- Email: [email protected]
Deformation and fracture of a long-rod projectile induced by an oblique moving plate: Numerical simulations
International Journal of Impact Engineering 40-41 (2012) 35e45
Contents lists available at SciVerse ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Deformation and fracture of a long-rod projectile induced by an oblique moving plate: Numerical simulations E. Lidén a, *, S. Mousavi a, A. Helte a, B. Lundberg b a b
FOI, Swedish Defence Research Agency, Defence and Security Systems and Technology Division, SE-164 90 Stockholm, Sweden The Ångström laboratory, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 September 2010 Received in revised form 31 August 2011 Accepted 5 September 2011 Available online 14 October 2011
Simulations have been performed to evaluate the possibility of reproducing the fragmentation of a long-rod projectile impacted by a moving oblique plate. When the moving plate slides along the projectile, fractures due to shear loading may occur in the projectile. Therefore, a fracture model suggested by XueeWierzbicki was used for the projectile together with the JohnsoneCook strength model. This fracture model is based on an equivalent plastic strain of fracture which depends on a stress triaxiality and a deviatoric stress parameter. The results of the simulations were compared with experimental results of a preceding study in which the impact conditions were varied in such a way that the projectile fractured in some but not all tests. The comparisons show that the simulations reproduced the fractures in the projectile well. Also, the transition from a deformed non-fractured to a severely fractured projectile was captured. The benefit of including the deviatoric stress parameter and the mechanisms leading to fragmentation of the projectile are discussed. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Reactive armour Oblique plate Long-rod projectile Fracture model Numerical simulation
1. Introduction In order to understand and evaluate the effects of weapons and armour systems, one has to study the interaction between different kinds of penetrators and armour modules. The high penetration capability of modern KE projectiles has resulted in the need of armour components that are able to disturb or fracture a projectile before it hits the main armour on the target. As fracture is a principal defeat mechanism, it is essential to know the influence of different parameters on the ability of the armour to fracture the projectile. An important basis for such knowledge is provided by numerical simulations that properly describe the interaction of projectile and armour and the fractures of the projectile. It is important to make use of a realistic fracture model for the projectile material under relevant load conditions. However, the lack of such a model has often resulted in simulations of interactions between projectiles and armour components using numerical erosion rather than fracture criteria. See for instance [1e5]. If the interest is focussed on disturbing the projectile as much as possible, it may sometimes be reasonable to perform simulations without use of a realistic fracture criterion. When the interaction between
* Corresponding author. Tel.: þ46 8 55503982; fax: þ46 8 55504080. E-mail address: [email protected] (E. Lidén). 0734-743X/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2011.09.003
the projectile and subsequent armour is of interest, however, the occurrence of fractures strongly influences the deformation and rotation of the fragments of the projectile. This, in turn, strongly influences the capability of the fragments to penetrate subsequent armour. Therefore, in this case, it is essential to accurately describe how the projectile fractures. In a preceding companion paper [6], 22 different cases of interactions of long-rod projectiles and oblique moving plates were investigated. Projectile length-to-diameter ratio, plate thickness, projectile velocity and plate velocity were varied under wellcontrolled experimental conditions with the aim to assess how the disturbance of the projectile depends on the impact parameters. In the present numerical study, four of the experimental cases were simulated with the intention to assess the disturbance of the projectile. The main emphasis was on assessing whether and how fragmentation of the projectile occurs. The transition from loading conditions that lead to a deformed essentially non-fractured projectile to loading conditions that produce a severely fractured projectile was studied in two pairs of cases. The first concerned the influence of the direction and the second the magnitude of the plate velocity. A fracture model suggested by XueeWierzbicki (XW) [7] and the JohnsoneCook (JC) strength model [8] were used for the projectile. This fracture model has not been commonly used in simulations of ballistic events and is not implemented in common
36
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
Nomenclature
Greek
a A, B, C a, c0 C1eC4 cp Df D1eD5 G m, n p S1eS3 t T v
material parameters in the JohnsoneCook strength model material parameters in Gruneisen equation of state material parameters in the XueeWierzbicki fracture model specific heat damage parameter material parameters in the JohnsoneCook fracture model shear modulus of elasticity material parameters in the JohnsoneCook strength model material parameter in the XueeWierzbicki fracture model material parameters in Gruneisen equation of state time from impact temperature velocity
simulation codes. It is based on an equivalent plastic strain of fracture that depends on parameters representing both stress triaxiality and deviatoric stress. This circumstance was assumed to allow accurate description of the initial consumption of the projectile nose as well as of subsequent fractures along the projectile, presumably due to shear loading.
Mie-Grüneisens equation of state and von Mises’s yield criterion with associated flow role [9] were used for both the plate and the projectile. The yield stress was taken as
sY 3 ; 3_ ; T ¼ ðA þ B3 n Þ 1 þ Cln 3_ =_3 0 1 ððT Tr Þ=ðTm Tr ÞÞm
(1) according to the JC strength model [8]. Here 3 is the equivalent plastic strain, 3_ is the equivalent plastic strain rate, T is the temperature, 3_ 0 is a reference strain rate, Tm is the melting temperature, Tr is a reference temperature, and A, B, C, n and m are material parameters which can be obtained from material tests or calibration against impact tests. The suitability of this model in the present application is supported by, e.g. [10]. Fractures in the projectile and the plate are postulated to occur when the damage parameter
Df ¼
X
D3 =3 f
3 3_
g0 h r s x
Subscripts eff effective f fracture m melt proj projectile r reference Y yield
while for the projectile it is defined by the XW fracture model [7] with XW
3f
1=p p ðh; xÞ ¼ C1 eC2 h C1 eC2 h C3 eC4 h 1 x :
(5)
In both models, the equivalent plastic strain at fracture depends on the stress triaxiality parameter
h ¼ smean =seff :
2. Models for deformation and fracture
obliquity, angle between normal of plate and axis of projectile equivalent plastic strain equivalent plastic strain rate material parameters in Gruneisen equation of state stress triaxiality parameter density stress deviatoric stress parameter
(6)
In the XW fracture model, it also depends on the deviatoric stress parameter
x ¼ ð27=2Þðs1 smean Þðs2 smean Þðs3 smean Þ=s3eff :
(7)
Here, s1, s2 and s3 are the principal stresses,
smean ¼ ð1=3Þðs1 þ s2 þ s3 Þ
(8)
is the mean stress, and
(2)
attains the value one. Here
D3 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2=3ÞD3 ij D3 ij
(3)
is the incremental equivalent plastic strain and 3 f is the equivalent plastic strain at fracture. When Df ¼ 1, the yield stress is set to zero so that the material can withstand only hydrostatic pressure. For the plate, the critical value of the equivalent plastic strain is defined by the JC fracture model [11] with 3
JC h; 3_ ; T f
¼
D1 þ D2 eD3 h
1 þ D4 ln 3_ =_3 0
½1 þ D5 ðT Tr Þ=ðTm Tr Þ;
(4)
Fig. 1. Dependence of equivalent plastic strain at fracture 3 XW of tungsten Y925 on f stress triaxiality parameter h and deviatoric stress parameter x according to the XW fracture model with the parameters used in this study.
