View source for Damage and stress state influence on the Bauschinger effect in aluminum alloys

==Abstract==
In this paper, we show that the Bauschinger effect is intimately tied not only to plasticity as is historically understood but to the damage state as well. We illustrate the plasticity and damage influence on the Bauschinger effect by employing different definitions (Bauschinger stress parameter, Bauschinger effect parameter, the ratio of forward-to-reverse yield, and the ratio of kinematic-to-isotropic hardening) for two differently processed aluminum alloys (rolled and cast) in which specimens were tested to different prestrain levels under tension and compression. Damage progression from second phase particles and inclusions that were generally equiaxed for the cast A356-T6 Al alloy and elongated for the rolled 7075 Al alloy was quantified from interrupted experiments. Observations showed that the Bauschinger effect had larger values for compression prestrains when compared to tension. The Bauschinger effect was also found to be a function of damage to particles/inclusions, dislocation/particle interaction, and the work hardening rate.<ref>[Jordon, J.B.; Horstemeyer, M.F.; Solanki, K.; Xue, Y. Damage and stress state influence on the Bauschinger effect in aluminum alloys. Mechanics of Materials 2007, 39, 920–931]</ref>

Authors: J.B. Jordan, [http://www.hpc.msstate.edu/directory/information.php?eid=63 Mark F. Horstemeyer], K. Solanki, Y.Xue

Corresponding Author: [mailto:bjordan@cavs.msstate.edu J.B. Jordan]

==Methodology==
For metals that experience plastic deformation, the mechanical response depends on its deformation history not just on its current stress state. This history can manifest changes in the mechanical response, like a difference between the yield stress in tension and compression as illustrated in Fig. 1, a schematic stress–strain curve for a ductile material. The stresses are the forward and reversed flow stresses, respectively. If both the forward and reverse flow stresses are equal, the material behaves in an isotropic manner. However, experiments have shown that in many ductile materials, the flow stress in the reverse direction has more permanent softening than the flow stress of forward direction. This reduced reverse flow is known as the Bauschinger effect.

[[Image: Bauschinger_effect_Stress-Strain_curve.jpg|thumb|right|500px| Figure 1: A stress–strain curve that exhibits the Bauschinger effect for typical metallic alloys.]]

The definition of the yield point can affect the Bauschinger effect. Specifically, a yield criterion based on the deviation from proportionality or a very small strain offset will imply a large Bauschinger effect while one that is based on a relatively large strain offset or on a backward extrapolated yield stress can lead to a barely noticeable Bauschinger effect.

The Bauschinger effect has been found to be a function of several parameters (loading path, strain rate, temperature and texture). The physical sources attributed to the Bauschinger effect during the reverse loading of a material can be generalized into long-range and short-range transients. The long-range transients include dislocation interactions, dislocation pile-ups at grain boundaries, and Orowan loops around strong precipitates. The short-range transients include dislocation resistance to motion or annihilation. The Bauschinger effect can also be attributed to the influence of microstructural features such as void volume fraction and the size and shape of second phase particles.

In this paper, we illustrate the importance of the damage state on the Bauschinger effect and employ an internal state variable plasticity damage model to capture the behavior for the rolled 7075-T651 aluminum alloy and cast A356-T6 aluminum alloy. Literature to date has focused on plasticity related to dislocation motion. Several authors have employed internal state variables to model the Bauschinger effect in ductile metals. To the best of the authors’ knowledge, there exists no model that captures the Bauschinger effect in terms of plasticity as a function of dislocation motion and damage accumulation.

The internal state variable (ISV) plasticity-damage model has been used to predict the plastic deformation of many types of metals under various loading conditions. This plasticity-damage model has been implemented into several finite element codes and applied to various industrial applications. The ISV model is a physically based plasticity and damage constitutive model that includes microstructural content and is consistent with continuum level kinematics, kinetics, and thermodynamics. As a result, the ISV model can capture the Bauschinger effect because of the plasticity (kinematic and isotropic hardening) and damage (arising from cracked or debonded particles).

The model parameters V, Y, H, Rd, Rs, h, rd, rs, are functions of temperature, stress state and strain rate and are correlated from experiments. The damage state is defined in terms of the change in the ratio of the volume of an element in the elastically unloaded state from the initial reference state. Embedded within this total damage state are three ISVs: void nucleation, growth arising from particles and from pre-existing pores, and coalescence. The damage model is able to capture the nucleation in terms of decohesion of the material matrix or particle fracture. In addition, the model allows void growth to occur at different sides of the crack or debonded particle with the size of the newly initiated void assumed to be the size of a second phase particle. Finally, the void coalescence is added to the model to capture the event of multiple voids growing together and joining and eventually resulting in ductile failure.

