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].

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.

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