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
37
Table 1 Cases studied. Case
Experimental case in [6]
Direction of plate velocity
Plate velocity [m/s]
Plate thickness [mm]
I II III IV
14 16 6 7
Backward Forward Forward Forward
200 200 200 300
6 6 3 3
pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs1 s2 Þ2 þðs2 s3 Þ2 þðs3 s1 Þ2
seff ¼ 1= 2
(9)
is the effective stress. Thus, dependence on the strain rate 3_ and the temperature T is included in the JC fracture model but not in the XW fracture model. The models involve material parameters D1eD5 and C1eC4, p, respectively, which can be obtained from material tests or from calibration against impact tests. The hardening exponent p is such that 1/p is an even integer [12]. The deviatoric stress parameter x depends on the principal stresses s1 s2 s3 . It follows from Eqs. (7) and (8) that this parameter is invariant with regard to both scaling ksi and translation si þ k. By use of these properties, it can be shown that 0 jxj 1. In particular, x ¼ 0 if s2 ¼ ðs1 þ s3 Þ=2, x ¼ 1 if s1 ¼ s2 , and x ¼ 1 if s2 ¼ s3 . Thus, disregarding a superimposed hydrostatic state of stress s1 ¼ s2 ¼ s3 ¼ s, the values x ¼ 0, 1 and 1 correspond to pure shear, uniaxial tension and uniaxial compression, respectively. If the influences of the strain rate 3_ and the temperature T in the JC fracture model are excluded and D1 ¼ 0, the XW fracture model with x ¼ 0 or x ¼ 1 has the same type of exponential dependence XW
ðh; 0Þ ¼ C3 eC4 h
(10)
XW
ðh; 1Þ ¼ C1 eC2 h
(11)
3f
3f
of the equivalent plastic strain at fracture on the stress triaxiality parameter h as the JC fracture model. The equivalent plastic strain at fracture of the XW fracture model is an even function of x and is ðh; 0Þ 3 XW ðh; xÞ 3 XW ðh; 1Þ designed to satisfy the inequalities 3 XW f f f for 1 x 1. Fig. 1 shows the influence of stress triaxiality and deviatoric stress parameters on the equivalent plastic strain at fracture for the XW fracture model with the material parameters used in this study.
a
Fig. 3. Original and refined mesh for the projectile in the simulations.
3. Simulations of projectile-plate interaction Two pairs of cases (14, 16) and (6, 7) from the experimental study [6] were simulated. Here, they will be referred to as Cases (I, II) and (III, IV), respectively. See Table 1. Each pair illustrates a transition from a loading condition that results in a deformed essentially non-fractured projectile to a loading condition that leads to a severely fractured projectile. The geometrical configurations, the material and the velocities used in the simulations correspond to those used in the experimental study [6]. The projectile was a flat-ended tungsten cylinder, with diameter 3 mm, length 90 mm (length-to-diameter ratio 30) and impact velocity 2000 m/s. The plates were made of steel and had width 20 mm, length 56 mm, thickness 3 or 6 mm, and magnitude of velocity 200 or 300 m/s. In the experiments, each plate was an integrated part of a sabot which had a stiffening effect. In the simulations, therefore, edge stiffeners were used as illustrated in Fig. 2. The influence of friction is complex. However, the contact between projectile and plate is expected to result in a thin
Table 2 Material parameters used in the simulations. Material properties
Steel plate
Tungsten projectile
r0 [kg/m3]
7820 79 477
17,700 160 134
G [GPa] cp [J/(kg K)] Mie-Gruneisen equation of state 4578 c0 [m/s] 1.33 S1 0 S2 0 S3 g0 1.67 a 0.43 JC strength model A [MPa] B [MPa] n C m Tm [K] Tr ¼ T(0) [K] 1 3_ 0 [s ]
b
Fig. 2. (a) Plate integrated in sabot for reverse impact experiments and (b) plate for simulations.
813 601 0.28 0.0139 1.04 1723 293 1.0
JC fracture model D1 D2 D3 D4 D5 XW fracture model C1 C2 C3 C4 p
4029 1.237 0 0 1.54 0
0 3.02 1.89 0.009 0.90 e e e e e
631 1258 0.092 0.014 0.94 1723 293 1.0
e e e e e 0.28 5.56 0.04 0.65 0.50
38
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45 Case II,Thick forwards moving plate
Case I, Thick backwards moving plate 200 m/s
60°
60° 2000 m/s
2000 m/s
Experiments
Reference
200 m/s
Numerical simulations
Fig. 4. Results of simulations compared with experimental results 150 ms after impact for Cases I and II with opposite plate flight directions. An undisturbed free-flight projectile is shown as a reference.
Case IV, Thin forwards moving plate
Case III,Thin forwards moving plate
60°
60°
2000 m/s
Reference
200 m/s
Experiments
2000 m/s 300 m/s
Numerical simulations
Fig. 5. Results of simulations compared with experimental results 150 ms after impact for Cases III and IV with the same plate flight directions but different plate velocities. An undisturbed free-flight projectile is shown as a reference.
melted region which serves as a lubricant and decreases the influence of friction. This can be justified by similar considerations as in [13], which show that in the assumed absence of melting, surface temperatures far above the melting points of the materials involved would be reached. In the simulations therefore, the effects of friction were neglected. In Cases I and II, a plate with thickness 6 mm and obliquity (angle between the normal of the plate and the axis of the projectile) 60 was considered. The velocity of the plate is referred to as “positive” or “backwards” if its component on the axis of the projectile is opposite to the velocity of the projectile (cf. the front plate of a reactive armour). Otherwise the velocity of the plate is referred to as “negative” or “forwards” (cf. the rear plate of a reactive armour). In Case I, the plate moved backwards (velocity 200 m/ s), while in Case II it moved forwards (velocity 200 m/s). In the corresponding experimental tests, the projectiles were heavily bent as a result of their interactions with the plates. The backwards moving plate caused no fragmentation, while the forwards moving plate severely fragmented the projectile. In Cases III and IV, a plate with thickness 3 mm, obliquity 60 and forward flight direction (as the rear plate in a reactive armour) was considered. The plate velocity was 200 and 300 m/s, respectively. In the corresponding experimental tests, the projectiles were only slightly bent as a result of their interactions with the plates. The slower plate caused almost no fragmentation, while the faster plate fractured the projectile into a number of almost straight fragments. The simulations were performed with LS-DYNA version 971 [9] in which the XueeWierzbicki fracture model was implemented for this study. A butterfly mesh was used for the circular cross section of the projectile. The element size was 0.25 mm in the plate and approximately 0.30 mm in the projectile. The dependence on mesh resolution was studied for Cases I and II by performing simulations with the element size in the projectile reduced to 0.10 mm. The ordinary and refined meshes are shown in Fig. 3. The number of
Fig. 6. Results for Cases I and II from study of mesh resolution. The contour plots indicate the level of damage in the symmetry plane. Blue to red corresponds to Df ¼ 0e1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
elements in the projectile with the original and refined meshes was 32,400 and 100,800, respectively. The parameters used in the strength and fracture models were determined from material testing and are shown in Table 2. Test specimens with five different geometries were used to estimate the parameters C1eC4 of the XW fracture model, Eq. (5). To achieve axially symmetrical states of stress with different values of the stress triaxiality parameter h, cylindrical specimens without notches and with notches of different sizes were used in quasistatic tests. To achieve pure shear fractures, quasi-static compression tests with notched hat-shaped specimens were also performed. Simulations of the tests were performed to estimate the parameters h and x at fracture at points in the specimens where crack initiation was observed to occur. The parameter p was set to 0.5 and the set of parameters C1eC4 were evaluated by minimising P 2 the function f ðC1 ; C2 ; C3 ; C4 Þ ¼ k ½Df ðC1 ; C2 ; C3 ; C4 ; pÞ 1 . As the XW fracture model does not include dependence on strain rate, the
39
quasi-static fracture surface obtained was adjusted on the basis of servo-hydraulic tests with smooth specimens (nominal strain rates 104, 102, 1, 10 and 400 s1) and on Split Hopkinson Bar tests with hat-shaped specimens (strain rate 104 s1). In order to avoid numerical interlocking at the sliding contact, a Lagrange formulation was used for the projectile, while a multimaterial ALE formulation was used for the plate (steel and vacuum). The element types used were “constant stress solid” (Type 1) for the Lagrange part and “1-point ALE multi-material element” (Type 11) for the Euler part. For the Lagrangian part of the model, fully damaged elements were not eroded in order to withstand compressive pressure loads. Instead a numerical erosion criterion set at 150% plastic strain was used to eliminate heavily deformed elements. To evaluate the benefit of including the deviatoric stress parameter x in the fracture model, some simulations were performed with this parameter set to 0 and 1 as in Eqs. (10) and (11),
Fig. 7. Results for Cases I and II from a study of the influence of the deviatoric stress parameter x of the fracture model. The contour plots indicate the level of damage in the symmetry plane. Blue to red corresponds to Df ¼ 0e1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
40
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
respectively. Also, in an attempt to understand the conditions that lead to fragmentation of the projectile, the stresses due to the interaction between the plate and the projectile were evaluated along the upper and lower sides of the projectile. 4. Results Figs. 4 and 5 show the geometry and position of the projectile 150 ms after impact, according to experiments and simulations, relative to the position of a free-flight projectile. The transition from deformed essentially non-fractured to severely fragmented projectile, in Cases I and II and Cases III and IV, can be seen in the results from both experiments and simulations. The influence of mesh resolution can be seen in Fig. 6, which shows that the results from the simulations with refined meshes are in good overall agreement with those of the original simulations.
