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Journal of Applied Mechanics | Volume 80 | Issue 3 | Terminal Ballistics and Impact Physics
Terminal Ballistics and Impact Physics

Influence of Length Scale on the Transition From Interface Defeat to Penetration in Unconfined Ceramic Targets

[+] Author and Article Information
Patrik Lundberg

e-mail: patrik.lundberg@foi.se

Olof Andersson

FOI,
Swedish Defence Research Agency,
SE-164 90 Stockholm, Sweden

Manuscript received June 29, 2012; final manuscript received December 2, 2012; accepted manuscript posted January 9, 2013; published online April 19, 2013. Assoc. Editor: Bo S. G. Janzon.

J. Appl. Mech 80(3), 031804 (Apr 19, 2013) (9 pages) Paper No: JAM-12-1282; doi: 10.1115/1.4023345 History: Received June 29, 2012; Revised December 02, 2012; Accepted January 09, 2013
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One observation from interface defeat experiments with thick ceramic targets is that confinement and prestress becomes less important if the test scale is reduced. A small unconfined target can show similar transition velocity as a large and heavily confined target. A possible explanation for this behavior is that the transition velocity depends on the formation and growth of macro cracks. Since the crack resistance increases with decreasing length scale, the extension of a crack in a small-scale target will need a stronger stress field, viz., a higher impact velocity, in order to propagate. An analytical model for the relation between projectile load, corresponding stress field, and the propagation of a cone-shaped crack under a state of interface defeat has been formulated. It is based on the assumption that the transition from interface defeat to penetration is controlled by the growth of the cone crack to a critical length. The model is compared to experimentally determined transition velocities for ceramic targets in different sizes, representing a linear scale factor of ten. The model shows that the projectile pressure at transition is proportional to one over the square root of the length scale. The experiments with small targets follow this relation as long as the projectile pressure at transition exceeds the bound of tensile failure of the ceramic. For larger targets, the transition will become independent of length scale and only depend on the tensile strength of the ceramic material. Both the experiments and the model indicate that scaling of interface defeat needs to be done with caution and that experimental data from one length scale needs to be examined carefully before extrapolating to another.
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References

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Lundberg, P., and Lundberg, B., 2005, “Transition Between Interface Defeat and Penetration for Tungsten Projectiles and Four Silicon Carbide Materials,” Int. J. Impact Eng., 31, pp. 781–792. [CrossRef]
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Anderson, O., Lundberg, P., and Renström, R., 2007, “Influence of Confinement on the Transition Velocity of Silicon Carbide,” Proc. 23rd International Symposium on Ballistics, Tarragona, Spain, April 16–20, Vol. 2, pp. 1273–1280.
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Behner, T., Anderson, C. E., Jr., Holmquist, T. J., Wickert, M., and Templeton, D. W., 2008, “Interface Defeat for Unconfined SiC Ceramics,” Proc. 24th International Symposium on Ballistics, New Orleans, LA, September 22–26, Vol. 1, pp. 35–42.
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Anderson, C. E., Jr., Behner, T., Orphal, D. L., Nicholls, A. E., Holmquist, T. J., and Wickert, M., 2008, “Long Rod Penetration Into Intact and Pre-damaged SiC Ceramic,” Proc. 24th International Symposium on Ballistics, New Orleans, LA, September 22–26, Vol. 2, pp. 822–829.
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Anderson, C. E., Jr., Behner, T., Holmquist, T. J., Orphal, D. L., and WickertM., 2009, “Dwell, Interface Defeat, and Penetration of Long Rods Impacting Silicon Carbide,” Southwest Research Institute Technical Report No. 18.12544/008.
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LaSalvia, J. C., Horwath, E. J., Rapacki, E. J., Shih, C. J., and Meyers, M. A., 2001, “Microstructural and Micromechanical Aspects of Ceramic/Long-Rod Projectiles Interactions: Dwell/Penetration Transitions,” Fundamental Issues and Applications of Shock-Wave and High-Strain-Rate Phenomena, K. P.Staudhammer, K. P.Murr, and M. A.Meyers, eds., Elsevier Science, New York, pp. 437–446.
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Figures

Grahic Jump Location
Fig. 1

Projectile shape (solid line), axisymmetric pressure of projectile (dotted line), and crack trajectory in target (gray) during a state of interface defeat

+Projectile shape (solid line), axisymmetric pressure of projectile (dotted line), and crack trajectory in target (gray) during a state of interface defeat
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Projectile shape (solid line), axisymmetric pressure of projectile (dotted line), and crack trajectory in target (gray) during a state of interface defeat
Grahic Jump Location
Fig. 2

Illustration of the relation between the principal stress σ¯1 (solid line) and the critical stress σ¯c (dotted and dashed line) in Eq. (13) versus the crack length r¯ for two different levels of projectile pressure. (i) p0 < p0* (dotted line): the crack will stop before passing the smallest double root r¯c1 = r¯c2 of Eq. (13). (ii) p0 = p0* (dashed line): the crack is determined by the double root close to the distinct minimum of the principal stress and a constant root r¯c, where the crack will stop. There are also larger single root solutions for a projectile pressure p0 = p0*, which satisfies Eq. (11).

