figure 23 and Figure 24. Note that the strain range of the UA plot
is larger than PA and OA. As was seen in Figure 18 and Figure 19, the Bauschinger
effect is much larger in the OA condition than the UA condition.
There is a dramatic
difference between the UA and OA tempers for all three important Bauschinger
features—reverse yield stress, hardening transient and permanent offset. In between
these two extremes, the PA condition is an intermediate transition state, and as such, it
displays some features from both the OA and UA conditions.
The most severe reduction in the yield point occurs in the OA temper. The Δσ
plots also show the OA temper has a significantly longer strain transient, which is a
strong function of the prestrain. The OA temper saturates to the same stress level as the
monotonic flow curve, producing no permanent offset. In the UA condition, the UTS for
each of the material is a function of the prestrain. Materials with higher prestrains
produce a larger permanent stress offset. Figure 23a clearly shows the evolution of this
offset as a function of prestrain for 2524 UA. For 6013 UA, there is an initial offset
which is a function of prestrain which eventually disappears and breaks at a similar stress
as monotonic tension.
The reverse loading curves for the largest prestrained samples of each temper are
shown in Figure 25. These curves clearly show the increase in the transient length
associated with increasing aging, with the PA materials showing an intermediate
response. In 2524, the PA temper has a similar initial yielding behavior to UA, but then
like OA shows no permanent offset. In 6013, the curve for PA lies within the bounds of
the UA and OA samples. The long-term transient shown in the OA tempers is much
longer than the initial prestrain, and can not easily be explained by any residual elastic
stress argument.
Because 2524 OA shows the most severe Bauschinger effect of the alloys and
tempers studied in this work, a more detailed study of the effect of prestrain was
performed on this condition. Figure 26 shows the monotonic tension curve along with
the tensile reloading curves of various compression-tension tests. In this figure, the
reloading curves have been shifted by their prestrain values, so that all the curves start at
the origin and can be compared to one another. The most obvious observation from this
figure is that as the prestrain increases, the length of the transient increases. Another
interesting feature is the curves are similar up until a plastic strain of about 0.003, after
which, they begin to deviate and show disparate behavior. This strain magnitude is
similar to that of the elastic continuum effects caused by the elastic mismatch of the
precipitates and the matrix mentioned earlier (Gan, 2005). After this period of common
behavior, increasing aging leads to a drop in the reverse flow stress. This suggests the
different nature of the curves is attributable to non-continuum interactions with the
precipitates.
Numerical Modeling of the Bauschinger Effect
Multiple Nonlinear Kinematic Hardening
An implementation of a multiple nonlinear kinematic hardening model was done
using a UMAT user subroutine in the commercial finite element program ABAQUS. In
this model, the evolution equation for the backstress is decoupled into three backstresses,
one of which is linear to allow for translation of the outer bounding surface and hence
permanent softening after the load reversal. The other two backstresses can be adjusted
independent of one another. One can evolve relatively rapidly to describe the initial yield
effects due to eleastic and slip-plane anisotropy effects. The other backstress evolves
less rapidly and seeks to reproduce the long-term transient produced by dislocation
evolution.
dα  dα 1  dα 2  dα 3
dα 1 
dα 2 
dα 3 
C1
e
C2
e
C3
e
σ  α 1 dp   1 α 1 dp
σ  α 2 dp   2 α 2 dp
(13)
σ  α 3 dp
For uniaxial deformation the flow stress is the sum of the backstress and the yield
surface radius. Once the terms in the backstress are known, it can be numerically
integrated as a function of strain, and the size of the yield surface can be calculated and
used to determine the isotropic hardening modulus from the monotonic flow curves.
The most straightforward parameter in this equation to fit is C3, which is related
to the permanent stress offset. The final offset, measured at the UTS of the reverse flow
curve, is plotted for 2524 UA in Figure 27. The permanent stress offset is exactly α3, so
the slope of this curve as a function of plastic strain is C3.