The influence of the deviatoric stress parameter x in the XW fracture model is shown in Figs. 7 and 8, where the results of simulation for variable x, Eq. (5), are compared with those for x ¼ 1 (uniaxial stress) and x ¼ 0 (pure shear), Eqs. (11) and (10), and also with experimental results. In Case I, variable x results in two fractures at the rear end of the projectile which did not occur in the experiment. A predisposition to such fractures can be seen in the simulations with x ¼ 1. For x ¼ 0, the rear end of the projectile is severely damaged and additional fractures can be seen. In Case II, variable x leads to five fractures instead of six in the experiment. For x ¼ 1 there are only two fractures. For x ¼ 0 the projectile is subjected to multiple fractures and severe damage. In Case III, both variable x and x ¼ 1 result in a single fracture in agreement with experiment, while x ¼ 0 results in three fractures. In Case IV, finally, where the experiment resulted in six fractures between parts on the projectile, four fractures and predisposition to
Fig. 8. Results for Cases III and IV from a study of the influence of the deviatoric stress parameter x of the fracture model. The contour plots indicate the level of damage in the symmetry plane. Blue to red corresponds to Df ¼ 0e1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
two additional fractures can be seen for variable x, while x ¼ 1 results in two fractures and propensity to two additional fractures. For x ¼ 0, again, the projectile is subjected to multiple fractures and severe damage. Figs. 9e12 illustrate, for the unconstrained XW fracture model with variable x, the different mechanisms involved when the
41
projectile penetrates a backwards and a forwards moving plate, Cases I and II, respectively. Fig. 9 shows sequences of damage of the projectile when it penetrates the plate. Also, the difference in contact geometry between the two cases during the penetration can be seen. Fig. 10 shows the lateral stress sxx on the upper and lower side of the mid-section of the projectile versus time. This
Fig. 9. Sequences of damage plots for the projectile during its penetration of the plate in Cases I and II. The contour plots indicate the level of damage. Blue to red corresponds to Df ¼ 0e1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
42
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
Fig. 10. Lateral stress sxx versus time t from impact at the upper and lower side of the mid-section of the projectile during the sliding perforation of the plate in Cases I and II. The contour plots indicate the level of stress in the symmetry plane. Blue to red corresponds to approximately sxx ¼ 0e3.5 GPa. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
illustrates the difference between the two cases in the loading on the projectile when the plate slides along its envelope. Fig. 11 shows the effective stress seff versus time t from impact along the projectile in the two cases. In these figures, where colours represent levels of effective stress, dark blue colour indicates zero stress and represents either an uninfluenced element (in the back of the projectile at an early time) or an eroded element (eroded nose elements and elements where fracture appears). Fig. 12 shows the shear stress szx versus time t from impact at the side of the projectile where the fractures are initiated, i.e. at the upper side of the projectile in Case I and at the lower side of the projectile in Case II. The localised regions of high shear stress in Case II agree well with the localised regions of high effective stress in Fig. 11(d). 5. Discussion Figs. 4e6 show that by use of the XW fracture model, and the well-established JC strength model, it was possible to accurately reproduce the experimental results for the interaction between a projectile and a moving oblique plate. Furthermore, this could be done without use of an extremely fine mesh. The transition from a non-fractured to a severely fractured projectile was captured, and the deformation and rotation of the projectile and its fragments, and the locations of fractures, agree well with the experimental
results. The material parameters used had been determined from simple material tests. A more sophisticated parameter evaluation would probably improve the agreement. Both the influence of the direction and of the magnitude of the plate velocity on the fragmentation was captured as can be seen in Figs. 4 and 5, respectively. For the different magnitudes of plate velocity in Cases III and IV, the type of load on the projectile is the same; only the magnitude of the load is different. For the different directions of plate velocity in Cases I and II, however, the type of load on the projectile is different. It is interesting to study this difference as our experimental results in [6] show that backwards moving plates did not fragment projectiles while forwards moving plates did if the load was adequate. Next, the interactions between projectile and plate in these two cases will be discussed on the basis of the results shown in Figs. 9e12. When the nose of the projectile impacts the plate as shown in Fig. 9, a penetration channel with greater diameter than that of the projectile is created. Due to the oblique orientation of the plate, the entrance part of the penetration channel expands more on the upper than on the lower side of the projectile. The reason for this lack of symmetry is that the projectile is supported by more material on its lower than on its upper side at the entrance. It can be seen that in Case I of a backwards moving plate, the plate pushes the projectile upwards. This immediately results in a sliding contact on the lower part of the penetration channel. In Case II of a forwards
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
43
Fig. 11. Effective stress seff versus time t from impact along the projectile in Cases I and II. (a) Case I, upper side. (b) Case II, upper side. (c) Case I, lower side. (d) Case II, lower side. The contour plots indicate the level of effective stress. The contour plots of the projectile below, indicating were fractures occur, are shown as a reference. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
moving plate, the plate approaches the projectile downwards. Because of the asymmetry of the penetration channel, the establishment of contact between the projectile and the plate along the upper part of the penetration channel is delayed. With the plate and projectile velocities used, the contact is not established until the nose of the projectile has reached the rear side of the plate (approximately 22 ms). The difference in time between Cases I and II for establishment of contact can be seen also in Fig. 11. Fig. 11 illustrates, in both Cases I and II, that when the projectile impacts the plate, large stresses arise at the nose of the projectile, which is continuously consumed during perforation, and an elastic compressive wave propagates along the projectile and reflects as an elastic tensile wave at the free rear end of the projectile. It can be seen that a smaller part of the nose of the projectile is consumed in
Case I of a backwards moving plate than in Case II of a forwards moving plate. This agrees well with our experimental results in [6], where the relative decrease in length was found to be 0.12 and 0.18, respectively. The front of the elastic compressive wave can be seen as the first sloping straight line in the diagrams. In both cases, the velocity of the wave is about 4500 m/s. A sloping dark red region, representing high effective stress, with a different slope can also be seen in the diagrams. It represents the sliding contact with the plate. The sliding velocity with respect to the projectile vslide ¼ vproj þ vplate =cos a is 2400 m/s in Case I of a backwards moving plate and 1600 m/s in Case II of a forwards moving plate. These sliding velocities imply that the contact between the projectile and the plate ends after about 43 and 64 ms, respectively, which agrees well with the high-stress regions in the diagrams. In
44
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
Fig. 12. Shear stress szx versus time t from impact at the side of the projectile where the fractures are initiated, i.e. at the upper side of the projectile in Case I and at the lower side of the projectile in Case II. The contour plots indicate the level of shear stress.