+Illustration of the relation between the principal stress σ¯1 (solid line) and the critical stress σ¯c (dotted and dashed line) in Eq. (13) versus the crack length r¯ for two different levels of projectile pressure. (i) p0 < p0* (dotted line): the crack will stop before passing the smallest double root r¯c1 = r¯c2 of Eq. (13). (ii) p0 = p0* (dashed line): the crack is determined by the double root close to the distinct minimum of the principal stress and a constant root r¯c, where the crack will stop. There are also larger single root solutions for a projectile pressure p0 = p0*, which satisfies Eq. (11).
View Large  |  Save Figure  |  Download Slide (.ppt)
Illustration of the relation between the principal stress σ¯1 (solid line) and the critical stress σ¯c (dotted and dashed line) in Eq. (13) versus the crack length r¯ for two different levels of projectile pressure. (i) p0 < p0* (dotted line): the crack will stop before passing the smallest double root r¯c1 = r¯c2 of Eq. (13). (ii) p0 = p0* (dashed line): the crack is determined by the double root close to the distinct minimum of the principal stress and a constant root r¯c, where the crack will stop. There are also larger single root solutions for a projectile pressure p0 = p0*, which satisfies Eq. (11).
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Fig. 3

Experimental setup in reverse mode: diameter 1 mm projectile mounted in Divinycell fixture and diameter 10 mm SiC-B cylinder mounted in a Lexan sabot

+Experimental setup in reverse mode: diameter 1 mm projectile mounted in Divinycell fixture and diameter 10 mm SiC-B cylinder mounted in a Lexan sabot
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Experimental setup in reverse mode: diameter 1 mm projectile mounted in Divinycell fixture and diameter 10 mm SiC-B cylinder mounted in a Lexan sabot
Grahic Jump Location
Fig. 4

Dimensions of ceramic cylinders and copper covers used. Dimensions are in mm.

+Dimensions of ceramic cylinders and copper covers used. Dimensions are in mm.
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Dimensions of ceramic cylinders and copper covers used. Dimensions are in mm.
Grahic Jump Location
Fig. 5

X-ray pictures of Y925 projectile (a =  0.25 mm) and SiC-B target at two different times after impact. (a) Impact velocity v0 =  1445 m/s, slightly below transition. (b) Impact velocity v0 =  1501 m/s, slightly above transition. The transition velocity is v0* =  1470 ± 25 m/s.

+X-ray pictures of Y925 projectile (a =  0.25 mm) and SiC-B target at two different times after impact. (a) Impact velocity v0 =  1445 m/s, slightly below transition. (b) Impact velocity v0 =  1501 m/s, slightly above transition. The transition velocity is v0* =  1470 ± 25 m/s.
View Large  |  Save Figure  |  Download Slide (.ppt)
X-ray pictures of Y925 projectile (a =  0.25 mm) and SiC-B target at two different times after impact. (a) Impact velocity v0 =  1445 m/s, slightly below transition. (b) Impact velocity v0 =  1501 m/s, slightly above transition. The transition velocity is v0* =  1470 ± 25 m/s.
Grahic Jump Location
Fig. 6

X-ray pictures of Y925 projectile (a = 0.5 mm) and SiC-B target at two different times after impact. (a) Impact velocity v0 = 1244 m/s, slightly below transition. (b) Impact velocity v0 = 1275 m/s, slightly above transition. The transition velocity is v0* = 1260 ± 16 m/s.

+X-ray pictures of Y925 projectile (a = 0.5 mm) and SiC-B target at two different times after impact. (a) Impact velocity v0 = 1244 m/s, slightly below transition. (b) Impact velocity v0 = 1275 m/s, slightly above transition. The transition velocity is v0* = 1260 ± 16 m/s.
View Large  |  Save Figure  |  Download Slide (.ppt)
X-ray pictures of Y925 projectile (a = 0.5 mm) and SiC-B target at two different times after impact. (a) Impact velocity v0 = 1244 m/s, slightly below transition. (b) Impact velocity v0 = 1275 m/s, slightly above transition. The transition velocity is v0* = 1260 ± 16 m/s.
Grahic Jump Location
Fig. 7

X-ray pictures of Y925 projectile (a = 1.0 mm) and SiC-B target at two different times after impact. (a) Impact velocity v0 = 1004 m/s, slightly below transition. (b) Impact velocity v0 = 1051 m/s, slightly above transition. The transition velocity is v0* = 1028 ± 23 m/s.