The history effect relating to the reverse yield tests can be modeled by the damage accumulation and the kinematic and isotropic internal variables. In order to quantify the Bauschinger effect in the experimental results, a von Mises yield surface is employed by inverting the kinetic equation, and ignoring the temperature and strain rate effects. The amount of kinematic and isotropic hardening can be quantified from the experimental data by using the definition of the yield surface.

==Results==

[[Image: 7075-T651_aluminum_alloy_grain_structure.jpg|thumb|right|500px| Figure 2: (a) Optical micrograph of the grain structure in the rolling direction of 7075-T651 aluminum alloy. (b) Optical micrograph A356-T6 cast aluminum alloy which shows second phase particles.]]

The materials experimentally tested in this research were 7075-T651 aluminum alloy and A356-T6 aluminum alloy. The 7075-T651 is a fairly high strength aluminum alloy used heavily in the aircraft industry. The alloy was examined in its as-received condition using a scanning electron microscope and an optical microscope to quantify the particle size and grain size. The grains of this wrought material were found to be pancake shaped as displayed in Fig. 2a. The pancake shaped grains were aligned in the rolling direction of the wrought plate and scattered throughout the alloy were Al7Cu2Fe and Fe3SiAl12 constituent particles. The mean size of these particles was approximately 2 μm, with a range of 1–25 μm. The aspect ratio for the particles was 4.5 with an average size of 18 by 4 μm and a particle volume fraction of approximately of 2%. In addition, Mg2Si constituent phases were also present with an approximate size of equal to or less than the Al7Cu2Fe particles. Also, many of these particles were broken during manufacturing process and distributed along the rolling direction creating a non-uniform distribution.

Similar to the wrought alloy, the cast A356-T6 was metallographically examined in the billet form using a scanning electron microscope and an optical microscope (Fig. 2b). Because no rolling procedure was applied to the material, the cast alloy was more isotropic compared to the wrought material as observed from the equiaxed particles and secondary dendrite arms. Second phase particles within the matrix ranged from 3 to 10 μm, and the silicon particles ranged from 4 to 70 μm. The aspect ratio for the particles was 1.0 with a mean size of 4 μm and a particle volume fraction of approximately of 7%.

Cylindrical low cycle fatigue type specimens with a uniform gage length were used for both alloys and were designed based on ASTM standard E606. For the A356-T6, the specimens were machined from the chilled end of the cast billet and had an outer diameter of 9.53 mm. Mechanical experiments were conducted with a strain rate of 0.0001/s in an ambient laboratory environment. Specimens with an outer diameter of 10.14 mm were used to test the 7075-T651 and were machined from the longitudinal direction of the two-inch-thick plate. The strain rate was 0.001/s and the temperature was ambient. 

[[Image: Aluminum_alloys_BSP_vs_Prestrain.jpg|thumb|left|500px| Figure 3: The Bauschinger stress parameter (BSP) and the ratio of kinematic and isotropic hardening (RKI) plotted versus the maximum forward prestrain reached before reversal for 7075-T651 and A356-T6.]]

Two types of experiments to observe the Bauschinger effects were conducted. First the cylindrical specimens were prestrained in tension, then uniaxially reloaded in compression. The second kind included a different set of specimens that were prestrained in compression, then uniaxially reloaded in tension. For the wrought material, three strain levels were tested: 1%, 2.5%, 5%. For the cast material, three strain levels were tested: 2%, 3%, 5%.

In addition, interrupted experiments were performed to determine the void nucleation rate of the cast and wrought alloys. Tensile and compression specimens were loaded monotonically to pre-determined strain levels and sectioned for void and/or crack density quantification. The cracked and debonded particles were then quantified as a function of effective strain.