accord with the discussion of Fig. 9 above, the high effective stress region due to the sliding contact appears immediately at impact in Case I of a backwards moving plate, while in Case II of a forwards moving plate it does not appear until the projectile has perforated the plate. When the projectile perforates the oblique plate, the exit part of the penetration channel expands more at its lower side where there is less supporting material than at its upper side. As the lower side in Case I of a backwards moving plate is the main contact surface, the sliding contact between the projectile and the penetration channel is released before the exit, see Fig. 9. In this case, there is contact with the plate both on the lower sliding side of the projectile and on the upper side of the nose. In Case II, there is no contact between the projectile and the lower side of the penetration channel during perforation. Yet, successive high-damage regions arise on the lower side of the projectile at the exit of the plate which subsequently leads to the initiation of fractures. For the first two fractures, this can be seen from a comparison of Figs. 4 and 9. After perforation, the projectile slides against the plate and creates the typical keyhole-shaped penetration channel observed in our preceding experimental study [6]. The contact during this sliding interaction occurs on the lower side of the channel in Case I and on the upper side in Case II. As a result, the history of load on the projectile at a fixed section is different in the two cases. Fig. 10 shows the lateral stress sxx versus time on the contact surface at the mid-section of the projectile during the sliding contact. In Case I of a backwards moving plate the magnitude of the stress abruptly increases to a maximum at the arrival of the plate. Then, it decreases to approximately half its value, till it abruptly drops to zero when the plate leaves. In Case II of a forwards moving plate there are also corresponding abrupt changes in the magnitude of the stress. However, the magnitude of the stress increases, doubles, and is maximal when the plate leaves the mid-section of the
projectile. At certain sections, this gives rise to high-damage regions in the projectile which subsequently result in fractures. The different characters of the stress histories between Cases I and II are due to the different thicknesses of supporting material on the upper and lower sides of the oblique plate. The figure also shows that the duration of contact is longer in Case II than in Case I, 7.5 and 5.0 ms, respectively. This is due to the different sliding velocities (1600 and 2400 m/s, respectively). The longer contact duration in Case II than in Case I, is illustrated also by the thicker high effective stress regions in Fig. 11(b) than in Fig. 11(c). In Case I, the sliding contact on the lower side of the projectile results in bending such that the projectile gets a convex shape upwards as shown in Fig. 4. This gives rise to large axial tensile stresses on the upper side of the projectile and large compressive stresses on the lower side. Correspondingly, large regions of high effective stress are formed on the upper and lower sides of the projectile. Because of the influence of the lateral stress (compressive on the lower side and zero on the upper side) on the effective stress, these high effective stress regions are larger on the upper side of the projectile than on the lower side. Compare the redcoloured regions in Fig. 11. The relatively uniform distribution of effective stress on the upper side of the projectile seems to be related to the absence of fractures in Case I. In Case II, similarly, the sliding contact on the upper side of the projectile results in bending such that the projectile gets a convex shape downwards as shown in Fig. 4. This gives rise to large axial tensile stresses on the lower side of the projectile and large compressive stresses on the upper side. On the lower side of the projectile this leads, as can be seen in Fig. 11(d), to localised regions of high effective stress and damage at which fractures are initiated. Fig. 9 illustrates that each of these regions first appears at the exit of the plate where there is an abrupt drop in the lateral load on the projectile (Fig. 10).
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
Fig. 12 shows for Case II the shear stress szx versus time from impact at the lower side of the projectile, where the fractures are initiated. The localised regions of high shear stress in this diagram agree well with the localised regions of high effective stress in Fig. 11(d) as well as with the positions of the fractures in Fig. 6. This indicates that the fractures originate from shear stress and supports the introduction of the deviatoric stress parameter x in the fracture model so that the sensitivity of the material to shear stresses can be taken into consideration. For example, Fig. 1 shows that with the material parameters used in the XW fracture model for tungsten Y925 in the simulations, the equivalent plastic strain at fracture increases with about 30% from pure shear (x ¼ 0, h ¼ 0) to uniaxial tension (x ¼ 1, h ¼ 1/3). Figs. 7 and 8 show that in Cases I and III, where the projectiles are mainly deformed in the experimental tests, the results of simulation with variable x and with fixed x ¼ 1 agree equally well with the experimental results, while those with fixed x ¼ 0 result in too much damage and fractures. In Cases II and IV, where the projectile is severely fractured in the experimental tests, the number of fragments is best reproduced in the simulations with fixed x ¼ 0 although the overall damage of the projectile is overestimated. In these cases, the simulations with variable x and with fixed x ¼ 1 miss some of the fractures. However, more of the fractures are captured with variable x, and those missing are represented by high levels of damage. Clearly, the deviatoric stress parameter x of the XW model has significant influence on the results of simulation. It allows fractures due to shear loading without extensive damage of the entire projectile. Introduction of the deviatoric stress parameter in the fracture model means that tests are required to determine the parameter p of the model. In studies where it is essential to capture fractures, such additional testing may be worthwhile. 6. Conclusions The main conclusions of this study can be summarised as follows. (i) It is possible to reproduce in simulations fractures in a tungsten long-rod projectile impacted by a moving oblique plate. (ii) This can be done without use of an extremely fine mesh. (iii) The transition from a deformed non-fractured projectile to a severely
45
fractured projectile due to a small increase in the magnitude of the plate velocity or a change from a backwards to a forwards moving plate can be captured. (iv) The larger effect on the projectile of a forwards moving plate than a backwards moving plate is due to the longer interaction time and the larger thickness of supporting plate material at the exit side of the plate. (v) The deviatoric stress parameter used in the XW model has significant influence and allows fractures due to shear loading without extensive damage of the entire projectile. References [1] Johnson GR, Cook WH. Lagrangian EPIC code computations for oblique, yawed-rod impacts onto thin-plate and spaced-plate targets at various velocities. Int J Impact Eng 1993;14:373e83. [2] Lidén E, Ottosson J, Holmberg L. WHA long rods penetrating stationary and moving oblique steel plates. In: Bailey HR, Holden L, editors. Proceedings 16th International Symposium on Ballistics, San Francisco. American Defense Preparedness Association. 1996;3:703–712. [3] Yoo Y-H, Shin H. Protection capability of dual flying plates against obliquely impacting long-rod penetrators. Int J Impact Eng 2004;30:55e68. [4] Yoo Y-H, Lee M. Protection effectiveness of an oblique plate against a long rod. Int J Impact Eng 2006;33:872e9. [5] Paik SH, Kim SJ, Yoo YH, Lee M. Protection performance of dual flying oblique plates against a yawed long-rod penetrator. Int J Impact Eng 2007;34: 1413e22. [6] Lidén E, Andersson O, Lundberg B. Deformation and fracture of a long rod projectile induced by a oblique moving plate. Experimental tests. Int J Impact Eng 2011;12:989e1000. [7] Wierzbicki T, Bao Y, Lee Y-W, Bai Y. Calibration and evaluation of seven fracture models. Int J Mech Sci 2005;47:719e43. [8] Johnson GR, Cook WH. A constitutive model and data for metals subjected to large strains, high strain rates, and high temperatures. In: Proceedings of the seventh international symposium on ballistics, The Hague. Royal Institute of Engineers in the Netherlands in co-operation with the American Defense Preparedness Association. 1983:541e547. [9] Hallquist JO. LS-DYNA theory manual. Livermore Software Technology Corporation; 2006. [10] Rohr I, Nahme H, Thoma K, Anderson CE. Material characterisation and constitutive modelling of a tungsten-sintered alloy for a wide range of strain rates. Int J Impact Eng 2008;35:811e9. [11] Johnson GR, Cook WH. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng Fract Mech 1985;21(1):31e48. [12] Bai Y, Wierzbicki T. A new model of metal plasticity and fracture with pressure and Lode dependence. Int J Plast 2008;24:1071e96. [13] Helte A, Lidén E. The role of KelvineHelmholtz instabilities on shaped charge jet interaction with reactive armor plates. J Appl Mech 2010;77:051805e11.
