Chapter IV: Shock Compression Behavior of Stainless Steel 316L Octet-Truss
4.3 Experimental Results and Discussion
4.3.1 Wave Definitions and Extraction
Similar to shock behavior in bulk materials [17], a two-wave structure consisting of (1) an elastic wave, and (2) a compaction (shock) wave was observed during experiments. Similar observation has been well documented in polymeric lattice structures on the sub-millimeter scale impacted around 200-600 m/s. [56, 57]. The elastic wave and shock wave may be defined and the corresponding locations can be extracted using full-field displacement and velocity measurements. Ravindran et al.
have used this technique to accurately define elastic and shock fronts in aluminum foams [93]. Full-field measurements were averaged over the center (middle) row of unit cells to approximate a uniaxial bulk material response. Displacement and velocity measurements were taken as the average over 30 y-pixel (vertical) positions corresponding to the center UC, for each x-pixel (horizontal) position.
The elastic wave front was defined using a 15 ππ displacement criterion based on the maximum observed DIC error (confidence interval) of 0.1 px and average image scale of 0.15 mm/px. The confidence interval for all experiments remained under 0.1 px during elastic deformation and therefore the criterion is larger than the experimental resolution. Wave definition and extraction for experiment #ππ152is shown in Fig. 4.4.
Figure 4.4(a) shows particle displacement plotted against horizontal position (La- grangian/undeformed coordinate) defined using the impact surface as π = 0 and time of impact as π‘ = 0. Each line represents a single time-instance taken from one experimental image and the marker represents the position of the elastic front.
A stricter or looser displacement criterion may slightly alter results, however the steepness and consistent spacing of displacement profiles imply similar results for criteria of 10-20 ππ.
The shock front and relevant parameters are defined in both position and time and visualization is shown in Fig. 4.4(b). Figure 4.4(c) shows the particle velocity(π’π) β time (π‘)history of all material points (x-pixel positions). Each material point of the specimen shows an initial low-amplitude particle velocity from the elastic wave followed by steep acceleration to a nearly constant particle velocity similar to the flyer speed. The shock front was defined for each material point as the point of largest change in velocityβequal to infinity in the case of a mathematically sharp shock. Based upon this definition, the shock front may be defined: with respect to time for one material point (pixel), or with respect to position for one time instance (image).
(c)
c)
0 50 100 150
Time [ s]
0 50 100 150 200 250 300
Velocity [m/s]
0 5 10 15 20 25 30 35 40 45 50
Undeformed Coordinate [mm]
0 50 100 150 200 250 300
Velocity [m/s]
(d) (b)
(a)
0 10 20 30 40 50
Undeformed Coordinate [mm]
-5 0 5 10 15 20 25
Displacement [ m]
xs xelas Us
xfly up
Xelas
ποΏ½ly πs+
ππ ποΏ½ly
ππ πs+
Figure 4.4: Wave definition and extraction for experiment #ππ152: (a) displacement position profiles with 15ππdisplacement criterion marking the location of the elastic wave front (πππ π π ); (b) visualization of the shock definition and relevant parameters; and DIC particle velocity (π’π) profiles with respect to (c) time and (d) undeformed (Lagrangian) coordinate, with the elastic wave (π₯Β€ππ π π ), flyer (π₯Β€π π π¦), velocity ahead of the shock (π₯Β€+
π ), and shock front (ππ ) marked. Every 10π‘ βline is highlighted to improve data visualization.
Figure 4.4(c) shows particle velocities with respect to time (πcurves of πdata points) and Fig. 4.4(d) shows particle velocities with respect to position (π curves ofπdata points) whereπ is the number of horizontal pixels in experimental images β 200 and π is the number of images analyzed β 90. Every 10π‘ β line is highlighted to improve data visualization. Determination of the shock front using π π’π/π π‘ caused
βlinesβ of front positions to emerge in Fig. 4.4(c) due to the low temporal resolution from the limited number of experimental images. Determination of the shock front usingπ π’π/π π improved extraction of the shock wave as shown in Fig. 4.4(d) due to an increased spatial resolution (π > π). The particle velocity ahead of the shock,
Β€ π₯+
π , was determined from the particle velocity-time history as the point of maximum change in curvature before the shock front.
The wave fronts were extracted and plotted against time as shown in Fig. 4.5(a) and visualized using experimental images in Supplementary Video S4. The elastic wave front (πππ π π ) was determined using the displacement criterion, the flyer front (π₯π π π¦) was determined using displacement measurements of the flyer, and the Lagrangian shock front (ππ ) was determined using the spatial derivative technique in Fig. 4.4(d).
The term βLagrangianβ is used to describe parameters defined using the undeformed coordinate system while the term βEulerianβ is used to describe parameters defined
using the deformed coordinate system. The Eulerian shock front (π₯π ) can be deter- mined by mapping the Lagrangian shock front to the deformed coordinate system using full-field displacement(πΏ)measurements: π₯π =ππ +πΏ. Flyer speed and shock velocities were computed from the front-time histories using a three-point central difference method. Figure 4.5(b) shows the calculated flyer speed, Lagrangian shock velocity (ππ ), and Eulerian shock velocity (π’π ) as a function of time with the steady shock defined from timeπ‘
1toπ‘
2to avoid edge or smoothing effects.
0 20 40 60 80 100 120 140
Time [ s]
0 100 200 300 400 500 600 700 800
Velocity [m/s]
0 20 40 60 80 100 120 140
Time [ s]
0 5 10 15 20 25 30 35 40 45
Front Position (X,x) [mm]
t1
xfly xelas Xs xs
(a) (b)
t2
Us us us xfly
t1 t2
Figure 4.5: Wave front position and velocity profiles: (a) elastic (πππ π π ), flyer (π₯π π π¦), La- grangian shock (ππ ), and Eulerian shock (π₯π ) front-time histories; and (b) corresponding flyer speed (π₯Β€π π π¦) and shock velocities in the Lagrangian (ππ ) and Eulerian (π’π ) configura- tions computed using a three-point central difference method. A 20% moving mean window was applied to smooth experimental scatter of Eulerian shock velocities (π’π ) indicated by the red solid line.
Deceleration of the flyer was observed in all experiments and scatter plots of the shock velocities showed large variations in values. These variations may largely be attributed to noise in experimental front locations combined with small timesteps used in numerical derivative calculations. Eulerian shock velocities were smoothed with a 20% moving mean window to eliminate scatter and resemble the smoothness of the flyer speed. Front positions appear mostly linear in time which supports the formation of a steady shock in the lattice specimens and justifies smoothing of data.
Experimental Elastic Wave Speeds
Elastic wave speeds were computed using a linear fit of the elastic front time history and error is reported as the 95% confidence interval of the fit. Results for all exper- iments are shown in Fig. 4.6. There is a clear trend showing positive correlation between elastic wave speed and relative density which agrees with positive stiffness
correlation with relative density found in lattice structures [4]. However, this trend inherently depends on manufacturing defects and geometry which control the stiff- ness and density of the specimen. Impact velocity does not appear to influence the wave speeds which agrees with results from other studies [57]. This also implies AM SS316L does not demonstrate strain-rate stiffening effects.
10 15 20 25
Relative Density [%]
1400 1600 1800 2000 2200 2400 2600
Elastic Wave Speed [m/s]
265m/s 295m/s 326m/s 378m/s
259m/s 285m/s 322m/s 378m/s
288m/s 364m/s
Figure 4.6: Elastic wave speed as a function of relative density for all experiments. Elastic wave speeds were computed using a linear fit of the elastic front time-history and error is reported as the 95% confidence interval of the fit.