#### 3.3. Initiation of Shear Bands

According to classical plasticity theory, shear banding depends on the initial orientation of the crystals [

19] and the occurrence of shear bands are strongly dependent on various factors such as material properties, geometry, loading conditions, and friction [

20]. A series of studies have been carried out to investigate the flow behavior of Ni-based alloy under uniaxial compression [

3,

17]. It is found that the shear band induced by uniaxial compression is much weaker than that induced by plain strain compression. It is worth noting that the Inconel 718 superalloy used in the present work has the similar material properties with those Ni-based superalloys investigated in the references [

17,

21], and mechanical testing was performed on similar experimental conditions by using Gleeble thermomechanical simulators. Therefore, the distinctive deformation mode of plane strain compression is expected to be the key factor for the intense occurrence of shear bands. To give a more complete insight into the initiation and evolution stages in the development of shear bands, a rigid-plastic FEM was developed to replicate the experimental results and observations.

The material parameters such as Young’s modulus, Poisson’s ratio and thermophysical parameters of Inconel 718 superalloy were quoted from Ref. [

15]. Specially, the constitutive model of the present alloy has been established in our previous work [

16], and it is given in the following type:

where

Z denotes the Zener–Hollomon parameter and can be represented by the following equation:

where

$\stackrel{.}{\mathsf{\epsilon}}$ is the strain rate,

T is the absolute temperature and

R is the universal gas constant,

R = 8.314 kJ/mol.

The failure of alloys during hot-working is mainly ascribed to additional tensile stress induced by inhomogeneous deformation [

22]. In the present study, the tensile stress is expressed in the form of a maximum tensile work criterion proposed by Cockroft and Latham [

23]:

where

C is the damage factor, σ

_{max} is the highest local tensile stress,

$\overline{\mathsf{\epsilon}}$ is the true strain and

${\overline{\mathsf{\epsilon}}}_{\mathrm{f}}$ is the true strain at fracture. When a cylindrical specimen is uniaxially compressed as shown in

Figure 4a, the additional tensile stress is localized at the equator of the bulge area (indicated by arrow). This additional stress, however, cannot give rise to the formation of shear band because the direction of the stress is along the circle and perpendicular to the primary stress axis. When the specimen is subjected to plane strain compression, as shown in

Figure 4b, the stress concentration occurs in an “X” type region of the specimen. The “X” type stress concentration is supposed to be related to surface tension of the specimen and the friction during deformation. It is clear that the stress concentration significantly promotes the occurrence of shear band due to the favorable orientation. As a result, the plane strain compression is easier to induce the initiation of shear bands in comparison with uniaxial compression.

The special deformation mode of plane strain compression leads to different features of state variables such as effective strain, effective stress, temperature, etc. In the present study, strain rate is chosen as the primary field quantity to inspect, because it gives the best indication of the instantaneous tendency toward shear band formation [

20]. The contours of effective strain rate for the two compression modes under the same deformation conditions are shown in

Figure 5. For the uniaxial compression, a relatively small gradient was found in the strain rate contours across the cross-section at the initial stage of deformation (5% height reduction in

Figure 5). In this case, the strain rate in the center area of the specimen is smaller than 15 s

^{−1}. Plastic deformation yielding 30% thickness reduction a maximum strain rate of 20 s

^{−1} a rather rectangle region of uniform plastic deformation (in comparison to the more round strain rate distribution in

Figure 5a).

On the other hand, the distribution of strain rate resulted from plane strain compression shows large gradients across the cross-section at the initial stage of deformation in

Figure 5d. The distribution of strain rate is extremely nonuniform and the X-shape region can be observed through the plane strain compression process (from 5% up to 30% thickness reduction). The high strain rate region begins to rotate away from the ordinate at the very beginning of deformation, which is considered to be related with the initiation of shear band according to previous studies [

20].

The interfacial friction between the die and workpiece plays an important role in the occurrence of shear bands in the compression process. Firstly, the level of friction between the die and workpiece usually increases with the increasing of deformation temperature for metals [

24,

25]. Therefore, the friction coefficient (

f) at a high temperature cannot be ignored. According to the FEM simulation results shown in

Figure 6, with the increasing of friction coefficient, the initial strain for the shear bands tends to form from the corners and center of uniaxial compression specimens, while the degree of shear banding enhances in the plane strain compression specimens. Secondly, Li et al. has found that although the friction coefficient was a constant at lower strain level, the instantaneous friction coefficient increased approximately exponentially with the true strain at higher strain level [

26]. Therefore, the effect of friction on the tendency to form flow localization is enhanced in the plane strain compression process, thus promoting the occurrence of shear bands. Thirdly, as shown in

Figure 4, the damage factor distributes at the equator of the cylindrical specimen (perpendicular to the primary stress axis) after uniaxial compression, while the plane strain compression leads to an “X” type stress concentration area from which the shear bands tend to initiate. This difference is associated with the frictional shear factor at the interface of die-workpiece. For uniaxial compression, a high level of friction may lead to the appearance of “dead” zones just below the die, and this can result in the “X” shear bands. However, the effect of friction will be balanced by DRX. If high strain rate, low deformation temperature or large strain is chosen, shear bands may appear. For plane strain compression, the deformation along the length direction is restricted and a large shear factor will be induced at the corners of contact surface between die and workpiece. Therefore, the “dead” zones are easier to form below the die, which promote the occurrence of shear bands.