Study of spatial-time inhomogeneity of serrated plastic flow Al-Mg alloy: using DIC-technique

The aim of the present paper is the investigation of temporal instabilities and spatial localization due to the Lüders behavior, the Portevin–Le Chatelier effect and the shoulder or necking effect during uniaxial tension tests of aluminum-magnesium alloy. This paper presents the brief description of the test procedure and experimental results of carrying out research by the combined use of a servo-hydraulic biaxial test system Instron 8850 and a non-contact 3-D digital image correlation measurement system Vic-3D. The digital image correlation is a highly effective computer-vision-based technique, which provides estimation of the displacement and strain fields on specimen surface by matching the reference subsets in the undeformed image (before loading) with the target subsets in the deformed images (captured during test). The evolution of inhomogeneous axial strain and axial strain rate fields has been illustrated for each stage of material’s deformation. To estimate the kinematics of serrated or jerky flow due to the strain bands propagation, the strain versus time curves and strain diagrams are given here. The experimental results show the recurrence in the strain distribution leveling along the specimen gauge. The changing between the macroscopic localization of the plastic flow, namely the running of the Lüders and PLC bands and the recovery of strain field homogeneity, has been observed.


INTRODUCTION
or projecting and numerical model's development of structures should be taken into the account not only mechanical and strength characteristics of materials, but also its behavior singularity. There are a lot of studies concerning the deformation and fracture processes in materials, occurring irregularly on all scales of observation: micro-, meso-and macroscopic scales [1,2]. Theoretical and experimental research of temporal instabilities and spatial localization during the tensile tests of different metals and alloys has been conducted for more than two hundred years all over the world [1][2][3][4]. The main types of macroscopic occurrences of the plastic deformation inhomogeneity are: the Lüders bands nucleation at the stage of the yield drop and plateau forming; an irregular plastic flow appearing either as the staircase phenomenon on the stress versus strain curves -the so-called Savart-Masson effect during force loading -or the serrated or jerky flow due to the Portevin-Le Chatelier (PLC) effect during kinematic loading [5][6][7][8][9][10][11][12]. Another widespread example of spatial inhomogeneity is the shoulder or necking effect at the postcritical deformation stage, which F manifests itself as local thinning of the specimen's transverse [13]. Analysis of fundamental and current scientific literature has revealed the relevance of the issue despite its long history [14]. Furthermore, appearance of the advanced test equipment, high-effect measuring systems and high-accuracy facilities for carrying out basic and applied research caused a great increase of scientists' concern focused on the aspects of the macroscopic localization of plastic flow, especially the local strain bands propagation, the influence of strain rate and temperature regimes, the chemical composition, specimen geometry, grain size and orientation, etc. on the occurrence of the PLC behavior [15][16][17][18][19]. In this work a technique based on the digital image correlation (DIC) method has been used for study of spatial-time inhomogeneity of serrated plastic flow Al-Mg alloy. The DIC is a highly effective non-contact computer-vision-based technique, which provides estimation of the displacement and strain fields on specimen surface by matching the reference subsets in the undeformed image (before loading) with the target subsets in the deformed images (captured during test) [20]. This paper presents, in the first part, the brief description of the test procedure of carrying out experimental investigations by the combined use of a servo-hydraulic biaxial test system Instron 8850 and a non-contact 3-D digital image correlation measurement system Vic-3D [21]. The Vic-3D system can be used for problem solving of deformable solid mechanics: experimental investigation of non-uniform strain fields and analysis of failure conditions in bodies with concentrators of different geometry, research of inelastic material deformation processes in complex strain-stress conditions, study of displacement and strain fields evolution during crack initiation, damage accumulation and material failure, etс. In the following, the representative load (P, kN) versus displacement (u, mm) curve observed during tensile tests on an Al-Mg alloy sheet specimens is shown. Then, a detailed description of the Lüders bands, the PLC bands initiation and propagation, the correspondence between the deformation bands and the serrations on the P-u curves is given. The macroscopic localization of axial strain due to the necking effect at the post-critical deformation stage is illustrated as well.
In conclusion, the recurrence in the strain distribution leveling along the specimen gauge is shown. The change between the macroscopic localization of the plastic flow, namely the running of the Lüders and PLC bands and the recovery of strain field homogeneity, has been observed.  All mechanical tests on uniaxial tension were performed in a servo-hydraulic biaxial test system Instron 8850 with constant loading rate in the range of 0.5 to 10.0 mm/min throughout the experiment at room temperature. The Instron 8850 is intended for static tests on tension, torsion, compression, flexure and combined tests on tension-torsion with the axial force capacity up to ±100 kN, the torque capacity up to ±1000 Nm and fatigue tests with various wave shapes and frequency up to 30 Hz; the loading rate from 0.1 mm/min up to 240 mm/s.  The registration of strain fields' evolution was conducted by the non-contact 3-D digital image correlation measurement system Vic-3D with the recording rate of 15 images per second and DCP cameras resolution of 4.0 Mp. It is the multicamera system which can be used for problem solving of deformable solid mechanics: experimental investigation of nonuniform strain fields and analysis of failure conditions in bodies with concentrators of different geometry, research of inelastic material deformation processes in complex strain-stress conditions, study of displacement and strain fields propagation during crack initiation, damage accumulation and material failure, etc. The procedure of the uniaxial tension loading experiment with the measuring of surface deformations includes several steps: preparation of specimen's surfaces by coating with white and black spray paint to generate random pattern; attaching the specimen to the hydraulic fixtures with flat specimen platens; calibration of the stereovision system with the set of target grids; synchronization of the imaging and loading data by using the image acquisition system Vic-Snap and a data collector. All analyses were performed by the software Vic-3D with a subset size of 19×19 pixels 2 and with a step size of 4 pixels between subset centers. Data extraction through image analysis was carried out by using the NSSD criterion (normalized sum of squared difference). The displacement data was converted into strain values by using the Lagrangian strain tensor.

