NUMERICAL MODELING OF SUPERPLASTIC DEFORMATION MECHANISMS

N. CHANDRA and P. DANG
Department of Mechanical Engineering, FAMU/FSU College of Engineering,
Florida A&M University, Florida State University
Tallahassee, FL 32316-2175, U.S.A.

ABSTRACT

A numerical model based on micromechanics is proposed to predict the superplastic behavior of materials in all the three regions of the flow stress vs. strain rate ( - ) plot and to predict the presence or absence of superplasticity in a given material. The model incorporates a threshold stress term for diffusional flow at the atomic level which manifests as the experimentally observed threshold stress at the macro level. The model is applied to both conventional superplastic materials (7475 aluminum alloy and Al-Zn-Mg-Cu alloy) and a high strain-rate superplastic (HSRS) material (IN905XL). With the introduction of the threshold stress, the influence of temperature and grain size on the behavior of these materials can be predicted over a wider range of strain rates. In addition the strain-rate sensitivity as a function of strain rate can be fairly accurately predicted. The variation of threshold stress with respect to temperature is also studied.

INTRODUCTION

The mechanical behavior of superplastic materials is characterized by a sigmoidal curve (regions I, II and III) spanning about seven to eight decades of strain rate in log - log plot, as shown in Figure 1 [1]. The presence of a plateau in region I is generally attributed to the presence of a threshold stress. Though the threshold stress behavior is exhibited in all superplastic materials, the magnitude of threshold stress is experimentally observed to be much higher for HSRS materials than that for conventional superplastic materials [2]. Threshold stress not only affects the material behavior in region I but also has significant influence on region II. Most of the superplastic deformation models cover only the superplastic region II, over a small range of strain rate. In this paper, the micromechanical polycrystalline model developed earlier [3,4] is extended to describe the material behavior in both conventional superplastic and HSRS materials and all the three regions by incorporating a threshold stress term. With the introduction of the threshold stress for diffusional flow at the atomic level which manifests as the experimentally observed threshold stress at the macro level, the model is applied to two conventional superplastic materials (Al7475 and Al-Zn-Mg-Cu) and an HSRS material (IN905XL). The influence of temperature and grain size on the superplastic behavior is predicted over a wide range of strain rate.

NUMERICAL MODEL BASED ON MICROMECHANICS

Grain boundary sliding (GBS) is believed to be the dominant strain producing mechanism during superplastic deformation. Most of the existing models relate the strain rate of the accommodation processes of GBS to the macroscopically measured superplastic strain rate. Such a relationship implies that the total strain in a superplastic material is governed purely by the accommodation processes. Based on this concept, Chandra and co-workers [3,4] have developed a micromechanical polycrystalline model to predict the behavior of superplastic deformation with various grain sizes and temperatures. This model significantly differs from the existing models in that it is developed from the grain level to the level of the aggregate in an explicit manner.

The relationship between the macroscopic strain rate and the accommodation processes can be expressed simply as follows:

 

It is assumed that the proportionality remains constant in the ranges of strain rate, temperature and grain size considered. Even though this equation describes the macroscopic behavior of the material, similar form is adopted in the development of micromechanical model at the grain level. Diffusional and dislocational movements of atoms are considered to be the major sources for accommodation, as suggested by Ashby-Verrall model [5]. The overall strain rate, , will be the sum of the strain rates contributed by each process as shown below.

 

Diffusional flow is contributed from diffusion through crystal lattice (bulk or volume diffusion) and diffusion along grain boundaries (boundary diffusion). In general, diffusional flow is the diffusion of atoms within the grains and grain boundaries and depends on the magnitude of normal stress. In the micromechanical model, the diffusion of atoms is influenced by the normal stress acting on the slip plane. Weertman [6] has shown the influence of the normal stress acting on a slip plane of a crystal on the diffusion process. Thus, the macro behavior of the superplastic material can be modeled from the level of crystals. In order to model the grain boundary diffusional flow, grain boundary can be considered as made of dislocations [7]. Thus the normal stress acting on the slip system can also be attributed to the grain boundary diffusional flow. In our micromechanical model, diffusion is the dominant rate controlling mechanism at low stress region. Hence the threshold stress is introduced in the diffusion processes in both the boundary and lattice diffusion (denoted by in the slip plane level) in order to describe the experimentally observed macro level threshold stresses (denoted by ). The total accommodation due to diffusional flow for kth slip system incorporating threshold stress is then given by

 

Here, is the activation energy of lattice diffusion, is the activation energy of grain boundary diffusion, T is the absolute temperature, d is the grain diameter, R is the universal gas constant, and and r are material constants which will be determined from experimental superplastic deformation data.