+X-ray pictures of Y925 projectile (a = 1.0 mm) and SiC-B target at two different times after impact. (a) Impact velocity v0 = 1004 m/s, slightly below transition. (b) Impact velocity v0 = 1051 m/s, slightly above transition. The transition velocity is v0* = 1028 ± 23 m/s.
View Large  |  Save Figure  |  Download Slide (.ppt)
X-ray pictures of Y925 projectile (a = 1.0 mm) and SiC-B target at two different times after impact. (a) Impact velocity v0 = 1004 m/s, slightly below transition. (b) Impact velocity v0 = 1051 m/s, slightly above transition. The transition velocity is v0* = 1028 ± 23 m/s.
Grahic Jump Location
Fig. 10

(a) Crack path z¯, (b) crack length c¯, and (c) principal tensile stress σ¯1 versus radial crack extension r¯

+(a) Crack path z¯, (b) crack length c¯, and (c) principal tensile stress σ¯1 versus radial crack extension r¯
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(a) Crack path z¯, (b) crack length c¯, and (c) principal tensile stress σ¯1 versus radial crack extension r¯
Grahic Jump Location
Fig. 11

(a) Calculated principal tensile stress σ¯1 (solid line) and critical stress σ¯c (dashed line) along the crack for a projectile pressure p0 = p0*. The arrows indicate the point where the crack will stop, i.e., the critical crack length r¯c. Log-scale is used for the x-axis in order to highlight the region close to r¯ = 2. (b) The critical crack length r¯c versus length scale represented by projectile radius a.

+(a) Calculated principal tensile stress σ¯1 (solid line) and critical stress σ¯c (dashed line) along the crack for a projectile pressure p0 = p0*. The arrows indicate the point where the crack will stop, i.e., the critical crack length r¯c. Log-scale is used for the x-axis in order to highlight the region close to r¯ = 2. (b) The critical crack length r¯c versus length scale represented by projectile radius a.
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(a) Calculated principal tensile stress σ¯1 (solid line) and critical stress σ¯c (dashed line) along the crack for a projectile pressure p0 = p0*. The arrows indicate the point where the crack will stop, i.e., the critical crack length r¯c. Log-scale is used for the x-axis in order to highlight the region close to r¯ = 2. (b) The critical crack length r¯c versus length scale represented by projectile radius a.
Grahic Jump Location
Fig. 12

(a) Transition velocity v0* and (b) estimated projectile pressure at transition p0* versus length scale represented by projectile radius a. Solid line corresponds to Eq. (14) and filled symbols to the SiC-B targets. Error bars indicate maximum and minimum values for projectile velocity and projectile pressure, respectively. Dashed lines correspond to the three bounds: tensile failure (TF), incipient plastic yield (IY), and full plastic yield (FY).

+(a) Transition velocity v0* and (b) estimated projectile pressure at transition p0* versus length scale represented by projectile radius a. Solid line corresponds to Eq. (14) and filled symbols to the SiC-B targets. Error bars indicate maximum and minimum values for projectile velocity and projectile pressure, respectively. Dashed lines correspond to the three bounds: tensile failure (TF), incipient plastic yield (IY), and full plastic yield (FY).
View Large  |  Save Figure  |  Download Slide (.ppt)
(a) Transition velocity v0* and (b) estimated projectile pressure at transition p0* versus length scale represented by projectile radius a. Solid line corresponds to Eq. (14) and filled symbols to the SiC-B targets. Error bars indicate maximum and minimum values for projectile velocity and projectile pressure, respectively. Dashed lines correspond to the three bounds: tensile failure (TF), incipient plastic yield (IY), and full plastic yield (FY).
Grahic Jump Location
Fig. 8

X-ray pictures of Y925 projectile (a =  2.5 mm) and SiC-B target, impact velocity v0 =  1148 m/s. In this test, a short period of interface defeat occurred before the penetration started.

+X-ray pictures of Y925 projectile (a =  2.5 mm) and SiC-B target, impact velocity v0 =  1148 m/s. In this test, a short period of interface defeat occurred before the penetration started.
View Large  |  Save Figure  |  Download Slide (.ppt)
X-ray pictures of Y925 projectile (a =  2.5 mm) and SiC-B target, impact velocity v0 =  1148 m/s. In this test, a short period of interface defeat occurred before the penetration started.
Grahic Jump Location
Fig. 9

(a) Position of maximum radial stress on target surface r¯i and (b) corresponding radial surface stress σ¯rr at the point r¯i versus Poisson's ratio ν

+(a) Position of maximum radial stress on target surface r¯i and (b) corresponding radial surface stress σ¯rr at the point r¯i versus Poisson's ratio ν
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(a) Position of maximum radial stress on target surface r¯i and (b) corresponding radial surface stress σ¯rr at the point r¯i versus Poisson's ratio ν

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