Contents lists available at SciVerse ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Deformation and fracture of a long-rod projectile induced by an oblique moving plate: Numerical simulations E. Lidén a, *, S. Mousavi a, A. Helte a, B. Lundberg b a b
FOI, Swedish Defence Research Agency, Defence and Security Systems and Technology Division, SE-164 90 Stockholm, Sweden The Ångström laboratory, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 September 2010 Received in revised form 31 August 2011 Accepted 5 September 2011 Available online 14 October 2011
Simulations have been performed to evaluate the possibility of reproducing the fragmentation of a long-rod projectile impacted by a moving oblique plate. When the moving plate slides along the projectile, fractures due to shear loading may occur in the projectile. Therefore, a fracture model suggested by XueeWierzbicki was used for the projectile together with the JohnsoneCook strength model. This fracture model is based on an equivalent plastic strain of fracture which depends on a stress triaxiality and a deviatoric stress parameter. The results of the simulations were compared with experimental results of a preceding study in which the impact conditions were varied in such a way that the projectile fractured in some but not all tests. The comparisons show that the simulations reproduced the fractures in the projectile well. Also, the transition from a deformed non-fractured to a severely fractured projectile was captured. The benefit of including the deviatoric stress parameter and the mechanisms leading to fragmentation of the projectile are discussed. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Reactive armour Oblique plate Long-rod projectile Fracture model Numerical simulation
1. Introduction In order to understand and evaluate the effects of weapons and armour systems, one has to study the interaction between different kinds of penetrators and armour modules. The high penetration capability of modern KE projectiles has resulted in the need of armour components that are able to disturb or fracture a projectile before it hits the main armour on the target. As fracture is a principal defeat mechanism, it is essential to know the influence of different parameters on the ability of the armour to fracture the projectile. An important basis for such knowledge is provided by numerical simulations that properly describe the interaction of projectile and armour and the fractures of the projectile. It is important to make use of a realistic fracture model for the projectile material under relevant load conditions. However, the lack of such a model has often resulted in simulations of interactions between projectiles and armour components using numerical erosion rather than fracture criteria. See for instance [1e5]. If the interest is focussed on disturbing the projectile as much as possible, it may sometimes be reasonable to perform simulations without use of a realistic fracture criterion. When the interaction between
* Corresponding author. Tel.: þ46 8 55503982; fax: þ46 8 55504080. E-mail address: [email protected] (E. Lidén). 0734-743X/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2011.09.003
the projectile and subsequent armour is of interest, however, the occurrence of fractures strongly influences the deformation and rotation of the fragments of the projectile. This, in turn, strongly influences the capability of the fragments to penetrate subsequent armour. Therefore, in this case, it is essential to accurately describe how the projectile fractures. In a preceding companion paper [6], 22 different cases of interactions of long-rod projectiles and oblique moving plates were investigated. Projectile length-to-diameter ratio, plate thickness, projectile velocity and plate velocity were varied under wellcontrolled experimental conditions with the aim to assess how the disturbance of the projectile depends on the impact parameters. In the present numerical study, four of the experimental cases were simulated with the intention to assess the disturbance of the projectile. The main emphasis was on assessing whether and how fragmentation of the projectile occurs. The transition from loading conditions that lead to a deformed essentially non-fractured projectile to loading conditions that produce a severely fractured projectile was studied in two pairs of cases. The first concerned the influence of the direction and the second the magnitude of the plate velocity. A fracture model suggested by XueeWierzbicki (XW) [7] and the JohnsoneCook (JC) strength model [8] were used for the projectile. This fracture model has not been commonly used in simulations of ballistic events and is not implemented in common
36
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
Nomenclature
Greek
a A, B, C a, c0 C1eC4 cp Df D1eD5 G m, n p S1eS3 t T v
material parameters in the JohnsoneCook strength model material parameters in Gruneisen equation of state material parameters in the XueeWierzbicki fracture model specific heat damage parameter material parameters in the JohnsoneCook fracture model shear modulus of elasticity material parameters in the JohnsoneCook strength model material parameter in the XueeWierzbicki fracture model material parameters in Gruneisen equation of state time from impact temperature velocity
simulation codes. It is based on an equivalent plastic strain of fracture that depends on parameters representing both stress triaxiality and deviatoric stress. This circumstance was assumed to allow accurate description of the initial consumption of the projectile nose as well as of subsequent fractures along the projectile, presumably due to shear loading.
Mie-Grüneisens equation of state and von Mises’s yield criterion with associated flow role [9] were used for both the plate and the projectile. The yield stress was taken as
sY 3 ; 3_ ; T ¼ ðA þ B3 n Þ 1 þ Cln 3_ =_3 0 1 ððT Tr Þ=ðTm Tr ÞÞm
(1) according to the JC strength model [8]. Here 3 is the equivalent plastic strain, 3_ is the equivalent plastic strain rate, T is the temperature, 3_ 0 is a reference strain rate, Tm is the melting temperature, Tr is a reference temperature, and A, B, C, n and m are material parameters which can be obtained from material tests or calibration against impact tests. The suitability of this model in the present application is supported by, e.g. [10]. Fractures in the projectile and the plate are postulated to occur when the damage parameter
Df ¼
X
D3 =3 f
3 3_
g0 h r s x
Subscripts eff effective f fracture m melt proj projectile r reference Y yield
while for the projectile it is defined by the XW fracture model [7] with XW
3f
1=p p ðh; xÞ ¼ C1 eC2 h C1 eC2 h C3 eC4 h 1 x :
(5)
In both models, the equivalent plastic strain at fracture depends on the stress triaxiality parameter
h ¼ smean =seff :
2. Models for deformation and fracture
obliquity, angle between normal of plate and axis of projectile equivalent plastic strain equivalent plastic strain rate material parameters in Gruneisen equation of state stress triaxiality parameter density stress deviatoric stress parameter
(6)
In the XW fracture model, it also depends on the deviatoric stress parameter
x ¼ ð27=2Þðs1 smean Þðs2 smean Þðs3 smean Þ=s3eff :
(7)
Here, s1, s2 and s3 are the principal stresses,
smean ¼ ð1=3Þðs1 þ s2 þ s3 Þ
(8)
is the mean stress, and
(2)
attains the value one. Here
D3 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2=3ÞD3 ij D3 ij
(3)
is the incremental equivalent plastic strain and 3 f is the equivalent plastic strain at fracture. When Df ¼ 1, the yield stress is set to zero so that the material can withstand only hydrostatic pressure. For the plate, the critical value of the equivalent plastic strain is defined by the JC fracture model [11] with 3
JC h; 3_ ; T f
¼
D1 þ D2 eD3 h
1 þ D4 ln 3_ =_3 0
½1 þ D5 ðT Tr Þ=ðTm Tr Þ;
(4)
Fig. 1. Dependence of equivalent plastic strain at fracture 3 XW of tungsten Y925 on f stress triaxiality parameter h and deviatoric stress parameter x according to the XW fracture model with the parameters used in this study.