RESULTS
ig. 3 shows the representative load-displacement curve of uniaxial tension test on the flat dog-bone tensile specimen with a displacement rate of 5.0 mm/min, which corresponds to an average strain rate of 0.1 min -1 . The curve includes the following stages: the linear elastic stage; the stage of yield drop and plateau forming; the extended stage of material's hardening; and the post-critical deformation stage. It is important to note that there are a great number of local drops of load or "serrations" on the load versus displacement curves called the Portevin-Le Chatelier effect (PLC) [9].

Elastic Deformation Stage
To estimate the kinematics and irregularity of plastic flow during uniaxial tensile tests of Al-Mg alloy sheet, the axial strain fields ( yy  ) and the strain rate fields ( yy  ) have been determined on the specimens surfaces. Fig. 4 contains the singled out elasto-plastic deformation stage of the P-u curve (Fig. 3) for the detailed description of the material behavior, especially during the Lüders band nucleation. Consequently, the stage of yield drop and yield plateau forming is under observation.

Stage of Yield Plateau Forming
The rapid jump of the axial strain level up to 0.93% When the front reached the opposite side of the specimen, the configuration of the axial strain fields became almost homogeneous (Fig. 8). It is important to note that in the region where the front of the localized strain had passed, the material's deformation processes stopped until the next deformation stage -the material hardening stage. To conduct numerical analysis and show regularities in the Lüders band motion, the diagrams of axial strain and the axial strain rate were calculated along the central line of specimen (in the line of loading). The curves I I V t t  , shown in Fig. 9, correspond to points I-VI of the P-u curve (Fig. 4). The velocity of the strain band propagation was about 7.7 mm/s or 462 mm/min and remained quite stable during the whole stage of the yield plateau forming. It is known that during the Lüders band motion the slope of the load-displacement curve is approximately zero, in other words the load remained at the level of 2.4 kN. The macroscopic increase of specimen was provided by the localized deformation in the region of the strain band.