The BSP, BEP, and RKI were calculated for A356-T6 and 7075-T651 from the experimental results of the tension-followed-by-compression and compression-followed-by-tension sequences for 0.2% strain offset yield definitions. The BSP and RKI are plotted in Fig. 3. The trends of the RKI parameter increase with an increasing applied strain for both alloys regardless of the loading sequence (tension-followed-by-compression versus compression-followed-by-tension). However, the BSP increases as the prestrain increases for the 7075-T651 for both loading sequences, but the BSP for the A356-T6 has different trend. As the prestrain increases, the BSP increases for the tension-followed-by-compression, but decreases for the compression-followed-by-tension. Another observation from the data shows that the A356-T6 has a higher RKI than the 7075-T651. 
[[Image: BSP_vs_Prestrain_Aluminum_Comparison.jpg|thumb|right|500px| Figure 4: A plot of the Bauschinger stress parameter (BSP) as a function of the maximum forward prestrain reached before reversal.]]
The last observation from the results is that the BEP is inversely related to the RKI, thus indicating that the BEP is correlated to the ratio of isotropic-to-kinematic hardening (RIK). This argument is further strengthened when comparing the definitions of the these parameters. The BEP parameter quantifies the amount of anisotropy associated with a given material and the RIK is the ratio of isotropic-to-kinematic hardening. Again the relation of BEP to RIK has not been observed in the literature.

To further illustrate the comparison of the data generated for this study with that in the literature, the BSP data from a cast 3x series aluminum alloy is plotted with the A356-T6 and 7075-T651 as shown in Fig. 4. Clearly, the tension-followed-by-compression BSP values for both alloys correlate well prior results (tension-followed-by-compression).

[[Image: Bauschinger_effect_True_Stress_&_True_Strain_curves.jpg|thumb|left|500px| Figure 5: A comparison of the internal state variable plasticity-damage model and experimental data to illustrate the Bauschinger effect for (a) rolled 7075-T651 aluminum alloy and (b) cast A356-T6 aluminum alloy.]]

[[Image: Void_Density_vs_Effective_Strain_Comparison_for_Aluminum.jpg|thumb|right|500px| Figure 6: A comparison of the damage nucleation model and experimental data of the void/crack density versus strain for A356-T6 and 7075-T651. The plot shows greater void/crack nucleation for tension than compression and greater for wrought 7075-T651 al than for the cast A356 Al alloy.]]


The ISV plasticity-damage model constants were determined from experimental data by using a least sum-squared best-fit method. The model-experiment correlation process produced different constants for A356-T6 and 7075-T651. The constants for the kinematic and isotropic hardening equations were selected to produce the best fit of the experimental tension-followed-by compression and the compression-followed-by-tension data. The comparison of the model to the experimental results for both the 7075-T651 and A356-T6 alloy are shown in Fig. 5, respectively.

Fig. 6 displays the experimental results of damage progression in terms of void nucleation versus effective strain (mises strain) and the ISV Model nucleation rate fit of A356-T6 and 7075-T651, respectively. The damage characterization study showed that void nucleation occurred in the A356-T6 due to silicon fracture and debonding of the aluminum-silicon interface within the dendrite cells. Previous studies have shown that silicon fracture would occur when the defect density was high in the particles, but interface debonding would occur when the defect density was small in the particles. For the 7075-T651 aluminum alloy, particle fracture and interface debonding of the iron-rich second phase were observed to be the main sources of void nucleation. For both alloys, the damage rate shown in Fig. 6 was the “averaged” rate for both fracture and debonding mechanism. The constants for the damage equations were selected to produce the best fit of experimental monotonic tension and compression experiments.

Clearly several trends can be discerned when studying the Bauschinger effect on these aluminum alloys. The RKI, BSP, and BEP for 7075-T651 aluminum and RKI and BSP for A356 aluminum were greater for compression-followed-by-tension than tension-followed-by-compression experiments. As the prestrain increased, the reverse yield stress increased for the A356-T6 aluminum alloy but decreased for the 7075-T651 aluminum alloy. In addition, as the prestrain increased, whether tension or compression, the RKI increased, but the BEP decreased. Since the BEP has an inverse relation with RKI, it can be thought to correlate to the ratio of isotropic hardening to kinematic hardening (RIK). Also, since RKI increases with increasing prestrain, the growth of the kinematic hardening is greater than the isotropic hardening, suggesting that anisotropic hardening is growing more than isotropic hardening for these two alloys. This finding is a bit surprising because aluminum alloys are FCC structures and have high stacking fault energies leading to enhanced cross-slip activity. If the BEP is the most appropriate definition from Bauschinger’s original work, the BSP and RKI represent 7075-T651 more accurately but the BEP represents A356-T6 more accurately. As such, different definitions represent different materials much better than others.