Initially, assume that α2 is zero. In this case the parameter γ1 controls the
saturation rate of the backstress, which governs both the rate at which the magnitude of
the backstress develops with prestrain, as well as the length of the reverse transient
hardening. In the discussion of the Bauschinger parameter γ, Figure 15 showed that γ is
well represented by a linear function of the inverse of the prestrain. For each temper, γ1
was plotted as a function of prestrain and the slope and intercept values were calculated.
Once C3 and the strain dependent γ1 are known, C1 can be found by numerically
computing the backstress and fitting this constant to the experimental data.
To produce a reasonable fit, some method must be made to account for the
disparity between the rapid evolution of the Bauschinger effect with prestrain and the
long-term transient behavior on reversal. The method utilized here, is to introduce a final
backstress term that evolves very rapidly with strain. This is preferable to defining an
initial value for the backstress, αo because it evolves in the proper direction with
prestrain. The cofficients for this final backstress are chosen by assuming a very large γ2
and then choosing the appropriate value for C2 to fit the backstress at the initial values.
Once these assumptions are made, the analytical representation of the backstress is fit to
the experimental results by changing C1 and C2, as is shown in Figure 28. Table 2 shows
the final values for the kinematic hardening variable used to model the three tempers of
2524.
Results of the Material Model.
These kinetic variables and the monotonic flow curve, were input into the UMAT
and used in ABAQUS to model tension and compression for four shell elements. Figure
29 shows the comparison of the measured compression-tension tests to the results
obtained from ABAQUS. In general, the material model is able to reproduce the major
features of the flow curve. The largest discrepancy is OA 2524, in which the hardening is
not well reproduced for the larger prestrains because the experimental hardening curve is
more linear that the exponential function used in Chaboche’s kinematic hardening model.
The other two tempers, where the reverse hardening curve is better represented by the
exponential function show very good agreement for both the reverse yield and hardening
behavior.
Application of Constitutive Model
This non-isotropic hardening has significant ramifications on the behavior of the
material. Figure 30 shows a general strain path applied to a group of elements in a finite
element analysis. Figure 31 shows the resultant hardening curve for this strain path using
both isotropic and the implemented multiple-backstress nonlinear kinematic hardening.
As seen in this figure the difference in the results can be quite significant. The isotopic
model assumes the material element continually hardens, whereas the non-linear
kinematic model shows the effects of the directional hardening.
To further assess the effects this UMAT might have on the results of metal
forming analysis, the 2524 UA, PA and OA material models were input into a model of a
drawbead, and the results using the UMAT were compared to the same model assuming
isotropic hardening. In this simulation, the sheet is modeled by 1000 shell elements,
which are clamped and drawn through the drawbead. An image of the final deformed
shape after being pulled through the drawbead is shown in Figure 32. The maximum
plastic strain in the part is approximately 0.22. The representative history of the force
needed to pull the sample through the bead is shown in Figure 33. After some initial
period, a steady state value for the drawing force is established. The variation is due to
changing contact conditions and can be reduced by mesh refinement. The average draw
force was computed for each case by taking the average of the points from 1.25 to 1.75
seconds. These forces are shown in Table 3. These results indicate a significant change
in the drawing force predicted by the material model used in this work and that predicted
assuming isotropic hardening. This improvement does come at a computational cost.
The difference in the computation times for these analyses is shown in Table 4. These
differences are significant, but are reasonable in light of the drastic difference in the
simulated forces.
The draw-bend test has been proven to be a valuable tool for examining
springback of formed parts because it closely mirrors the actual conditions in a forming
operation, as the sheet deforms by a combination of bending and stretching. In this test, a
strip of material is drawn over a tooling radius under the constant influence of a backforce, which resists the drawing motion (Geng et al, 2002a; 2002b; Wang 2004; Li et al
2002). As shown in the schematic in Figure 34, the strip, which is initially 90°, springs
back an angle, θ, the springback angle.
Accurate prediction of the springback angle requires careful numerical methods
(Li et al, 2002) and accurate material representations of both the initial yield anisotropy
and the hardening evolution (Geng and Wagoner, 2002). The implementation of the
model in this work was used to simulate the draw-bend test. The results of this
simulation and simulations assuming isotropic hardening were then compared to
experimental results.