Accommodation due to dislocation is available through the movement of dislocations which has been modeled as glide of dislocations on the slip plane [8]. This glide of dislocations is directly influenced by the resolved shear stress acting on the slip plane. Using this approach, the accommodation rate due to dislocations is derived as,

 

where and n are material constants which will be determined from experimental superplastic deformation data. Hence, there are four constants ( , n and ) embedded in the proportionality factor K mentioned in equation (1). Since K contains grain size d and temperature T, K=k(d,T). The dependencies of d and T in superplasticity are specifically described in equations (3) and (4).

Once the strain rate of all slip systems are calculated, the polycrystal is then modeled by self-consistent relation [3,4]. Stress distribution inside the polycrystal is highly heterogeneous, primarily due to the variation of grain orientation. Stress redistribution occurs continuously as the deformation proceeds. Since the accommodation activity of a constituent grain depends directly on its local stress, the superplastic deformation of the aggregate depends only on the level of accommodation in the constituent grains. The required principle of stress redistribution among the constituent grains is derived from the self-consistent relation. The detail procedures were described elsewhere [3,4].

RESULTS AND DISCUSSION

The sigmoidal curve showing the relationship between the applied stress and strain rate during superplastic deformation is obtained by numerical simulation of step strain rate test. The polycrystal material is superplastically deformed at a constant strain rate and temperature until the applied stress value reaches a steady state. This steady state stress value is the flow stress at that particular strain rate and temperature. Then the material is further deformed at an increased strain rate, which results in a new corresponding flow stress. Thus by numerical simulation, at a particular constant temperature, the flow stress can be determined for a wide range of strain rates, including the specific strain rates reported in the experimental results. The appropriate material constants , , n and are chosen such that the simulation results match the experimental data for one specific temperature and a grain size. It should be noted that for a given material only one - curve at a specific temperature and grain size is used in the evaluation of constants , and n. However, the threshold stress, , is found to be strongly dependent on temperature.

The first material that is considered for the study of temperature effect is Al7475, a statically recrystallized aluminum alloy. The thermo-mechanical properties of this material has been reported by Hamilton et al [9]. The grain size was controlled by a thermo-mechanical treatment of the material. The step strain rate tests were reported for K, K, K and K. The numerical simulation is carried out for the temperature at K and grain size at 14 m, and the material constants , , n and are obtained. The numerical simulation is subsequently carried out for other temperatures by varying only the threshold stress, and keeping , and n as constants, as shown in Figure 2. It is seen in this figure that the two sets of data are in fairly good agreement. Consistent with the experimental observations, the threshold stress, is temperature dependent parameter, and their relations will be shown later in this paper. The second material studied is an Al-6.05Zn-1.91Mg-1.46Cu-0.15Cr-0.09Mn-0.12Fe alloy tested by Malek [10], where chromium and magnesium rich particles were found to be distributed as second phase. Though the micromechanical model assumes the material as single phase, it describes the material behavior of this alloy successfully, as shown in Figure 3. The HSRS material that is considered for the study of temperature effect is IN905XL alloy with a chemical composition of Al-4.0wt%Mg-1.5wt%Li-1.2wt%C-0.4wt%O tested by Higashi et al [2]. The material exhibits superplasticity at very high strain rates (over 1 ) and the region I with a plateau is apparently seen on the sigmoidal plot of stress vs. strain rate. The numerical simulation is carried out for seven temperatures between K and K by varying only the threshold stress, , and keeping , and n as constants. The simulation results (curves) compared with the experimental data (symbols) are given in Figure 4. It is seen that the overall agreement between two sets of data for all temperatures is reasonably acceptable. Furthermore, the results shown in Figure 4 clearly indicate that the proposed model is able to model region I of high strain-rate materials.

The model is then applied to study the effect of varying grain size on the superplastic behavior. Hamilton et al [9] have reported the effect of grain size on the superplastic behavior of Al7475 aluminum alloy. Various grain sizes (12.3, 16.3, 22, 69.4 and 156.3 microns) were obtained by thermo-mechanical treatment and the experiments were conducted at K. For numerical simulation, the same material constants including the threshold stress at K, obtained previously in this material is then applied for different grain sizes. Figure 5 shows the numerical predictions and experimental values. The effect of grain size in the micromechanical model depends on the grain size as given in equation (3). Considering that no modifications to the equation or to the constants are necessary to predict the grain size effect, the theoretical predictions are excellent. Since the value of threshold stress at K is used for all grain sizes, the threshold stress is considered to be independent of grain size. Obviously macroscopic threshold stress depends on grain size whereas slip level threshold stress does not.