In discussing the results of our work, we cannot neglect the role of the strengthening phases of these alloys. These nano-sized precipitates are the primary obstacles of dislocation movement within the matrix. However, grain boundaries (7075-T651), secondary dendrite arms (A356-T6), and larger constituents can provide (although small) additional barriers that further strengthen the alloys. As previously mentioned, these obstacles create barriers for dislocations that result in anisotropic hardening and produce, as displayed by both the wrought and the cast alloys, a Bauschinger effect. In addition, as the forward strain increases, the dislocation density increases, resulting in an accumulation of damage that is reflected in the flow stress.

In terms of quantifying the damage state for this work, we only looked at the larger cracked or debonded particles (>1 μm). Several conclusions can be made about the damage state. Dislocation buildup at the constituent particles is similar in compression and tension, but it is the local tensile stress state that fractures and/or debonds the particles moreso than the remote compressive stress state. As the forward strain increases, dislocations pileup at these constituents resulting in an increased local stress that either exceeds the interfacial strength of the particle matrix or the local fracture strength of the particle. This damage accumulation allows the internal stresses to relax and thus reduces the magnitude of the back stress.

[[Image: Aluminum_alloys_RKI_vs_Prestrain.jpg|thumb|left|500px| Figure 7: The experimental and finite element model of the ratio of kinematic and isotropic hardening (RKI) plotted versus the maximum forward pre-strain reached before reversal for 7075-T651 and A356-T6.]]

For both alloys the tension prestrain incurred a larger damage nucleation rate and a lesser work hardening rate than the compression prestrain as shown in Fig. 7. This tension-compression asymmetry demarks that these aluminum alloys are history dependent and that material models should include history dependence if path dependence is experienced. Although the non-linear, anistropic plasticity coupled with damage evolution give different complex responses in nonmonotonic loading sequences, the internal state variable plasticity model given by Bammann, 1990; Bammann et al., 1993 ;  Bammann et al., 1996 and updated with damage by Horstemeyer and Gokhale (1999) represent the mechanical behavior fairly well. The experimental data illustrated in Fig. 5 can be used to assess plasticity-damage models.

The difference observed between the tension and compression prestrain is due to the intimate relationship of the damage nucleation difference coupled with the hardening rate difference. The local dislocation density that built up at particles relaxed as particles cracked or debonded in turn affecting the work hardening rate. For the A356 aluminum alloy, the damage nucleation rate was higher and the work hardening rate was lower when comparing tension to compression. Also, the damage nucleation rate was very similar for A356-T6 and 7075-T651 under tension even though the average particle size was different, the aspect ratio of the particles was different, and the volume fraction was different. Unfortunately, at the time of publication the compression nucleation rate data for the 7075-T651 was unavailable. However, for this study, the compression damage nucleation rate for the 7075-T651 was assumed to be the same as the A356-T6.

The purpose of this work was to show that damage accumulation, which evolves at a different rate in tension versus compression, affects the Bauschinger effect. Other researchers have modeled the Bauschinger effect using only hardening laws. However, we have presented experimental evidence that shows that the stress state is also dependent on damage nucleation resulting from particles fracturing or debonding from the aluminum matrix. Furthermore, we describe an internal state variable model that can capture the Bauschinger effect by using isotropic and kinematic hardening variables and void/crack nucleation, growth, and coalescence variables.

Experiments in tension-followed-by-compression and compression-followed-by-tension were performed on rolled and cast aluminum alloys. A new definition as well as existing definitions were employed to quantify the degree of the Bauschinger effect. To better understand the contribution of the kinematic and isotropic hardening to the plastic flow stress, the ratio of the kinematic-to-isotropic hardening parameter was employed. Based on the experimental work, both alloys (7075-T651 and A356-T651) displayed higher ratio of kinematic to isotropic hardening and Bauschinger stress parameter in compression-followed-by-tension compared to tension-followed-by-compression. Experimental damage nucleation results showed a path dependence that effects the stress/strain asymmetry and suggested a link between the Ratio of Kinematic and Isotropic and damage nucleation rate.

No longer can we model the Bauschinger effect without the inclusion of a void/crack nucleation model that can distinguish between tension and compression and apply to different material microstructures. An internal state variable plasticity-damage model was introduced, along with pertinent equations and assumptions, and was able to reproduce the evolving Bauschinger effect as the damage rate evolves.

==References==
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[[Category: Research Paper]]
[[Category: Aluminum]]
[[Category: Microscale]]