Both the simulations and the experiments pulled the sheet 101.6 mm over a tool
of radius 19.05mm. The model used to simulate the draw-bend process consisted of 2100
shell elements, with 51 integration points through the thickness of the sheet. The
simulation run time using the UMAT material model was approximately three times
longer than the isotropic hardening model.
The results of the draw-bend tests and simulations are shown in Figure 35. The
springback angle is plotted against the back force normalized by the yield force. The
transition in the springback angle at intermediate back forces is caused by the
development of anticlastic curvature, and is not expected to be reasonably captured with
the simple Von Mises yield function assumed in this implementation (Wang, 2004). At
lower forces, this curvature does not develop, and these simulations are expected to
produce reasonable results. The 2524 UA samples show the unexpected result that the
springback results are better explained by the isotropic hardening model. The OA and
PA tempers show a much larger Bauschinger effect than the UA samples, and these
samples show much better agreement between the experimental results and the
simulation using the new UMAT. In both these cases, the new model is able to predict
the experimentally measured angle in the lower back-force region within 1.5°, while the
isotropic hardening assumption differs by over 10°.
Conclusions
This work sought to understand the effect of changing precipitate structure on the
mechanical properties of age-hardenable aluminum alloys subjected to non-proportional
deformation, and incorporate these findings into a finite element program. The following
conclusions were made about the measurement and analysis of the Bauschinger effect:
1. Standard scalar measurements of the Bauschinger stress change were found to
depend significantly on the offset strain used to define the reverse yield point. An
offset greater than 0.004 strain was determined to be necessary to avoid noise in
the measurement associated with experimental error and small elastic continuum
effects.
2. A measurement of the transient length, , can be computed from the strain
necessary for the reverse flow stress to regain a fraction of the original forward
stress was found to be a simple and reliable way to measure the transient. If the
monotonic curve is available and certain assumptions are made for the kinematic
hardening of the material, another parameter γ can be calculated. This parameter
was found to be a linear function of the inverse of the prestrain, and can be used
to help fit the kinematic hardening model.
3. For both 2524 and 6013, the Bauschinger effect increases as the sample is
artificially aged, and reaches a maximum in the over-aged state, after peakhardness had been attained.
4. Because initially the Bauschinger effect evolves relatively slowly with aging, the
SHT condition was not studied extensively because the UA state was found have
sufficiently similar behavior, while being more metallurgically stable.
5. All three of the main features identified with the Bauschinger effect—reverse
yield point, hardening transient and offset—are significantly different in the UA
and OA tempers. The PA temper shows an intermediate response between the
two.

The revere yield point for the OA material is reduced more than in the UA
case.

The UA temper contains a prestrain dependent offset, while the OA does not.

The OA material also shows a much longer period of transient hardening after
the stress reversal., which is a strong function of prestrain.
6. The initial plastic behavior up to 0.004 strain was very similar for 2524 OA
samples with various amounts of prestrain. After this initial period, which is on
the order of the strains associated with continuum elastic effects, the Bauschinger
effect was found to be a strong function of prestrain.
7. Experimental tests show transients that persist for reverse strain over 10% strain,
suggesting explanations for the Bauschinger effect based on residual elasticity or
Orowan dislocations are insufficient.
8. The yield surface appears to change rapidly with deformation. Because the
change is so rapid, it can be approximated for modeling purposes by an initial
backstress, αo; however, the directionality of this backstress does not assure the
correct behavior for multi-axial loading, so it is preferable instead to model this
behavior with a non-linear kinematic hardening component that evolves very
rapidly with strain.
9. A material model was created and implemented into a user material subroutine for
the commercial finite element program ABAQUS. This material model was fit
for 2524 using the results from the compression-tension experiments and is able
to reproduce the experimentally observed behavior except for the reverse
hardening of the OA temper, which is not well fit by the exponential function
assumed in the non-linear kinematic hardening model used..
10. A strip of material drawn through a drawbead was simulated in ABAQUS using
the material model created in this work and an isotropic hardening assumption.
Using the non-linear kinematic hardening model increases the computation time
for this problem by 61% for UA, 34% for PA and 40% for OA. However, using
the model significantly changed the computed steady-state drawing force by 10%
in the UA case and as much as 24% for OA.
