Stress relaxation behavior of annealed aluminum-carbon nanotube composite

3.2.1 Stress relaxation and hardening

According to Choudhry and Ashraf [14], stress relaxation Δσ(t) and hardening component Δσ′(t) can be determined using Eqs. (5) and (6), respectively:

The values of Δσ(t) and Δσ′(t) were calculated using experimental stress relaxation curves and plotted against corresponding stresses (σ_o) to find the trends of stress relaxation and hardening in both materials under investigation (Figure 4). The plots of both materials showed that the hardening component was initially more than stress relaxation; however, as the initial stress increased, a gradual decrease in hardening components occurred. A comparison of the stress relaxation and hardening component values of aluminum and Al-CNT composite is given in Table 3. It was observed that the composite exhibited lower values of stress relaxation and higher values of hardening component, as compared to aluminum.

Figure 4:

Plots of stress relaxation Δσ=σ_o−σ(t), hardening component Δσ′=σ′(t)−σ_o, and their sum Δσ*=Δσ+Δσ′ manual="yes; between initial stress σ_o, for a relaxation time of 300 s.

In case of aluminum, the hardening component (Δσ′) could be attributed to the migration of point defects toward the core of edge dislocations and/or pinning [5]. However, the behavior of the composite was different. Under given strain conditions, dislocation loops were generated around the obstacles (i.e. the nanotubes). Continuing strain at point B (inset of Figure 3) resulted in dislocation multiplication and breakaway; hence, the dislocation density was increased and the hardening component emerged, as manifested by hardening at point C. The hardening component (Δσ′) increased with the strain to a certain tensile stress (σ_o); afterwards, it decreased and stress relaxation (Δσ) dominated, which was the limit of deformation hardening. The limit of deformation hardening indicated that the nanotubes were effective obstacles to the glide of dislocations until that strain level; afterward, at higher strain levels, dislocation-to-dislocation interaction increased and dominated the deformation mechanism [15, 16]. At low strain conditions, the stress relaxation and deformation hardening of the composite material were similar to aluminum in behavior, although the deformation hardening was twice as much that of aluminum. However, in higher stress conditions, the deformation hardening component decreased rapidly and diminished (Figure 4). This observation suggested that the strengthening in the composite was induced by deformation hardening (i.e. dislocation-to-dislocation interactions) at lower stresses; however, at higher stresses, the dislocation interaction-based hardening mechanism was replaced by strengthening induced by the interaction of the nanotubes itself with the matrix, which doubled the mechanical strength of the material. This phenomenon is known as the shear-lag model, where the load is transferred to the nanotubes from the matrix through interfacial shear stress. Hence, the stiffness of the nanotubes is directly involved in strengthening [17].

3.2.2 Thermally activated barrier

In a crystalline material, the flow stresses could be divided into two components: (i) long-range elastic interactions of the moving dislocations with the microstructure and (ii) the minimum stress required to overcome the opposing local energy barriers, which could be of various types, i.e. an intrinsic lattice resistance, small obstacles (precipitates), or unfavorable dislocation core.

The first component is elastic in nature and fades out as temperature increases; however, the second component dominates the high-temperature deformation behavior of the material. Short-range interaction of dislocations with energy barriers takes place in such a small volume that it is strongly influenced by thermal vibrations. Thermal activation helps dislocations to overcome these barriers, thus resulting in a reduction of stress as the temperature rises. These short-range thermally activated barriers govern almost all temperature-dependent mechanical properties of materials.

These processes are obviously of fundamental importance for the understanding and modeling of strength of structural materials. New materials have rather complex structures, in which dislocation mechanisms are more difficult to identify than in single-phase metals and alloys [18]. The strength of the barriers could be calculated using stress relaxation test data as shown below.

To calculate the strength of thermally activated barrier, graphs of stress relaxation (Δσ) vs. time were plotted at different stress levels (σ_o). A logarithmic scale was used to express the time axis, which helped reduce the skewness of the data (Figure 5). The slope of the each graph (S) represents the relaxation rate at a particular initial stress level (σ_o). Therefore, the graphs between initial stress (σ_o) and slope (S) were plotted, as shown in Figure 6. It is obvious in Figure 6 that the rate of stress relaxation is less in the Al-CNT composite than in aluminum, which suggests that the Al-CNT composite presents more resistance to relaxation in stressed condition than aluminum. Res et al. [19] proposed a single barrier model, where the height of the thermally activated barrier can be calculated using the relaxation rate (S) and initial stress level (σ_o), according to Eq. (7):

Figure 5:

Graphs of stress relaxation data. Note that the time is plotted in logarithmic scale. The graphs have different initial stress (σ_o) values, which are written on the right margins of the graphs.

Figure 6:

Rate of stress relaxation (S) as a function of initial tensile stress (σ_o) for aluminum and Al-CNT composite.

where U_o is intrinsic height of the thermally activated barrier, m is a constant (~25), K is Boltzmann constant (0.8617×10⁻⁴ eV K⁻¹), and T is room temperature (298 K). The values of dσ_o/dS are calculated from Figure 6 (i.e. 46 and 102 s⁻¹ for aluminum and the composite, respectively). The corresponding values of the strength of the thermally activated barriers for aluminum and the composite are 1.7 and 2.6 eV, respectively. Similar types of results were also deduced by Butt et al. [5] and Choudhry and Ashraf [14] for aluminum and AA7075, respectively, depicting the magnitude in order for processes like cross-slip, vacancy formation, and mutual inhalation of dislocations. The higher values of U_o in case of the composite can be attributed to the impeding of dislocation pile-ups due to fine grain size and the nanotubes, seizing the cross-slip phenomenon and increasing the dislocation density.