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
37
Table 1 Cases studied. Case
Experimental case in [6]
Direction of plate velocity
Plate velocity [m/s]
Plate thickness [mm]
I II III IV
14 16 6 7
Backward Forward Forward Forward
200 200 200 300
6 6 3 3
pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs1 s2 Þ2 þðs2 s3 Þ2 þðs3 s1 Þ2
seff ¼ 1= 2
(9)
is the effective stress. Thus, dependence on the strain rate 3_ and the temperature T is included in the JC fracture model but not in the XW fracture model. The models involve material parameters D1eD5 and C1eC4, p, respectively, which can be obtained from material tests or from calibration against impact tests. The hardening exponent p is such that 1/p is an even integer [12]. The deviatoric stress parameter x depends on the principal stresses s1 s2 s3 . It follows from Eqs. (7) and (8) that this parameter is invariant with regard to both scaling ksi and translation si þ k. By use of these properties, it can be shown that 0 jxj 1. In particular, x ¼ 0 if s2 ¼ ðs1 þ s3 Þ=2, x ¼ 1 if s1 ¼ s2 , and x ¼ 1 if s2 ¼ s3 . Thus, disregarding a superimposed hydrostatic state of stress s1 ¼ s2 ¼ s3 ¼ s, the values x ¼ 0, 1 and 1 correspond to pure shear, uniaxial tension and uniaxial compression, respectively. If the influences of the strain rate 3_ and the temperature T in the JC fracture model are excluded and D1 ¼ 0, the XW fracture model with x ¼ 0 or x ¼ 1 has the same type of exponential dependence XW
ðh; 0Þ ¼ C3 eC4 h
(10)
XW
ðh; 1Þ ¼ C1 eC2 h
(11)
3f
3f
of the equivalent plastic strain at fracture on the stress triaxiality parameter h as the JC fracture model. The equivalent plastic strain at fracture of the XW fracture model is an even function of x and is ðh; 0Þ 3 XW ðh; xÞ 3 XW ðh; 1Þ designed to satisfy the inequalities 3 XW f f f for 1 x 1. Fig. 1 shows the influence of stress triaxiality and deviatoric stress parameters on the equivalent plastic strain at fracture for the XW fracture model with the material parameters used in this study.
a
Fig. 3. Original and refined mesh for the projectile in the simulations.
3. Simulations of projectile-plate interaction Two pairs of cases (14, 16) and (6, 7) from the experimental study [6] were simulated. Here, they will be referred to as Cases (I, II) and (III, IV), respectively. See Table 1. Each pair illustrates a transition from a loading condition that results in a deformed essentially non-fractured projectile to a loading condition that leads to a severely fractured projectile. The geometrical configurations, the material and the velocities used in the simulations correspond to those used in the experimental study [6]. The projectile was a flat-ended tungsten cylinder, with diameter 3 mm, length 90 mm (length-to-diameter ratio 30) and impact velocity 2000 m/s. The plates were made of steel and had width 20 mm, length 56 mm, thickness 3 or 6 mm, and magnitude of velocity 200 or 300 m/s. In the experiments, each plate was an integrated part of a sabot which had a stiffening effect. In the simulations, therefore, edge stiffeners were used as illustrated in Fig. 2. The influence of friction is complex. However, the contact between projectile and plate is expected to result in a thin
Table 2 Material parameters used in the simulations. Material properties
Steel plate
Tungsten projectile
r0 [kg/m3]
7820 79 477
17,700 160 134
G [GPa] cp [J/(kg K)] Mie-Gruneisen equation of state 4578 c0 [m/s] 1.33 S1 0 S2 0 S3 g0 1.67 a 0.43 JC strength model A [MPa] B [MPa] n C m Tm [K] Tr ¼ T(0) [K] 1 3_ 0 [s ]
b
Fig. 2. (a) Plate integrated in sabot for reverse impact experiments and (b) plate for simulations.
813 601 0.28 0.0139 1.04 1723 293 1.0
JC fracture model D1 D2 D3 D4 D5 XW fracture model C1 C2 C3 C4 p
4029 1.237 0 0 1.54 0
0 3.02 1.89 0.009 0.90 e e e e e
631 1258 0.092 0.014 0.94 1723 293 1.0
e e e e e 0.28 5.56 0.04 0.65 0.50
38
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45 Case II,Thick forwards moving plate
Case I, Thick backwards moving plate 200 m/s
60°
60° 2000 m/s
2000 m/s
Experiments
Reference
200 m/s
Numerical simulations
Fig. 4. Results of simulations compared with experimental results 150 ms after impact for Cases I and II with opposite plate flight directions. An undisturbed free-flight projectile is shown as a reference.
Case IV, Thin forwards moving plate
Case III,Thin forwards moving plate
60°
60°
2000 m/s
Reference
200 m/s
Experiments
2000 m/s 300 m/s
Numerical simulations
Fig. 5. Results of simulations compared with experimental results 150 ms after impact for Cases III and IV with the same plate flight directions but different plate velocities. An undisturbed free-flight projectile is shown as a reference.
melted region which serves as a lubricant and decreases the influence of friction. This can be justified by similar considerations as in [13], which show that in the assumed absence of melting, surface temperatures far above the melting points of the materials involved would be reached. In the simulations therefore, the effects of friction were neglected. In Cases I and II, a plate with thickness 6 mm and obliquity (angle between the normal of the plate and the axis of the projectile) 60 was considered. The velocity of the plate is referred to as “positive” or “backwards” if its component on the axis of the projectile is opposite to the velocity of the projectile (cf. the front plate of a reactive armour). Otherwise the velocity of the plate is referred to as “negative” or “forwards” (cf. the rear plate of a reactive armour). In Case I, the plate moved backwards (velocity 200 m/ s), while in Case II it moved forwards (velocity 200 m/s). In the corresponding experimental tests, the projectiles were heavily bent as a result of their interactions with the plates. The backwards moving plate caused no fragmentation, while the forwards moving plate severely fragmented the projectile. In Cases III and IV, a plate with thickness 3 mm, obliquity 60 and forward flight direction (as the rear plate in a reactive armour) was considered. The plate velocity was 200 and 300 m/s, respectively. In the corresponding experimental tests, the projectiles were only slightly bent as a result of their interactions with the plates. The slower plate caused almost no fragmentation, while the faster plate fractured the projectile into a number of almost straight fragments. The simulations were performed with LS-DYNA version 971 [9] in which the XueeWierzbicki fracture model was implemented for this study. A butterfly mesh was used for the circular cross section of the projectile. The element size was 0.25 mm in the plate and approximately 0.30 mm in the projectile. The dependence on mesh resolution was studied for Cases I and II by performing simulations with the element size in the projectile reduced to 0.10 mm. The ordinary and refined meshes are shown in Fig. 3. The number of
Fig. 6. Results for Cases I and II from study of mesh resolution. The contour plots indicate the level of damage in the symmetry plane. Blue to red corresponds to Df ¼ 0e1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
elements in the projectile with the original and refined meshes was 32,400 and 100,800, respectively. The parameters used in the strength and fracture models were determined from material testing and are shown in Table 2. Test specimens with five different geometries were used to estimate the parameters C1eC4 of the XW fracture model, Eq. (5). To achieve axially symmetrical states of stress with different values of the stress triaxiality parameter h, cylindrical specimens without notches and with notches of different sizes were used in quasistatic tests. To achieve pure shear fractures, quasi-static compression tests with notched hat-shaped specimens were also performed. Simulations of the tests were performed to estimate the parameters h and x at fracture at points in the specimens where crack initiation was observed to occur. The parameter p was set to 0.5 and the set of parameters C1eC4 were evaluated by minimising P 2 the function f ðC1 ; C2 ; C3 ; C4 Þ ¼ k ½Df ðC1 ; C2 ; C3 ; C4 ; pÞ 1 . As the XW fracture model does not include dependence on strain rate, the
39
quasi-static fracture surface obtained was adjusted on the basis of servo-hydraulic tests with smooth specimens (nominal strain rates 104, 102, 1, 10 and 400 s1) and on Split Hopkinson Bar tests with hat-shaped specimens (strain rate 104 s1). In order to avoid numerical interlocking at the sliding contact, a Lagrange formulation was used for the projectile, while a multimaterial ALE formulation was used for the plate (steel and vacuum). The element types used were “constant stress solid” (Type 1) for the Lagrange part and “1-point ALE multi-material element” (Type 11) for the Euler part. For the Lagrangian part of the model, fully damaged elements were not eroded in order to withstand compressive pressure loads. Instead a numerical erosion criterion set at 150% plastic strain was used to eliminate heavily deformed elements. To evaluate the benefit of including the deviatoric stress parameter x in the fracture model, some simulations were performed with this parameter set to 0 and 1 as in Eqs. (10) and (11),
Fig. 7. Results for Cases I and II from a study of the influence of the deviatoric stress parameter x of the fracture model. The contour plots indicate the level of damage in the symmetry plane. Blue to red corresponds to Df ¼ 0e1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
40
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
respectively. Also, in an attempt to understand the conditions that lead to fragmentation of the projectile, the stresses due to the interaction between the plate and the projectile were evaluated along the upper and lower sides of the projectile. 4. Results Figs. 4 and 5 show the geometry and position of the projectile 150 ms after impact, according to experiments and simulations, relative to the position of a free-flight projectile. The transition from deformed essentially non-fractured to severely fragmented projectile, in Cases I and II and Cases III and IV, can be seen in the results from both experiments and simulations. The influence of mesh resolution can be seen in Fig. 6, which shows that the results from the simulations with refined meshes are in good overall agreement with those of the original simulations.