11. Simulations of the drawbead test using the new material model have been
performed and require an increase the run time by a factor of three compared to
isotropic hardening models. While there is some disagreement between the
simulated and experimental results in 2524 UA, for the region where agreement
was expected, the simulated angles for 2524 OA and PA were within 1.5° of the
experimental results.
Acknowledgments
This project was funded by the National Science Foundation under project DMR013945. Thanks are also extended to Alcoa for providing the aluminum used in this
work.
List of Tables
Table 1: Nominal compositions for aluminum alloys 2524 and 6013 .
Table 2: Values of kinematic hardening variable, γ, for each test condition.
Table 3: Comparison of steady-state forces computed in ABAQUS drawbead simulation
for 2524.
Table 4: Comparison of CPU time for ABAQUS drawbead simulation for 2524.
List of Figures
Figure 0: Rotated reverse loading curve and monotonic curve.
Figure 2: Continuum effects of precipitates on the Bauschinger effect [10].
Figure 3: Schematic of the Orowan mechanism [2].
Figure 4: Aging curves for aluminum alloys 6013 and 2524.
Figure 5: TEM images of 2524 UA (a), PA (b) and OA (c).
Figure 6: TEM images of 6013 UA (a), PA (b) and OA (c).
Figure 7: Monotonic tension and compression curves for Al 2524 showing assumed
monotonic curve for PA (a) and OA (b) tempers.
Figure 8: Monotonic tension and compression curves for Al 6013 showing assumed
monotonic curve for PA (a) and OA (b) tempers.
Figure 9: Tension-compression and compression-tension curves for 2524 UA (a), PA (b)
and OA (c) tempers.
Figure 10: Tension-compression and compression-tension curves for 6013 UA (a), PA
(b) and OA (c) tempers.
Figure 11: β values as a function of aging time for compression-tension tests of 2524,
computed using various plastic offset strains.
Figure 12: Schematic showing the determination of the Bauschinger parameter 
Figure 13: as a function of prestrain for two stress threshold criteria for 2524 OA.
Figure 14: Δσ as function of εp for 2524 OA after the indicated prestrains.
Figure 15: Bauschinger parameter γ as a function of prestrain for 2524 OA.
Figure 16: Monotonic tensile curves for various tempers of Al 2524.
Figure 17: Monotonic tensile curves for various tempers of Al 6013.
Figure 18: Yield and Bauschinger evolution of 2524 with aging at 230°C using the strain
based () and stress based (β) parameters.
Figure 19: Yield and Bauschinger evolution of 2524 with aging at 230°C.
Figure 20: β measurement for 2524 (a) and 6013 (b) tension-compression tests.
Figure 21: Compression-Tension results for 2524 UA (a), PA (b) and OA (c) tempers.
Figure 22: Compression-Tension results for 6013 UA (a), PA (b) and OA (c) tempers.
Figure 23: Δσ versus reverse plastic strain for 2524 UA (a), PA (b) and OA (c).
Figure 24: Δσ versus reverse plastic strain for 6013 UA (a), PA (b) and OA (c).
Figure 25: Δσ versus εp for largest prestrains tested for2524(a) and 6013(b).
Figure 26: Monotonic and reverse loading curves of 2524 OA with various prestrains.
Figure 27: Amount of permanent stress offset measured from the compression-tension of
2524 UA.
Figure 28: Experimental and numerical representation of the backstress evolution for
2524.
Figure 29: Comparison of experimental and simulated tension-compression results for
2524 .UA (a), PA (b), and OA (c).
Figure 30: Strain path used for non-monotonic constitutive equation test.
Figure 31: Comparison of stress-strain response for isotropic and UMAT material
models under strain path defined in Figure 30.
Figure 32: ABAQUS model for the draw-bead simulation.
Figure 33: Simulated force vs. time for 2524 OA assuming non-linear kinematic
hardening.
Figure 34: Schematic of the draw-bend test [97].
Figure 35: Results of the draw-bend test for 2524 UA (a), PA (b) and OA (c).
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