As mentioned earlier, the superplastic strain rate is accommodated by lattice and boundary diffusional and dislocational movement of atoms. The relative contribution of each of these processes can be studied by computing the ratio strain rate of each of the processes to the total strain rate, expressed as percentage. Let and represent the contribution of lattice and grain boundary accommodation towards the total strain rate . and can be computed from the two terms in equation (3) by summing over all possible slip systems for all grains. Similarly , the dislocational contribution to the strain rate, can be calculated from equation (4). Figure 6 shows the relative contributions of lattice and grain boundary diffusion and dislocation accommodations in Al7475 alloy at K and K. As can be seen from the results, the diffusion contribution to the strain rate is dominant at low strain rate regions. The strain rate due to lattice diffusion is higher than that from boundary diffusion. The dislocational contribution becomes larger as the strain rate increases. Also, at low temperature the relative contribution due to dislocation increases. Since threshold stress is present at low strain rate region (region I), it is concluded from Figure 6 that the threshold stress arises from the diffusion process. This is the reason that we have introduced threshold stress only in the diffusion equation (3) but not in the dislocation equation (4).

The sigmoidal curve shown in Figure 1 is usually divided into three regions based on the strain-rate sensitivity m, defined as the slope of the curve, i.e., . As can be seen from the curve, m continuously varies throughout the regions, with a maximum value of m occurring in region II (superplastic region). Region I and region III, at the lowest and highest ranges of respectively, exhibit low values of m. Figures 7 compares the typically predicted m curves with the corresponding experimental m curves for Al7475 and Al-Zn-Mg-Cu alloys. As can be seen from the figure, the agreement of the entire m curve between numerical simulation and experiment is fairly good.

In agreement with the experimental observation of the aggregate threshold stress, the micro-level threshold stress ( ) used in the model was found to be strongly dependent on temperature. Figure 8 shows the effect of temperature on the threshold stress in all three materials studied in this work. It is seen that the dependence of micro-level threshold stress on temperature can be expressed by a similar formulation given by Mohamed [11] for aggregate threshold stress as

where is a constant and is the activation energy at the slip plane level for micro-level threshold stress.

SUMMARY AND CONCLUSIONS

A micromechanical model based on grain-level diffusion and dislocation and using the concept of threshold stress has been proposed to model the superplastic deformation process. The model is successfully applied to aluminum-based conventional superplastic and HSRS materials. The material constants including the threshold stresses ( ) are evaluated from experimental data. These constants (except ) are found to be true constants for a particular material irrespective of temperature and grain size. The flow stress and the strain-rate sensitivity have been successfully predicted in a wide range of strain rates (10 to 10 ). The threshold stress ( ) was found to be strongly dependent on temperature and independent of grain size.

ACKNOWLEDGMENTS

The authors wish to acknowledge Army Research Office (Drs. Andrew Crowson and Wilbur Simmons) for financing the project.

REFERENCES

1. J. Pilling and N. Ridley, Superplasticity in Crystalline Solids, The Institute of Metals, London, UK, 1989.
2. K. Higashi, T. Okada, T. Mukai, and N. Tanimura, Scripta Metallurgica et Materialia, 26, 761 (1992).
3. K. Murali and N. Chandra, Acta Metallurgica et Materialia, 43, 1783 (1995).
4. N. Chandra and K. Murali, Scripta Metallurgica et Materialia, 32, 1429 (1995).
5. M.F. Ashby and R.A. Verrall, Acta Metallurgica, 21, 149 (1973).
6. J. Weertman,Philosophical Magazine, 11, 1217 (1965).
7. D. McLean, Grain Boundaries in Metals, Oxford University Press, London, UK, 1957.
8. J. Weertman, Journal of Applied Physics, 26, 1213 (1955).
9. C.H. Hamilton, C.C. Bampton and N.E. Paton, Superplasticity in High Strength Aluminium Alloys, in N.E. Paton and C.H. Hamilton, editors, Superplastic Forming of Structural Alloys, pages 173-189, Philadelphia, 1982. TMS.
10. P. Malek, Material Science and Engineering, A137, 21 (1991).
11. F. A. Mohamed, Journal of Material Science Letters, 7, 215 (1988).

• About this document ...