The influence of the deviatoric stress parameter x in the XW fracture model is shown in Figs. 7 and 8, where the results of simulation for variable x, Eq. (5), are compared with those for x ¼ 1 (uniaxial stress) and x ¼ 0 (pure shear), Eqs. (11) and (10), and also with experimental results. In Case I, variable x results in two fractures at the rear end of the projectile which did not occur in the experiment. A predisposition to such fractures can be seen in the simulations with x ¼ 1. For x ¼ 0, the rear end of the projectile is severely damaged and additional fractures can be seen. In Case II, variable x leads to five fractures instead of six in the experiment. For x ¼ 1 there are only two fractures. For x ¼ 0 the projectile is subjected to multiple fractures and severe damage. In Case III, both variable x and x ¼ 1 result in a single fracture in agreement with experiment, while x ¼ 0 results in three fractures. In Case IV, finally, where the experiment resulted in six fractures between parts on the projectile, four fractures and predisposition to
Fig. 8. Results for Cases III and IV from a study of the influence of the deviatoric stress parameter x of the fracture model. The contour plots indicate the level of damage in the symmetry plane. Blue to red corresponds to Df ¼ 0e1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
two additional fractures can be seen for variable x, while x ¼ 1 results in two fractures and propensity to two additional fractures. For x ¼ 0, again, the projectile is subjected to multiple fractures and severe damage. Figs. 9e12 illustrate, for the unconstrained XW fracture model with variable x, the different mechanisms involved when the
41
projectile penetrates a backwards and a forwards moving plate, Cases I and II, respectively. Fig. 9 shows sequences of damage of the projectile when it penetrates the plate. Also, the difference in contact geometry between the two cases during the penetration can be seen. Fig. 10 shows the lateral stress sxx on the upper and lower side of the mid-section of the projectile versus time. This
Fig. 9. Sequences of damage plots for the projectile during its penetration of the plate in Cases I and II. The contour plots indicate the level of damage. Blue to red corresponds to Df ¼ 0e1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
42
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
Fig. 10. Lateral stress sxx versus time t from impact at the upper and lower side of the mid-section of the projectile during the sliding perforation of the plate in Cases I and II. The contour plots indicate the level of stress in the symmetry plane. Blue to red corresponds to approximately sxx ¼ 0e3.5 GPa. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
illustrates the difference between the two cases in the loading on the projectile when the plate slides along its envelope. Fig. 11 shows the effective stress seff versus time t from impact along the projectile in the two cases. In these figures, where colours represent levels of effective stress, dark blue colour indicates zero stress and represents either an uninfluenced element (in the back of the projectile at an early time) or an eroded element (eroded nose elements and elements where fracture appears). Fig. 12 shows the shear stress szx versus time t from impact at the side of the projectile where the fractures are initiated, i.e. at the upper side of the projectile in Case I and at the lower side of the projectile in Case II. The localised regions of high shear stress in Case II agree well with the localised regions of high effective stress in Fig. 11(d). 5. Discussion Figs. 4e6 show that by use of the XW fracture model, and the well-established JC strength model, it was possible to accurately reproduce the experimental results for the interaction between a projectile and a moving oblique plate. Furthermore, this could be done without use of an extremely fine mesh. The transition from a non-fractured to a severely fractured projectile was captured, and the deformation and rotation of the projectile and its fragments, and the locations of fractures, agree well with the experimental
results. The material parameters used had been determined from simple material tests. A more sophisticated parameter evaluation would probably improve the agreement. Both the influence of the direction and of the magnitude of the plate velocity on the fragmentation was captured as can be seen in Figs. 4 and 5, respectively. For the different magnitudes of plate velocity in Cases III and IV, the type of load on the projectile is the same; only the magnitude of the load is different. For the different directions of plate velocity in Cases I and II, however, the type of load on the projectile is different. It is interesting to study this difference as our experimental results in [6] show that backwards moving plates did not fragment projectiles while forwards moving plates did if the load was adequate. Next, the interactions between projectile and plate in these two cases will be discussed on the basis of the results shown in Figs. 9e12. When the nose of the projectile impacts the plate as shown in Fig. 9, a penetration channel with greater diameter than that of the projectile is created. Due to the oblique orientation of the plate, the entrance part of the penetration channel expands more on the upper than on the lower side of the projectile. The reason for this lack of symmetry is that the projectile is supported by more material on its lower than on its upper side at the entrance. It can be seen that in Case I of a backwards moving plate, the plate pushes the projectile upwards. This immediately results in a sliding contact on the lower part of the penetration channel. In Case II of a forwards
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
43
Fig. 11. Effective stress seff versus time t from impact along the projectile in Cases I and II. (a) Case I, upper side. (b) Case II, upper side. (c) Case I, lower side. (d) Case II, lower side. The contour plots indicate the level of effective stress. The contour plots of the projectile below, indicating were fractures occur, are shown as a reference. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
moving plate, the plate approaches the projectile downwards. Because of the asymmetry of the penetration channel, the establishment of contact between the projectile and the plate along the upper part of the penetration channel is delayed. With the plate and projectile velocities used, the contact is not established until the nose of the projectile has reached the rear side of the plate (approximately 22 ms). The difference in time between Cases I and II for establishment of contact can be seen also in Fig. 11. Fig. 11 illustrates, in both Cases I and II, that when the projectile impacts the plate, large stresses arise at the nose of the projectile, which is continuously consumed during perforation, and an elastic compressive wave propagates along the projectile and reflects as an elastic tensile wave at the free rear end of the projectile. It can be seen that a smaller part of the nose of the projectile is consumed in
Case I of a backwards moving plate than in Case II of a forwards moving plate. This agrees well with our experimental results in [6], where the relative decrease in length was found to be 0.12 and 0.18, respectively. The front of the elastic compressive wave can be seen as the first sloping straight line in the diagrams. In both cases, the velocity of the wave is about 4500 m/s. A sloping dark red region, representing high effective stress, with a different slope can also be seen in the diagrams. It represents the sliding contact with the plate. The sliding velocity with respect to the projectile vslide ¼ vproj þ vplate =cos a is 2400 m/s in Case I of a backwards moving plate and 1600 m/s in Case II of a forwards moving plate. These sliding velocities imply that the contact between the projectile and the plate ends after about 43 and 64 ms, respectively, which agrees well with the high-stress regions in the diagrams. In
44
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
Fig. 12. Shear stress szx versus time t from impact at the side of the projectile where the fractures are initiated, i.e. at the upper side of the projectile in Case I and at the lower side of the projectile in Case II. The contour plots indicate the level of shear stress.