Material Hardening Stage
With further increase in load, the PLC phenomenon characterized by serrations in the load-displacement curve due to the repeated initiation and propagation of localized plastic strain bands along the specimen during tensile test is observed. Under kinematic loading the serrations appeared as repeated oscillations of the applied stress. To study the PLC behavior, interrelation between numerous serrations and the strain band distribution has been performed. The change in configuration of axial strain fields (Fig. 10, a) and axial strain rate fields (Fig. 10, b) caused by distribution of the particular PLC band is illustrated as follows. The time gap ( t  ) between captured pictures was 0.3 second.
(a) (b) Figure 10: Evolution of axial strain (a) and axial strain rate fields (b) due to propagation of the PLC band during the time period 1 6 t t  . The results of the experiments indicated that jerky flow in the tensile Al-Mg specimen happened by interchange of continuous propagation of a single band and stochastic nucleation of bands (Fig. 11). The evolution of deformation fields during the time period 1 6 t t  corresponding to the flat region in the curve, the slope of the load-displacement curve is insignificant (Fig. 3). From the side of the top grip, the localized plastic strain band nucleated and started to run toward the bottom grip. The angle between the specimen axis and the strain band was approximately 59° [8]. The PLC band passed lengthwise the specimen surface with constant rate of about 20.7 mm/s or 1242.0 mm/min (Fig.  12). The diagrams of strain and strain rate were extracted along the specimen axis corresponding to the time period 1 6 t t  . Similarly to the Lüders band propagation, the material actively deformed only in the region of the localized strain band, at the PLC band front.   Table 3: Values of strain and strain rate calculated for the time period 1 6 t t  .
When the strain band reached the opposite side of the sample, the stochastic nucleation of localized plastic strain bands has been observed on the specimen surface during the time period 7 1 2 t t  , as mentioned above (Fig. 13). The time gap ( t  ) between captured pictures was 0.3 second. The angle between the specimen axis and the strain bands repeatedly changed in the range of ±59°. It is necessary to point out that in the region where the previous band passed, material deformation stopped; thus, specimen elongation took place due to the deformation of peripheral regions of gauge length (close to the grips) (Fig. 14). Besides, it is clearly seen that the plastic deformation was happening by jerks (Fig. 14, b). As shown in Tab. 4, the local axial strain rate changed in the range of 65.0 to 105.0 %/min.  Figure 14: Diagrams of axial strain (a) and the axial strain rate (b) during the time period 6 7 12 , t t t  .
The time moment 12 t corresponds to the recovery of the strain field homogeneity on the specimen surface. The recurrence in the strain distribution leveling along the specimen gauge was observed during the material hardening stage. To estimate the regularities of this behavior, the next PLC band's nucleation and propagation has been studied as well. Therefore, Fig. 15 represents the continuous propagation of the single strain band (the time period 1 7 t t    ). The time gap ( t  ) between captured pictures was 0.3 second. When the strain band passed the specimen gauge, the stochastic initiation of the bands was observed repeatedly. Further, the changing between the macroscopic localization of the plastic flow and the recovery of strain field homogeneity was registered.
(a) (b) Figure 15: Diagrams of axial strain (a) and the axial strain rate (b) during the time period 12 t , 1 7 t t    .

Material Softening Stage
It is well established that during tensile test of plastic materials the 'shoulder' or 'necking' effect at the material softening stage or the co-called postcritical deformation stage, which manifests itself as local thinning of the specimen cross-section. When the strain bands' propagation had faded away, the increase of plastic strain localization occurred in the central part of the specimen. For example, the evolution of the axial strain rate fields at the stage of the 'necking effect' initiation is illustrated in Fig. 16, the load level of 4.75 kN. The time gap ( t  ) between captured pictures was 0.15 second. The moment of the angle change between the specimen axis and the strain band is shown. To analyse the spatial inhomogeneity at the stage of the necking effect evolution, the diagrams of axial strain are calculated for the time period * * 1 6 t t  . The time gap ( t  ) between captured pictures was 1.57 second. As shown in Tab. 5, value of the local axial strain rate rapidly increased with the increase of the localized plastic strain value in the central part of specimen gauge.    The macroscopic failure of the specimen was at the load level of 4.26 kN, the average value of axial strain of 24.43 % and the local axial strain of 70.44 %. The picture of the flat dog-bone specimen with the crack after tension test is represented in Fig. 16 (b).

CONCLUSIONS
he present investigation has shown that the non-contact 3-D digital image correlation measurement system Vic-3D is a highly effective computer-vision-based technique, which provides estimation of the temporal instabilities and spatial localization due to the Lüders behavior, the Portevin-Le Chatelier effect and the shoulder or necking effect during uniaxial tension tests of aluminum-magnesium alloy. The evolution of inhomogeneous axial strain and axial strain rate fields has been illustrated for each stage of material's deformation. To estimate the kinematics of serrated or jerky flow due to the strain bands propagation, the strain versus time curves and strain diagrams are given here. The experimental results show the recurrence in the strain distribution leveling along the specimen gauge. The changing between the macroscopic localization of the plastic flow, namely the running of the Lüders and PLC bands and the recovery of strain field homogeneity, has been observed. The results provide an important data base for the development of the theoretical and numerical description of the material behavior in conditions of the serrated flow appearance, especially of the mechanisms and regularities of the Lüders and PLC bands nucleation and propagation.

ACKNOWLEDGMENTS
he work was carried out in the Center of Experimental Mechanics of the Perm National Research Polytechnic University with support of the Russian Foundation for Basic Research (grant № 13-08-00304, № 13-08-96016).

NOMENCLATURE L
Total length of the specimen (mm) l Gauge length of the specimen (mm) B Total width of the specimen (mm) b Width of the specimen (mm) a Thickness of the specimen (mm) r Transition radius from the grip part to the gauge length of the specimen (mm) P Load (kN) u Displacement (mm)