accord with the discussion of Fig. 9 above, the high effective stress region due to the sliding contact appears immediately at impact in Case I of a backwards moving plate, while in Case II of a forwards moving plate it does not appear until the projectile has perforated the plate. When the projectile perforates the oblique plate, the exit part of the penetration channel expands more at its lower side where there is less supporting material than at its upper side. As the lower side in Case I of a backwards moving plate is the main contact surface, the sliding contact between the projectile and the penetration channel is released before the exit, see Fig. 9. In this case, there is contact with the plate both on the lower sliding side of the projectile and on the upper side of the nose. In Case II, there is no contact between the projectile and the lower side of the penetration channel during perforation. Yet, successive high-damage regions arise on the lower side of the projectile at the exit of the plate which subsequently leads to the initiation of fractures. For the first two fractures, this can be seen from a comparison of Figs. 4 and 9. After perforation, the projectile slides against the plate and creates the typical keyhole-shaped penetration channel observed in our preceding experimental study [6]. The contact during this sliding interaction occurs on the lower side of the channel in Case I and on the upper side in Case II. As a result, the history of load on the projectile at a fixed section is different in the two cases. Fig. 10 shows the lateral stress sxx versus time on the contact surface at the mid-section of the projectile during the sliding contact. In Case I of a backwards moving plate the magnitude of the stress abruptly increases to a maximum at the arrival of the plate. Then, it decreases to approximately half its value, till it abruptly drops to zero when the plate leaves. In Case II of a forwards moving plate there are also corresponding abrupt changes in the magnitude of the stress. However, the magnitude of the stress increases, doubles, and is maximal when the plate leaves the mid-section of the
projectile. At certain sections, this gives rise to high-damage regions in the projectile which subsequently result in fractures. The different characters of the stress histories between Cases I and II are due to the different thicknesses of supporting material on the upper and lower sides of the oblique plate. The figure also shows that the duration of contact is longer in Case II than in Case I, 7.5 and 5.0 ms, respectively. This is due to the different sliding velocities (1600 and 2400 m/s, respectively). The longer contact duration in Case II than in Case I, is illustrated also by the thicker high effective stress regions in Fig. 11(b) than in Fig. 11(c). In Case I, the sliding contact on the lower side of the projectile results in bending such that the projectile gets a convex shape upwards as shown in Fig. 4. This gives rise to large axial tensile stresses on the upper side of the projectile and large compressive stresses on the lower side. Correspondingly, large regions of high effective stress are formed on the upper and lower sides of the projectile. Because of the influence of the lateral stress (compressive on the lower side and zero on the upper side) on the effective stress, these high effective stress regions are larger on the upper side of the projectile than on the lower side. Compare the redcoloured regions in Fig. 11. The relatively uniform distribution of effective stress on the upper side of the projectile seems to be related to the absence of fractures in Case I. In Case II, similarly, the sliding contact on the upper side of the projectile results in bending such that the projectile gets a convex shape downwards as shown in Fig. 4. This gives rise to large axial tensile stresses on the lower side of the projectile and large compressive stresses on the upper side. On the lower side of the projectile this leads, as can be seen in Fig. 11(d), to localised regions of high effective stress and damage at which fractures are initiated. Fig. 9 illustrates that each of these regions first appears at the exit of the plate where there is an abrupt drop in the lateral load on the projectile (Fig. 10).
E. Lidén et al. / International Journal of Impact Engineering 40-41 (2012) 35e45
Fig. 12 shows for Case II the shear stress szx versus time from impact at the lower side of the projectile, where the fractures are initiated. The localised regions of high shear stress in this diagram agree well with the localised regions of high effective stress in Fig. 11(d) as well as with the positions of the fractures in Fig. 6. This indicates that the fractures originate from shear stress and supports the introduction of the deviatoric stress parameter x in the fracture model so that the sensitivity of the material to shear stresses can be taken into consideration. For example, Fig. 1 shows that with the material parameters used in the XW fracture model for tungsten Y925 in the simulations, the equivalent plastic strain at fracture increases with about 30% from pure shear (x ¼ 0, h ¼ 0) to uniaxial tension (x ¼ 1, h ¼ 1/3). Figs. 7 and 8 show that in Cases I and III, where the projectiles are mainly deformed in the experimental tests, the results of simulation with variable x and with fixed x ¼ 1 agree equally well with the experimental results, while those with fixed x ¼ 0 result in too much damage and fractures. In Cases II and IV, where the projectile is severely fractured in the experimental tests, the number of fragments is best reproduced in the simulations with fixed x ¼ 0 although the overall damage of the projectile is overestimated. In these cases, the simulations with variable x and with fixed x ¼ 1 miss some of the fractures. However, more of the fractures are captured with variable x, and those missing are represented by high levels of damage. Clearly, the deviatoric stress parameter x of the XW model has significant influence on the results of simulation. It allows fractures due to shear loading without extensive damage of the entire projectile. Introduction of the deviatoric stress parameter in the fracture model means that tests are required to determine the parameter p of the model. In studies where it is essential to capture fractures, such additional testing may be worthwhile. 6. Conclusions The main conclusions of this study can be summarised as follows. (i) It is possible to reproduce in simulations fractures in a tungsten long-rod projectile impacted by a moving oblique plate. (ii) This can be done without use of an extremely fine mesh. (iii) The transition from a deformed non-fractured projectile to a severely
45
fractured projectile due to a small increase in the magnitude of the plate velocity or a change from a backwards to a forwards moving plate can be captured. (iv) The larger effect on the projectile of a forwards moving plate than a backwards moving plate is due to the longer interaction time and the larger thickness of supporting plate material at the exit side of the plate. (v) The deviatoric stress parameter used in the XW model has significant influence and allows fractures due to shear loading without extensive damage of the entire projectile. References [1] Johnson GR, Cook WH. Lagrangian EPIC code computations for oblique, yawed-rod impacts onto thin-plate and spaced-plate targets at various velocities. Int J Impact Eng 1993;14:373e83. [2] Lidén E, Ottosson J, Holmberg L. WHA long rods penetrating stationary and moving oblique steel plates. In: Bailey HR, Holden L, editors. Proceedings 16th International Symposium on Ballistics, San Francisco. American Defense Preparedness Association. 1996;3:703–712. [3] Yoo Y-H, Shin H. Protection capability of dual flying plates against obliquely impacting long-rod penetrators. Int J Impact Eng 2004;30:55e68. [4] Yoo Y-H, Lee M. Protection effectiveness of an oblique plate against a long rod. Int J Impact Eng 2006;33:872e9. [5] Paik SH, Kim SJ, Yoo YH, Lee M. Protection performance of dual flying oblique plates against a yawed long-rod penetrator. Int J Impact Eng 2007;34: 1413e22. [6] Lidén E, Andersson O, Lundberg B. Deformation and fracture of a long rod projectile induced by a oblique moving plate. Experimental tests. Int J Impact Eng 2011;12:989e1000. [7] Wierzbicki T, Bao Y, Lee Y-W, Bai Y. Calibration and evaluation of seven fracture models. Int J Mech Sci 2005;47:719e43. [8] Johnson GR, Cook WH. A constitutive model and data for metals subjected to large strains, high strain rates, and high temperatures. In: Proceedings of the seventh international symposium on ballistics, The Hague. Royal Institute of Engineers in the Netherlands in co-operation with the American Defense Preparedness Association. 1983:541e547. [9] Hallquist JO. LS-DYNA theory manual. Livermore Software Technology Corporation; 2006. [10] Rohr I, Nahme H, Thoma K, Anderson CE. Material characterisation and constitutive modelling of a tungsten-sintered alloy for a wide range of strain rates. Int J Impact Eng 2008;35:811e9. [11] Johnson GR, Cook WH. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng Fract Mech 1985;21(1):31e48. [12] Bai Y, Wierzbicki T. A new model of metal plasticity and fracture with pressure and Lode dependence. Int J Plast 2008;24:1071e96. [13] Helte A, Lidén E. The role of KelvineHelmholtz instabilities on shaped charge jet interaction with reactive armor plates. J Appl Mech 2010;77:051805e11.